%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP243^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Feb 28 09:10:01 AM UTC 2026
% Result : Theorem 66.57s 16.16s
% Output : Refutation 66.57s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 7
% Syntax : Number of formulae : 37 ( 37 unt; 0 typ; 0 def)
% Number of atoms : 137 ( 40 equ; 0 cnn)
% Maximal formula atoms : 1 ( 3 avg)
% Number of connectives : 418 ( 6 ~; 0 |; 18 &; 369 @)
% ( 0 <=>; 18 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 2 avg)
% Number of types : 90 ( 89 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 941 ( 937 usr; 60 con; 0-8 aty)
% ( 7 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 61 ( 49 ^; 12 !; 0 ?; 61 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
real: $tType ).
thf(type_def_6,type,
int: $tType ).
thf(type_def_8,type,
rat: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
num: $tType ).
thf(type_def_11,type,
product_prod_int_int: $tType ).
thf(type_def_12,type,
product_prod_nat_nat: $tType ).
thf(type_def_13,type,
option_num: $tType ).
thf(type_def_14,type,
product_prod_num_num: $tType ).
thf(type_def_15,type,
code_integer: $tType ).
thf(type_def_16,type,
produc6271795597528267376eger_o: $tType ).
thf(type_def_17,type,
produc8923325533196201883nteger: $tType ).
thf(type_def_18,type,
set_fi4554929511873752355omplex: $tType ).
thf(type_def_19,type,
filter6041513312241820739omplex: $tType ).
thf(type_def_20,type,
set_fi7789364187291644575l_real: $tType ).
thf(type_def_21,type,
filter2146258269922977983l_real: $tType ).
thf(type_def_22,type,
set_real: $tType ).
thf(type_def_23,type,
set_int: $tType ).
thf(type_def_24,type,
complex: $tType ).
thf(type_def_25,type,
filter_real: $tType ).
thf(type_def_26,type,
extended_enat: $tType ).
thf(type_def_27,type,
filter_nat: $tType ).
thf(type_def_28,type,
set_Pr5085853215250843933omplex: $tType ).
thf(type_def_29,type,
set_Pr6218003697084177305l_real: $tType ).
thf(type_def_30,type,
set_o: $tType ).
thf(type_def_31,type,
set_complex: $tType ).
thf(type_def_32,type,
set_list_nat: $tType ).
thf(type_def_33,type,
set_nat: $tType ).
thf(type_def_34,type,
set_Product_unit: $tType ).
thf(type_def_35,type,
set_char: $tType ).
thf(type_def_36,type,
set_list_o: $tType ).
thf(type_def_37,type,
set_list_complex: $tType ).
thf(type_def_38,type,
set_list_int: $tType ).
thf(type_def_39,type,
set_list_VEBT_VEBT: $tType ).
thf(type_def_40,type,
set_num: $tType ).
thf(type_def_41,type,
set_rat: $tType ).
thf(type_def_42,type,
set_set_complex: $tType ).
thf(type_def_43,type,
set_set_int: $tType ).
thf(type_def_44,type,
set_set_nat: $tType ).
thf(type_def_45,type,
set_VEBT_VEBT: $tType ).
thf(type_def_46,type,
char: $tType ).
thf(type_def_47,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(type_def_48,type,
list_nat: $tType ).
thf(type_def_49,type,
produc859450856879609959at_nat: $tType ).
thf(type_def_50,type,
list_o: $tType ).
thf(type_def_51,type,
list_int: $tType ).
thf(type_def_52,type,
option_nat: $tType ).
thf(type_def_53,type,
vEBT_VEBT: $tType ).
thf(type_def_54,type,
set_Pr958786334691620121nt_int: $tType ).
thf(type_def_55,type,
list_complex: $tType ).
thf(type_def_56,type,
list_list_nat: $tType ).
thf(type_def_57,type,
list_P8469869581646625389at_nat: $tType ).
thf(type_def_58,type,
list_real: $tType ).
thf(type_def_59,type,
list_VEBT_VEBT: $tType ).
thf(type_def_60,type,
list_num: $tType ).
thf(type_def_61,type,
list_P4002435161011370285od_o_o: $tType ).
thf(type_def_62,type,
product_prod_o_o: $tType ).
thf(type_def_63,type,
list_P3795440434834930179_o_int: $tType ).
thf(type_def_64,type,
product_prod_o_int: $tType ).
thf(type_def_65,type,
list_P6285523579766656935_o_nat: $tType ).
thf(type_def_66,type,
product_prod_o_nat: $tType ).
thf(type_def_67,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(type_def_68,type,
produc2504756804600209347T_VEBT: $tType ).
thf(type_def_69,type,
list_P1726324292696863441at_num: $tType ).
thf(type_def_70,type,
product_prod_nat_num: $tType ).
thf(type_def_71,type,
list_P3744719386663036955um_num: $tType ).
thf(type_def_72,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(type_def_73,type,
produc334124729049499915VEBT_o: $tType ).
thf(type_def_74,type,
list_P4547456442757143711BT_int: $tType ).
thf(type_def_75,type,
produc4894624898956917775BT_int: $tType ).
thf(type_def_76,type,
list_P7037539587688870467BT_nat: $tType ).
thf(type_def_77,type,
produc9072475918466114483BT_nat: $tType ).
thf(type_def_78,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(type_def_79,type,
produc8243902056947475879T_VEBT: $tType ).
thf(type_def_80,type,
list_P7333126701944960589_nat_o: $tType ).
thf(type_def_81,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(type_def_82,type,
option4927543243414619207at_nat: $tType ).
thf(type_def_83,type,
set_Code_integer: $tType ).
thf(type_def_84,type,
produc4953844613479565601on_nat: $tType ).
thf(type_def_85,type,
produc2233624965454879586on_nat: $tType ).
thf(type_def_86,type,
produc8306885398267862888on_nat: $tType ).
thf(type_def_87,type,
produc3447558737645232053on_num: $tType ).
thf(type_def_88,type,
produc7036089656553540234on_num: $tType ).
thf(type_def_89,type,
produc1193250871479095198on_num: $tType ).
thf(type_def_90,type,
produc6121120109295599847at_nat: $tType ).
thf(type_def_91,type,
produc5491161045314408544at_nat: $tType ).
thf(type_def_92,type,
produc5542196010084753463at_nat: $tType ).
thf(type_def_93,type,
produc4411394909380815293omplex: $tType ).
thf(type_def_94,type,
produc2422161461964618553l_real: $tType ).
thf(func_def_0,type,
produc5542196010084753463at_nat: $tType ).
thf(func_def_1,type,
produc5491161045314408544at_nat: $tType ).
thf(func_def_2,type,
produc1193250871479095198on_num: $tType ).
thf(func_def_3,type,
produc8306885398267862888on_nat: $tType ).
thf(func_def_4,type,
produc6121120109295599847at_nat: $tType ).
thf(func_def_5,type,
produc7036089656553540234on_num: $tType ).
thf(func_def_6,type,
produc2233624965454879586on_nat: $tType ).
thf(func_def_7,type,
list_P8469869581646625389at_nat: $tType ).
thf(func_def_8,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(func_def_9,type,
produc859450856879609959at_nat: $tType ).
thf(func_def_10,type,
set_fi4554929511873752355omplex: $tType ).
thf(func_def_11,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(func_def_12,type,
produc3447558737645232053on_num: $tType ).
thf(func_def_13,type,
produc4953844613479565601on_nat: $tType ).
thf(func_def_14,type,
set_fi7789364187291644575l_real: $tType ).
thf(func_def_15,type,
filter6041513312241820739omplex: $tType ).
thf(func_def_16,type,
list_P7037539587688870467BT_nat: $tType ).
thf(func_def_17,type,
list_P4547456442757143711BT_int: $tType ).
thf(func_def_18,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(func_def_19,type,
produc8243902056947475879T_VEBT: $tType ).
thf(func_def_20,type,
set_Pr5085853215250843933omplex: $tType ).
thf(func_def_21,type,
produc8923325533196201883nteger: $tType ).
thf(func_def_22,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(func_def_23,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(func_def_24,type,
filter2146258269922977983l_real: $tType ).
thf(func_def_25,type,
option4927543243414619207at_nat: $tType ).
thf(func_def_26,type,
set_Pr6218003697084177305l_real: $tType ).
thf(func_def_27,type,
list_P3744719386663036955um_num: $tType ).
thf(func_def_28,type,
list_P1726324292696863441at_num: $tType ).
thf(func_def_29,type,
produc9072475918466114483BT_nat: $tType ).
thf(func_def_30,type,
produc4894624898956917775BT_int: $tType ).
thf(func_def_31,type,
set_Pr958786334691620121nt_int: $tType ).
thf(func_def_32,type,
produc4411394909380815293omplex: $tType ).
thf(func_def_33,type,
list_P7333126701944960589_nat_o: $tType ).
thf(func_def_34,type,
list_P6285523579766656935_o_nat: $tType ).
thf(func_def_35,type,
list_P3795440434834930179_o_int: $tType ).
thf(func_def_36,type,
set_list_VEBT_VEBT: $tType ).
thf(func_def_37,type,
produc334124729049499915VEBT_o: $tType ).
thf(func_def_38,type,
produc2504756804600209347T_VEBT: $tType ).
thf(func_def_39,type,
produc6271795597528267376eger_o: $tType ).
thf(func_def_40,type,
produc2422161461964618553l_real: $tType ).
thf(func_def_41,type,
product_prod_num_num: $tType ).
thf(func_def_42,type,
product_prod_nat_num: $tType ).
thf(func_def_43,type,
product_prod_nat_nat: $tType ).
thf(func_def_44,type,
product_prod_int_int: $tType ).
thf(func_def_45,type,
list_P4002435161011370285od_o_o: $tType ).
thf(func_def_46,type,
set_list_complex: $tType ).
thf(func_def_47,type,
set_set_complex: $tType ).
thf(func_def_48,type,
list_list_nat: $tType ).
thf(func_def_49,type,
list_VEBT_VEBT: $tType ).
thf(func_def_50,type,
set_list_nat: $tType ).
thf(func_def_51,type,
set_list_int: $tType ).
thf(func_def_52,type,
product_prod_o_nat: $tType ).
thf(func_def_53,type,
product_prod_o_int: $tType ).
thf(func_def_54,type,
set_VEBT_VEBT: $tType ).
thf(func_def_55,type,
set_set_nat: $tType ).
thf(func_def_56,type,
set_set_int: $tType ).
thf(func_def_57,type,
set_Code_integer: $tType ).
thf(func_def_58,type,
set_Product_unit: $tType ).
thf(func_def_59,type,
list_complex: $tType ).
thf(func_def_60,type,
set_list_o: $tType ).
thf(func_def_61,type,
product_prod_o_o: $tType ).
thf(func_def_62,type,
set_complex: $tType ).
thf(func_def_63,type,
filter_real: $tType ).
thf(func_def_64,type,
option_num: $tType ).
thf(func_def_65,type,
option_nat: $tType ).
thf(func_def_66,type,
filter_nat: $tType ).
thf(func_def_67,type,
set_char: $tType ).
thf(func_def_68,type,
list_real: $tType ).
thf(func_def_69,type,
set_real: $tType ).
thf(func_def_70,type,
list_num: $tType ).
thf(func_def_71,type,
list_nat: $tType ).
thf(func_def_72,type,
list_int: $tType ).
thf(func_def_73,type,
vEBT_VEBT: $tType ).
thf(func_def_74,type,
set_rat: $tType ).
thf(func_def_75,type,
set_num: $tType ).
thf(func_def_76,type,
set_nat: $tType ).
thf(func_def_77,type,
set_int: $tType ).
thf(func_def_78,type,
code_integer: $tType ).
thf(func_def_79,type,
extended_enat: $tType ).
thf(func_def_80,type,
list_o: $tType ).
thf(func_def_81,type,
complex: $tType ).
thf(func_def_82,type,
set_o: $tType ).
thf(func_def_83,type,
char: $tType ).
thf(func_def_84,type,
real: $tType ).
thf(func_def_85,type,
rat: $tType ).
thf(func_def_86,type,
num: $tType ).
thf(func_def_87,type,
nat: $tType ).
thf(func_def_88,type,
int: $tType ).
thf(func_def_89,type,
archim7802044766580827645g_real: real > int ).
thf(func_def_90,type,
archim3151403230148437115or_rat: rat > int ).
thf(func_def_91,type,
archim6058952711729229775r_real: real > int ).
thf(func_def_92,type,
archim2898591450579166408c_real: real > real ).
thf(func_def_93,type,
archim7778729529865785530nd_rat: rat > int ).
thf(func_def_94,type,
archim8280529875227126926d_real: real > int ).
thf(func_def_95,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_96,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_97,type,
bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
thf(func_def_98,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_99,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_100,type,
bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
thf(func_def_101,type,
bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
thf(func_def_102,type,
bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(func_def_103,type,
bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
thf(func_def_104,type,
bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_105,type,
bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
thf(func_def_106,type,
bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_107,type,
bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
thf(func_def_108,type,
bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_109,type,
bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
thf(func_def_110,type,
bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
thf(func_def_111,type,
bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
thf(func_def_112,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_113,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_114,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_115,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_116,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_117,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_118,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_119,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_120,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_121,type,
bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
thf(func_def_122,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_123,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_124,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_125,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_126,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_127,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_128,type,
binomial: nat > nat > nat ).
thf(func_def_129,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_130,type,
bit_and_not_num: num > num > option_num ).
thf(func_def_131,type,
bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_132,type,
bit_concat_bit: nat > int > int > int ).
thf(func_def_133,type,
bit_or_not_num_neg: num > num > num ).
thf(func_def_134,type,
bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_135,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(func_def_136,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(func_def_137,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(func_def_138,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(func_def_139,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(func_def_140,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(func_def_141,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(func_def_142,type,
bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
thf(func_def_143,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(func_def_144,type,
bit_se2161824704523386999it_nat: nat > nat > nat ).
thf(func_def_145,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(func_def_146,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(func_def_147,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(func_def_148,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(func_def_149,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(func_def_150,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(func_def_151,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(func_def_152,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(func_def_153,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(func_def_154,type,
bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
thf(func_def_155,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(func_def_156,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(func_def_157,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(func_def_158,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(func_def_159,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(func_def_160,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(func_def_161,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(func_def_162,type,
bit_se9216721137139052372nteger: code_integer > nat > $o ).
thf(func_def_163,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(func_def_164,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(func_def_165,type,
bit_take_bit_num: nat > num > option_num ).
thf(func_def_166,type,
bit_un1837492267222099188nd_num: num > num > option_num ).
thf(func_def_167,type,
bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_168,type,
bit_un6178654185764691216or_num: num > num > option_num ).
thf(func_def_169,type,
bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_170,type,
bit_un7362597486090784418nd_num: num > num > option_num ).
thf(func_def_171,type,
bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_172,type,
bit_un2480387367778600638or_num: num > num > option_num ).
thf(func_def_173,type,
bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
thf(func_def_174,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(func_def_175,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_176,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_177,type,
code_int_of_integer: code_integer > int ).
thf(func_def_178,type,
code_integer_of_int: int > code_integer ).
thf(func_def_179,type,
code_integer_of_num: num > code_integer ).
thf(func_def_180,type,
code_nat_of_integer: code_integer > nat ).
thf(func_def_181,type,
code_negative: num > code_integer ).
thf(func_def_182,type,
code_num_of_integer: code_integer > num ).
thf(func_def_183,type,
code_positive: num > code_integer ).
thf(func_def_184,type,
code_Target_negative: num > int ).
thf(func_def_185,type,
code_Target_positive: num > int ).
thf(func_def_186,type,
comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
thf(func_def_187,type,
comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
thf(func_def_188,type,
comple4887499456419720421f_real: set_real > real ).
thf(func_def_189,type,
complete_Sup_Sup_int: set_int > int ).
thf(func_def_190,type,
comple1385675409528146559p_real: set_real > real ).
thf(func_def_191,type,
arg: complex > real ).
thf(func_def_192,type,
cis: real > complex ).
thf(func_def_193,type,
cnj: complex > complex ).
thf(func_def_194,type,
complex2: real > real > complex ).
thf(func_def_195,type,
im: complex > real ).
thf(func_def_196,type,
re: complex > real ).
thf(func_def_197,type,
csqrt: complex > complex ).
thf(func_def_198,type,
imaginary_unit: complex ).
thf(func_def_199,type,
rcis: real > real > complex ).
thf(func_def_200,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(func_def_201,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(func_def_202,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_203,type,
has_ve631408500373753343e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_204,type,
adjust_div: product_prod_int_int > int ).
thf(func_def_205,type,
adjust_mod: int > int > int ).
thf(func_def_206,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_207,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(func_def_208,type,
unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
thf(func_def_209,type,
unique6319869463603278526ux_int: product_prod_int_int > $o ).
thf(func_def_210,type,
unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
thf(func_def_211,type,
unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
thf(func_def_212,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(func_def_213,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(func_def_214,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_215,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(func_def_216,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_217,type,
euclid4774559944035922753ze_int: int > nat ).
thf(func_def_218,type,
euclid3395696857347342551nt_int: int > int ).
thf(func_def_219,type,
euclid3398187327856392827nt_nat: nat > nat ).
thf(func_def_220,type,
extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).
thf(func_def_221,type,
comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
thf(func_def_222,type,
comm_s2602460028002588243omplex: complex > nat > complex ).
thf(func_def_223,type,
comm_s4660882817536571857er_int: int > nat > int ).
thf(func_def_224,type,
comm_s4663373288045622133er_nat: nat > nat > nat ).
thf(func_def_225,type,
comm_s4028243227959126397er_rat: rat > nat > rat ).
thf(func_def_226,type,
comm_s7457072308508201937r_real: real > nat > real ).
thf(func_def_227,type,
semiri3624122377584611663nteger: nat > code_integer ).
thf(func_def_228,type,
semiri5044797733671781792omplex: nat > complex ).
thf(func_def_229,type,
semiri1406184849735516958ct_int: nat > int ).
thf(func_def_230,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(func_def_231,type,
semiri773545260158071498ct_rat: nat > rat ).
thf(func_def_232,type,
semiri2265585572941072030t_real: nat > real ).
thf(func_def_233,type,
invers8013647133539491842omplex: complex > complex ).
thf(func_def_234,type,
inverse_inverse_rat: rat > rat ).
thf(func_def_235,type,
inverse_inverse_real: real > real ).
thf(func_def_236,type,
at_bot_real: filter_real ).
thf(func_def_237,type,
at_top_nat: filter_nat ).
thf(func_def_238,type,
at_top_real: filter_real ).
thf(func_def_239,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_240,type,
eventually_real: ( real > $o ) > filter_real > $o ).
thf(func_def_241,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_242,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(func_def_243,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(func_def_244,type,
filtermap_real_real: ( real > real ) > filter_real > filter_real ).
thf(func_def_245,type,
princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
thf(func_def_246,type,
princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
thf(func_def_247,type,
finite_card_o: set_o > nat ).
thf(func_def_248,type,
finite_card_complex: set_complex > nat ).
thf(func_def_249,type,
finite_card_int: set_int > nat ).
thf(func_def_250,type,
finite_card_list_nat: set_list_nat > nat ).
thf(func_def_251,type,
finite_card_nat: set_nat > nat ).
thf(func_def_252,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(func_def_253,type,
finite_card_char: set_char > nat ).
thf(func_def_254,type,
finite_finite_o: set_o > $o ).
thf(func_def_255,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(func_def_256,type,
finite_finite_int: set_int > $o ).
thf(func_def_257,type,
finite_finite_list_o: set_list_o > $o ).
thf(func_def_258,type,
finite8712137658972009173omplex: set_list_complex > $o ).
thf(func_def_259,type,
finite3922522038869484883st_int: set_list_int > $o ).
thf(func_def_260,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(func_def_261,type,
finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
thf(func_def_262,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_263,type,
finite_finite_num: set_num > $o ).
thf(func_def_264,type,
finite_finite_rat: set_rat > $o ).
thf(func_def_265,type,
finite_finite_real: set_real > $o ).
thf(func_def_266,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(func_def_267,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(func_def_268,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(func_def_269,type,
finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
thf(func_def_270,type,
bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(func_def_271,type,
bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
thf(func_def_272,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_273,type,
comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_274,type,
comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_275,type,
comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).
thf(func_def_276,type,
comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
thf(func_def_277,type,
comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
thf(func_def_278,type,
comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).
thf(func_def_279,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(func_def_280,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_281,type,
inj_on_nat_char: ( nat > char ) > set_nat > $o ).
thf(func_def_282,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(func_def_283,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_284,type,
map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
thf(func_def_285,type,
strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_286,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(func_def_287,type,
fun_pair_leq: set_Pr8693737435421807431at_nat ).
thf(func_def_288,type,
fun_pair_less: set_Pr8693737435421807431at_nat ).
thf(func_def_289,type,
gcd_Gcd_int: set_int > int ).
thf(func_def_290,type,
gcd_Gcd_nat: set_nat > nat ).
thf(func_def_291,type,
bezw: nat > nat > product_prod_int_int ).
thf(func_def_292,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_293,type,
gcd_gcd_int: int > int > int ).
thf(func_def_294,type,
gcd_gcd_nat: nat > nat > nat ).
thf(func_def_295,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(func_def_296,type,
abs_abs_complex: complex > complex ).
thf(func_def_297,type,
abs_abs_int: int > int ).
thf(func_def_298,type,
abs_abs_rat: rat > rat ).
thf(func_def_299,type,
abs_abs_real: real > real ).
thf(func_def_300,type,
minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
thf(func_def_301,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_302,type,
minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(func_def_303,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_304,type,
minus_4666906530369457071_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat > $o ).
thf(func_def_305,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(func_def_306,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(func_def_307,type,
minus_minus_complex: complex > complex > complex ).
thf(func_def_308,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_309,type,
minus_minus_int: int > int > int ).
thf(func_def_310,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_311,type,
minus_minus_rat: rat > rat > rat ).
thf(func_def_312,type,
minus_minus_real: real > real > real ).
thf(func_def_313,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(func_def_314,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(func_def_315,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(func_def_316,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_317,type,
minus_8321449233255521966at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_318,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(func_def_319,type,
one_one_Code_integer: code_integer ).
thf(func_def_320,type,
one_one_complex: complex ).
thf(func_def_321,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(func_def_322,type,
one_one_int: int ).
thf(func_def_323,type,
one_one_nat: nat ).
thf(func_def_324,type,
one_one_rat: rat ).
thf(func_def_325,type,
one_one_real: real ).
thf(func_def_326,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(func_def_327,type,
plus_plus_complex: complex > complex > complex ).
thf(func_def_328,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_329,type,
plus_plus_int: int > int > int ).
thf(func_def_330,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_331,type,
plus_plus_num: num > num > num ).
thf(func_def_332,type,
plus_plus_rat: rat > rat > rat ).
thf(func_def_333,type,
plus_plus_real: real > real > real ).
thf(func_def_334,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(func_def_335,type,
sgn_sgn_complex: complex > complex ).
thf(func_def_336,type,
sgn_sgn_int: int > int ).
thf(func_def_337,type,
sgn_sgn_rat: rat > rat ).
thf(func_def_338,type,
sgn_sgn_real: real > real ).
thf(func_def_339,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(func_def_340,type,
times_times_complex: complex > complex > complex ).
thf(func_def_341,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_342,type,
times_times_int: int > int > int ).
thf(func_def_343,type,
times_times_nat: nat > nat > nat ).
thf(func_def_344,type,
times_times_num: num > num > num ).
thf(func_def_345,type,
times_times_rat: rat > rat > rat ).
thf(func_def_346,type,
times_times_real: real > real > real ).
thf(func_def_347,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(func_def_348,type,
uminus1482373934393186551omplex: complex > complex ).
thf(func_def_349,type,
uminus_uminus_int: int > int ).
thf(func_def_350,type,
uminus_uminus_rat: rat > rat ).
thf(func_def_351,type,
uminus_uminus_real: real > real ).
thf(func_def_352,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(func_def_353,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(func_def_354,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(func_def_355,type,
zero_z3403309356797280102nteger: code_integer ).
thf(func_def_356,type,
zero_zero_complex: complex ).
thf(func_def_357,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(func_def_358,type,
zero_zero_int: int ).
thf(func_def_359,type,
zero_zero_nat: nat ).
thf(func_def_360,type,
zero_zero_rat: rat ).
thf(func_def_361,type,
zero_zero_real: real ).
thf(func_def_362,type,
groups6621422865394947399nteger: ( complex > code_integer ) > set_complex > code_integer ).
thf(func_def_363,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_364,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(func_def_365,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(func_def_366,type,
groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).
thf(func_def_367,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(func_def_368,type,
groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).
thf(func_def_369,type,
groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
thf(func_def_370,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(func_def_371,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(func_def_372,type,
groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_373,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(func_def_374,type,
groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(func_def_375,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(func_def_376,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(func_def_377,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_378,type,
groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_379,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(func_def_380,type,
groups7963564606188603611omplex: ( produc859450856879609959at_nat > complex ) > set_Pr8693737435421807431at_nat > complex ).
thf(func_def_381,type,
groups1900718384385340925at_nat: ( produc859450856879609959at_nat > nat ) > set_Pr8693737435421807431at_nat > nat ).
thf(func_def_382,type,
groups7817252763086508633t_real: ( produc859450856879609959at_nat > real ) > set_Pr8693737435421807431at_nat > real ).
thf(func_def_383,type,
groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).
thf(func_def_384,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(func_def_385,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(func_def_386,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(func_def_387,type,
groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).
thf(func_def_388,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(func_def_389,type,
groups8682486955453173170nteger: ( complex > code_integer ) > set_complex > code_integer ).
thf(func_def_390,type,
groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_391,type,
groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).
thf(func_def_392,type,
groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).
thf(func_def_393,type,
groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).
thf(func_def_394,type,
groups766887009212190081x_real: ( complex > real ) > set_complex > real ).
thf(func_def_395,type,
groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).
thf(func_def_396,type,
groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).
thf(func_def_397,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(func_def_398,type,
groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).
thf(func_def_399,type,
groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_400,type,
groups2316167850115554303t_real: ( int > real ) > set_int > real ).
thf(func_def_401,type,
groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(func_def_402,type,
groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).
thf(func_def_403,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(func_def_404,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_405,type,
groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_406,type,
groups129246275422532515t_real: ( nat > real ) > set_nat > real ).
thf(func_def_407,type,
groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).
thf(func_def_408,type,
groups713298508707869441omplex: ( real > complex ) > set_real > complex ).
thf(func_def_409,type,
groups4694064378042380927al_int: ( real > int ) > set_real > int ).
thf(func_def_410,type,
groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).
thf(func_def_411,type,
groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).
thf(func_def_412,type,
groups1681761925125756287l_real: ( real > real ) > set_real > real ).
thf(func_def_413,type,
groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
thf(func_def_414,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(func_def_415,type,
the_int: ( int > $o ) > int ).
thf(func_def_416,type,
the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
thf(func_def_417,type,
the_real: ( real > $o ) > real ).
thf(func_def_418,type,
if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).
thf(func_def_419,type,
if_nat_int_int: $o > ( nat > int > int ) > ( nat > int > int ) > nat > int > int ).
thf(func_def_420,type,
if_nat_nat_nat: $o > ( nat > nat > nat ) > ( nat > nat > nat ) > nat > nat > nat ).
thf(func_def_421,type,
if_nat_rat: $o > ( nat > rat ) > ( nat > rat ) > nat > rat ).
thf(func_def_422,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(func_def_423,type,
if_complex: $o > complex > complex > complex ).
thf(func_def_424,type,
if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
thf(func_def_425,type,
if_int: $o > int > int > int ).
thf(func_def_426,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(func_def_427,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(func_def_428,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_429,type,
if_num: $o > num > num > num ).
thf(func_def_430,type,
if_option_nat: $o > option_nat > option_nat > option_nat ).
thf(func_def_431,type,
if_option_num: $o > option_num > option_num > option_num ).
thf(func_def_432,type,
if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(func_def_433,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_434,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(func_def_435,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_436,type,
if_rat: $o > rat > rat > rat ).
thf(func_def_437,type,
if_real: $o > real > real > real ).
thf(func_def_438,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(func_def_439,type,
if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_440,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(func_def_441,type,
abs_Integ: product_prod_nat_nat > int ).
thf(func_def_442,type,
rep_Integ: int > product_prod_nat_nat ).
thf(func_def_443,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(func_def_444,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(func_def_445,type,
intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_446,type,
nat2: int > nat ).
thf(func_def_447,type,
pcr_int: product_prod_nat_nat > int > $o ).
thf(func_def_448,type,
power_int_real: real > int > real ).
thf(func_def_449,type,
ring_1_Ints_real: set_real ).
thf(func_def_450,type,
ring_18347121197199848620nteger: int > code_integer ).
thf(func_def_451,type,
ring_17405671764205052669omplex: int > complex ).
thf(func_def_452,type,
ring_1_of_int_int: int > int ).
thf(func_def_453,type,
ring_1_of_int_rat: int > rat ).
thf(func_def_454,type,
ring_1_of_int_real: int > real ).
thf(func_def_455,type,
inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_456,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_457,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_458,type,
sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_459,type,
sup_sup_nat: nat > nat > nat ).
thf(func_def_460,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_461,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(func_def_462,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(func_def_463,type,
at_infinity_real: filter_real ).
thf(func_def_464,type,
append_int: list_int > list_int > list_int ).
thf(func_def_465,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_466,type,
drop_nat: nat > list_nat > list_nat ).
thf(func_def_467,type,
last_nat: list_nat > nat ).
thf(func_def_468,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(func_def_469,type,
cons_int: int > list_int > list_int ).
thf(func_def_470,type,
cons_nat: nat > list_nat > list_nat ).
thf(func_def_471,type,
nil_int: list_int ).
thf(func_def_472,type,
nil_nat: list_nat ).
thf(func_def_473,type,
hd_nat: list_nat > nat ).
thf(func_def_474,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(func_def_475,type,
set_o2: list_o > set_o ).
thf(func_def_476,type,
set_complex2: list_complex > set_complex ).
thf(func_def_477,type,
set_int2: list_int > set_int ).
thf(func_def_478,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(func_def_479,type,
set_nat2: list_nat > set_nat ).
thf(func_def_480,type,
set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_481,type,
set_real2: list_real > set_real ).
thf(func_def_482,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_483,type,
size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(func_def_484,type,
list_update_o: list_o > nat > $o > list_o ).
thf(func_def_485,type,
list_update_complex: list_complex > nat > complex > list_complex ).
thf(func_def_486,type,
list_update_int: list_int > nat > int > list_int ).
thf(func_def_487,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(func_def_488,type,
list_u5003261594476800725at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat > list_P8469869581646625389at_nat ).
thf(func_def_489,type,
list_update_real: list_real > nat > real > list_real ).
thf(func_def_490,type,
list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_491,type,
nth_o: list_o > nat > $o ).
thf(func_def_492,type,
nth_complex: list_complex > nat > complex ).
thf(func_def_493,type,
nth_int: list_int > nat > int ).
thf(func_def_494,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(func_def_495,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_496,type,
nth_num: list_num > nat > num ).
thf(func_def_497,type,
nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
thf(func_def_498,type,
nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
thf(func_def_499,type,
nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
thf(func_def_500,type,
nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
thf(func_def_501,type,
nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
thf(func_def_502,type,
nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
thf(func_def_503,type,
nth_Pr6744343527793145070at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat ).
thf(func_def_504,type,
nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
thf(func_def_505,type,
nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
thf(func_def_506,type,
nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
thf(func_def_507,type,
nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
thf(func_def_508,type,
nth_real: list_real > nat > real ).
thf(func_def_509,type,
nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_510,type,
product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
thf(func_def_511,type,
product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
thf(func_def_512,type,
product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
thf(func_def_513,type,
product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
thf(func_def_514,type,
product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
thf(func_def_515,type,
product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
thf(func_def_516,type,
produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
thf(func_def_517,type,
product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
thf(func_def_518,type,
product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
thf(func_def_519,type,
produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
thf(func_def_520,type,
produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
thf(func_def_521,type,
produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
thf(func_def_522,type,
replicate_int: nat > int > list_int ).
thf(func_def_523,type,
replicate_nat: nat > nat > list_nat ).
thf(func_def_524,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_525,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(func_def_526,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(func_def_527,type,
take_nat: nat > list_nat > list_nat ).
thf(func_def_528,type,
upt: nat > nat > list_nat ).
thf(func_def_529,type,
upto: int > int > list_int ).
thf(func_def_530,type,
upto_aux: int > int > list_int > list_int ).
thf(func_def_531,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_532,type,
suc: nat > nat ).
thf(func_def_533,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_534,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(func_def_535,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_536,type,
case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
thf(func_def_537,type,
pred: nat > nat ).
thf(func_def_538,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(func_def_539,type,
semiri8010041392384452111omplex: nat > complex ).
thf(func_def_540,type,
semiri4216267220026989637d_enat: nat > extended_enat ).
thf(func_def_541,type,
semiri1314217659103216013at_int: nat > int ).
thf(func_def_542,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(func_def_543,type,
semiri681578069525770553at_rat: nat > rat ).
thf(func_def_544,type,
semiri5074537144036343181t_real: nat > real ).
thf(func_def_545,type,
semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
thf(func_def_546,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(func_def_547,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(func_def_548,type,
semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
thf(func_def_549,type,
semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
thf(func_def_550,type,
size_size_list_o: list_o > nat ).
thf(func_def_551,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(func_def_552,type,
size_size_list_int: list_int > nat ).
thf(func_def_553,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(func_def_554,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_555,type,
size_size_list_num: list_num > nat ).
thf(func_def_556,type,
size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
thf(func_def_557,type,
size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
thf(func_def_558,type,
size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
thf(func_def_559,type,
size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
thf(func_def_560,type,
size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
thf(func_def_561,type,
size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
thf(func_def_562,type,
size_s3679842834875189465at_nat: list_P8469869581646625389at_nat > nat ).
thf(func_def_563,type,
size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
thf(func_def_564,type,
size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
thf(func_def_565,type,
size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
thf(func_def_566,type,
size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
thf(func_def_567,type,
size_size_list_real: list_real > nat ).
thf(func_def_568,type,
size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
thf(func_def_569,type,
size_size_num: num > nat ).
thf(func_def_570,type,
size_size_option_nat: option_nat > nat ).
thf(func_def_571,type,
size_size_option_num: option_num > nat ).
thf(func_def_572,type,
size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
thf(func_def_573,type,
size_size_VEBT_VEBT: vEBT_VEBT > nat ).
thf(func_def_574,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(func_def_575,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_576,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(func_def_577,type,
nat_set_decode: nat > set_nat ).
thf(func_def_578,type,
nat_set_encode: set_nat > nat ).
thf(func_def_579,type,
nat_triangle: nat > nat ).
thf(func_def_580,type,
root: nat > real > real ).
thf(func_def_581,type,
sqrt: real > real ).
thf(func_def_582,type,
bitM: num > num ).
thf(func_def_583,type,
inc: num > num ).
thf(func_def_584,type,
neg_nu8804712462038260780nteger: code_integer > code_integer ).
thf(func_def_585,type,
neg_nu7009210354673126013omplex: complex > complex ).
thf(func_def_586,type,
neg_numeral_dbl_int: int > int ).
thf(func_def_587,type,
neg_numeral_dbl_rat: rat > rat ).
thf(func_def_588,type,
neg_numeral_dbl_real: real > real ).
thf(func_def_589,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(func_def_590,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(func_def_591,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(func_def_592,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(func_def_593,type,
neg_nu6075765906172075777c_real: real > real ).
thf(func_def_594,type,
neg_nu5831290666863070958nteger: code_integer > code_integer ).
thf(func_def_595,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(func_def_596,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(func_def_597,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(func_def_598,type,
neg_nu8295874005876285629c_real: real > real ).
thf(func_def_599,type,
neg_numeral_sub_int: num > num > int ).
thf(func_def_600,type,
bit0: num > num ).
thf(func_def_601,type,
bit1: num > num ).
thf(func_def_602,type,
one: num ).
thf(func_def_603,type,
case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
thf(func_def_604,type,
size_num: num > nat ).
thf(func_def_605,type,
num_of_nat: nat > num ).
thf(func_def_606,type,
numera6620942414471956472nteger: num > code_integer ).
thf(func_def_607,type,
numera6690914467698888265omplex: num > complex ).
thf(func_def_608,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(func_def_609,type,
numeral_numeral_int: num > int ).
thf(func_def_610,type,
numeral_numeral_nat: num > nat ).
thf(func_def_611,type,
numeral_numeral_rat: num > rat ).
thf(func_def_612,type,
numeral_numeral_real: num > real ).
thf(func_def_613,type,
pow: num > num > num ).
thf(func_def_614,type,
pred_numeral: num > nat ).
thf(func_def_615,type,
sqr: num > num ).
thf(func_def_616,type,
none_nat: option_nat ).
thf(func_def_617,type,
none_num: option_num ).
thf(func_def_618,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(func_def_619,type,
some_nat: nat > option_nat ).
thf(func_def_620,type,
some_num: num > option_num ).
thf(func_def_621,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(func_def_622,type,
case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
thf(func_def_623,type,
case_option_int_num: int > ( num > int ) > option_num > int ).
thf(func_def_624,type,
case_option_num_num: num > ( num > num ) > option_num > num ).
thf(func_def_625,type,
case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
thf(func_def_626,type,
map_option_num_num: ( num > num ) > option_num > option_num ).
thf(func_def_627,type,
size_option_nat: ( nat > nat ) > option_nat > nat ).
thf(func_def_628,type,
size_option_num: ( num > nat ) > option_num > nat ).
thf(func_def_629,type,
size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
thf(func_def_630,type,
the_nat: option_nat > nat ).
thf(func_def_631,type,
the_num: option_num > num ).
thf(func_def_632,type,
the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
thf(func_def_633,type,
bot_bot_nat_o: nat > $o ).
thf(func_def_634,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(func_def_635,type,
bot_bot_nat: nat ).
thf(func_def_636,type,
bot_bot_set_complex: set_complex ).
thf(func_def_637,type,
bot_bot_set_int: set_int ).
thf(func_def_638,type,
bot_bot_set_nat: set_nat ).
thf(func_def_639,type,
bot_bot_set_num: set_num ).
thf(func_def_640,type,
bot_bot_set_rat: set_rat ).
thf(func_def_641,type,
bot_bot_set_real: set_real ).
thf(func_def_642,type,
bot_bot_set_set_int: set_set_int ).
thf(func_def_643,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(func_def_644,type,
ord_Least_real: ( real > $o ) > real ).
thf(func_def_645,type,
ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(func_def_646,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_647,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_648,type,
ord_le7432393201274343594_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > $o ).
thf(func_def_649,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_650,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(func_def_651,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(func_def_652,type,
ord_less_int: int > int > $o ).
thf(func_def_653,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_654,type,
ord_less_num: num > num > $o ).
thf(func_def_655,type,
ord_less_rat: rat > rat > $o ).
thf(func_def_656,type,
ord_less_real: real > real > $o ).
thf(func_def_657,type,
ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_658,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(func_def_659,type,
ord_less_set_int: set_int > set_int > $o ).
thf(func_def_660,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_661,type,
ord_less_set_num: set_num > set_num > $o ).
thf(func_def_662,type,
ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_663,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(func_def_664,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_665,type,
ord_less_set_set_int: set_set_int > set_set_int > $o ).
thf(func_def_666,type,
ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
thf(func_def_667,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_668,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_669,type,
ord_le7858099551454983350_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > $o ).
thf(func_def_670,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_671,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(func_def_672,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(func_def_673,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(func_def_674,type,
ord_le4104064031414453916r_real: filter_real > filter_real > $o ).
thf(func_def_675,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_676,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_677,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_678,type,
ord_less_eq_rat: rat > rat > $o ).
thf(func_def_679,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_680,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(func_def_681,type,
ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).
thf(func_def_682,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(func_def_683,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(func_def_684,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(func_def_685,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_686,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(func_def_687,type,
ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_688,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(func_def_689,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_690,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(func_def_691,type,
ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_692,type,
ord_max_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_693,type,
ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_694,type,
ord_max_int: int > int > int ).
thf(func_def_695,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_696,type,
ord_max_num: num > num > num ).
thf(func_def_697,type,
ord_max_rat: rat > rat > rat ).
thf(func_def_698,type,
ord_max_real: real > real > real ).
thf(func_def_699,type,
ord_max_set_int: set_int > set_int > set_int ).
thf(func_def_700,type,
ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_701,type,
ord_min_nat: nat > nat > nat ).
thf(func_def_702,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(func_def_703,type,
order_9091379641038594480t_real: ( nat > real ) > $o ).
thf(func_def_704,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(func_def_705,type,
order_mono_nat_real: ( nat > real ) > $o ).
thf(func_def_706,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(func_def_707,type,
order_7092887310737990675l_real: ( real > real ) > $o ).
thf(func_def_708,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(func_def_709,type,
top_top_set_o: set_o ).
thf(func_def_710,type,
top_top_set_nat: set_nat ).
thf(func_def_711,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(func_def_712,type,
top_top_set_real: set_real ).
thf(func_def_713,type,
top_top_set_char: set_char ).
thf(func_def_714,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(func_def_715,type,
power_power_complex: complex > nat > complex ).
thf(func_def_716,type,
power_power_int: int > nat > int ).
thf(func_def_717,type,
power_power_nat: nat > nat > nat ).
thf(func_def_718,type,
power_power_rat: rat > nat > rat ).
thf(func_def_719,type,
power_power_real: real > nat > real ).
thf(func_def_720,type,
produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
thf(func_def_721,type,
produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
thf(func_def_722,type,
produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
thf(func_def_723,type,
produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
thf(func_def_724,type,
produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
thf(func_def_725,type,
produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
thf(func_def_726,type,
product_Pair_o_o: $o > $o > product_prod_o_o ).
thf(func_def_727,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(func_def_728,type,
product_Pair_o_nat: $o > nat > product_prod_o_nat ).
thf(func_def_729,type,
produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
thf(func_def_730,type,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(func_def_731,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_732,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(func_def_733,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_734,type,
product_Pair_nat_num: nat > num > product_prod_nat_num ).
thf(func_def_735,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(func_def_736,type,
produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
thf(func_def_737,type,
produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
thf(func_def_738,type,
produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
thf(func_def_739,type,
produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
thf(func_def_740,type,
produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
thf(func_def_741,type,
produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
thf(func_def_742,type,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
thf(func_def_743,type,
produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
thf(func_def_744,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_745,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(func_def_746,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
thf(func_def_747,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(func_def_748,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
thf(func_def_749,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_750,type,
produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
thf(func_def_751,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(func_def_752,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_753,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_754,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_755,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_756,type,
produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
thf(func_def_757,type,
produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
thf(func_def_758,type,
produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
thf(func_def_759,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_760,type,
produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
thf(func_def_761,type,
produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
thf(func_def_762,type,
produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
thf(func_def_763,type,
produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
thf(func_def_764,type,
produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
thf(func_def_765,type,
product_fst_int_int: product_prod_int_int > int ).
thf(func_def_766,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_767,type,
product_snd_int_int: product_prod_int_int > int ).
thf(func_def_768,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_769,type,
abs_Rat: product_prod_int_int > rat ).
thf(func_def_770,type,
fract: int > int > rat ).
thf(func_def_771,type,
frct: product_prod_int_int > rat ).
thf(func_def_772,type,
rep_Rat: rat > product_prod_int_int ).
thf(func_def_773,type,
field_5140801741446780682s_real: set_real ).
thf(func_def_774,type,
field_7254667332652039916t_real: rat > real ).
thf(func_def_775,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(func_def_776,type,
pcr_rat: product_prod_int_int > rat > $o ).
thf(func_def_777,type,
positive: rat > $o ).
thf(func_def_778,type,
quotient_of: rat > product_prod_int_int ).
thf(func_def_779,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_780,type,
real2: ( nat > rat ) > real ).
thf(func_def_781,type,
cauchy: ( nat > rat ) > $o ).
thf(func_def_782,type,
cr_real: ( nat > rat ) > real > $o ).
thf(func_def_783,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(func_def_784,type,
positive2: real > $o ).
thf(func_def_785,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(func_def_786,type,
rep_real: real > nat > rat ).
thf(func_def_787,type,
vanishes: ( nat > rat ) > $o ).
thf(func_def_788,type,
real_V2521375963428798218omplex: set_complex ).
thf(func_def_789,type,
real_V5970128139526366754l_real: ( real > real ) > $o ).
thf(func_def_790,type,
real_V3694042436643373181omplex: complex > complex > real ).
thf(func_def_791,type,
real_V975177566351809787t_real: real > real > real ).
thf(func_def_792,type,
real_V1022390504157884413omplex: complex > real ).
thf(func_def_793,type,
real_V7735802525324610683m_real: real > real ).
thf(func_def_794,type,
real_V4546457046886955230omplex: real > complex ).
thf(func_def_795,type,
real_V2046097035970521341omplex: real > complex > complex ).
thf(func_def_796,type,
real_V1485227260804924795R_real: real > real > real ).
thf(func_def_797,type,
transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
thf(func_def_798,type,
algebr932160517623751201me_int: int > int > $o ).
thf(func_def_799,type,
algebr934650988132801477me_nat: nat > nat > $o ).
thf(func_def_800,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(func_def_801,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(func_def_802,type,
divide_divide_int: int > int > int ).
thf(func_def_803,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_804,type,
divide_divide_rat: rat > rat > rat ).
thf(func_def_805,type,
divide_divide_real: real > real > real ).
thf(func_def_806,type,
dvd_dvd_Code_integer: code_integer > code_integer > $o ).
thf(func_def_807,type,
dvd_dvd_complex: complex > complex > $o ).
thf(func_def_808,type,
dvd_dvd_int: int > int > $o ).
thf(func_def_809,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_810,type,
dvd_dvd_rat: rat > rat > $o ).
thf(func_def_811,type,
dvd_dvd_real: real > real > $o ).
thf(func_def_812,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(func_def_813,type,
modulo_modulo_int: int > int > int ).
thf(func_def_814,type,
modulo_modulo_nat: nat > nat > nat ).
thf(func_def_815,type,
zero_n356916108424825756nteger: $o > code_integer ).
thf(func_def_816,type,
zero_n1201886186963655149omplex: $o > complex ).
thf(func_def_817,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(func_def_818,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(func_def_819,type,
zero_n2052037380579107095ol_rat: $o > rat ).
thf(func_def_820,type,
zero_n3304061248610475627l_real: $o > real ).
thf(func_def_821,type,
suminf_complex: ( nat > complex ) > complex ).
thf(func_def_822,type,
suminf_int: ( nat > int ) > int ).
thf(func_def_823,type,
suminf_nat: ( nat > nat ) > nat ).
thf(func_def_824,type,
suminf_real: ( nat > real ) > real ).
thf(func_def_825,type,
summable_complex: ( nat > complex ) > $o ).
thf(func_def_826,type,
summable_int: ( nat > int ) > $o ).
thf(func_def_827,type,
summable_nat: ( nat > nat ) > $o ).
thf(func_def_828,type,
summable_real: ( nat > real ) > $o ).
thf(func_def_829,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(func_def_830,type,
sums_int: ( nat > int ) > int > $o ).
thf(func_def_831,type,
sums_nat: ( nat > nat ) > nat > $o ).
thf(func_def_832,type,
sums_real: ( nat > real ) > real > $o ).
thf(func_def_833,type,
collect_o: ( $o > $o ) > set_o ).
thf(func_def_834,type,
collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
thf(func_def_835,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(func_def_836,type,
collect_int: ( int > $o ) > set_int ).
thf(func_def_837,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(func_def_838,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
thf(func_def_839,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(func_def_840,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(func_def_841,type,
collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
thf(func_def_842,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_843,type,
collect_num: ( num > $o ) > set_num ).
thf(func_def_844,type,
collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
thf(func_def_845,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(func_def_846,type,
collec7088162979684241874at_nat: ( produc859450856879609959at_nat > $o ) > set_Pr8693737435421807431at_nat ).
thf(func_def_847,type,
collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
thf(func_def_848,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(func_def_849,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_850,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(func_def_851,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(func_def_852,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_853,type,
collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).
thf(func_def_854,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_855,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(func_def_856,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_857,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(func_def_858,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(func_def_859,type,
image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
thf(func_def_860,type,
image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
thf(func_def_861,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_862,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(func_def_863,type,
insert_int: int > set_int > set_int ).
thf(func_def_864,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_865,type,
insert_real: real > set_real > set_real ).
thf(func_def_866,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_867,type,
set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
thf(func_def_868,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(func_def_869,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(func_def_870,type,
set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
thf(func_def_871,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(func_def_872,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(func_def_873,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(func_def_874,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(func_def_875,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(func_def_876,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(func_def_877,type,
set_or370866239135849197et_int: set_int > set_int > set_set_int ).
thf(func_def_878,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(func_def_879,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(func_def_880,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(func_def_881,type,
set_ord_atLeast_real: real > set_real ).
thf(func_def_882,type,
set_ord_atMost_int: int > set_int ).
thf(func_def_883,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_884,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(func_def_885,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(func_def_886,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(func_def_887,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(func_def_888,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(func_def_889,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(func_def_890,type,
set_or5849166863359141190n_real: real > set_real ).
thf(func_def_891,type,
set_ord_lessThan_int: int > set_int ).
thf(func_def_892,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_893,type,
set_ord_lessThan_num: num > set_num ).
thf(func_def_894,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(func_def_895,type,
set_or5984915006950818249n_real: real > set_real ).
thf(func_def_896,type,
ascii_of: char > char ).
thf(func_def_897,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(func_def_898,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(func_def_899,type,
integer_of_char: char > code_integer ).
thf(func_def_900,type,
unique3096191561947761185of_nat: nat > char ).
thf(func_def_901,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(func_def_902,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(func_def_903,type,
topolo4899668324122417113eq_int: ( nat > int ) > $o ).
thf(func_def_904,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(func_def_905,type,
topolo1459490580787246023eq_num: ( nat > num ) > $o ).
thf(func_def_906,type,
topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
thf(func_def_907,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(func_def_908,type,
topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
thf(func_def_909,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(func_def_910,type,
topolo7531315842566124627t_real: ( nat > real ) > $o ).
thf(func_def_911,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(func_def_912,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(func_def_913,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
thf(func_def_914,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
thf(func_def_915,type,
arccos: real > real ).
thf(func_def_916,type,
arcosh_real: real > real ).
thf(func_def_917,type,
arcsin: real > real ).
thf(func_def_918,type,
arctan: real > real ).
thf(func_def_919,type,
arsinh_real: real > real ).
thf(func_def_920,type,
artanh_real: real > real ).
thf(func_def_921,type,
cos_complex: complex > complex ).
thf(func_def_922,type,
cos_real: real > real ).
thf(func_def_923,type,
cos_coeff: nat > real ).
thf(func_def_924,type,
cosh_real: real > real ).
thf(func_def_925,type,
cot_real: real > real ).
thf(func_def_926,type,
diffs_complex: ( nat > complex ) > nat > complex ).
thf(func_def_927,type,
diffs_int: ( nat > int ) > nat > int ).
thf(func_def_928,type,
diffs_rat: ( nat > rat ) > nat > rat ).
thf(func_def_929,type,
diffs_real: ( nat > real ) > nat > real ).
thf(func_def_930,type,
exp_complex: complex > complex ).
thf(func_def_931,type,
exp_real: real > real ).
thf(func_def_932,type,
ln_ln_real: real > real ).
thf(func_def_933,type,
log: real > real > real ).
thf(func_def_934,type,
pi: real ).
thf(func_def_935,type,
powr_real: real > real > real ).
thf(func_def_936,type,
sin_complex: complex > complex ).
thf(func_def_937,type,
sin_real: real > real ).
thf(func_def_938,type,
sin_coeff: nat > real ).
thf(func_def_939,type,
sinh_real: real > real ).
thf(func_def_940,type,
tan_real: real > real ).
thf(func_def_941,type,
tanh_real: real > real ).
thf(func_def_942,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(func_def_943,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_944,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(func_def_945,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(func_def_946,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(func_def_947,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_948,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(func_def_949,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(func_def_950,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_951,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(func_def_952,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_953,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(func_def_954,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_955,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(func_def_956,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_957,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(func_def_958,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(func_def_959,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_960,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_961,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(func_def_962,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(func_def_963,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_964,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_965,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(func_def_966,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_967,type,
vEBT_VEBT_add: option_nat > option_nat > option_nat ).
thf(func_def_968,type,
vEBT_VEBT_greater: option_nat > option_nat > $o ).
thf(func_def_969,type,
vEBT_VEBT_less: option_nat > option_nat > $o ).
thf(func_def_970,type,
vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
thf(func_def_971,type,
vEBT_VEBT_max_in_set: set_nat > nat > $o ).
thf(func_def_972,type,
vEBT_VEBT_min_in_set: set_nat > nat > $o ).
thf(func_def_973,type,
vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
thf(func_def_974,type,
vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
thf(func_def_975,type,
vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
thf(func_def_976,type,
vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
thf(func_def_977,type,
vEBT_VEBT_power: option_nat > option_nat > option_nat ).
thf(func_def_978,type,
vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
thf(func_def_979,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_980,type,
vEBT_vebt_mint: vEBT_VEBT > option_nat ).
thf(func_def_981,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_982,type,
vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
thf(func_def_983,type,
vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
thf(func_def_984,type,
vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_985,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(func_def_986,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_987,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_988,type,
accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
thf(func_def_989,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(func_def_990,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(func_def_991,type,
fChoice_real: ( real > $o ) > real ).
thf(func_def_992,type,
member_o: $o > set_o > $o ).
thf(func_def_993,type,
member_complex: complex > set_complex > $o ).
thf(func_def_994,type,
member_int: int > set_int > $o ).
thf(func_def_995,type,
member_list_o: list_o > set_list_o > $o ).
thf(func_def_996,type,
member_list_int: list_int > set_list_int > $o ).
thf(func_def_997,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(func_def_998,type,
member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
thf(func_def_999,type,
member_nat: nat > set_nat > $o ).
thf(func_def_1000,type,
member_num: num > set_num > $o ).
thf(func_def_1001,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_1002,type,
member_rat: rat > set_rat > $o ).
thf(func_def_1003,type,
member_real: real > set_real > $o ).
thf(func_def_1004,type,
member_set_int: set_int > set_set_int > $o ).
thf(func_def_1005,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_1006,type,
deg: nat ).
thf(func_def_1007,type,
m: nat ).
thf(func_def_1008,type,
ma: nat ).
thf(func_def_1009,type,
mi: nat ).
thf(func_def_1010,type,
na: nat ).
thf(func_def_1011,type,
sc: nat ).
thf(func_def_1012,type,
summary: vEBT_VEBT ).
thf(func_def_1013,type,
treeList: list_VEBT_VEBT ).
thf(func_def_1014,type,
xa: nat ).
thf(func_def_1033,type,
sK0: nat ).
thf(func_def_1034,type,
sK1: nat > nat > nat ).
thf(func_def_1035,type,
sK2: set_nat > nat > nat ).
thf(func_def_1036,type,
sP3: nat > $o ).
thf(func_def_1037,type,
sP4: $o > $o ).
thf(func_def_1038,type,
sP5: nat > $o ).
thf(func_def_1039,type,
sP6: nat > $o ).
thf(func_def_1040,type,
sP7: nat > $o ).
thf(func_def_1041,type,
sP8: nat > $o ).
thf(func_def_1042,type,
sP9: option_nat > $o ).
thf(func_def_1043,type,
ph10:
!>[X0: $tType] : X0 ).
thf(f30628,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f30627]) ).
thf(f30627,plain,
$true = $false,
inference(backward_demodulation,[status(thm)],[f30579,f30626]) ).
thf(f30626,plain,
( ( sP4 @ $true )
= $false ),
inference(boolean_simplification,[status(thm)],[f30624]) ).
thf(f30624,plain,
( $false
= ( sP4
@ ( ( some_nat @ sc )
= ( some_nat @ sc ) ) ) ),
inference(superposition,[status(thm)],[f30617,f30599]) ).
thf(f30599,plain,
( ( some_nat @ sc )
= ( vEBT_vebt_succ @ summary @ ( divide_divide_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f30514]) ).
thf(f30514,plain,
( ( some_nat @ sc )
= ( vEBT_vebt_succ @ summary
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ xa
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f30387,f30432]) ).
thf(f30432,plain,
( vEBT_VEBT_high
= ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f10262]) ).
thf(f10262,plain,
( vEBT_VEBT_high
= ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f10261]) ).
thf(f10261,plain,
( ( ^ [X0: nat,X1: nat] : ( divide_divide_nat @ X0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X1 ) ) )
= vEBT_VEBT_high ),
inference(rectify,[status(thm)],[f11]) ).
thf(f11,axiom,
( ( ^ [X5: nat,X6: nat] : ( divide_divide_nat @ X5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X6 ) ) )
= vEBT_VEBT_high ),
file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_10_high__def) ).
thf(f30387,plain,
( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( some_nat @ sc ) ),
inference(cnf_transformation,[status(thm)],[f21923]) ).
thf(f21923,plain,
( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( some_nat @ sc ) ),
inference(fool_elimination,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
= ( some_nat @ sc ) ),
file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_1__092_060open_062vebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_092_060close_062) ).
thf(f30617,plain,
( $false
= ( sP4
@ ( ( vEBT_vebt_succ @ summary @ ( divide_divide_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
= ( some_nat @ sc ) ) ) ),
inference(superposition,[status(thm)],[f30607,f30592]) ).
thf(f30592,plain,
! [X0: nat,X1: nat] :
( ( ( member_nat @ X1 @ ( collect_nat @ ( vEBT_vebt_member @ summary ) ) )
& ( ord_less_nat @ X0 @ X1 )
& ( !! @ nat
@ ^ [Y0: nat] :
( ( member_nat @ Y0 @ ( collect_nat @ ( vEBT_vebt_member @ summary ) ) )
=> ( ( ord_less_nat @ X0 @ Y0 )
=> ( ord_less_eq_nat @ X1 @ Y0 ) ) ) ) )
= ( ( vEBT_vebt_succ @ summary @ X0 )
= ( some_nat @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f30549]) ).
thf(f30549,plain,
! [X0: nat,X1: nat] :
( ( ( vEBT_vebt_succ @ summary @ X0 )
= ( some_nat @ X1 ) )
= ( ^ [Y0: set_nat,Y1: nat,Y2: nat] :
( ( member_nat @ Y2 @ Y0 )
& ( ord_less_nat @ Y1 @ Y2 )
& ( !! @ nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ Y0 )
=> ( ( ord_less_nat @ Y1 @ Y3 )
=> ( ord_less_eq_nat @ Y2 @ Y3 ) ) ) ) )
@ ( ^ [Y0: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ Y0 ) )
@ summary )
@ X0
@ X1 ) ),
inference(definition_unfolding,[status(thm)],[f30458,f30500,f30501]) ).
thf(f30501,plain,
( vEBT_VEBT_set_vebt
= ( ^ [Y0: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ Y0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f19206]) ).
thf(f19206,plain,
( vEBT_VEBT_set_vebt
= ( ^ [Y0: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ Y0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f19205]) ).
thf(f19205,plain,
( ( ^ [X0: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ X0 ) ) )
= vEBT_VEBT_set_vebt ),
inference(rectify,[status(thm)],[f3177]) ).
thf(f3177,axiom,
( ( ^ [X170: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ X170 ) ) )
= vEBT_VEBT_set_vebt ),
file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_3176_set__vebt_H__def) ).
thf(f30500,plain,
( vEBT_is_succ_in_set
= ( ^ [Y0: set_nat,Y1: nat,Y2: nat] :
( ( member_nat @ Y2 @ Y0 )
& ( ord_less_nat @ Y1 @ Y2 )
& ( !! @ nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ Y0 )
=> ( ( ord_less_nat @ Y1 @ Y3 )
=> ( ord_less_eq_nat @ Y2 @ Y3 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f21418]) ).
thf(f21418,plain,
( vEBT_is_succ_in_set
= ( ^ [Y0: set_nat,Y1: nat,Y2: nat] :
( ( member_nat @ Y2 @ Y0 )
& ( ord_less_nat @ Y1 @ Y2 )
& ( !! @ nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ Y0 )
=> ( ( ord_less_nat @ Y1 @ Y3 )
=> ( ord_less_eq_nat @ Y2 @ Y3 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f21417]) ).
thf(f21417,plain,
( ( ^ [X0: set_nat,X1: nat,X2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ X0 )
=> ( ( ord_less_nat @ X1 @ X3 )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) )
& ( ord_less_nat @ X1 @ X2 )
& ( member_nat @ X2 @ X0 ) ) )
= vEBT_is_succ_in_set ),
inference(rectify,[status(thm)],[f230]) ).
thf(f230,axiom,
( ( ^ [X10: set_nat,X5: nat,X11: nat] :
( ! [X29: nat] :
( ( member_nat @ X29 @ X10 )
=> ( ( ord_less_nat @ X5 @ X29 )
=> ( ord_less_eq_nat @ X11 @ X29 ) ) )
& ( ord_less_nat @ X5 @ X11 )
& ( member_nat @ X11 @ X10 ) ) )
= vEBT_is_succ_in_set ),
file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_229_is__succ__in__set__def) ).
thf(f30458,plain,
! [X0: nat,X1: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X0 @ X1 )
= ( ( vEBT_vebt_succ @ summary @ X0 )
= ( some_nat @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f26650]) ).
thf(f26650,plain,
! [X0: nat,X1: nat] :
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X0 @ X1 )
= ( ( vEBT_vebt_succ @ summary @ X0 )
= ( some_nat @ X1 ) ) ),
inference(fool_elimination,[status(thm)],[f1]) ).
thf(f1,axiom,
! [X0: nat,X1: nat] :
( ( ( vEBT_vebt_succ @ summary @ X0 )
= ( some_nat @ X1 ) )
= ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X0 @ X1 ) ),
file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_0__C4_Ohyps_C_I3_J) ).
thf(f30607,plain,
( ( sP4
@ ( ( member_nat @ sc @ ( collect_nat @ ( vEBT_vebt_member @ summary ) ) )
& ( ord_less_nat @ ( divide_divide_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ sc )
& ( !! @ nat
@ ^ [Y0: nat] :
( ( member_nat @ Y0 @ ( collect_nat @ ( vEBT_vebt_member @ summary ) ) )
=> ( ( ord_less_nat @ ( divide_divide_nat @ xa @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Y0 )
=> ( ord_less_eq_nat @ sc @ Y0 ) ) ) ) ) )
= $false ),
inference(beta_eta_normalization,[status(thm)],[f30578]) ).
thf(f30578,plain,
( $false
= ( sP4
@ ( ^ [Y0: set_nat,Y1: nat,Y2: nat] :
( ( member_nat @ Y2 @ Y0 )
& ( ord_less_nat @ Y1 @ Y2 )
& ( !! @ nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ Y0 )
=> ( ( ord_less_nat @ Y1 @ Y3 )
=> ( ord_less_eq_nat @ Y2 @ Y3 ) ) ) ) )
@ ( ^ [Y0: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ Y0 ) )
@ summary )
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ xa
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
@ sc ) ) ),
introduced(definition,[new_symbols(naming,[sP4])],[inequality_splitting_name_introduction]) ).
thf(f30579,plain,
( $true
= ( sP4 @ $true ) ),
inference(inequality_splitting,[status(thm)],[f30509,f30578]) ).
thf(f30509,plain,
( $true
!= ( ^ [Y0: set_nat,Y1: nat,Y2: nat] :
( ( member_nat @ Y2 @ Y0 )
& ( ord_less_nat @ Y1 @ Y2 )
& ( !! @ nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ Y0 )
=> ( ( ord_less_nat @ Y1 @ Y3 )
=> ( ord_less_eq_nat @ Y2 @ Y3 ) ) ) ) )
@ ( ^ [Y0: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ Y0 ) )
@ summary )
@ ( ^ [Y0: nat,Y1: nat] : ( divide_divide_nat @ Y0 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y1 ) )
@ xa
@ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
@ sc ) ),
inference(definition_unfolding,[status(thm)],[f30378,f30500,f30501,f30432]) ).
thf(f30378,plain,
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc )
!= $true ),
inference(cnf_transformation,[status(thm)],[f30237]) ).
thf(f30237,plain,
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc )
!= $true ),
inference(flattening,[status(thm)],[f12065]) ).
thf(f12065,plain,
( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc )
!= $true ),
inference(fool_elimination,[status(thm)],[f12064]) ).
thf(f12064,plain,
~ ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ),
inference(rectify,[status(thm)],[f10257]) ).
thf(f10257,negated_conjecture,
~ ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ),
inference(negated_conjecture,[status(cth)],[f10256]) ).
thf(f10256,conjecture,
vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc,
file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP243^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.35 % Computer : n008.cluster.edu
% 0.16/0.35 % Model : x86_64 x86_64
% 0.16/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.35 % Memory : 8042.1875MB
% 0.16/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.35 % CPULimit : 300
% 0.16/0.35 % WCLimit : 300
% 0.16/0.35 % DateTime : Thu Feb 26 20:02:54 EST 2026
% 0.16/0.35 % CPUTime :
% 0.16/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.47 ---- Original DTF file ---
% 0.20/0.48 thf(spec,logic,$$dhol).
% 0.20/0.48 %------------------------------------------------------------------------------
% 0.20/0.48 % File : ITP243^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.20/0.48 % Domain : Interactive Theorem Proving
% 0.20/0.48 % Problem : Sledgehammer problem VEBT_Succ 00352_020667
% 0.20/0.48 % Version : [Des22] axioms.
% 0.20/0.48 % English :
% 0.20/0.48
% 0.20/0.48 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.48 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.20/0.48 % Source : [Des22]
% 0.20/0.48 % Names : 0070_VEBT_Succ_00352_020667 [Des22]
% 0.20/0.48
% 0.20/0.48 % Status : Theorem
% 0.20/0.48 % Rating : ? v9.2.1
% 0.20/0.48 % Syntax : Number of formulae : 11271 (5697 unt;1015 typ; 0 def)
% 0.20/0.48 % Number of atoms : 28678 (12041 equ; 0 cnn)
% 0.20/0.48 % Maximal formula atoms : 71 ( 2 avg)
% 0.20/0.48 % Number of connectives : 112193 (2670 ~; 531 |;1772 &;96387 @)
% 0.20/0.48 % ( 0 <=>;10833 =>; 0 <=; 0 <~>)
% 0.20/0.48 % Maximal formula depth : 39 ( 6 avg)
% 0.20/0.48 % Number of types : 90 ( 89 usr)
% 0.20/0.48 % Number of type conns : 4291 (4291 >; 0 *; 0 +; 0 <<)
% 0.20/0.48 % Number of symbols : 929 ( 926 usr; 58 con; 0-8 aty)
% 0.20/0.48 % Number of variables : 25678 (2245 ^;22702 !; 731 ?;25678 :)
% 0.20/0.48 % SPC : TH0_THM_EQU_NAR
% 0.20/0.48
% 0.20/0.48 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.20/0.48 % from the van Emde Boas Trees session in the Archive of Formal
% 0.20/0.48 % proofs -
% 0.20/0.48 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.20/0.48 % 2022-02-18 00:20:26.849
% 0.20/0.48 % Bugfixes : Added ()s around negated sides of equations.
% 0.20/0.48 %------------------------------------------------------------------------------
% 0.20/0.48 % Could-be-implicit typings (89)
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% 0.20/0.48 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,
% 0.20/0.48 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 ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
% 0.20/0.48
% 0.20/0.48 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 0.20/0.48 bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 0.20/0.48
% 0.20/0.48 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 0.20/0.48 bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 0.20/0.48 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 0.20/0.48
% 0.20/0.48 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,
% 0.20/0.48 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 0.32/0.48 bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.48 bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 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 ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Binomial_Obinomial,type,
% 0.32/0.48 binomial: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 0.32/0.48 bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 0.32/0.48 bit_and_not_num: num > num > option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 0.32/0.48 bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 0.32/0.48 bit_concat_bit: nat > int > int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 0.32/0.48 bit_or_not_num_neg: num > num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 0.32/0.48 bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 0.32/0.48 bit_ri7919022796975470100ot_int: int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
% 0.32/0.48 bit_ri631733984087533419it_int: nat > int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
% 0.32/0.48 bit_se725231765392027082nd_int: int > int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
% 0.32/0.48 bit_se727722235901077358nd_nat: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
% 0.32/0.48 bit_se8568078237143864401it_int: nat > int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
% 0.32/0.48 bit_se8570568707652914677it_nat: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
% 0.32/0.48 bit_se2159334234014336723it_int: nat > int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
% 0.32/0.48 bit_se2161824704523386999it_nat: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
% 0.32/0.48 bit_se2000444600071755411sk_int: nat > int ).
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% 0.32/0.48 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.32/0.48 princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.48 finite_card_list_nat: set_list_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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% 0.32/0.48 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.32/0.48 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 0.32/0.48 bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.32/0.48 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.32/0.48
% 0.32/0.48 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,
% 0.32/0.48 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
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% 0.32/0.48 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Fun_Omap__fun_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.32/0.48 map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).
% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48
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% 0.32/0.48 thf(sy_c_Fun__Def_Opair__less,type,
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% 0.32/0.48 thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
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% 0.32/0.48 thf(sy_c_GCD_Obezw,type,
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% 0.32/0.48 thf(sy_c_GCD_Obezw__rel,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
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% 0.32/0.48 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
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% 0.32/0.48 thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_If_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_If_001t__Num__Onum,type,
% 0.32/0.48 if_num: $o > num > num > num ).
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% 0.32/0.48 thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.32/0.48 if_option_nat: $o > option_nat > option_nat > option_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_If_001t__Option__Ooption_It__Num__Onum_J,type,
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% 0.32/0.48
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% 0.32/0.48 thf(sy_c_If_001t__Real__Oreal,type,
% 0.32/0.48 if_real: $o > real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
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% 0.32/0.48 thf(sy_c_If_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.48 if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
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% 0.32/0.48
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% 0.32/0.48 abs_Integ: product_prod_nat_nat > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Int_ORep__Integ,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Int_Oint__ge__less__than2,type,
% 0.32/0.48 int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Int_Ointrel,type,
% 0.32/0.48 intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
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% 0.32/0.48 thf(sy_c_Int_Onat,type,
% 0.32/0.48 nat2: int > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Int_Opcr__int,type,
% 0.32/0.48 pcr_int: product_prod_nat_nat > int > $o ).
% 0.32/0.48
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% 0.32/0.48 power_int_real: real > int > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
% 0.32/0.48 ring_1_Ints_real: set_real ).
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% 0.32/0.48 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
% 0.32/0.48 lattic8265883725875713057ax_nat: set_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
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% 0.32/0.48
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% 0.32/0.48 at_infinity_real: filter_real ).
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% 0.32/0.48 thf(sy_c_List_Oappend_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 0.32/0.48 drop_nat: nat > list_nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olast_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 0.32/0.48 linord2614967742042102400et_nat: set_nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 0.32/0.48 cons_int: int > list_int > list_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 0.32/0.48 cons_nat: nat > list_nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 0.32/0.48 hd_nat: list_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_Oset_001_Eo,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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% 0.32/0.48 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist__update_001_Eo,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist__update_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
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% 0.32/0.48 thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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% 0.32/0.48 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Onth_001t__Int__Oint,type,
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% 0.32/0.48 thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Onth_001t__Nat__Onat,type,
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% 0.32/0.48 thf(sy_c_List_Onth_001t__Num__Onum,type,
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% 0.32/0.48 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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% 0.32/0.48 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Onth_001t__Real__Oreal,type,
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% 0.32/0.48 thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
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% 0.32/0.48 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
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% 0.32/0.48 thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
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% 0.32/0.48 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 0.32/0.48 replicate_int: nat > int > list_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 0.32/0.48 replicate_nat: nat > nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.48 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 0.32/0.48 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 0.32/0.48 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 0.32/0.48 take_nat: nat > list_nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oupt,type,
% 0.32/0.48 upt: nat > nat > list_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oupto,type,
% 0.32/0.48 upto: int > int > list_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oupto__aux,type,
% 0.32/0.48 upto_aux: int > int > list_int > list_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_List_Oupto__rel,type,
% 0.32/0.48 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_OSuc,type,
% 0.32/0.48 suc: nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.32/0.48 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.32/0.48 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.32/0.48 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Onat_Opred,type,
% 0.32/0.48 pred: nat > nat ).
% 0.32/0.48
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% 0.32/0.48 semiri4939895301339042750nteger: nat > code_integer ).
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% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
% 0.32/0.48 semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
% 0.32/0.48 semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.32/0.48 size_size_list_o: list_o > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
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% 0.32/0.48 size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
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% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 0.32/0.48 size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
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% 0.32/0.48 size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.32/0.48 size_size_list_real: list_real > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.32/0.48 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 0.32/0.48 size_size_num: num > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.32/0.48 size_size_option_nat: option_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.32/0.48 size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
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% 0.32/0.48 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.32/0.48 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oprod__encode,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oset__decode,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.32/0.48 nat_set_encode: set_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.32/0.48 nat_triangle: nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_NthRoot_Oroot,type,
% 0.32/0.48 root: nat > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_NthRoot_Osqrt,type,
% 0.32/0.48 sqrt: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_OBitM,type,
% 0.32/0.48 bitM: num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oinc,type,
% 0.32/0.48 inc: num > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 0.32/0.48 neg_nu7009210354673126013omplex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 0.32/0.48 neg_numeral_dbl_int: int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 0.32/0.48 neg_numeral_dbl_rat: rat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 0.32/0.48 neg_numeral_dbl_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 0.32/0.48 neg_nu6511756317524482435omplex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 0.32/0.48 neg_nu3811975205180677377ec_int: int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
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% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
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% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 0.32/0.48 neg_nu8557863876264182079omplex: complex > complex ).
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% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
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% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
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% 0.32/0.48 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
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% 0.32/0.48 thf(sy_c_Num_Onum_OBit0,type,
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% 0.32/0.48 thf(sy_c_Num_Onum_OOne,type,
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% 0.32/0.48 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
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% 0.32/0.48 thf(sy_c_Num_Onum_Osize__num,type,
% 0.32/0.48 size_num: num > nat ).
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% 0.32/0.48 thf(sy_c_Num_Onum__of__nat,type,
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% 0.32/0.48 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 power_8256067586552552935nteger: code_integer > nat > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
% 0.32/0.48 power_power_complex: complex > nat > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 0.32/0.48 power_power_int: int > nat > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 0.32/0.48 power_power_nat: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
% 0.32/0.48 power_power_rat: rat > nat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
% 0.32/0.48 power_power_real: real > nat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 0.32/0.48 produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 0.32/0.48 produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 0.32/0.48 produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 0.32/0.48 produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.32/0.48 produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.32/0.48 produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
% 0.32/0.48 product_Pair_o_o: $o > $o > product_prod_o_o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 0.32/0.48 product_Pair_o_int: $o > int > product_prod_o_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
% 0.32/0.48 product_Pair_o_nat: $o > nat > product_prod_o_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.48 produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
% 0.32/0.48 produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.48 product_Pair_int_int: int > int > product_prod_int_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
% 0.32/0.48 product_Pair_nat_num: nat > num > product_prod_nat_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
% 0.32/0.48 product_Pair_num_num: num > num > product_prod_num_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.32/0.48 produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.32/0.48 produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.32/0.48 produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 0.32/0.48 produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 0.32/0.48 produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 0.32/0.48 produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.48 produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 0.32/0.48 produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 0.32/0.48 produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 0.32/0.48 produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 0.32/0.48 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.32/0.48 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 0.32/0.48 produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 0.32/0.48 produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.48 produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 0.32/0.48 produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.32/0.48 produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 0.32/0.48 produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 0.32/0.48 produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 0.32/0.48 produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 0.32/0.48 produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 0.32/0.48 produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 0.32/0.48 produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 0.32/0.48 produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 0.32/0.48 produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.48 product_fst_int_int: product_prod_int_int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 product_fst_nat_nat: product_prod_nat_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 0.32/0.48 product_snd_int_int: product_prod_int_int > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.32/0.48 product_snd_nat_nat: product_prod_nat_nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_OAbs__Rat,type,
% 0.32/0.48 abs_Rat: product_prod_int_int > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_OFract,type,
% 0.32/0.48 fract: int > int > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_OFrct,type,
% 0.32/0.48 frct: product_prod_int_int > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_ORep__Rat,type,
% 0.32/0.48 rep_Rat: rat > product_prod_int_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 0.32/0.48 field_5140801741446780682s_real: set_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 0.32/0.48 field_7254667332652039916t_real: rat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Onormalize,type,
% 0.32/0.48 normalize: product_prod_int_int > product_prod_int_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Opcr__rat,type,
% 0.32/0.48 pcr_rat: product_prod_int_int > rat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Opositive,type,
% 0.32/0.48 positive: rat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Oquotient__of,type,
% 0.32/0.48 quotient_of: rat > product_prod_int_int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Rat_Oratrel,type,
% 0.32/0.48 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_OReal,type,
% 0.32/0.48 real2: ( nat > rat ) > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Ocauchy,type,
% 0.32/0.48 cauchy: ( nat > rat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Ocr__real,type,
% 0.32/0.48 cr_real: ( nat > rat ) > real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Opcr__real,type,
% 0.32/0.48 pcr_real: ( nat > rat ) > real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Opositive,type,
% 0.32/0.48 positive2: real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Orealrel,type,
% 0.32/0.48 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Orep__real,type,
% 0.32/0.48 rep_real: real > nat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real_Ovanishes,type,
% 0.32/0.48 vanishes: ( nat > rat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 0.32/0.48 real_V2521375963428798218omplex: set_complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.48 real_V5970128139526366754l_real: ( real > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 0.32/0.48 real_V3694042436643373181omplex: complex > complex > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 0.32/0.48 real_V975177566351809787t_real: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 0.32/0.48 real_V1022390504157884413omplex: complex > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
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% 0.32/0.48
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 0.32/0.48 set_ord_lessThan_rat: rat > set_rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
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% 0.32/0.48 ascii_of: char > char ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ochar_OChar,type,
% 0.32/0.48 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 0.32/0.48 comm_s629917340098488124ar_nat: char > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ointeger__of__char,type,
% 0.32/0.48 integer_of_char: char > code_integer ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 0.32/0.48 unique3096191561947761185of_nat: nat > char ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.48 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 0.32/0.48 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 0.32/0.48 topolo4899668324122417113eq_int: ( nat > int ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 0.32/0.48 topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
% 0.32/0.48 topolo1459490580787246023eq_num: ( nat > num ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
% 0.32/0.48 topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 0.32/0.48 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
% 0.32/0.48 topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
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% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 0.32/0.48 topolo2815343760600316023s_real: real > filter_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 0.32/0.48 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 0.32/0.48 topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 0.32/0.48 topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarccos,type,
% 0.32/0.48 arccos: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 0.32/0.48 arcosh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarcsin,type,
% 0.32/0.48 arcsin: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarctan,type,
% 0.32/0.48 arctan: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 0.32/0.48 arsinh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 0.32/0.48 artanh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 0.32/0.48 cos_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 0.32/0.48 cos_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocos__coeff,type,
% 0.32/0.48 cos_coeff: nat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 0.32/0.48 cosh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 0.32/0.48 cot_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 0.32/0.48 diffs_complex: ( nat > complex ) > nat > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
% 0.32/0.48 diffs_int: ( nat > int ) > nat > int ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
% 0.32/0.48 diffs_rat: ( nat > rat ) > nat > rat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 0.32/0.48 diffs_real: ( nat > real ) > nat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 0.32/0.48 exp_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 0.32/0.48 exp_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 0.32/0.48 ln_ln_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Olog,type,
% 0.32/0.48 log: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Opi,type,
% 0.32/0.48 pi: real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 0.32/0.48 powr_real: real > real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 0.32/0.48 sin_complex: complex > complex ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 0.32/0.48 sin_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osin__coeff,type,
% 0.32/0.48 sin_coeff: nat > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 0.32/0.48 sinh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 0.32/0.48 tan_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 0.32/0.48 tanh_real: real > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 0.32/0.48 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 0.32/0.48 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 0.32/0.48 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 0.32/0.48 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 0.32/0.48 vEBT_VEBT_high: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 0.32/0.48 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 0.32/0.48 vEBT_VEBT_low: nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 0.32/0.48 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 0.32/0.48 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 0.32/0.48 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 0.32/0.48 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 0.32/0.48 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 0.32/0.48 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 0.32/0.48 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 0.32/0.48 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 0.32/0.48 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 0.32/0.48 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 0.32/0.48 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 0.32/0.48 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 0.32/0.48 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 0.32/0.48 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 0.32/0.48 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 0.32/0.48 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 0.32/0.48 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 0.32/0.48 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 0.32/0.48 vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 0.32/0.48 vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 0.32/0.48 vEBT_VEBT_less: option_nat > option_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 0.32/0.48 vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 0.32/0.48 vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 0.32/0.48 vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 0.32/0.48 vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 0.32/0.48 vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 0.32/0.48 vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 0.32/0.48 vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 0.32/0.48 vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 0.32/0.48 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 0.32/0.48 vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 0.32/0.48 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 0.32/0.48 vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 0.32/0.48 vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 0.32/0.48 vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 0.32/0.48 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.32/0.48 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.32/0.48 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 0.32/0.48 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.32/0.48 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.48 accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_fChoice_001t__Real__Oreal,type,
% 0.32/0.48 fChoice_real: ( real > $o ) > real ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001_Eo,type,
% 0.32/0.48 member_o: $o > set_o > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 0.32/0.48 member_complex: complex > set_complex > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Int__Oint,type,
% 0.32/0.48 member_int: int > set_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 0.32/0.48 member_list_o: list_o > set_list_o > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 0.32/0.48 member_list_int: list_int > set_list_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 0.32/0.48 member_list_nat: list_nat > set_list_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.32/0.48 member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Nat__Onat,type,
% 0.32/0.48 member_nat: nat > set_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Num__Onum,type,
% 0.32/0.48 member_num: num > set_num > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.32/0.48 member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Rat__Orat,type,
% 0.32/0.48 member_rat: rat > set_rat > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Real__Oreal,type,
% 0.32/0.48 member_real: real > set_real > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 0.32/0.48 member_set_int: set_int > set_set_int > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.32/0.48 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_deg____,type,
% 0.32/0.48 deg: nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_m____,type,
% 0.32/0.48 m: nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_ma____,type,
% 0.32/0.48 ma: nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_mi____,type,
% 0.32/0.48 mi: nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_na____,type,
% 0.32/0.48 na: nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_sc____,type,
% 0.32/0.48 sc: nat ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_summary____,type,
% 0.32/0.48 summary: vEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_treeList____,type,
% 0.32/0.48 treeList: list_VEBT_VEBT ).
% 0.32/0.48
% 0.32/0.48 thf(sy_v_xa____,type,
% 0.32/0.48 xa: nat ).
% 0.32/0.48
% 0.32/0.48 % Relevant facts (10209)
% 0.32/0.48 thf(fact_0__C4_Ohyps_C_I3_J,axiom,
% 0.32/0.48 ! [X: nat,Sx: nat] :
% 0.32/0.48 ( ( ( vEBT_vebt_succ @ summary @ X )
% 0.32/0.48 = ( some_nat @ Sx ) )
% 0.32/0.48 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X @ Sx ) ) ).
% 0.32/0.48
% 0.32/0.48 % "4.hyps"(3)
% 0.32/0.48 thf(fact_1__092_060open_062vebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_092_060close_062,axiom,
% 0.32/0.48 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.48 = ( some_nat @ sc ) ) ).
% 0.32/0.48
% 0.32/0.48 % \<open>vebt_succ summary (high x (deg div 2)) = Some sc\<close>
% 0.32/0.48 thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062sc_O_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.32/0.48 ~ ! [Sc: nat] :
% 0.32/0.48 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.48 != ( some_nat @ Sc ) ) ).
% 0.32/0.48
% 0.32/0.48 % \<open>\<And>thesis. (\<And>sc. vebt_succ summary (high x (deg div 2)) = Some sc \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.32/0.48 thf(fact_3_False,axiom,
% 0.32/0.48 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.48 != none_nat ) ).
% 0.32/0.48
% 0.32/0.48 % False
% 0.32/0.48 thf(fact_4__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 0.32/0.48 ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = na ) ).
% 0.32/0.48
% 0.32/0.48 % \<open>deg div 2 = n\<close>
% 0.32/0.48 thf(fact_5_semiring__norm_I85_J,axiom,
% 0.32/0.48 ! [M: num] :
% 0.32/0.48 ( ( bit0 @ M )
% 0.32/0.48 != one ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(85)
% 0.32/0.48 thf(fact_6_semiring__norm_I83_J,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( one
% 0.32/0.48 != ( bit0 @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(83)
% 0.32/0.48 thf(fact_7__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
% 0.32/0.48 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 0.32/0.48
% 0.32/0.48 % \<open>2 \<le> deg\<close>
% 0.32/0.48 thf(fact_8_numeral__Bit0__div__2,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_Bit0_div_2
% 0.32/0.48 thf(fact_9_numeral__Bit0__div__2,axiom,
% 0.32/0.48 ! [N: num] :
% 0.32/0.48 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.48 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_Bit0_div_2
% 0.32/0.48 thf(fact_10_high__def,axiom,
% 0.32/0.48 ( vEBT_VEBT_high
% 0.32/0.48 = ( ^ [X2: nat,N2: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % high_def
% 0.32/0.48 thf(fact_11_divide__numeral__1,axiom,
% 0.32/0.48 ! [A: complex] :
% 0.32/0.48 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % divide_numeral_1
% 0.32/0.48 thf(fact_12_divide__numeral__1,axiom,
% 0.32/0.48 ! [A: real] :
% 0.32/0.48 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % divide_numeral_1
% 0.32/0.48 thf(fact_13_divide__numeral__1,axiom,
% 0.32/0.48 ! [A: rat] :
% 0.32/0.48 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.32/0.48 = A ) ).
% 0.32/0.48
% 0.32/0.48 % divide_numeral_1
% 0.32/0.48 thf(fact_14_verit__eq__simplify_I8_J,axiom,
% 0.32/0.48 ! [X22: num,Y2: num] :
% 0.32/0.48 ( ( ( bit0 @ X22 )
% 0.32/0.48 = ( bit0 @ Y2 ) )
% 0.32/0.48 = ( X22 = Y2 ) ) ).
% 0.32/0.48
% 0.32/0.48 % verit_eq_simplify(8)
% 0.32/0.48 thf(fact_15_semiring__norm_I87_J,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( bit0 @ M )
% 0.32/0.48 = ( bit0 @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % semiring_norm(87)
% 0.32/0.48 thf(fact_16_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numera6690914467698888265omplex @ M )
% 0.32/0.48 = ( numera6690914467698888265omplex @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_17_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numeral_numeral_real @ M )
% 0.32/0.48 = ( numeral_numeral_real @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_18_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numeral_numeral_rat @ M )
% 0.32/0.48 = ( numeral_numeral_rat @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_19_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numeral_numeral_nat @ M )
% 0.32/0.48 = ( numeral_numeral_nat @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_20_numeral__eq__iff,axiom,
% 0.32/0.48 ! [M: num,N: num] :
% 0.32/0.48 ( ( ( numeral_numeral_int @ M )
% 0.32/0.48 = ( numeral_numeral_int @ N ) )
% 0.32/0.48 = ( M = N ) ) ).
% 0.32/0.48
% 0.32/0.48 % numeral_eq_iff
% 0.32/0.48 thf(fact_21__C4_Ohyps_C_I5_J,axiom,
% 0.32/0.48 m = na ).
% 0.32/0.48
% 0.32/0.48 % "4.hyps"(5)
% 0.32/0.48 thf(fact_22_min__in__set__def,axiom,
% 0.32/0.48 ( vEBT_VEBT_min_in_set
% 0.32/0.48 = ( ^ [Xs: set_nat,X2: nat] :
% 0.32/0.48 ( ( member_nat @ X2 @ Xs )
% 0.32/0.48 & ! [Y: nat] :
% 0.32/0.48 ( ( member_nat @ Y @ Xs )
% 0.32/0.48 => ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 % min_in_set_def
% 0.32/0.49 thf(fact_23_max__in__set__def,axiom,
% 0.32/0.49 ( vEBT_VEBT_max_in_set
% 0.32/0.49 = ( ^ [Xs: set_nat,X2: nat] :
% 0.32/0.49 ( ( member_nat @ X2 @ Xs )
% 0.32/0.49 & ! [Y: nat] :
% 0.32/0.49 ( ( member_nat @ Y @ Xs )
% 0.32/0.49 => ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % max_in_set_def
% 0.32/0.49 thf(fact_24_power__shift,axiom,
% 0.32/0.49 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.49 ( ( ( power_power_nat @ X @ Y3 )
% 0.32/0.49 = Z )
% 0.32/0.49 = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y3 ) )
% 0.32/0.49 = ( some_nat @ Z ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_shift
% 0.32/0.49 thf(fact_25_numeral__le__iff,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_le_iff
% 0.32/0.49 thf(fact_26_numeral__le__iff,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_le_iff
% 0.32/0.49 thf(fact_27_numeral__le__iff,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_le_iff
% 0.32/0.49 thf(fact_28_numeral__le__iff,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_le_iff
% 0.32/0.49 thf(fact_29_lesseq__shift,axiom,
% 0.32/0.49 ( ord_less_eq_nat
% 0.32/0.49 = ( ^ [X2: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % lesseq_shift
% 0.32/0.49 thf(fact_30_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.49 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(2)
% 0.32/0.49 thf(fact_31_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.49 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(2)
% 0.32/0.49 thf(fact_32_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.49 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(2)
% 0.32/0.49 thf(fact_33_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.49 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(2)
% 0.32/0.49 thf(fact_34_verit__comp__simplify1_I2_J,axiom,
% 0.32/0.49 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(2)
% 0.32/0.49 thf(fact_35_verit__la__disequality,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 | ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.49 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_la_disequality
% 0.32/0.49 thf(fact_36_verit__la__disequality,axiom,
% 0.32/0.49 ! [A: num,B: num] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 | ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.49 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_la_disequality
% 0.32/0.49 thf(fact_37_verit__la__disequality,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 | ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.49 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_la_disequality
% 0.32/0.49 thf(fact_38_verit__la__disequality,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 | ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.49 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_la_disequality
% 0.32/0.49 thf(fact_39_div__le__dividend,axiom,
% 0.32/0.49 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.32/0.49
% 0.32/0.49 % div_le_dividend
% 0.32/0.49 thf(fact_40_div__le__mono,axiom,
% 0.32/0.49 ! [M: nat,N: nat,K: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.49 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_le_mono
% 0.32/0.49 thf(fact_41_verit__eq__simplify_I10_J,axiom,
% 0.32/0.49 ! [X22: num] :
% 0.32/0.49 ( one
% 0.32/0.49 != ( bit0 @ X22 ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_eq_simplify(10)
% 0.32/0.49 thf(fact_42__092_060open_062mi_A_092_060le_062_Ax_092_060close_062,axiom,
% 0.32/0.49 ord_less_eq_nat @ mi @ xa ).
% 0.32/0.49
% 0.32/0.49 % \<open>mi \<le> x\<close>
% 0.32/0.49 thf(fact_43_not__None__eq,axiom,
% 0.32/0.49 ! [X: option_nat] :
% 0.32/0.49 ( ( X != none_nat )
% 0.32/0.49 = ( ? [Y: nat] :
% 0.32/0.49 ( X
% 0.32/0.49 = ( some_nat @ Y ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % not_None_eq
% 0.32/0.49 thf(fact_44_not__None__eq,axiom,
% 0.32/0.49 ! [X: option4927543243414619207at_nat] :
% 0.32/0.49 ( ( X != none_P5556105721700978146at_nat )
% 0.32/0.49 = ( ? [Y: product_prod_nat_nat] :
% 0.32/0.49 ( X
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % not_None_eq
% 0.32/0.49 thf(fact_45_not__None__eq,axiom,
% 0.32/0.49 ! [X: option_num] :
% 0.32/0.49 ( ( X != none_num )
% 0.32/0.49 = ( ? [Y: num] :
% 0.32/0.49 ( X
% 0.32/0.49 = ( some_num @ Y ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % not_None_eq
% 0.32/0.49 thf(fact_46_not__Some__eq,axiom,
% 0.32/0.49 ! [X: option_nat] :
% 0.32/0.49 ( ( ! [Y: nat] :
% 0.32/0.49 ( X
% 0.32/0.49 != ( some_nat @ Y ) ) )
% 0.32/0.49 = ( X = none_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % not_Some_eq
% 0.32/0.49 thf(fact_47_not__Some__eq,axiom,
% 0.32/0.49 ! [X: option4927543243414619207at_nat] :
% 0.32/0.49 ( ( ! [Y: product_prod_nat_nat] :
% 0.32/0.49 ( X
% 0.32/0.49 != ( some_P7363390416028606310at_nat @ Y ) ) )
% 0.32/0.49 = ( X = none_P5556105721700978146at_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % not_Some_eq
% 0.32/0.49 thf(fact_48_not__Some__eq,axiom,
% 0.32/0.49 ! [X: option_num] :
% 0.32/0.49 ( ( ! [Y: num] :
% 0.32/0.49 ( X
% 0.32/0.49 != ( some_num @ Y ) ) )
% 0.32/0.49 = ( X = none_num ) ) ).
% 0.32/0.49
% 0.32/0.49 % not_Some_eq
% 0.32/0.49 thf(fact_49_power2__nat__le__imp__le,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_nat_le_imp_le
% 0.32/0.49 thf(fact_50_power2__nat__le__eq__le,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.49 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_nat_le_eq_le
% 0.32/0.49 thf(fact_51_self__le__ge2__pow,axiom,
% 0.32/0.49 ! [K: nat,M: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % self_le_ge2_pow
% 0.32/0.49 thf(fact_52_power__minus__is__div,axiom,
% 0.32/0.49 ! [B: nat,A: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.49 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.49 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_minus_is_div
% 0.32/0.49 thf(fact_53_pow__sum,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.49 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % pow_sum
% 0.32/0.49 thf(fact_54__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
% 0.32/0.49 ord_less_eq_nat @ one_one_nat @ na ).
% 0.32/0.49
% 0.32/0.49 % \<open>1 \<le> n\<close>
% 0.32/0.49 thf(fact_55__C4_Ohyps_C_I10_J,axiom,
% 0.32/0.49 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.32/0.49
% 0.32/0.49 % "4.hyps"(10)
% 0.32/0.49 thf(fact_56_option_Oinject,axiom,
% 0.32/0.49 ! [X22: nat,Y2: nat] :
% 0.32/0.49 ( ( ( some_nat @ X22 )
% 0.32/0.49 = ( some_nat @ Y2 ) )
% 0.32/0.49 = ( X22 = Y2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.inject
% 0.32/0.49 thf(fact_57_option_Oinject,axiom,
% 0.32/0.49 ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 0.32/0.49 ( ( ( some_P7363390416028606310at_nat @ X22 )
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ Y2 ) )
% 0.32/0.49 = ( X22 = Y2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.inject
% 0.32/0.49 thf(fact_58_option_Oinject,axiom,
% 0.32/0.49 ! [X22: num,Y2: num] :
% 0.32/0.49 ( ( ( some_num @ X22 )
% 0.32/0.49 = ( some_num @ Y2 ) )
% 0.32/0.49 = ( X22 = Y2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.inject
% 0.32/0.49 thf(fact_59_zdiv__numeral__Bit0,axiom,
% 0.32/0.49 ! [V: num,W: num] :
% 0.32/0.49 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.32/0.49 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % zdiv_numeral_Bit0
% 0.32/0.49 thf(fact_60__C4_Ohyps_C_I9_J,axiom,
% 0.32/0.49 ord_less_eq_nat @ mi @ ma ).
% 0.32/0.49
% 0.32/0.49 % "4.hyps"(9)
% 0.32/0.49 thf(fact_61__C4_Ohyps_C_I6_J,axiom,
% 0.32/0.49 ( deg
% 0.32/0.49 = ( plus_plus_nat @ na @ m ) ) ).
% 0.32/0.49
% 0.32/0.49 % "4.hyps"(6)
% 0.32/0.49 thf(fact_62_high__bound__aux,axiom,
% 0.32/0.49 ! [Ma: nat,N: nat,M: nat] :
% 0.32/0.49 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.32/0.49 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % high_bound_aux
% 0.32/0.49 thf(fact_63_numeral__less__iff,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_less_iff
% 0.32/0.49 thf(fact_64_numeral__less__iff,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_less_iff
% 0.32/0.49 thf(fact_65_numeral__less__iff,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_less_iff
% 0.32/0.49 thf(fact_66_numeral__less__iff,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_less_iff
% 0.32/0.49 thf(fact_67_add__numeral__left,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: complex] :
% 0.32/0.49 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.32/0.49 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_numeral_left
% 0.32/0.49 thf(fact_68_add__numeral__left,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: real] :
% 0.32/0.49 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.32/0.49 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_numeral_left
% 0.32/0.49 thf(fact_69_add__numeral__left,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: rat] :
% 0.32/0.49 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.32/0.49 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_numeral_left
% 0.32/0.49 thf(fact_70_add__numeral__left,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: nat] :
% 0.32/0.49 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.32/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_numeral_left
% 0.32/0.49 thf(fact_71_add__numeral__left,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: int] :
% 0.32/0.49 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.32/0.49 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_numeral_left
% 0.32/0.49 thf(fact_72_numeral__plus__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_numeral
% 0.32/0.49 thf(fact_73_numeral__plus__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_numeral
% 0.32/0.49 thf(fact_74_numeral__plus__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_numeral
% 0.32/0.49 thf(fact_75_numeral__plus__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_numeral
% 0.32/0.49 thf(fact_76_numeral__plus__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_numeral
% 0.32/0.49 thf(fact_77_mem__Collect__eq,axiom,
% 0.32/0.49 ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
% 0.32/0.49 ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
% 0.32/0.49 = ( P @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % mem_Collect_eq
% 0.32/0.49 thf(fact_78_mem__Collect__eq,axiom,
% 0.32/0.49 ! [A: complex,P: complex > $o] :
% 0.32/0.49 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 0.32/0.49 = ( P @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % mem_Collect_eq
% 0.32/0.49 thf(fact_79_mem__Collect__eq,axiom,
% 0.32/0.49 ! [A: real,P: real > $o] :
% 0.32/0.49 ( ( member_real @ A @ ( collect_real @ P ) )
% 0.32/0.49 = ( P @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % mem_Collect_eq
% 0.32/0.49 thf(fact_80_mem__Collect__eq,axiom,
% 0.32/0.49 ! [A: list_nat,P: list_nat > $o] :
% 0.32/0.49 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.32/0.49 = ( P @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % mem_Collect_eq
% 0.32/0.49 thf(fact_81_mem__Collect__eq,axiom,
% 0.32/0.49 ! [A: nat,P: nat > $o] :
% 0.32/0.49 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.32/0.49 = ( P @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % mem_Collect_eq
% 0.32/0.49 thf(fact_82_mem__Collect__eq,axiom,
% 0.32/0.49 ! [A: int,P: int > $o] :
% 0.32/0.49 ( ( member_int @ A @ ( collect_int @ P ) )
% 0.32/0.49 = ( P @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % mem_Collect_eq
% 0.32/0.49 thf(fact_83_Collect__mem__eq,axiom,
% 0.32/0.49 ! [A2: set_Pr8693737435421807431at_nat] :
% 0.32/0.49 ( ( collec7088162979684241874at_nat
% 0.32/0.49 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A2 ) )
% 0.32/0.49 = A2 ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_mem_eq
% 0.32/0.49 thf(fact_84_Collect__mem__eq,axiom,
% 0.32/0.49 ! [A2: set_complex] :
% 0.32/0.49 ( ( collect_complex
% 0.32/0.49 @ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
% 0.32/0.49 = A2 ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_mem_eq
% 0.32/0.49 thf(fact_85_Collect__mem__eq,axiom,
% 0.32/0.49 ! [A2: set_real] :
% 0.32/0.49 ( ( collect_real
% 0.32/0.49 @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
% 0.32/0.49 = A2 ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_mem_eq
% 0.32/0.49 thf(fact_86_Collect__mem__eq,axiom,
% 0.32/0.49 ! [A2: set_list_nat] :
% 0.32/0.49 ( ( collect_list_nat
% 0.32/0.49 @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A2 ) )
% 0.32/0.49 = A2 ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_mem_eq
% 0.32/0.49 thf(fact_87_Collect__mem__eq,axiom,
% 0.32/0.49 ! [A2: set_nat] :
% 0.32/0.49 ( ( collect_nat
% 0.32/0.49 @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
% 0.32/0.49 = A2 ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_mem_eq
% 0.32/0.49 thf(fact_88_Collect__mem__eq,axiom,
% 0.32/0.49 ! [A2: set_int] :
% 0.32/0.49 ( ( collect_int
% 0.32/0.49 @ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
% 0.32/0.49 = A2 ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_mem_eq
% 0.32/0.49 thf(fact_89_Collect__cong,axiom,
% 0.32/0.49 ! [P: complex > $o,Q: complex > $o] :
% 0.32/0.49 ( ! [X3: complex] :
% 0.32/0.49 ( ( P @ X3 )
% 0.32/0.49 = ( Q @ X3 ) )
% 0.32/0.49 => ( ( collect_complex @ P )
% 0.32/0.49 = ( collect_complex @ Q ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_cong
% 0.32/0.49 thf(fact_90_Collect__cong,axiom,
% 0.32/0.49 ! [P: real > $o,Q: real > $o] :
% 0.32/0.49 ( ! [X3: real] :
% 0.32/0.49 ( ( P @ X3 )
% 0.32/0.49 = ( Q @ X3 ) )
% 0.32/0.49 => ( ( collect_real @ P )
% 0.32/0.49 = ( collect_real @ Q ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_cong
% 0.32/0.49 thf(fact_91_Collect__cong,axiom,
% 0.32/0.49 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.32/0.49 ( ! [X3: list_nat] :
% 0.32/0.49 ( ( P @ X3 )
% 0.32/0.49 = ( Q @ X3 ) )
% 0.32/0.49 => ( ( collect_list_nat @ P )
% 0.32/0.49 = ( collect_list_nat @ Q ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_cong
% 0.32/0.49 thf(fact_92_Collect__cong,axiom,
% 0.32/0.49 ! [P: nat > $o,Q: nat > $o] :
% 0.32/0.49 ( ! [X3: nat] :
% 0.32/0.49 ( ( P @ X3 )
% 0.32/0.49 = ( Q @ X3 ) )
% 0.32/0.49 => ( ( collect_nat @ P )
% 0.32/0.49 = ( collect_nat @ Q ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_cong
% 0.32/0.49 thf(fact_93_Collect__cong,axiom,
% 0.32/0.49 ! [P: int > $o,Q: int > $o] :
% 0.32/0.49 ( ! [X3: int] :
% 0.32/0.49 ( ( P @ X3 )
% 0.32/0.49 = ( Q @ X3 ) )
% 0.32/0.49 => ( ( collect_int @ P )
% 0.32/0.49 = ( collect_int @ Q ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Collect_cong
% 0.32/0.49 thf(fact_94_power__one,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( power_power_rat @ one_one_rat @ N )
% 0.32/0.49 = one_one_rat ) ).
% 0.32/0.49
% 0.32/0.49 % power_one
% 0.32/0.49 thf(fact_95_power__one,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( power_power_nat @ one_one_nat @ N )
% 0.32/0.49 = one_one_nat ) ).
% 0.32/0.49
% 0.32/0.49 % power_one
% 0.32/0.49 thf(fact_96_power__one,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( power_power_real @ one_one_real @ N )
% 0.32/0.49 = one_one_real ) ).
% 0.32/0.49
% 0.32/0.49 % power_one
% 0.32/0.49 thf(fact_97_power__one,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( power_power_int @ one_one_int @ N )
% 0.32/0.49 = one_one_int ) ).
% 0.32/0.49
% 0.32/0.49 % power_one
% 0.32/0.49 thf(fact_98_power__one,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( power_power_complex @ one_one_complex @ N )
% 0.32/0.49 = one_one_complex ) ).
% 0.32/0.49
% 0.32/0.49 % power_one
% 0.32/0.49 thf(fact_99_power__one__right,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( power_power_nat @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % power_one_right
% 0.32/0.49 thf(fact_100_power__one__right,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( power_power_real @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % power_one_right
% 0.32/0.49 thf(fact_101_power__one__right,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( power_power_int @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % power_one_right
% 0.32/0.49 thf(fact_102_power__one__right,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( power_power_complex @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % power_one_right
% 0.32/0.49 thf(fact_103_semiring__norm_I71_J,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.49 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % semiring_norm(71)
% 0.32/0.49 thf(fact_104_semiring__norm_I68_J,axiom,
% 0.32/0.49 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.32/0.49
% 0.32/0.49 % semiring_norm(68)
% 0.32/0.49 thf(fact_105_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numera6690914467698888265omplex @ N )
% 0.32/0.49 = one_one_complex )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_106_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numeral_numeral_real @ N )
% 0.32/0.49 = one_one_real )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_107_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numeral_numeral_rat @ N )
% 0.32/0.49 = one_one_rat )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_108_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numeral_numeral_nat @ N )
% 0.32/0.49 = one_one_nat )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_109_numeral__eq__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ( numeral_numeral_int @ N )
% 0.32/0.49 = one_one_int )
% 0.32/0.49 = ( N = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_eq_one_iff
% 0.32/0.49 thf(fact_110_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_complex
% 0.32/0.49 = ( numera6690914467698888265omplex @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_111_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_real
% 0.32/0.49 = ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_112_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_rat
% 0.32/0.49 = ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_113_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_nat
% 0.32/0.49 = ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_114_one__eq__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( one_one_int
% 0.32/0.49 = ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( one = N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_eq_numeral_iff
% 0.32/0.49 thf(fact_115_power__inject__exp,axiom,
% 0.32/0.49 ! [A: real,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.49 => ( ( ( power_power_real @ A @ M )
% 0.32/0.49 = ( power_power_real @ A @ N ) )
% 0.32/0.49 = ( M = N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_inject_exp
% 0.32/0.49 thf(fact_116_power__inject__exp,axiom,
% 0.32/0.49 ! [A: rat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.49 => ( ( ( power_power_rat @ A @ M )
% 0.32/0.49 = ( power_power_rat @ A @ N ) )
% 0.32/0.49 = ( M = N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_inject_exp
% 0.32/0.49 thf(fact_117_power__inject__exp,axiom,
% 0.32/0.49 ! [A: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.49 => ( ( ( power_power_nat @ A @ M )
% 0.32/0.49 = ( power_power_nat @ A @ N ) )
% 0.32/0.49 = ( M = N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_inject_exp
% 0.32/0.49 thf(fact_118_power__inject__exp,axiom,
% 0.32/0.49 ! [A: int,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.49 => ( ( ( power_power_int @ A @ M )
% 0.32/0.49 = ( power_power_int @ A @ N ) )
% 0.32/0.49 = ( M = N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_inject_exp
% 0.32/0.49 thf(fact_119_semiring__norm_I69_J,axiom,
% 0.32/0.49 ! [M: num] :
% 0.32/0.49 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.32/0.49
% 0.32/0.49 % semiring_norm(69)
% 0.32/0.49 thf(fact_120_one__less__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( ord_less_num @ one @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_less_numeral_iff
% 0.32/0.49 thf(fact_121_one__less__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( ord_less_num @ one @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_less_numeral_iff
% 0.32/0.49 thf(fact_122_one__less__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( ord_less_num @ one @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_less_numeral_iff
% 0.32/0.49 thf(fact_123_one__less__numeral__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( ord_less_num @ one @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_less_numeral_iff
% 0.32/0.49 thf(fact_124_numeral__plus__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 0.32/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_one
% 0.32/0.49 thf(fact_125_numeral__plus__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.32/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_one
% 0.32/0.49 thf(fact_126_numeral__plus__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.32/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_one
% 0.32/0.49 thf(fact_127_numeral__plus__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.32/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_one
% 0.32/0.49 thf(fact_128_numeral__plus__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.32/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_plus_one
% 0.32/0.49 thf(fact_129_one__plus__numeral,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.49 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral
% 0.32/0.49 thf(fact_130_one__plus__numeral,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral
% 0.32/0.49 thf(fact_131_one__plus__numeral,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral
% 0.32/0.49 thf(fact_132_one__plus__numeral,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral
% 0.32/0.49 thf(fact_133_one__plus__numeral,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral
% 0.32/0.49 thf(fact_134_power__strict__increasing__iff,axiom,
% 0.32/0.49 ! [B: real,X: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.49 => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y3 ) )
% 0.32/0.49 = ( ord_less_nat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_strict_increasing_iff
% 0.32/0.49 thf(fact_135_power__strict__increasing__iff,axiom,
% 0.32/0.49 ! [B: rat,X: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ B )
% 0.32/0.49 => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y3 ) )
% 0.32/0.49 = ( ord_less_nat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_strict_increasing_iff
% 0.32/0.49 thf(fact_136_power__strict__increasing__iff,axiom,
% 0.32/0.49 ! [B: nat,X: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ B )
% 0.32/0.49 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y3 ) )
% 0.32/0.49 = ( ord_less_nat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_strict_increasing_iff
% 0.32/0.49 thf(fact_137_power__strict__increasing__iff,axiom,
% 0.32/0.49 ! [B: int,X: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ B )
% 0.32/0.49 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y3 ) )
% 0.32/0.49 = ( ord_less_nat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_strict_increasing_iff
% 0.32/0.49 thf(fact_138_greater__shift,axiom,
% 0.32/0.49 ( ord_less_nat
% 0.32/0.49 = ( ^ [Y: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % greater_shift
% 0.32/0.49 thf(fact_139_one__add__one,axiom,
% 0.32/0.49 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 0.32/0.49 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_add_one
% 0.32/0.49 thf(fact_140_one__add__one,axiom,
% 0.32/0.49 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 0.32/0.49 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_add_one
% 0.32/0.49 thf(fact_141_one__add__one,axiom,
% 0.32/0.49 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 0.32/0.49 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_add_one
% 0.32/0.49 thf(fact_142_one__add__one,axiom,
% 0.32/0.49 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.32/0.49 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_add_one
% 0.32/0.49 thf(fact_143_one__add__one,axiom,
% 0.32/0.49 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 0.32/0.49 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_add_one
% 0.32/0.49 thf(fact_144_power__increasing__iff,axiom,
% 0.32/0.49 ! [B: real,X: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.49 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y3 ) )
% 0.32/0.49 = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_increasing_iff
% 0.32/0.49 thf(fact_145_power__increasing__iff,axiom,
% 0.32/0.49 ! [B: rat,X: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ B )
% 0.32/0.49 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y3 ) )
% 0.32/0.49 = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_increasing_iff
% 0.32/0.49 thf(fact_146_power__increasing__iff,axiom,
% 0.32/0.49 ! [B: nat,X: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ B )
% 0.32/0.49 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y3 ) )
% 0.32/0.49 = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_increasing_iff
% 0.32/0.49 thf(fact_147_power__increasing__iff,axiom,
% 0.32/0.49 ! [B: int,X: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ B )
% 0.32/0.49 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y3 ) )
% 0.32/0.49 = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_increasing_iff
% 0.32/0.49 thf(fact_148_add__self__div__2,axiom,
% 0.32/0.49 ! [M: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = M ) ).
% 0.32/0.49
% 0.32/0.49 % add_self_div_2
% 0.32/0.49 thf(fact_149_numeral__le__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.32/0.49 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_le_one_iff
% 0.32/0.49 thf(fact_150_numeral__le__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.32/0.49 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_le_one_iff
% 0.32/0.49 thf(fact_151_numeral__le__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.32/0.49 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_le_one_iff
% 0.32/0.49 thf(fact_152_numeral__le__one__iff,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.32/0.49 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_le_one_iff
% 0.32/0.49 thf(fact_153__C4_Ohyps_C_I2_J,axiom,
% 0.32/0.49 vEBT_invar_vebt @ summary @ m ).
% 0.32/0.49
% 0.32/0.49 % "4.hyps"(2)
% 0.32/0.49 thf(fact_154_local_Opower__def,axiom,
% 0.32/0.49 ( vEBT_VEBT_power
% 0.32/0.49 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % local.power_def
% 0.32/0.49 thf(fact_155_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ~ ( ord_less_real @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(1)
% 0.32/0.49 thf(fact_156_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ~ ( ord_less_rat @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(1)
% 0.32/0.49 thf(fact_157_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.49 ! [A: num] :
% 0.32/0.49 ~ ( ord_less_num @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(1)
% 0.32/0.49 thf(fact_158_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ~ ( ord_less_nat @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(1)
% 0.32/0.49 thf(fact_159_verit__comp__simplify1_I1_J,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ~ ( ord_less_int @ A @ A ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(1)
% 0.32/0.49 thf(fact_160_is__num__normalize_I1_J,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_num_normalize(1)
% 0.32/0.49 thf(fact_161_is__num__normalize_I1_J,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_num_normalize(1)
% 0.32/0.49 thf(fact_162_is__num__normalize_I1_J,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_num_normalize(1)
% 0.32/0.49 thf(fact_163_less__numeral__extra_I4_J,axiom,
% 0.32/0.49 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.32/0.49
% 0.32/0.49 % less_numeral_extra(4)
% 0.32/0.49 thf(fact_164_less__numeral__extra_I4_J,axiom,
% 0.32/0.49 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.32/0.49
% 0.32/0.49 % less_numeral_extra(4)
% 0.32/0.49 thf(fact_165_less__numeral__extra_I4_J,axiom,
% 0.32/0.49 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.32/0.49
% 0.32/0.49 % less_numeral_extra(4)
% 0.32/0.49 thf(fact_166_less__numeral__extra_I4_J,axiom,
% 0.32/0.49 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.32/0.49
% 0.32/0.49 % less_numeral_extra(4)
% 0.32/0.49 thf(fact_167_power__less__imp__less__exp,axiom,
% 0.32/0.49 ! [A: real,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.49 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.32/0.49 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_less_imp_less_exp
% 0.32/0.49 thf(fact_168_power__less__imp__less__exp,axiom,
% 0.32/0.49 ! [A: rat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.49 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.32/0.49 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_less_imp_less_exp
% 0.32/0.49 thf(fact_169_power__less__imp__less__exp,axiom,
% 0.32/0.49 ! [A: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.49 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.49 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_less_imp_less_exp
% 0.32/0.49 thf(fact_170_power__less__imp__less__exp,axiom,
% 0.32/0.49 ! [A: int,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.49 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.32/0.49 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_less_imp_less_exp
% 0.32/0.49 thf(fact_171_power__strict__increasing,axiom,
% 0.32/0.49 ! [N: nat,N3: nat,A: real] :
% 0.32/0.49 ( ( ord_less_nat @ N @ N3 )
% 0.32/0.49 => ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.49 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_strict_increasing
% 0.32/0.49 thf(fact_172_power__strict__increasing,axiom,
% 0.32/0.49 ! [N: nat,N3: nat,A: rat] :
% 0.32/0.49 ( ( ord_less_nat @ N @ N3 )
% 0.32/0.49 => ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.49 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_strict_increasing
% 0.32/0.49 thf(fact_173_power__strict__increasing,axiom,
% 0.32/0.49 ! [N: nat,N3: nat,A: nat] :
% 0.32/0.49 ( ( ord_less_nat @ N @ N3 )
% 0.32/0.49 => ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.49 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_strict_increasing
% 0.32/0.49 thf(fact_174_power__strict__increasing,axiom,
% 0.32/0.49 ! [N: nat,N3: nat,A: int] :
% 0.32/0.49 ( ( ord_less_nat @ N @ N3 )
% 0.32/0.49 => ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.49 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_strict_increasing
% 0.32/0.49 thf(fact_175_discrete,axiom,
% 0.32/0.49 ( ord_less_nat
% 0.32/0.49 = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % discrete
% 0.32/0.49 thf(fact_176_discrete,axiom,
% 0.32/0.49 ( ord_less_int
% 0.32/0.49 = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % discrete
% 0.32/0.49 thf(fact_177_one__plus__numeral__commute,axiom,
% 0.32/0.49 ! [X: num] :
% 0.32/0.49 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 0.32/0.49 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral_commute
% 0.32/0.49 thf(fact_178_one__plus__numeral__commute,axiom,
% 0.32/0.49 ! [X: num] :
% 0.32/0.49 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.32/0.49 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral_commute
% 0.32/0.49 thf(fact_179_one__plus__numeral__commute,axiom,
% 0.32/0.49 ! [X: num] :
% 0.32/0.49 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.32/0.49 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral_commute
% 0.32/0.49 thf(fact_180_one__plus__numeral__commute,axiom,
% 0.32/0.49 ! [X: num] :
% 0.32/0.49 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.32/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral_commute
% 0.32/0.49 thf(fact_181_one__plus__numeral__commute,axiom,
% 0.32/0.49 ! [X: num] :
% 0.32/0.49 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.32/0.49 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_plus_numeral_commute
% 0.32/0.49 thf(fact_182_not__numeral__less__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.32/0.49
% 0.32/0.49 % not_numeral_less_one
% 0.32/0.49 thf(fact_183_not__numeral__less__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.32/0.49
% 0.32/0.49 % not_numeral_less_one
% 0.32/0.49 thf(fact_184_not__numeral__less__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.32/0.49
% 0.32/0.49 % not_numeral_less_one
% 0.32/0.49 thf(fact_185_not__numeral__less__one,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.32/0.49
% 0.32/0.49 % not_numeral_less_one
% 0.32/0.49 thf(fact_186_power__le__imp__le__exp,axiom,
% 0.32/0.49 ! [A: real,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.49 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_le_imp_le_exp
% 0.32/0.49 thf(fact_187_power__le__imp__le__exp,axiom,
% 0.32/0.49 ! [A: rat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.49 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_le_imp_le_exp
% 0.32/0.49 thf(fact_188_power__le__imp__le__exp,axiom,
% 0.32/0.49 ! [A: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.49 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_le_imp_le_exp
% 0.32/0.49 thf(fact_189_power__le__imp__le__exp,axiom,
% 0.32/0.49 ! [A: int,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.49 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_le_imp_le_exp
% 0.32/0.49 thf(fact_190_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.49 ! [B2: real,A4: real] :
% 0.32/0.49 ( ( ~ ( ord_less_eq_real @ B2 @ A4 ) )
% 0.32/0.49 = ( ord_less_real @ A4 @ B2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(3)
% 0.32/0.49 thf(fact_191_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.49 ! [B2: rat,A4: rat] :
% 0.32/0.49 ( ( ~ ( ord_less_eq_rat @ B2 @ A4 ) )
% 0.32/0.49 = ( ord_less_rat @ A4 @ B2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(3)
% 0.32/0.49 thf(fact_192_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.49 ! [B2: num,A4: num] :
% 0.32/0.49 ( ( ~ ( ord_less_eq_num @ B2 @ A4 ) )
% 0.32/0.49 = ( ord_less_num @ A4 @ B2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(3)
% 0.32/0.49 thf(fact_193_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.49 ! [B2: nat,A4: nat] :
% 0.32/0.49 ( ( ~ ( ord_less_eq_nat @ B2 @ A4 ) )
% 0.32/0.49 = ( ord_less_nat @ A4 @ B2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(3)
% 0.32/0.49 thf(fact_194_verit__comp__simplify1_I3_J,axiom,
% 0.32/0.49 ! [B2: int,A4: int] :
% 0.32/0.49 ( ( ~ ( ord_less_eq_int @ B2 @ A4 ) )
% 0.32/0.49 = ( ord_less_int @ A4 @ B2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % verit_comp_simplify1(3)
% 0.32/0.49 thf(fact_195_le__num__One__iff,axiom,
% 0.32/0.49 ! [X: num] :
% 0.32/0.49 ( ( ord_less_eq_num @ X @ one )
% 0.32/0.49 = ( X = one ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_num_One_iff
% 0.32/0.49 thf(fact_196_le__numeral__extra_I4_J,axiom,
% 0.32/0.49 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.32/0.49
% 0.32/0.49 % le_numeral_extra(4)
% 0.32/0.49 thf(fact_197_le__numeral__extra_I4_J,axiom,
% 0.32/0.49 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.32/0.49
% 0.32/0.49 % le_numeral_extra(4)
% 0.32/0.49 thf(fact_198_le__numeral__extra_I4_J,axiom,
% 0.32/0.49 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.32/0.49
% 0.32/0.49 % le_numeral_extra(4)
% 0.32/0.49 thf(fact_199_le__numeral__extra_I4_J,axiom,
% 0.32/0.49 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.32/0.49
% 0.32/0.49 % le_numeral_extra(4)
% 0.32/0.49 thf(fact_200_ex__power__ivl1,axiom,
% 0.32/0.49 ! [B: nat,K: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.49 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.32/0.49 => ? [N4: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
% 0.32/0.49 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ex_power_ivl1
% 0.32/0.49 thf(fact_201_ex__power__ivl2,axiom,
% 0.32/0.49 ! [B: nat,K: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.49 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.49 => ? [N4: nat] :
% 0.32/0.49 ( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
% 0.32/0.49 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ex_power_ivl2
% 0.32/0.49 thf(fact_202_one__le__power,axiom,
% 0.32/0.49 ! [A: real,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/0.49 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_le_power
% 0.32/0.49 thf(fact_203_one__le__power,axiom,
% 0.32/0.49 ! [A: rat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.32/0.49 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_le_power
% 0.32/0.49 thf(fact_204_one__le__power,axiom,
% 0.32/0.49 ! [A: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/0.49 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_le_power
% 0.32/0.49 thf(fact_205_one__le__power,axiom,
% 0.32/0.49 ! [A: int,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/0.49 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_le_power
% 0.32/0.49 thf(fact_206_power__one__over,axiom,
% 0.32/0.49 ! [A: complex,N: nat] :
% 0.32/0.49 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 0.32/0.49 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_one_over
% 0.32/0.49 thf(fact_207_power__one__over,axiom,
% 0.32/0.49 ! [A: real,N: nat] :
% 0.32/0.49 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 0.32/0.49 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_one_over
% 0.32/0.49 thf(fact_208_power__one__over,axiom,
% 0.32/0.49 ! [A: rat,N: nat] :
% 0.32/0.49 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 0.32/0.49 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_one_over
% 0.32/0.49 thf(fact_209_nat__1__add__1,axiom,
% 0.32/0.49 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.32/0.49 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_1_add_1
% 0.32/0.49 thf(fact_210_numeral__Bit0,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.32/0.49 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_Bit0
% 0.32/0.49 thf(fact_211_numeral__Bit0,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.32/0.49 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_Bit0
% 0.32/0.49 thf(fact_212_numeral__Bit0,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.32/0.49 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_Bit0
% 0.32/0.49 thf(fact_213_numeral__Bit0,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.32/0.49 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_Bit0
% 0.32/0.49 thf(fact_214_numeral__Bit0,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.32/0.49 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_Bit0
% 0.32/0.49 thf(fact_215_power__increasing,axiom,
% 0.32/0.49 ! [N: nat,N3: nat,A: real] :
% 0.32/0.49 ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.49 => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/0.49 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_increasing
% 0.32/0.49 thf(fact_216_power__increasing,axiom,
% 0.32/0.49 ! [N: nat,N3: nat,A: rat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.49 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.32/0.49 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_increasing
% 0.32/0.49 thf(fact_217_power__increasing,axiom,
% 0.32/0.49 ! [N: nat,N3: nat,A: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.49 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/0.49 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_increasing
% 0.32/0.49 thf(fact_218_power__increasing,axiom,
% 0.32/0.49 ! [N: nat,N3: nat,A: int] :
% 0.32/0.49 ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.49 => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/0.49 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_increasing
% 0.32/0.49 thf(fact_219_one__le__numeral,axiom,
% 0.32/0.49 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_le_numeral
% 0.32/0.49 thf(fact_220_one__le__numeral,axiom,
% 0.32/0.49 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_le_numeral
% 0.32/0.49 thf(fact_221_one__le__numeral,axiom,
% 0.32/0.49 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_le_numeral
% 0.32/0.49 thf(fact_222_one__le__numeral,axiom,
% 0.32/0.49 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % one_le_numeral
% 0.32/0.49 thf(fact_223_numeral__One,axiom,
% 0.32/0.49 ( ( numera6690914467698888265omplex @ one )
% 0.32/0.49 = one_one_complex ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_One
% 0.32/0.49 thf(fact_224_numeral__One,axiom,
% 0.32/0.49 ( ( numeral_numeral_real @ one )
% 0.32/0.49 = one_one_real ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_One
% 0.32/0.49 thf(fact_225_numeral__One,axiom,
% 0.32/0.49 ( ( numeral_numeral_rat @ one )
% 0.32/0.49 = one_one_rat ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_One
% 0.32/0.49 thf(fact_226_numeral__One,axiom,
% 0.32/0.49 ( ( numeral_numeral_nat @ one )
% 0.32/0.49 = one_one_nat ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_One
% 0.32/0.49 thf(fact_227_numeral__One,axiom,
% 0.32/0.49 ( ( numeral_numeral_int @ one )
% 0.32/0.49 = one_one_int ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_One
% 0.32/0.49 thf(fact_228_numerals_I1_J,axiom,
% 0.32/0.49 ( ( numeral_numeral_nat @ one )
% 0.32/0.49 = one_one_nat ) ).
% 0.32/0.49
% 0.32/0.49 % numerals(1)
% 0.32/0.49 thf(fact_229_is__succ__in__set__def,axiom,
% 0.32/0.49 ( vEBT_is_succ_in_set
% 0.32/0.49 = ( ^ [Xs: set_nat,X2: nat,Y: nat] :
% 0.32/0.49 ( ( member_nat @ Y @ Xs )
% 0.32/0.49 & ( ord_less_nat @ X2 @ Y )
% 0.32/0.49 & ! [Z2: nat] :
% 0.32/0.49 ( ( member_nat @ Z2 @ Xs )
% 0.32/0.49 => ( ( ord_less_nat @ X2 @ Z2 )
% 0.32/0.49 => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % is_succ_in_set_def
% 0.32/0.49 thf(fact_230_power2__commute,axiom,
% 0.32/0.49 ! [X: complex,Y3: complex] :
% 0.32/0.49 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_commute
% 0.32/0.49 thf(fact_231_power2__commute,axiom,
% 0.32/0.49 ! [X: real,Y3: real] :
% 0.32/0.49 ( ( power_power_real @ ( minus_minus_real @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( power_power_real @ ( minus_minus_real @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_commute
% 0.32/0.49 thf(fact_232_power2__commute,axiom,
% 0.32/0.49 ! [X: rat,Y3: rat] :
% 0.32/0.49 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_commute
% 0.32/0.49 thf(fact_233_power2__commute,axiom,
% 0.32/0.49 ! [X: int,Y3: int] :
% 0.32/0.49 ( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( power_power_int @ ( minus_minus_int @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_commute
% 0.32/0.49 thf(fact_234_less__exp,axiom,
% 0.32/0.49 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_exp
% 0.32/0.49 thf(fact_235_one__power2,axiom,
% 0.32/0.49 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = one_one_rat ) ).
% 0.32/0.49
% 0.32/0.49 % one_power2
% 0.32/0.49 thf(fact_236_one__power2,axiom,
% 0.32/0.49 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = one_one_nat ) ).
% 0.32/0.49
% 0.32/0.49 % one_power2
% 0.32/0.49 thf(fact_237_one__power2,axiom,
% 0.32/0.49 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = one_one_real ) ).
% 0.32/0.49
% 0.32/0.49 % one_power2
% 0.32/0.49 thf(fact_238_one__power2,axiom,
% 0.32/0.49 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = one_one_int ) ).
% 0.32/0.49
% 0.32/0.49 % one_power2
% 0.32/0.49 thf(fact_239_one__power2,axiom,
% 0.32/0.49 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = one_one_complex ) ).
% 0.32/0.49
% 0.32/0.49 % one_power2
% 0.32/0.49 thf(fact_240_diff__le__diff__pow,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.49 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % diff_le_diff_pow
% 0.32/0.49 thf(fact_241_power__divide,axiom,
% 0.32/0.49 ! [A: complex,B: complex,N: nat] :
% 0.32/0.49 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 0.32/0.49 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_divide
% 0.32/0.49 thf(fact_242_power__divide,axiom,
% 0.32/0.49 ! [A: real,B: real,N: nat] :
% 0.32/0.49 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 0.32/0.49 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_divide
% 0.32/0.49 thf(fact_243_power__divide,axiom,
% 0.32/0.49 ! [A: rat,B: rat,N: nat] :
% 0.32/0.49 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 0.32/0.49 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_divide
% 0.32/0.49 thf(fact_244_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 0.32/0.49 ( ( ( X = none_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: nat,B3: nat] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_nat @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_nat @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_245_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.32/0.49 ( ( ( X = none_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: nat,B3: product_prod_nat_nat] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_nat @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_246_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 0.32/0.49 ( ( ( X = none_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_num )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: nat,B3: num] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_nat @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_num @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_247_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 0.32/0.49 ( ( ( X = none_P5556105721700978146at_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: product_prod_nat_nat,B3: nat] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_nat @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_248_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.32/0.49 ( ( ( X = none_P5556105721700978146at_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_249_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 0.32/0.49 ( ( ( X = none_P5556105721700978146at_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_num )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: product_prod_nat_nat,B3: num] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_num @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_250_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 0.32/0.49 ( ( ( X = none_num )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: num,B3: nat] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_num @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_nat @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_251_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.32/0.49 ( ( ( X = none_num )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: num,B3: product_prod_nat_nat] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_num @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_252_combine__options__cases,axiom,
% 0.32/0.49 ! [X: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 0.32/0.49 ( ( ( X = none_num )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ( ( Y3 = none_num )
% 0.32/0.49 => ( P @ X @ Y3 ) )
% 0.32/0.49 => ( ! [A5: num,B3: num] :
% 0.32/0.49 ( ( X
% 0.32/0.49 = ( some_num @ A5 ) )
% 0.32/0.49 => ( ( Y3
% 0.32/0.49 = ( some_num @ B3 ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) )
% 0.32/0.49 => ( P @ X @ Y3 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_options_cases
% 0.32/0.49 thf(fact_253_split__option__all,axiom,
% 0.32/0.49 ( ( ^ [P2: option_nat > $o] :
% 0.32/0.49 ! [X4: option_nat] : ( P2 @ X4 ) )
% 0.32/0.49 = ( ^ [P3: option_nat > $o] :
% 0.32/0.49 ( ( P3 @ none_nat )
% 0.32/0.49 & ! [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % split_option_all
% 0.32/0.49 thf(fact_254_split__option__all,axiom,
% 0.32/0.49 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.32/0.49 ! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
% 0.32/0.49 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.32/0.49 ( ( P3 @ none_P5556105721700978146at_nat )
% 0.32/0.49 & ! [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % split_option_all
% 0.32/0.49 thf(fact_255_split__option__all,axiom,
% 0.32/0.49 ( ( ^ [P2: option_num > $o] :
% 0.32/0.49 ! [X4: option_num] : ( P2 @ X4 ) )
% 0.32/0.49 = ( ^ [P3: option_num > $o] :
% 0.32/0.49 ( ( P3 @ none_num )
% 0.32/0.49 & ! [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % split_option_all
% 0.32/0.49 thf(fact_256_split__option__ex,axiom,
% 0.32/0.49 ( ( ^ [P2: option_nat > $o] :
% 0.32/0.49 ? [X4: option_nat] : ( P2 @ X4 ) )
% 0.32/0.49 = ( ^ [P3: option_nat > $o] :
% 0.32/0.49 ( ( P3 @ none_nat )
% 0.32/0.49 | ? [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % split_option_ex
% 0.32/0.49 thf(fact_257_split__option__ex,axiom,
% 0.32/0.49 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.32/0.49 ? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
% 0.32/0.49 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.32/0.49 ( ( P3 @ none_P5556105721700978146at_nat )
% 0.32/0.49 | ? [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % split_option_ex
% 0.32/0.49 thf(fact_258_split__option__ex,axiom,
% 0.32/0.49 ( ( ^ [P2: option_num > $o] :
% 0.32/0.49 ? [X4: option_num] : ( P2 @ X4 ) )
% 0.32/0.49 = ( ^ [P3: option_num > $o] :
% 0.32/0.49 ( ( P3 @ none_num )
% 0.32/0.49 | ? [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % split_option_ex
% 0.32/0.49 thf(fact_259_option_Oexhaust,axiom,
% 0.32/0.49 ! [Y3: option_nat] :
% 0.32/0.49 ( ( Y3 != none_nat )
% 0.32/0.49 => ~ ! [X23: nat] :
% 0.32/0.49 ( Y3
% 0.32/0.49 != ( some_nat @ X23 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.exhaust
% 0.32/0.49 thf(fact_260_option_Oexhaust,axiom,
% 0.32/0.49 ! [Y3: option4927543243414619207at_nat] :
% 0.32/0.49 ( ( Y3 != none_P5556105721700978146at_nat )
% 0.32/0.49 => ~ ! [X23: product_prod_nat_nat] :
% 0.32/0.49 ( Y3
% 0.32/0.49 != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.exhaust
% 0.32/0.49 thf(fact_261_option_Oexhaust,axiom,
% 0.32/0.49 ! [Y3: option_num] :
% 0.32/0.49 ( ( Y3 != none_num )
% 0.32/0.49 => ~ ! [X23: num] :
% 0.32/0.49 ( Y3
% 0.32/0.49 != ( some_num @ X23 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.exhaust
% 0.32/0.49 thf(fact_262_option_OdiscI,axiom,
% 0.32/0.49 ! [Option: option_nat,X22: nat] :
% 0.32/0.49 ( ( Option
% 0.32/0.49 = ( some_nat @ X22 ) )
% 0.32/0.49 => ( Option != none_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.discI
% 0.32/0.49 thf(fact_263_option_OdiscI,axiom,
% 0.32/0.49 ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 0.32/0.49 ( ( Option
% 0.32/0.49 = ( some_P7363390416028606310at_nat @ X22 ) )
% 0.32/0.49 => ( Option != none_P5556105721700978146at_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.discI
% 0.32/0.49 thf(fact_264_option_OdiscI,axiom,
% 0.32/0.49 ! [Option: option_num,X22: num] :
% 0.32/0.49 ( ( Option
% 0.32/0.49 = ( some_num @ X22 ) )
% 0.32/0.49 => ( Option != none_num ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.discI
% 0.32/0.49 thf(fact_265_option_Odistinct_I1_J,axiom,
% 0.32/0.49 ! [X22: nat] :
% 0.32/0.49 ( none_nat
% 0.32/0.49 != ( some_nat @ X22 ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.distinct(1)
% 0.32/0.49 thf(fact_266_option_Odistinct_I1_J,axiom,
% 0.32/0.49 ! [X22: product_prod_nat_nat] :
% 0.32/0.49 ( none_P5556105721700978146at_nat
% 0.32/0.49 != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.distinct(1)
% 0.32/0.49 thf(fact_267_option_Odistinct_I1_J,axiom,
% 0.32/0.49 ! [X22: num] :
% 0.32/0.49 ( none_num
% 0.32/0.49 != ( some_num @ X22 ) ) ).
% 0.32/0.49
% 0.32/0.49 % option.distinct(1)
% 0.32/0.49 thf(fact_268_add__shift,axiom,
% 0.32/0.49 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.49 ( ( ( plus_plus_nat @ X @ Y3 )
% 0.32/0.49 = Z )
% 0.32/0.49 = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y3 ) )
% 0.32/0.49 = ( some_nat @ Z ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_shift
% 0.32/0.49 thf(fact_269_less__shift,axiom,
% 0.32/0.49 ( ord_less_nat
% 0.32/0.49 = ( ^ [X2: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_shift
% 0.32/0.49 thf(fact_270_Nat_Oadd__diff__assoc,axiom,
% 0.32/0.49 ! [K: nat,J: nat,I: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.49 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.32/0.49 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Nat.add_diff_assoc
% 0.32/0.49 thf(fact_271_Nat_Oadd__diff__assoc2,axiom,
% 0.32/0.49 ! [K: nat,J: nat,I: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.49 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.32/0.49 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Nat.add_diff_assoc2
% 0.32/0.49 thf(fact_272_Nat_Odiff__diff__right,axiom,
% 0.32/0.49 ! [K: nat,J: nat,I: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.49 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.32/0.49 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Nat.diff_diff_right
% 0.32/0.49 thf(fact_273_le__add__diff__inverse,axiom,
% 0.32/0.49 ! [B: real,A: real] :
% 0.32/0.49 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.49 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_add_diff_inverse
% 0.32/0.49 thf(fact_274_le__add__diff__inverse,axiom,
% 0.32/0.49 ! [B: rat,A: rat] :
% 0.32/0.49 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.49 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_add_diff_inverse
% 0.32/0.49 thf(fact_275_le__add__diff__inverse,axiom,
% 0.32/0.49 ! [B: nat,A: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.49 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_add_diff_inverse
% 0.32/0.49 thf(fact_276_le__add__diff__inverse,axiom,
% 0.32/0.49 ! [B: int,A: int] :
% 0.32/0.49 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.49 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_add_diff_inverse
% 0.32/0.49 thf(fact_277_le__add__diff__inverse2,axiom,
% 0.32/0.49 ! [B: real,A: real] :
% 0.32/0.49 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.49 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_add_diff_inverse2
% 0.32/0.49 thf(fact_278_le__add__diff__inverse2,axiom,
% 0.32/0.49 ! [B: rat,A: rat] :
% 0.32/0.49 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.49 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_add_diff_inverse2
% 0.32/0.49 thf(fact_279_le__add__diff__inverse2,axiom,
% 0.32/0.49 ! [B: nat,A: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.49 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_add_diff_inverse2
% 0.32/0.49 thf(fact_280_le__add__diff__inverse2,axiom,
% 0.32/0.49 ! [B: int,A: int] :
% 0.32/0.49 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.49 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.32/0.49 = A ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_add_diff_inverse2
% 0.32/0.49 thf(fact_281__092_060open_062high_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
% 0.32/0.49 ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
% 0.32/0.49 & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % \<open>high x n < 2 ^ n \<and> low x n < 2 ^ n\<close>
% 0.32/0.49 thf(fact_282_div__exp__eq,axiom,
% 0.32/0.49 ! [A: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.49 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_exp_eq
% 0.32/0.49 thf(fact_283_div__exp__eq,axiom,
% 0.32/0.49 ! [A: int,M: nat,N: nat] :
% 0.32/0.49 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.49 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_exp_eq
% 0.32/0.49 thf(fact_284_field__less__half__sum,axiom,
% 0.32/0.49 ! [X: real,Y3: real] :
% 0.32/0.49 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.49 => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % field_less_half_sum
% 0.32/0.49 thf(fact_285_field__less__half__sum,axiom,
% 0.32/0.49 ! [X: rat,Y3: rat] :
% 0.32/0.49 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.49 => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % field_less_half_sum
% 0.32/0.49 thf(fact_286_succ__member,axiom,
% 0.32/0.49 ! [T: vEBT_VEBT,X: nat,Y3: nat] :
% 0.32/0.49 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y3 )
% 0.32/0.49 = ( ( vEBT_vebt_member @ T @ Y3 )
% 0.32/0.49 & ( ord_less_nat @ X @ Y3 )
% 0.32/0.49 & ! [Z2: nat] :
% 0.32/0.49 ( ( ( vEBT_vebt_member @ T @ Z2 )
% 0.32/0.49 & ( ord_less_nat @ X @ Z2 ) )
% 0.32/0.49 => ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % succ_member
% 0.32/0.49 thf(fact_287_high__inv,axiom,
% 0.32/0.49 ! [X: nat,N: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.49 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 0.32/0.49 = Y3 ) ) ).
% 0.32/0.49
% 0.32/0.49 % high_inv
% 0.32/0.49 thf(fact_288_add__def,axiom,
% 0.32/0.49 ( vEBT_VEBT_add
% 0.32/0.49 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_def
% 0.32/0.49 thf(fact_289_bit__split__inv,axiom,
% 0.32/0.49 ! [X: nat,D: nat] :
% 0.32/0.49 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
% 0.32/0.49 = X ) ).
% 0.32/0.49
% 0.32/0.49 % bit_split_inv
% 0.32/0.49 thf(fact_290_member__correct,axiom,
% 0.32/0.49 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.49 => ( ( vEBT_vebt_member @ T @ X )
% 0.32/0.49 = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % member_correct
% 0.32/0.49 thf(fact_291_set__vebt__set__vebt_H__valid,axiom,
% 0.32/0.49 ! [T: vEBT_VEBT,N: nat] :
% 0.32/0.49 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.49 => ( ( vEBT_set_vebt @ T )
% 0.32/0.49 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % set_vebt_set_vebt'_valid
% 0.32/0.49 thf(fact_292_member__bound,axiom,
% 0.32/0.49 ! [Tree: vEBT_VEBT,X: nat,N: nat] :
% 0.32/0.49 ( ( vEBT_vebt_member @ Tree @ X )
% 0.32/0.49 => ( ( vEBT_invar_vebt @ Tree @ N )
% 0.32/0.49 => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % member_bound
% 0.32/0.49 thf(fact_293_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.49 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_294_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.49 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_295_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.49 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_296_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.49 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_297_numeral__times__numeral,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.49 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % numeral_times_numeral
% 0.32/0.49 thf(fact_298_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 0.32/0.49 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_299_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: real] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 0.32/0.49 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_300_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 0.32/0.49 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_301_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 0.32/0.49 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_302_mult__numeral__left__semiring__numeral,axiom,
% 0.32/0.49 ! [V: num,W: num,Z: int] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 0.32/0.49 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_left_semiring_numeral
% 0.32/0.49 thf(fact_303_low__inv,axiom,
% 0.32/0.49 ! [X: nat,N: nat,Y3: nat] :
% 0.32/0.49 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.49 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 0.32/0.49 = X ) ) ).
% 0.32/0.49
% 0.32/0.49 % low_inv
% 0.32/0.49 thf(fact_304_bits__div__by__1,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % bits_div_by_1
% 0.32/0.49 thf(fact_305_bits__div__by__1,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( divide_divide_int @ A @ one_one_int )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % bits_div_by_1
% 0.32/0.49 thf(fact_306_div__by__1,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_307_div__by__1,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( divide_divide_real @ A @ one_one_real )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_308_div__by__1,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( divide_divide_rat @ A @ one_one_rat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_309_div__by__1,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ A @ one_one_nat )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_310_div__by__1,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( divide_divide_int @ A @ one_one_int )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % div_by_1
% 0.32/0.49 thf(fact_311_nat__add__left__cancel__less,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.32/0.49 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_add_left_cancel_less
% 0.32/0.49 thf(fact_312_nat__add__left__cancel__le,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.32/0.49 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_add_left_cancel_le
% 0.32/0.49 thf(fact_313_diff__diff__cancel,axiom,
% 0.32/0.49 ! [I: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ N )
% 0.32/0.49 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.32/0.49 = I ) ) ).
% 0.32/0.49
% 0.32/0.49 % diff_diff_cancel
% 0.32/0.49 thf(fact_314_diff__diff__left,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat] :
% 0.32/0.49 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.32/0.49 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % diff_diff_left
% 0.32/0.49 thf(fact_315_nat__1__eq__mult__iff,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( one_one_nat
% 0.32/0.49 = ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( ( M = one_one_nat )
% 0.32/0.49 & ( N = one_one_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_1_eq_mult_iff
% 0.32/0.49 thf(fact_316_nat__mult__eq__1__iff,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ M @ N )
% 0.32/0.49 = one_one_nat )
% 0.32/0.49 = ( ( M = one_one_nat )
% 0.32/0.49 & ( N = one_one_nat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_mult_eq_1_iff
% 0.32/0.49 thf(fact_317_semiring__norm_I6_J,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.49 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % semiring_norm(6)
% 0.32/0.49 thf(fact_318_semiring__norm_I78_J,axiom,
% 0.32/0.49 ! [M: num,N: num] :
% 0.32/0.49 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.49 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % semiring_norm(78)
% 0.32/0.49 thf(fact_319_semiring__norm_I75_J,axiom,
% 0.32/0.49 ! [M: num] :
% 0.32/0.49 ~ ( ord_less_num @ M @ one ) ).
% 0.32/0.49
% 0.32/0.49 % semiring_norm(75)
% 0.32/0.49 thf(fact_320_bit__concat__def,axiom,
% 0.32/0.49 ( vEBT_VEBT_bit_concat
% 0.32/0.49 = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % bit_concat_def
% 0.32/0.49 thf(fact_321_distrib__left__numeral,axiom,
% 0.32/0.49 ! [V: num,B: complex,C: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 0.32/0.49 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left_numeral
% 0.32/0.49 thf(fact_322_distrib__left__numeral,axiom,
% 0.32/0.49 ! [V: num,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left_numeral
% 0.32/0.49 thf(fact_323_distrib__left__numeral,axiom,
% 0.32/0.49 ! [V: num,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left_numeral
% 0.32/0.49 thf(fact_324_distrib__left__numeral,axiom,
% 0.32/0.49 ! [V: num,B: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left_numeral
% 0.32/0.49 thf(fact_325_distrib__left__numeral,axiom,
% 0.32/0.49 ! [V: num,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left_numeral
% 0.32/0.49 thf(fact_326_distrib__right__numeral,axiom,
% 0.32/0.49 ! [A: complex,B: complex,V: num] :
% 0.32/0.49 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.32/0.49 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right_numeral
% 0.32/0.49 thf(fact_327_distrib__right__numeral,axiom,
% 0.32/0.49 ! [A: real,B: real,V: num] :
% 0.32/0.49 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right_numeral
% 0.32/0.49 thf(fact_328_distrib__right__numeral,axiom,
% 0.32/0.49 ! [A: rat,B: rat,V: num] :
% 0.32/0.49 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right_numeral
% 0.32/0.49 thf(fact_329_distrib__right__numeral,axiom,
% 0.32/0.49 ! [A: nat,B: nat,V: num] :
% 0.32/0.49 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right_numeral
% 0.32/0.49 thf(fact_330_distrib__right__numeral,axiom,
% 0.32/0.49 ! [A: int,B: int,V: num] :
% 0.32/0.49 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right_numeral
% 0.32/0.49 thf(fact_331_left__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [A: complex,B: complex,V: num] :
% 0.32/0.49 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.32/0.49 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib_numeral
% 0.32/0.49 thf(fact_332_left__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [A: real,B: real,V: num] :
% 0.32/0.49 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib_numeral
% 0.32/0.49 thf(fact_333_left__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [A: rat,B: rat,V: num] :
% 0.32/0.49 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib_numeral
% 0.32/0.49 thf(fact_334_left__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [A: int,B: int,V: num] :
% 0.32/0.49 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib_numeral
% 0.32/0.49 thf(fact_335_right__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [V: num,B: complex,C: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 0.32/0.49 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib_numeral
% 0.32/0.49 thf(fact_336_right__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [V: num,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib_numeral
% 0.32/0.49 thf(fact_337_right__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [V: num,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib_numeral
% 0.32/0.49 thf(fact_338_right__diff__distrib__numeral,axiom,
% 0.32/0.49 ! [V: num,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib_numeral
% 0.32/0.49 thf(fact_339_semiring__norm_I2_J,axiom,
% 0.32/0.49 ( ( plus_plus_num @ one @ one )
% 0.32/0.49 = ( bit0 @ one ) ) ).
% 0.32/0.49
% 0.32/0.49 % semiring_norm(2)
% 0.32/0.49 thf(fact_340_semiring__norm_I76_J,axiom,
% 0.32/0.49 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % semiring_norm(76)
% 0.32/0.49 thf(fact_341_le__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [A: real,B: real,W: num] :
% 0.32/0.49 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.49 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_divide_eq_numeral1(1)
% 0.32/0.49 thf(fact_342_le__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [A: rat,B: rat,W: num] :
% 0.32/0.49 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.49 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_divide_eq_numeral1(1)
% 0.32/0.49 thf(fact_343_divide__le__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [B: real,W: num,A: real] :
% 0.32/0.49 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.32/0.49 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_le_eq_numeral1(1)
% 0.32/0.49 thf(fact_344_divide__le__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [B: rat,W: num,A: rat] :
% 0.32/0.49 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.32/0.49 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_le_eq_numeral1(1)
% 0.32/0.49 thf(fact_345_less__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [A: real,B: real,W: num] :
% 0.32/0.49 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.49 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_divide_eq_numeral1(1)
% 0.32/0.49 thf(fact_346_less__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [A: rat,B: rat,W: num] :
% 0.32/0.49 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.49 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_divide_eq_numeral1(1)
% 0.32/0.49 thf(fact_347_divide__less__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [B: real,W: num,A: real] :
% 0.32/0.49 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 0.32/0.49 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_less_eq_numeral1(1)
% 0.32/0.49 thf(fact_348_divide__less__eq__numeral1_I1_J,axiom,
% 0.32/0.49 ! [B: rat,W: num,A: rat] :
% 0.32/0.49 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 0.32/0.49 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % divide_less_eq_numeral1(1)
% 0.32/0.49 thf(fact_349_power__add__numeral,axiom,
% 0.32/0.49 ! [A: complex,M: num,N: num] :
% 0.32/0.49 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.49 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral
% 0.32/0.49 thf(fact_350_power__add__numeral,axiom,
% 0.32/0.49 ! [A: real,M: num,N: num] :
% 0.32/0.49 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.49 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral
% 0.32/0.49 thf(fact_351_power__add__numeral,axiom,
% 0.32/0.49 ! [A: rat,M: num,N: num] :
% 0.32/0.49 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.49 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral
% 0.32/0.49 thf(fact_352_power__add__numeral,axiom,
% 0.32/0.49 ! [A: nat,M: num,N: num] :
% 0.32/0.49 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.49 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral
% 0.32/0.49 thf(fact_353_power__add__numeral,axiom,
% 0.32/0.49 ! [A: int,M: num,N: num] :
% 0.32/0.49 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 0.32/0.49 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral
% 0.32/0.49 thf(fact_354_power__add__numeral2,axiom,
% 0.32/0.49 ! [A: complex,M: num,N: num,B: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.49 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral2
% 0.32/0.49 thf(fact_355_power__add__numeral2,axiom,
% 0.32/0.49 ! [A: real,M: num,N: num,B: real] :
% 0.32/0.49 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.49 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral2
% 0.32/0.49 thf(fact_356_power__add__numeral2,axiom,
% 0.32/0.49 ! [A: rat,M: num,N: num,B: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.49 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral2
% 0.32/0.49 thf(fact_357_power__add__numeral2,axiom,
% 0.32/0.49 ! [A: nat,M: num,N: num,B: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.49 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral2
% 0.32/0.49 thf(fact_358_power__add__numeral2,axiom,
% 0.32/0.49 ! [A: int,M: num,N: num,B: int] :
% 0.32/0.49 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 0.32/0.49 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add_numeral2
% 0.32/0.49 thf(fact_359_ring__class_Oring__distribs_I2_J,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ring_class.ring_distribs(2)
% 0.32/0.49 thf(fact_360_ring__class_Oring__distribs_I2_J,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ring_class.ring_distribs(2)
% 0.32/0.49 thf(fact_361_ring__class_Oring__distribs_I2_J,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ring_class.ring_distribs(2)
% 0.32/0.49 thf(fact_362_ring__class_Oring__distribs_I1_J,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ring_class.ring_distribs(1)
% 0.32/0.49 thf(fact_363_ring__class_Oring__distribs_I1_J,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ring_class.ring_distribs(1)
% 0.32/0.49 thf(fact_364_ring__class_Oring__distribs_I1_J,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ring_class.ring_distribs(1)
% 0.32/0.49 thf(fact_365_comm__semiring__class_Odistrib,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % comm_semiring_class.distrib
% 0.32/0.49 thf(fact_366_comm__semiring__class_Odistrib,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % comm_semiring_class.distrib
% 0.32/0.49 thf(fact_367_comm__semiring__class_Odistrib,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % comm_semiring_class.distrib
% 0.32/0.49 thf(fact_368_comm__semiring__class_Odistrib,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % comm_semiring_class.distrib
% 0.32/0.49 thf(fact_369_distrib__left,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left
% 0.32/0.49 thf(fact_370_distrib__left,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left
% 0.32/0.49 thf(fact_371_distrib__left,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left
% 0.32/0.49 thf(fact_372_distrib__left,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_left
% 0.32/0.49 thf(fact_373_distrib__right,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right
% 0.32/0.49 thf(fact_374_distrib__right,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right
% 0.32/0.49 thf(fact_375_distrib__right,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right
% 0.32/0.49 thf(fact_376_distrib__right,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % distrib_right
% 0.32/0.49 thf(fact_377_combine__common__factor,axiom,
% 0.32/0.49 ! [A: real,E: real,B: real,C: real] :
% 0.32/0.49 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_common_factor
% 0.32/0.49 thf(fact_378_combine__common__factor,axiom,
% 0.32/0.49 ! [A: rat,E: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_common_factor
% 0.32/0.49 thf(fact_379_combine__common__factor,axiom,
% 0.32/0.49 ! [A: nat,E: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_common_factor
% 0.32/0.49 thf(fact_380_combine__common__factor,axiom,
% 0.32/0.49 ! [A: int,E: int,B: int,C: int] :
% 0.32/0.49 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% 0.32/0.49
% 0.32/0.49 % combine_common_factor
% 0.32/0.49 thf(fact_381_left__diff__distrib,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib
% 0.32/0.49 thf(fact_382_left__diff__distrib,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib
% 0.32/0.49 thf(fact_383_left__diff__distrib,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib
% 0.32/0.49 thf(fact_384_right__diff__distrib,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib
% 0.32/0.49 thf(fact_385_right__diff__distrib,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib
% 0.32/0.49 thf(fact_386_right__diff__distrib,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib
% 0.32/0.49 thf(fact_387_left__diff__distrib_H,axiom,
% 0.32/0.49 ! [B: real,C: real,A: real] :
% 0.32/0.49 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib'
% 0.32/0.49 thf(fact_388_left__diff__distrib_H,axiom,
% 0.32/0.49 ! [B: rat,C: rat,A: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib'
% 0.32/0.49 thf(fact_389_left__diff__distrib_H,axiom,
% 0.32/0.49 ! [B: nat,C: nat,A: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 0.32/0.49 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib'
% 0.32/0.49 thf(fact_390_left__diff__distrib_H,axiom,
% 0.32/0.49 ! [B: int,C: int,A: int] :
% 0.32/0.49 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_diff_distrib'
% 0.32/0.49 thf(fact_391_right__diff__distrib_H,axiom,
% 0.32/0.49 ! [A: real,B: real,C: real] :
% 0.32/0.49 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.32/0.49 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib'
% 0.32/0.49 thf(fact_392_right__diff__distrib_H,axiom,
% 0.32/0.49 ! [A: rat,B: rat,C: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.49 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib'
% 0.32/0.49 thf(fact_393_right__diff__distrib_H,axiom,
% 0.32/0.49 ! [A: nat,B: nat,C: nat] :
% 0.32/0.49 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 0.32/0.49 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib'
% 0.32/0.49 thf(fact_394_right__diff__distrib_H,axiom,
% 0.32/0.49 ! [A: int,B: int,C: int] :
% 0.32/0.49 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.32/0.49 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % right_diff_distrib'
% 0.32/0.49 thf(fact_395_le__cube,axiom,
% 0.32/0.49 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_cube
% 0.32/0.49 thf(fact_396_le__square,axiom,
% 0.32/0.49 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_square
% 0.32/0.49 thf(fact_397_mult__le__mono,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ( ord_less_eq_nat @ K @ L2 )
% 0.32/0.49 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_le_mono
% 0.32/0.49 thf(fact_398_mult__le__mono1,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_le_mono1
% 0.32/0.49 thf(fact_399_mult__le__mono2,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_le_mono2
% 0.32/0.49 thf(fact_400_add__mult__distrib,axiom,
% 0.32/0.49 ! [M: nat,N: nat,K: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_mult_distrib
% 0.32/0.49 thf(fact_401_add__mult__distrib2,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_mult_distrib2
% 0.32/0.49 thf(fact_402_diff__mult__distrib,axiom,
% 0.32/0.49 ! [M: nat,N: nat,K: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.32/0.49 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % diff_mult_distrib
% 0.32/0.49 thf(fact_403_diff__mult__distrib2,axiom,
% 0.32/0.49 ! [K: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.49 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % diff_mult_distrib2
% 0.32/0.49 thf(fact_404_nat__mult__1,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( times_times_nat @ one_one_nat @ N )
% 0.32/0.49 = N ) ).
% 0.32/0.49
% 0.32/0.49 % nat_mult_1
% 0.32/0.49 thf(fact_405_nat__mult__1__right,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ( ( times_times_nat @ N @ one_one_nat )
% 0.32/0.49 = N ) ).
% 0.32/0.49
% 0.32/0.49 % nat_mult_1_right
% 0.32/0.49 thf(fact_406_two__realpow__ge__one,axiom,
% 0.32/0.49 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % two_realpow_ge_one
% 0.32/0.49 thf(fact_407_add__One__commute,axiom,
% 0.32/0.49 ! [N: num] :
% 0.32/0.49 ( ( plus_plus_num @ one @ N )
% 0.32/0.49 = ( plus_plus_num @ N @ one ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_One_commute
% 0.32/0.49 thf(fact_408_power__commutes,axiom,
% 0.32/0.49 ! [A: complex,N: nat] :
% 0.32/0.49 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 0.32/0.49 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commutes
% 0.32/0.49 thf(fact_409_power__commutes,axiom,
% 0.32/0.49 ! [A: real,N: nat] :
% 0.32/0.49 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 0.32/0.49 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commutes
% 0.32/0.49 thf(fact_410_power__commutes,axiom,
% 0.32/0.49 ! [A: rat,N: nat] :
% 0.32/0.49 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 0.32/0.49 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commutes
% 0.32/0.49 thf(fact_411_power__commutes,axiom,
% 0.32/0.49 ! [A: nat,N: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 0.32/0.49 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commutes
% 0.32/0.49 thf(fact_412_power__commutes,axiom,
% 0.32/0.49 ! [A: int,N: nat] :
% 0.32/0.49 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 0.32/0.49 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commutes
% 0.32/0.49 thf(fact_413_power__mult__distrib,axiom,
% 0.32/0.49 ! [A: complex,B: complex,N: nat] :
% 0.32/0.49 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 0.32/0.49 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult_distrib
% 0.32/0.49 thf(fact_414_power__mult__distrib,axiom,
% 0.32/0.49 ! [A: real,B: real,N: nat] :
% 0.32/0.49 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 0.32/0.49 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult_distrib
% 0.32/0.49 thf(fact_415_power__mult__distrib,axiom,
% 0.32/0.49 ! [A: rat,B: rat,N: nat] :
% 0.32/0.49 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 0.32/0.49 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult_distrib
% 0.32/0.49 thf(fact_416_power__mult__distrib,axiom,
% 0.32/0.49 ! [A: nat,B: nat,N: nat] :
% 0.32/0.49 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 0.32/0.49 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult_distrib
% 0.32/0.49 thf(fact_417_power__mult__distrib,axiom,
% 0.32/0.49 ! [A: int,B: int,N: nat] :
% 0.32/0.49 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 0.32/0.49 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult_distrib
% 0.32/0.49 thf(fact_418_power__commuting__commutes,axiom,
% 0.32/0.49 ! [X: complex,Y3: complex,N: nat] :
% 0.32/0.49 ( ( ( times_times_complex @ X @ Y3 )
% 0.32/0.49 = ( times_times_complex @ Y3 @ X ) )
% 0.32/0.49 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y3 )
% 0.32/0.49 = ( times_times_complex @ Y3 @ ( power_power_complex @ X @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commuting_commutes
% 0.32/0.49 thf(fact_419_power__commuting__commutes,axiom,
% 0.32/0.49 ! [X: real,Y3: real,N: nat] :
% 0.32/0.49 ( ( ( times_times_real @ X @ Y3 )
% 0.32/0.49 = ( times_times_real @ Y3 @ X ) )
% 0.32/0.49 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y3 )
% 0.32/0.49 = ( times_times_real @ Y3 @ ( power_power_real @ X @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commuting_commutes
% 0.32/0.49 thf(fact_420_power__commuting__commutes,axiom,
% 0.32/0.49 ! [X: rat,Y3: rat,N: nat] :
% 0.32/0.49 ( ( ( times_times_rat @ X @ Y3 )
% 0.32/0.49 = ( times_times_rat @ Y3 @ X ) )
% 0.32/0.49 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y3 )
% 0.32/0.49 = ( times_times_rat @ Y3 @ ( power_power_rat @ X @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commuting_commutes
% 0.32/0.49 thf(fact_421_power__commuting__commutes,axiom,
% 0.32/0.49 ! [X: nat,Y3: nat,N: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ X @ Y3 )
% 0.32/0.49 = ( times_times_nat @ Y3 @ X ) )
% 0.32/0.49 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y3 )
% 0.32/0.49 = ( times_times_nat @ Y3 @ ( power_power_nat @ X @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commuting_commutes
% 0.32/0.49 thf(fact_422_power__commuting__commutes,axiom,
% 0.32/0.49 ! [X: int,Y3: int,N: nat] :
% 0.32/0.49 ( ( ( times_times_int @ X @ Y3 )
% 0.32/0.49 = ( times_times_int @ Y3 @ X ) )
% 0.32/0.49 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y3 )
% 0.32/0.49 = ( times_times_int @ Y3 @ ( power_power_int @ X @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_commuting_commutes
% 0.32/0.49 thf(fact_423_power__mult,axiom,
% 0.32/0.49 ! [A: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult
% 0.32/0.49 thf(fact_424_power__mult,axiom,
% 0.32/0.49 ! [A: real,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult
% 0.32/0.49 thf(fact_425_power__mult,axiom,
% 0.32/0.49 ! [A: int,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult
% 0.32/0.49 thf(fact_426_power__mult,axiom,
% 0.32/0.49 ! [A: complex,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 0.32/0.49 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_mult
% 0.32/0.49 thf(fact_427_left__add__mult__distrib,axiom,
% 0.32/0.49 ! [I: nat,U: nat,J: nat,K: nat] :
% 0.32/0.49 ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_add_mult_distrib
% 0.32/0.49 thf(fact_428_div__mult2__eq,axiom,
% 0.32/0.49 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.49 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.32/0.49 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 0.32/0.49
% 0.32/0.49 % div_mult2_eq
% 0.32/0.49 thf(fact_429_less__1__mult,axiom,
% 0.32/0.49 ! [M: real,N: real] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ M )
% 0.32/0.49 => ( ( ord_less_real @ one_one_real @ N )
% 0.32/0.49 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_1_mult
% 0.32/0.49 thf(fact_430_less__1__mult,axiom,
% 0.32/0.49 ! [M: rat,N: rat] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ M )
% 0.32/0.49 => ( ( ord_less_rat @ one_one_rat @ N )
% 0.32/0.49 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_1_mult
% 0.32/0.49 thf(fact_431_less__1__mult,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ M )
% 0.32/0.49 => ( ( ord_less_nat @ one_one_nat @ N )
% 0.32/0.49 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_1_mult
% 0.32/0.49 thf(fact_432_less__1__mult,axiom,
% 0.32/0.49 ! [M: int,N: int] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ M )
% 0.32/0.49 => ( ( ord_less_int @ one_one_int @ N )
% 0.32/0.49 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_1_mult
% 0.32/0.49 thf(fact_433_eq__add__iff1,axiom,
% 0.32/0.49 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.49 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.49 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
% 0.32/0.49 = D ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_add_iff1
% 0.32/0.49 thf(fact_434_eq__add__iff1,axiom,
% 0.32/0.49 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.49 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.49 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
% 0.32/0.49 = D ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_add_iff1
% 0.32/0.49 thf(fact_435_eq__add__iff1,axiom,
% 0.32/0.49 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.49 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.49 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
% 0.32/0.49 = D ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_add_iff1
% 0.32/0.49 thf(fact_436_eq__add__iff2,axiom,
% 0.32/0.49 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.49 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.49 = ( C
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_add_iff2
% 0.32/0.49 thf(fact_437_eq__add__iff2,axiom,
% 0.32/0.49 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.49 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.49 = ( C
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_add_iff2
% 0.32/0.49 thf(fact_438_eq__add__iff2,axiom,
% 0.32/0.49 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.49 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.49 = ( C
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_add_iff2
% 0.32/0.49 thf(fact_439_square__diff__square__factored,axiom,
% 0.32/0.49 ! [X: real,Y3: real] :
% 0.32/0.49 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.32/0.49 = ( times_times_real @ ( plus_plus_real @ X @ Y3 ) @ ( minus_minus_real @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % square_diff_square_factored
% 0.32/0.49 thf(fact_440_square__diff__square__factored,axiom,
% 0.32/0.49 ! [X: rat,Y3: rat] :
% 0.32/0.49 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.32/0.49 = ( times_times_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( minus_minus_rat @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % square_diff_square_factored
% 0.32/0.49 thf(fact_441_square__diff__square__factored,axiom,
% 0.32/0.49 ! [X: int,Y3: int] :
% 0.32/0.49 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.32/0.49 = ( times_times_int @ ( plus_plus_int @ X @ Y3 ) @ ( minus_minus_int @ X @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % square_diff_square_factored
% 0.32/0.49 thf(fact_442_mult__diff__mult,axiom,
% 0.32/0.49 ! [X: real,Y3: real,A: real,B: real] :
% 0.32/0.49 ( ( minus_minus_real @ ( times_times_real @ X @ Y3 ) @ ( times_times_real @ A @ B ) )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_diff_mult
% 0.32/0.49 thf(fact_443_mult__diff__mult,axiom,
% 0.32/0.49 ! [X: rat,Y3: rat,A: rat,B: rat] :
% 0.32/0.49 ( ( minus_minus_rat @ ( times_times_rat @ X @ Y3 ) @ ( times_times_rat @ A @ B ) )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y3 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_diff_mult
% 0.32/0.49 thf(fact_444_mult__diff__mult,axiom,
% 0.32/0.49 ! [X: int,Y3: int,A: int,B: int] :
% 0.32/0.49 ( ( minus_minus_int @ ( times_times_int @ X @ Y3 ) @ ( times_times_int @ A @ B ) )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y3 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_diff_mult
% 0.32/0.49 thf(fact_445_ordered__ring__class_Ole__add__iff2,axiom,
% 0.32/0.49 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.49 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ordered_ring_class.le_add_iff2
% 0.32/0.49 thf(fact_446_ordered__ring__class_Ole__add__iff2,axiom,
% 0.32/0.49 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.49 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ordered_ring_class.le_add_iff2
% 0.32/0.49 thf(fact_447_ordered__ring__class_Ole__add__iff2,axiom,
% 0.32/0.49 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.49 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % ordered_ring_class.le_add_iff2
% 0.32/0.49 thf(fact_448_ordered__ring__class_Ole__add__iff1,axiom,
% 0.32/0.49 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.49 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.49
% 0.32/0.49 % ordered_ring_class.le_add_iff1
% 0.32/0.49 thf(fact_449_ordered__ring__class_Ole__add__iff1,axiom,
% 0.32/0.49 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.49 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.49
% 0.32/0.49 % ordered_ring_class.le_add_iff1
% 0.32/0.49 thf(fact_450_ordered__ring__class_Ole__add__iff1,axiom,
% 0.32/0.49 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.49 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.49
% 0.32/0.49 % ordered_ring_class.le_add_iff1
% 0.32/0.49 thf(fact_451_less__add__iff1,axiom,
% 0.32/0.49 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.49 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_add_iff1
% 0.32/0.49 thf(fact_452_less__add__iff1,axiom,
% 0.32/0.49 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.49 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_add_iff1
% 0.32/0.49 thf(fact_453_less__add__iff1,axiom,
% 0.32/0.49 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.49 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_add_iff1
% 0.32/0.49 thf(fact_454_less__add__iff2,axiom,
% 0.32/0.49 ! [A: real,E: real,C: real,B: real,D: real] :
% 0.32/0.49 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_add_iff2
% 0.32/0.49 thf(fact_455_less__add__iff2,axiom,
% 0.32/0.49 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 0.32/0.49 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_add_iff2
% 0.32/0.49 thf(fact_456_less__add__iff2,axiom,
% 0.32/0.49 ! [A: int,E: int,C: int,B: int,D: int] :
% 0.32/0.49 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 0.32/0.49 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_add_iff2
% 0.32/0.49 thf(fact_457_square__diff__one__factored,axiom,
% 0.32/0.49 ! [X: complex] :
% 0.32/0.49 ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 0.32/0.49 = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % square_diff_one_factored
% 0.32/0.49 thf(fact_458_square__diff__one__factored,axiom,
% 0.32/0.49 ! [X: real] :
% 0.32/0.49 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 0.32/0.49 = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % square_diff_one_factored
% 0.32/0.49 thf(fact_459_square__diff__one__factored,axiom,
% 0.32/0.49 ! [X: rat] :
% 0.32/0.49 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 0.32/0.49 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % square_diff_one_factored
% 0.32/0.49 thf(fact_460_square__diff__one__factored,axiom,
% 0.32/0.49 ! [X: int] :
% 0.32/0.49 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 0.32/0.49 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % square_diff_one_factored
% 0.32/0.49 thf(fact_461_mult__numeral__1__right,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1_right
% 0.32/0.49 thf(fact_462_mult__numeral__1__right,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1_right
% 0.32/0.49 thf(fact_463_mult__numeral__1__right,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1_right
% 0.32/0.49 thf(fact_464_mult__numeral__1__right,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1_right
% 0.32/0.49 thf(fact_465_mult__numeral__1__right,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1_right
% 0.32/0.49 thf(fact_466_mult__numeral__1,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1
% 0.32/0.49 thf(fact_467_mult__numeral__1,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1
% 0.32/0.49 thf(fact_468_mult__numeral__1,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1
% 0.32/0.49 thf(fact_469_mult__numeral__1,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1
% 0.32/0.49 thf(fact_470_mult__numeral__1,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 0.32/0.49 = A ) ).
% 0.32/0.49
% 0.32/0.49 % mult_numeral_1
% 0.32/0.49 thf(fact_471_left__right__inverse__power,axiom,
% 0.32/0.49 ! [X: complex,Y3: complex,N: nat] :
% 0.32/0.49 ( ( ( times_times_complex @ X @ Y3 )
% 0.32/0.49 = one_one_complex )
% 0.32/0.49 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 0.32/0.49 = one_one_complex ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_right_inverse_power
% 0.32/0.49 thf(fact_472_left__right__inverse__power,axiom,
% 0.32/0.49 ! [X: real,Y3: real,N: nat] :
% 0.32/0.49 ( ( ( times_times_real @ X @ Y3 )
% 0.32/0.49 = one_one_real )
% 0.32/0.49 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y3 @ N ) )
% 0.32/0.49 = one_one_real ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_right_inverse_power
% 0.32/0.49 thf(fact_473_left__right__inverse__power,axiom,
% 0.32/0.49 ! [X: rat,Y3: rat,N: nat] :
% 0.32/0.49 ( ( ( times_times_rat @ X @ Y3 )
% 0.32/0.49 = one_one_rat )
% 0.32/0.49 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 0.32/0.49 = one_one_rat ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_right_inverse_power
% 0.32/0.49 thf(fact_474_left__right__inverse__power,axiom,
% 0.32/0.49 ! [X: nat,Y3: nat,N: nat] :
% 0.32/0.49 ( ( ( times_times_nat @ X @ Y3 )
% 0.32/0.49 = one_one_nat )
% 0.32/0.49 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ N ) )
% 0.32/0.49 = one_one_nat ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_right_inverse_power
% 0.32/0.49 thf(fact_475_left__right__inverse__power,axiom,
% 0.32/0.49 ! [X: int,Y3: int,N: nat] :
% 0.32/0.49 ( ( ( times_times_int @ X @ Y3 )
% 0.32/0.49 = one_one_int )
% 0.32/0.49 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) )
% 0.32/0.49 = one_one_int ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_right_inverse_power
% 0.32/0.49 thf(fact_476_power__add,axiom,
% 0.32/0.49 ! [A: complex,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add
% 0.32/0.49 thf(fact_477_power__add,axiom,
% 0.32/0.49 ! [A: real,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add
% 0.32/0.49 thf(fact_478_power__add,axiom,
% 0.32/0.49 ! [A: rat,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add
% 0.32/0.49 thf(fact_479_power__add,axiom,
% 0.32/0.49 ! [A: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add
% 0.32/0.49 thf(fact_480_power__add,axiom,
% 0.32/0.49 ! [A: int,M: nat,N: nat] :
% 0.32/0.49 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.49 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_add
% 0.32/0.49 thf(fact_481_less__mult__imp__div__less,axiom,
% 0.32/0.49 ! [M: nat,I: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.32/0.49 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_mult_imp_div_less
% 0.32/0.49 thf(fact_482_div__times__less__eq__dividend,axiom,
% 0.32/0.49 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.32/0.49
% 0.32/0.49 % div_times_less_eq_dividend
% 0.32/0.49 thf(fact_483_times__div__less__eq__dividend,axiom,
% 0.32/0.49 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.32/0.49
% 0.32/0.49 % times_div_less_eq_dividend
% 0.32/0.49 thf(fact_484_power__less__power__Suc,axiom,
% 0.32/0.49 ! [A: real,N: nat] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.49 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_less_power_Suc
% 0.32/0.49 thf(fact_485_power__less__power__Suc,axiom,
% 0.32/0.49 ! [A: rat,N: nat] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.49 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_less_power_Suc
% 0.32/0.49 thf(fact_486_power__less__power__Suc,axiom,
% 0.32/0.49 ! [A: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.49 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_less_power_Suc
% 0.32/0.49 thf(fact_487_power__less__power__Suc,axiom,
% 0.32/0.49 ! [A: int,N: nat] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.49 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_less_power_Suc
% 0.32/0.49 thf(fact_488_power__gt1__lemma,axiom,
% 0.32/0.49 ! [A: real,N: nat] :
% 0.32/0.49 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.49 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_gt1_lemma
% 0.32/0.49 thf(fact_489_power__gt1__lemma,axiom,
% 0.32/0.49 ! [A: rat,N: nat] :
% 0.32/0.49 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.49 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_gt1_lemma
% 0.32/0.49 thf(fact_490_power__gt1__lemma,axiom,
% 0.32/0.49 ! [A: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.49 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_gt1_lemma
% 0.32/0.49 thf(fact_491_power__gt1__lemma,axiom,
% 0.32/0.49 ! [A: int,N: nat] :
% 0.32/0.49 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.49 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_gt1_lemma
% 0.32/0.49 thf(fact_492_linorder__neqE__linordered__idom,axiom,
% 0.32/0.49 ! [X: real,Y3: real] :
% 0.32/0.49 ( ( X != Y3 )
% 0.32/0.49 => ( ~ ( ord_less_real @ X @ Y3 )
% 0.32/0.49 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % linorder_neqE_linordered_idom
% 0.32/0.49 thf(fact_493_linorder__neqE__linordered__idom,axiom,
% 0.32/0.49 ! [X: rat,Y3: rat] :
% 0.32/0.49 ( ( X != Y3 )
% 0.32/0.49 => ( ~ ( ord_less_rat @ X @ Y3 )
% 0.32/0.49 => ( ord_less_rat @ Y3 @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % linorder_neqE_linordered_idom
% 0.32/0.49 thf(fact_494_linorder__neqE__linordered__idom,axiom,
% 0.32/0.49 ! [X: int,Y3: int] :
% 0.32/0.49 ( ( X != Y3 )
% 0.32/0.49 => ( ~ ( ord_less_int @ X @ Y3 )
% 0.32/0.49 => ( ord_less_int @ Y3 @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % linorder_neqE_linordered_idom
% 0.32/0.49 thf(fact_495_nat__diff__add__eq2,axiom,
% 0.32/0.49 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.49 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_diff_add_eq2
% 0.32/0.49 thf(fact_496_nat__diff__add__eq1,axiom,
% 0.32/0.49 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.49 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.49 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_diff_add_eq1
% 0.32/0.49 thf(fact_497_nat__le__add__iff2,axiom,
% 0.32/0.49 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.49 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_le_add_iff2
% 0.32/0.49 thf(fact_498_nat__le__add__iff1,axiom,
% 0.32/0.49 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.49 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.49 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_le_add_iff1
% 0.32/0.49 thf(fact_499_nat__eq__add__iff2,axiom,
% 0.32/0.49 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.49 = ( M
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_eq_add_iff2
% 0.32/0.49 thf(fact_500_nat__eq__add__iff1,axiom,
% 0.32/0.49 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.49 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.49 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.32/0.49 = N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_eq_add_iff1
% 0.32/0.49 thf(fact_501_nat__neq__iff,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( M != N )
% 0.32/0.49 = ( ( ord_less_nat @ M @ N )
% 0.32/0.49 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_neq_iff
% 0.32/0.49 thf(fact_502_less__not__refl,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ~ ( ord_less_nat @ N @ N ) ).
% 0.32/0.49
% 0.32/0.49 % less_not_refl
% 0.32/0.49 thf(fact_503_less__not__refl2,axiom,
% 0.32/0.49 ! [N: nat,M: nat] :
% 0.32/0.49 ( ( ord_less_nat @ N @ M )
% 0.32/0.49 => ( M != N ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_not_refl2
% 0.32/0.49 thf(fact_504_less__not__refl3,axiom,
% 0.32/0.49 ! [S: nat,T: nat] :
% 0.32/0.49 ( ( ord_less_nat @ S @ T )
% 0.32/0.49 => ( S != T ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_not_refl3
% 0.32/0.49 thf(fact_505_less__irrefl__nat,axiom,
% 0.32/0.49 ! [N: nat] :
% 0.32/0.49 ~ ( ord_less_nat @ N @ N ) ).
% 0.32/0.49
% 0.32/0.49 % less_irrefl_nat
% 0.32/0.49 thf(fact_506_nat__less__induct,axiom,
% 0.32/0.49 ! [P: nat > $o,N: nat] :
% 0.32/0.49 ( ! [N4: nat] :
% 0.32/0.49 ( ! [M2: nat] :
% 0.32/0.49 ( ( ord_less_nat @ M2 @ N4 )
% 0.32/0.49 => ( P @ M2 ) )
% 0.32/0.49 => ( P @ N4 ) )
% 0.32/0.49 => ( P @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_less_induct
% 0.32/0.49 thf(fact_507_infinite__descent,axiom,
% 0.32/0.49 ! [P: nat > $o,N: nat] :
% 0.32/0.49 ( ! [N4: nat] :
% 0.32/0.49 ( ~ ( P @ N4 )
% 0.32/0.49 => ? [M2: nat] :
% 0.32/0.49 ( ( ord_less_nat @ M2 @ N4 )
% 0.32/0.49 & ~ ( P @ M2 ) ) )
% 0.32/0.49 => ( P @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % infinite_descent
% 0.32/0.49 thf(fact_508_linorder__neqE__nat,axiom,
% 0.32/0.49 ! [X: nat,Y3: nat] :
% 0.32/0.49 ( ( X != Y3 )
% 0.32/0.49 => ( ~ ( ord_less_nat @ X @ Y3 )
% 0.32/0.49 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % linorder_neqE_nat
% 0.32/0.49 thf(fact_509_le__refl,axiom,
% 0.32/0.49 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.32/0.49
% 0.32/0.49 % le_refl
% 0.32/0.49 thf(fact_510_le__trans,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ( ord_less_eq_nat @ J @ K )
% 0.32/0.49 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_trans
% 0.32/0.49 thf(fact_511_eq__imp__le,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( M = N )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % eq_imp_le
% 0.32/0.49 thf(fact_512_le__antisym,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.49 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.49 => ( M = N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_antisym
% 0.32/0.49 thf(fact_513_nat__le__linear,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.49 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_le_linear
% 0.32/0.49 thf(fact_514_Nat_Oex__has__greatest__nat,axiom,
% 0.32/0.49 ! [P: nat > $o,K: nat,B: nat] :
% 0.32/0.49 ( ( P @ K )
% 0.32/0.49 => ( ! [Y4: nat] :
% 0.32/0.49 ( ( P @ Y4 )
% 0.32/0.49 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.32/0.49 => ? [X3: nat] :
% 0.32/0.49 ( ( P @ X3 )
% 0.32/0.49 & ! [Y5: nat] :
% 0.32/0.49 ( ( P @ Y5 )
% 0.32/0.49 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % Nat.ex_has_greatest_nat
% 0.32/0.49 thf(fact_515_diff__commute,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat] :
% 0.32/0.49 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.32/0.49 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.32/0.49
% 0.32/0.49 % diff_commute
% 0.32/0.49 thf(fact_516_left__add__twice,axiom,
% 0.32/0.49 ! [A: complex,B: complex] :
% 0.32/0.49 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.49 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_add_twice
% 0.32/0.49 thf(fact_517_left__add__twice,axiom,
% 0.32/0.49 ! [A: real,B: real] :
% 0.32/0.49 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.49 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_add_twice
% 0.32/0.49 thf(fact_518_left__add__twice,axiom,
% 0.32/0.49 ! [A: rat,B: rat] :
% 0.32/0.49 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.49 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_add_twice
% 0.32/0.49 thf(fact_519_left__add__twice,axiom,
% 0.32/0.49 ! [A: nat,B: nat] :
% 0.32/0.49 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.49 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_add_twice
% 0.32/0.49 thf(fact_520_left__add__twice,axiom,
% 0.32/0.49 ! [A: int,B: int] :
% 0.32/0.49 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.49 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.32/0.49
% 0.32/0.49 % left_add_twice
% 0.32/0.49 thf(fact_521_mult__2__right,axiom,
% 0.32/0.49 ! [Z: complex] :
% 0.32/0.49 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2_right
% 0.32/0.49 thf(fact_522_mult__2__right,axiom,
% 0.32/0.49 ! [Z: real] :
% 0.32/0.49 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2_right
% 0.32/0.49 thf(fact_523_mult__2__right,axiom,
% 0.32/0.49 ! [Z: rat] :
% 0.32/0.49 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2_right
% 0.32/0.49 thf(fact_524_mult__2__right,axiom,
% 0.32/0.49 ! [Z: nat] :
% 0.32/0.49 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2_right
% 0.32/0.49 thf(fact_525_mult__2__right,axiom,
% 0.32/0.49 ! [Z: int] :
% 0.32/0.49 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2_right
% 0.32/0.49 thf(fact_526_mult__2,axiom,
% 0.32/0.49 ! [Z: complex] :
% 0.32/0.49 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 0.32/0.49 = ( plus_plus_complex @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2
% 0.32/0.49 thf(fact_527_mult__2,axiom,
% 0.32/0.49 ! [Z: real] :
% 0.32/0.49 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 0.32/0.49 = ( plus_plus_real @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2
% 0.32/0.49 thf(fact_528_mult__2,axiom,
% 0.32/0.49 ! [Z: rat] :
% 0.32/0.49 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 0.32/0.49 = ( plus_plus_rat @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2
% 0.32/0.49 thf(fact_529_mult__2,axiom,
% 0.32/0.49 ! [Z: nat] :
% 0.32/0.49 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 0.32/0.49 = ( plus_plus_nat @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2
% 0.32/0.49 thf(fact_530_mult__2,axiom,
% 0.32/0.49 ! [Z: int] :
% 0.32/0.49 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 0.32/0.49 = ( plus_plus_int @ Z @ Z ) ) ).
% 0.32/0.49
% 0.32/0.49 % mult_2
% 0.32/0.49 thf(fact_531_power2__eq__square,axiom,
% 0.32/0.49 ! [A: complex] :
% 0.32/0.49 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( times_times_complex @ A @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_eq_square
% 0.32/0.49 thf(fact_532_power2__eq__square,axiom,
% 0.32/0.49 ! [A: real] :
% 0.32/0.49 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( times_times_real @ A @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_eq_square
% 0.32/0.49 thf(fact_533_power2__eq__square,axiom,
% 0.32/0.49 ! [A: rat] :
% 0.32/0.49 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( times_times_rat @ A @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_eq_square
% 0.32/0.49 thf(fact_534_power2__eq__square,axiom,
% 0.32/0.49 ! [A: nat] :
% 0.32/0.49 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( times_times_nat @ A @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_eq_square
% 0.32/0.49 thf(fact_535_power2__eq__square,axiom,
% 0.32/0.49 ! [A: int] :
% 0.32/0.49 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( times_times_int @ A @ A ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_eq_square
% 0.32/0.49 thf(fact_536_power4__eq__xxxx,axiom,
% 0.32/0.49 ! [X: complex] :
% 0.32/0.49 ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.49 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.49
% 0.32/0.49 % power4_eq_xxxx
% 0.32/0.49 thf(fact_537_power4__eq__xxxx,axiom,
% 0.32/0.49 ! [X: real] :
% 0.32/0.49 ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.49 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.49
% 0.32/0.49 % power4_eq_xxxx
% 0.32/0.49 thf(fact_538_power4__eq__xxxx,axiom,
% 0.32/0.49 ! [X: rat] :
% 0.32/0.49 ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.49 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.49
% 0.32/0.49 % power4_eq_xxxx
% 0.32/0.49 thf(fact_539_power4__eq__xxxx,axiom,
% 0.32/0.49 ! [X: nat] :
% 0.32/0.49 ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.49 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.49
% 0.32/0.49 % power4_eq_xxxx
% 0.32/0.49 thf(fact_540_power4__eq__xxxx,axiom,
% 0.32/0.49 ! [X: int] :
% 0.32/0.49 ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.49 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 0.32/0.49
% 0.32/0.49 % power4_eq_xxxx
% 0.32/0.49 thf(fact_541_power__even__eq,axiom,
% 0.32/0.49 ! [A: nat,N: nat] :
% 0.32/0.49 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.49 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_even_eq
% 0.32/0.49 thf(fact_542_power__even__eq,axiom,
% 0.32/0.49 ! [A: real,N: nat] :
% 0.32/0.49 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.49 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_even_eq
% 0.32/0.49 thf(fact_543_power__even__eq,axiom,
% 0.32/0.49 ! [A: int,N: nat] :
% 0.32/0.49 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.49 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_even_eq
% 0.32/0.49 thf(fact_544_power__even__eq,axiom,
% 0.32/0.49 ! [A: complex,N: nat] :
% 0.32/0.49 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.49 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power_even_eq
% 0.32/0.49 thf(fact_545_nat__less__add__iff2,axiom,
% 0.32/0.49 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.49 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_less_add_iff2
% 0.32/0.49 thf(fact_546_nat__less__add__iff1,axiom,
% 0.32/0.49 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.49 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.32/0.49 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_less_add_iff1
% 0.32/0.49 thf(fact_547_power2__sum,axiom,
% 0.32/0.49 ! [X: complex,Y3: complex] :
% 0.32/0.49 ( ( power_power_complex @ ( plus_plus_complex @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_sum
% 0.32/0.49 thf(fact_548_power2__sum,axiom,
% 0.32/0.49 ! [X: real,Y3: real] :
% 0.32/0.49 ( ( power_power_real @ ( plus_plus_real @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_sum
% 0.32/0.49 thf(fact_549_power2__sum,axiom,
% 0.32/0.49 ! [X: rat,Y3: rat] :
% 0.32/0.49 ( ( power_power_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_sum
% 0.32/0.49 thf(fact_550_power2__sum,axiom,
% 0.32/0.49 ! [X: nat,Y3: nat] :
% 0.32/0.49 ( ( power_power_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_sum
% 0.32/0.49 thf(fact_551_power2__sum,axiom,
% 0.32/0.49 ! [X: int,Y3: int] :
% 0.32/0.49 ( ( power_power_int @ ( plus_plus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.49 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % power2_sum
% 0.32/0.49 thf(fact_552_add__diff__add,axiom,
% 0.32/0.49 ! [A: real,C: real,B: real,D: real] :
% 0.32/0.49 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 0.32/0.49 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_diff_add
% 0.32/0.49 thf(fact_553_add__diff__add,axiom,
% 0.32/0.49 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.32/0.49 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 0.32/0.49 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_diff_add
% 0.32/0.49 thf(fact_554_add__diff__add,axiom,
% 0.32/0.49 ! [A: int,C: int,B: int,D: int] :
% 0.32/0.49 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 0.32/0.49 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_diff_add
% 0.32/0.49 thf(fact_555_nat__less__le,axiom,
% 0.32/0.49 ( ord_less_nat
% 0.32/0.49 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.49 & ( M3 != N2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % nat_less_le
% 0.32/0.49 thf(fact_556_less__imp__le__nat,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ M @ N )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_imp_le_nat
% 0.32/0.49 thf(fact_557_le__eq__less__or__eq,axiom,
% 0.32/0.49 ( ord_less_eq_nat
% 0.32/0.49 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.49 ( ( ord_less_nat @ M3 @ N2 )
% 0.32/0.49 | ( M3 = N2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_eq_less_or_eq
% 0.32/0.49 thf(fact_558_less__or__eq__imp__le,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ( ord_less_nat @ M @ N )
% 0.32/0.49 | ( M = N ) )
% 0.32/0.49 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_or_eq_imp_le
% 0.32/0.49 thf(fact_559_le__neq__implies__less,axiom,
% 0.32/0.49 ! [M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.49 => ( ( M != N )
% 0.32/0.49 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % le_neq_implies_less
% 0.32/0.49 thf(fact_560_less__mono__imp__le__mono,axiom,
% 0.32/0.49 ! [F: nat > nat,I: nat,J: nat] :
% 0.32/0.49 ( ! [I2: nat,J2: nat] :
% 0.32/0.49 ( ( ord_less_nat @ I2 @ J2 )
% 0.32/0.49 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.32/0.49 => ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.49 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_mono_imp_le_mono
% 0.32/0.49 thf(fact_561_add__lessD1,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat] :
% 0.32/0.49 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.32/0.49 => ( ord_less_nat @ I @ K ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_lessD1
% 0.32/0.49 thf(fact_562_add__less__mono,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.49 ( ( ord_less_nat @ I @ J )
% 0.32/0.49 => ( ( ord_less_nat @ K @ L2 )
% 0.32/0.49 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_less_mono
% 0.32/0.49 thf(fact_563_not__add__less1,axiom,
% 0.32/0.49 ! [I: nat,J: nat] :
% 0.32/0.49 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.32/0.49
% 0.32/0.49 % not_add_less1
% 0.32/0.49 thf(fact_564_not__add__less2,axiom,
% 0.32/0.49 ! [J: nat,I: nat] :
% 0.32/0.49 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.32/0.49
% 0.32/0.49 % not_add_less2
% 0.32/0.49 thf(fact_565_add__less__mono1,axiom,
% 0.32/0.49 ! [I: nat,J: nat,K: nat] :
% 0.32/0.49 ( ( ord_less_nat @ I @ J )
% 0.32/0.49 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % add_less_mono1
% 0.32/0.49 thf(fact_566_trans__less__add1,axiom,
% 0.32/0.49 ! [I: nat,J: nat,M: nat] :
% 0.32/0.49 ( ( ord_less_nat @ I @ J )
% 0.32/0.49 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % trans_less_add1
% 0.32/0.49 thf(fact_567_trans__less__add2,axiom,
% 0.32/0.49 ! [I: nat,J: nat,M: nat] :
% 0.32/0.49 ( ( ord_less_nat @ I @ J )
% 0.32/0.49 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % trans_less_add2
% 0.32/0.49 thf(fact_568_less__add__eq__less,axiom,
% 0.32/0.49 ! [K: nat,L2: nat,M: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_nat @ K @ L2 )
% 0.32/0.49 => ( ( ( plus_plus_nat @ M @ L2 )
% 0.32/0.49 = ( plus_plus_nat @ K @ N ) )
% 0.32/0.49 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.49
% 0.32/0.49 % less_add_eq_less
% 0.32/0.49 thf(fact_569_add__leE,axiom,
% 0.32/0.49 ! [M: nat,K: nat,N: nat] :
% 0.32/0.49 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.49 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.49 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_leE
% 0.32/0.50 thf(fact_570_le__add1,axiom,
% 0.32/0.50 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_add1
% 0.32/0.50 thf(fact_571_le__add2,axiom,
% 0.32/0.50 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_add2
% 0.32/0.50 thf(fact_572_add__leD1,axiom,
% 0.32/0.50 ! [M: nat,K: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_leD1
% 0.32/0.50 thf(fact_573_add__leD2,axiom,
% 0.32/0.50 ! [M: nat,K: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.32/0.50 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_leD2
% 0.32/0.50 thf(fact_574_le__Suc__ex,axiom,
% 0.32/0.50 ! [K: nat,L2: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ L2 )
% 0.32/0.50 => ? [N4: nat] :
% 0.32/0.50 ( L2
% 0.32/0.50 = ( plus_plus_nat @ K @ N4 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_Suc_ex
% 0.32/0.50 thf(fact_575_add__le__mono,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.50 => ( ( ord_less_eq_nat @ K @ L2 )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_mono
% 0.32/0.50 thf(fact_576_add__le__mono1,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_mono1
% 0.32/0.50 thf(fact_577_trans__le__add1,axiom,
% 0.32/0.50 ! [I: nat,J: nat,M: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.50 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % trans_le_add1
% 0.32/0.50 thf(fact_578_trans__le__add2,axiom,
% 0.32/0.50 ! [I: nat,J: nat,M: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.50 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % trans_le_add2
% 0.32/0.50 thf(fact_579_nat__le__iff__add,axiom,
% 0.32/0.50 ( ord_less_eq_nat
% 0.32/0.50 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.50 ? [K2: nat] :
% 0.32/0.50 ( N2
% 0.32/0.50 = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nat_le_iff_add
% 0.32/0.50 thf(fact_580_diff__less__mono2,axiom,
% 0.32/0.50 ! [M: nat,N: nat,L2: nat] :
% 0.32/0.50 ( ( ord_less_nat @ M @ N )
% 0.32/0.50 => ( ( ord_less_nat @ M @ L2 )
% 0.32/0.50 => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_less_mono2
% 0.32/0.50 thf(fact_581_less__imp__diff__less,axiom,
% 0.32/0.50 ! [J: nat,K: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_nat @ J @ K )
% 0.32/0.50 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_imp_diff_less
% 0.32/0.50 thf(fact_582_eq__diff__iff,axiom,
% 0.32/0.50 ! [K: nat,M: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.50 => ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.50 => ( ( ( minus_minus_nat @ M @ K )
% 0.32/0.50 = ( minus_minus_nat @ N @ K ) )
% 0.32/0.50 = ( M = N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_diff_iff
% 0.32/0.50 thf(fact_583_le__diff__iff,axiom,
% 0.32/0.50 ! [K: nat,M: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.50 => ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.50 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.32/0.50 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_diff_iff
% 0.32/0.50 thf(fact_584_Nat_Odiff__diff__eq,axiom,
% 0.32/0.50 ! [K: nat,M: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.50 => ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.50 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.32/0.50 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % Nat.diff_diff_eq
% 0.32/0.50 thf(fact_585_diff__le__mono,axiom,
% 0.32/0.50 ! [M: nat,N: nat,L2: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_le_mono
% 0.32/0.50 thf(fact_586_diff__le__self,axiom,
% 0.32/0.50 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.32/0.50
% 0.32/0.50 % diff_le_self
% 0.32/0.50 thf(fact_587_le__diff__iff_H,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ C )
% 0.32/0.50 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.50 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.32/0.50 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_diff_iff'
% 0.32/0.50 thf(fact_588_diff__le__mono2,axiom,
% 0.32/0.50 ! [M: nat,N: nat,L2: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_le_mono2
% 0.32/0.50 thf(fact_589_power2__diff,axiom,
% 0.32/0.50 ! [X: complex,Y3: complex] :
% 0.32/0.50 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % power2_diff
% 0.32/0.50 thf(fact_590_power2__diff,axiom,
% 0.32/0.50 ! [X: real,Y3: real] :
% 0.32/0.50 ( ( power_power_real @ ( minus_minus_real @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % power2_diff
% 0.32/0.50 thf(fact_591_power2__diff,axiom,
% 0.32/0.50 ! [X: rat,Y3: rat] :
% 0.32/0.50 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % power2_diff
% 0.32/0.50 thf(fact_592_power2__diff,axiom,
% 0.32/0.50 ! [X: int,Y3: int] :
% 0.32/0.50 ( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % power2_diff
% 0.32/0.50 thf(fact_593_Nat_Odiff__cancel,axiom,
% 0.32/0.50 ! [K: nat,M: nat,N: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.32/0.50 = ( minus_minus_nat @ M @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % Nat.diff_cancel
% 0.32/0.50 thf(fact_594_diff__cancel2,axiom,
% 0.32/0.50 ! [M: nat,K: nat,N: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.32/0.50 = ( minus_minus_nat @ M @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_cancel2
% 0.32/0.50 thf(fact_595_diff__add__inverse,axiom,
% 0.32/0.50 ! [N: nat,M: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.32/0.50 = M ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_inverse
% 0.32/0.50 thf(fact_596_diff__add__inverse2,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.32/0.50 = M ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_inverse2
% 0.32/0.50 thf(fact_597_add__le__add__imp__diff__le,axiom,
% 0.32/0.50 ! [I: real,K: real,N: real,J: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.32/0.50 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.32/0.50 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.32/0.50 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.32/0.50 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_add_imp_diff_le
% 0.32/0.50 thf(fact_598_add__le__add__imp__diff__le,axiom,
% 0.32/0.50 ! [I: rat,K: rat,N: rat,J: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.32/0.50 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.32/0.50 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.32/0.50 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.32/0.50 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_add_imp_diff_le
% 0.32/0.50 thf(fact_599_add__le__add__imp__diff__le,axiom,
% 0.32/0.50 ! [I: nat,K: nat,N: nat,J: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.32/0.50 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.32/0.50 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.32/0.50 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.32/0.50 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_add_imp_diff_le
% 0.32/0.50 thf(fact_600_add__le__add__imp__diff__le,axiom,
% 0.32/0.50 ! [I: int,K: int,N: int,J: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.32/0.50 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.32/0.50 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.32/0.50 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.32/0.50 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_add_imp_diff_le
% 0.32/0.50 thf(fact_601_add__le__imp__le__diff,axiom,
% 0.32/0.50 ! [I: real,K: real,N: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.32/0.50 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_diff
% 0.32/0.50 thf(fact_602_add__le__imp__le__diff,axiom,
% 0.32/0.50 ! [I: rat,K: rat,N: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.32/0.50 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_diff
% 0.32/0.50 thf(fact_603_add__le__imp__le__diff,axiom,
% 0.32/0.50 ! [I: nat,K: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.32/0.50 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_diff
% 0.32/0.50 thf(fact_604_add__le__imp__le__diff,axiom,
% 0.32/0.50 ! [I: int,K: int,N: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.32/0.50 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_diff
% 0.32/0.50 thf(fact_605_less__add__one,axiom,
% 0.32/0.50 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_add_one
% 0.32/0.50 thf(fact_606_less__add__one,axiom,
% 0.32/0.50 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_add_one
% 0.32/0.50 thf(fact_607_less__add__one,axiom,
% 0.32/0.50 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_add_one
% 0.32/0.50 thf(fact_608_less__add__one,axiom,
% 0.32/0.50 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_add_one
% 0.32/0.50 thf(fact_609_add__mono1,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono1
% 0.32/0.50 thf(fact_610_add__mono1,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono1
% 0.32/0.50 thf(fact_611_add__mono1,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_nat @ A @ B )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono1
% 0.32/0.50 thf(fact_612_add__mono1,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( ord_less_int @ A @ B )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono1
% 0.32/0.50 thf(fact_613_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ~ ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % linordered_semidom_class.add_diff_inverse
% 0.32/0.50 thf(fact_614_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ~ ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % linordered_semidom_class.add_diff_inverse
% 0.32/0.50 thf(fact_615_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ~ ( ord_less_nat @ A @ B )
% 0.32/0.50 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % linordered_semidom_class.add_diff_inverse
% 0.32/0.50 thf(fact_616_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ~ ( ord_less_int @ A @ B )
% 0.32/0.50 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.32/0.50 = A ) ) ).
% 0.32/0.50
% 0.32/0.50 % linordered_semidom_class.add_diff_inverse
% 0.32/0.50 thf(fact_617_mono__nat__linear__lb,axiom,
% 0.32/0.50 ! [F: nat > nat,M: nat,K: nat] :
% 0.32/0.50 ( ! [M4: nat,N4: nat] :
% 0.32/0.50 ( ( ord_less_nat @ M4 @ N4 )
% 0.32/0.50 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mono_nat_linear_lb
% 0.32/0.50 thf(fact_618_diff__less__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_nat @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.50 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_less_mono
% 0.32/0.50 thf(fact_619_less__diff__iff,axiom,
% 0.32/0.50 ! [K: nat,M: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.50 => ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.50 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.32/0.50 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_diff_iff
% 0.32/0.50 thf(fact_620_less__diff__conv,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat] :
% 0.32/0.50 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.32/0.50 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_diff_conv
% 0.32/0.50 thf(fact_621_add__diff__inverse__nat,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ~ ( ord_less_nat @ M @ N )
% 0.32/0.50 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.50 = M ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_inverse_nat
% 0.32/0.50 thf(fact_622_Nat_Ole__imp__diff__is__add,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.50 => ( ( ( minus_minus_nat @ J @ I )
% 0.32/0.50 = K )
% 0.32/0.50 = ( J
% 0.32/0.50 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % Nat.le_imp_diff_is_add
% 0.32/0.50 thf(fact_623_Nat_Odiff__add__assoc2,axiom,
% 0.32/0.50 ! [K: nat,J: nat,I: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.32/0.50 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % Nat.diff_add_assoc2
% 0.32/0.50 thf(fact_624_Nat_Odiff__add__assoc,axiom,
% 0.32/0.50 ! [K: nat,J: nat,I: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.32/0.50 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % Nat.diff_add_assoc
% 0.32/0.50 thf(fact_625_Nat_Ole__diff__conv2,axiom,
% 0.32/0.50 ! [K: nat,J: nat,I: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.50 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.32/0.50 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % Nat.le_diff_conv2
% 0.32/0.50 thf(fact_626_le__diff__conv,axiom,
% 0.32/0.50 ! [J: nat,K: nat,I: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.32/0.50 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_diff_conv
% 0.32/0.50 thf(fact_627_less__diff__conv2,axiom,
% 0.32/0.50 ! [K: nat,J: nat,I: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.50 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.32/0.50 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_diff_conv2
% 0.32/0.50 thf(fact_628_field__sum__of__halves,axiom,
% 0.32/0.50 ! [X: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = X ) ).
% 0.32/0.50
% 0.32/0.50 % field_sum_of_halves
% 0.32/0.50 thf(fact_629_field__sum__of__halves,axiom,
% 0.32/0.50 ! [X: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = X ) ).
% 0.32/0.50
% 0.32/0.50 % field_sum_of_halves
% 0.32/0.50 thf(fact_630_post__member__pre__member,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat,Y3: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.50 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.50 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y3 )
% 0.32/0.50 => ( ( vEBT_vebt_member @ T @ Y3 )
% 0.32/0.50 | ( X = Y3 ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % post_member_pre_member
% 0.32/0.50 thf(fact_631_mul__def,axiom,
% 0.32/0.50 ( vEBT_VEBT_mul
% 0.32/0.50 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % mul_def
% 0.32/0.50 thf(fact_632_misiz,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,M: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( some_nat @ M )
% 0.32/0.50 = ( vEBT_vebt_mint @ T ) )
% 0.32/0.50 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % misiz
% 0.32/0.50 thf(fact_633_maxt__sound,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 0.32/0.50 => ( ( vEBT_vebt_maxt @ T )
% 0.32/0.50 = ( some_nat @ X ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % maxt_sound
% 0.32/0.50 thf(fact_634_maxt__corr,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( vEBT_vebt_maxt @ T )
% 0.32/0.50 = ( some_nat @ X ) )
% 0.32/0.50 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % maxt_corr
% 0.32/0.50 thf(fact_635_mul__shift,axiom,
% 0.32/0.50 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.50 ( ( ( times_times_nat @ X @ Y3 )
% 0.32/0.50 = Z )
% 0.32/0.50 = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y3 ) )
% 0.32/0.50 = ( some_nat @ Z ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mul_shift
% 0.32/0.50 thf(fact_636_mint__sound,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 0.32/0.50 => ( ( vEBT_vebt_mint @ T )
% 0.32/0.50 = ( some_nat @ X ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mint_sound
% 0.32/0.50 thf(fact_637_mint__corr,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.32/0.50 = ( some_nat @ X ) )
% 0.32/0.50 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mint_corr
% 0.32/0.50 thf(fact_638_maxt__corr__help,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( vEBT_vebt_maxt @ T )
% 0.32/0.50 = ( some_nat @ Maxi ) )
% 0.32/0.50 => ( ( vEBT_vebt_member @ T @ X )
% 0.32/0.50 => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % maxt_corr_help
% 0.32/0.50 thf(fact_639_sum__squares__bound,axiom,
% 0.32/0.50 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % sum_squares_bound
% 0.32/0.50 thf(fact_640_sum__squares__bound,axiom,
% 0.32/0.50 ! [X: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % sum_squares_bound
% 0.32/0.50 thf(fact_641_mint__corr__help,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.32/0.50 = ( some_nat @ Mini ) )
% 0.32/0.50 => ( ( vEBT_vebt_member @ T @ X )
% 0.32/0.50 => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mint_corr_help
% 0.32/0.50 thf(fact_642_maxt__member,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( vEBT_vebt_maxt @ T )
% 0.32/0.50 = ( some_nat @ Maxi ) )
% 0.32/0.50 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % maxt_member
% 0.32/0.50 thf(fact_643_mint__member,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.32/0.50 = ( some_nat @ Maxi ) )
% 0.32/0.50 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mint_member
% 0.32/0.50 thf(fact_644_semiring__norm_I13_J,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.50 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % semiring_norm(13)
% 0.32/0.50 thf(fact_645_semiring__norm_I12_J,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( times_times_num @ one @ N )
% 0.32/0.50 = N ) ).
% 0.32/0.50
% 0.32/0.50 % semiring_norm(12)
% 0.32/0.50 thf(fact_646_semiring__norm_I11_J,axiom,
% 0.32/0.50 ! [M: num] :
% 0.32/0.50 ( ( times_times_num @ M @ one )
% 0.32/0.50 = M ) ).
% 0.32/0.50
% 0.32/0.50 % semiring_norm(11)
% 0.32/0.50 thf(fact_647_num__double,axiom,
% 0.32/0.50 ! [N: num] :
% 0.32/0.50 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 0.32/0.50 = ( bit0 @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % num_double
% 0.32/0.50 thf(fact_648_power__mult__numeral,axiom,
% 0.32/0.50 ! [A: nat,M: num,N: num] :
% 0.32/0.50 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.50 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % power_mult_numeral
% 0.32/0.50 thf(fact_649_power__mult__numeral,axiom,
% 0.32/0.50 ! [A: real,M: num,N: num] :
% 0.32/0.50 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.50 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % power_mult_numeral
% 0.32/0.50 thf(fact_650_power__mult__numeral,axiom,
% 0.32/0.50 ! [A: int,M: num,N: num] :
% 0.32/0.50 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.50 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % power_mult_numeral
% 0.32/0.50 thf(fact_651_power__mult__numeral,axiom,
% 0.32/0.50 ! [A: complex,M: num,N: num] :
% 0.32/0.50 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.50 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % power_mult_numeral
% 0.32/0.50 thf(fact_652_complete__real,axiom,
% 0.32/0.50 ! [S2: set_real] :
% 0.32/0.50 ( ? [X5: real] : ( member_real @ X5 @ S2 )
% 0.32/0.50 => ( ? [Z3: real] :
% 0.32/0.50 ! [X3: real] :
% 0.32/0.50 ( ( member_real @ X3 @ S2 )
% 0.32/0.50 => ( ord_less_eq_real @ X3 @ Z3 ) )
% 0.32/0.50 => ? [Y4: real] :
% 0.32/0.50 ( ! [X5: real] :
% 0.32/0.50 ( ( member_real @ X5 @ S2 )
% 0.32/0.50 => ( ord_less_eq_real @ X5 @ Y4 ) )
% 0.32/0.50 & ! [Z3: real] :
% 0.32/0.50 ( ! [X3: real] :
% 0.32/0.50 ( ( member_real @ X3 @ S2 )
% 0.32/0.50 => ( ord_less_eq_real @ X3 @ Z3 ) )
% 0.32/0.50 => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % complete_real
% 0.32/0.50 thf(fact_653_real__arch__pow,axiom,
% 0.32/0.50 ! [X: real,Y3: real] :
% 0.32/0.50 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.50 => ? [N4: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X @ N4 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % real_arch_pow
% 0.32/0.50 thf(fact_654_less__eq__real__def,axiom,
% 0.32/0.50 ( ord_less_eq_real
% 0.32/0.50 = ( ^ [X2: real,Y: real] :
% 0.32/0.50 ( ( ord_less_real @ X2 @ Y )
% 0.32/0.50 | ( X2 = Y ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_eq_real_def
% 0.32/0.50 thf(fact_655_four__x__squared,axiom,
% 0.32/0.50 ! [X: real] :
% 0.32/0.50 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.50 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % four_x_squared
% 0.32/0.50 thf(fact_656_L2__set__mult__ineq__lemma,axiom,
% 0.32/0.50 ! [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 ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % L2_set_mult_ineq_lemma
% 0.32/0.50 thf(fact_657_div__mult2__numeral__eq,axiom,
% 0.32/0.50 ! [A: nat,K: num,L2: num] :
% 0.32/0.50 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
% 0.32/0.50 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult2_numeral_eq
% 0.32/0.50 thf(fact_658_div__mult2__numeral__eq,axiom,
% 0.32/0.50 ! [A: int,K: num,L2: num] :
% 0.32/0.50 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
% 0.32/0.50 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult2_numeral_eq
% 0.32/0.50 thf(fact_659_real__average__minus__second,axiom,
% 0.32/0.50 ! [B: real,A: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.32/0.50 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % real_average_minus_second
% 0.32/0.50 thf(fact_660_real__average__minus__first,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 0.32/0.50 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % real_average_minus_first
% 0.32/0.50 thf(fact_661_minminNull,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT] :
% 0.32/0.50 ( ( ( vEBT_vebt_mint @ T )
% 0.32/0.50 = none_nat )
% 0.32/0.50 => ( vEBT_VEBT_minNull @ T ) ) ).
% 0.32/0.50
% 0.32/0.50 % minminNull
% 0.32/0.50 thf(fact_662_minNullmin,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT] :
% 0.32/0.50 ( ( vEBT_VEBT_minNull @ T )
% 0.32/0.50 => ( ( vEBT_vebt_mint @ T )
% 0.32/0.50 = none_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % minNullmin
% 0.32/0.50 thf(fact_663_enat__ord__number_I1_J,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.32/0.50 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % enat_ord_number(1)
% 0.32/0.50 thf(fact_664_enat__ord__number_I2_J,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.32/0.50 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % enat_ord_number(2)
% 0.32/0.50 thf(fact_665_add__diff__cancel__right_H,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right'
% 0.32/0.50 thf(fact_666_add__diff__cancel__right_H,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right'
% 0.32/0.50 thf(fact_667_add__diff__cancel__right_H,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right'
% 0.32/0.50 thf(fact_668_add__diff__cancel__right_H,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right'
% 0.32/0.50 thf(fact_669_add__diff__cancel__right,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 = ( minus_minus_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right
% 0.32/0.50 thf(fact_670_add__diff__cancel__right,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 = ( minus_minus_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right
% 0.32/0.50 thf(fact_671_add__diff__cancel__right,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.50 = ( minus_minus_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right
% 0.32/0.50 thf(fact_672_add__diff__cancel__right,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 = ( minus_minus_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_right
% 0.32/0.50 thf(fact_673_add__diff__cancel__left_H,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.32/0.50 = B ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left'
% 0.32/0.50 thf(fact_674_add__diff__cancel__left_H,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 0.32/0.50 = B ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left'
% 0.32/0.50 thf(fact_675_add__diff__cancel__left_H,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.32/0.50 = B ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left'
% 0.32/0.50 thf(fact_676_add__diff__cancel__left_H,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.32/0.50 = B ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left'
% 0.32/0.50 thf(fact_677_add__diff__cancel__left,axiom,
% 0.32/0.50 ! [C: real,A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.50 = ( minus_minus_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left
% 0.32/0.50 thf(fact_678_add__diff__cancel__left,axiom,
% 0.32/0.50 ! [C: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.50 = ( minus_minus_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left
% 0.32/0.50 thf(fact_679_add__diff__cancel__left,axiom,
% 0.32/0.50 ! [C: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.50 = ( minus_minus_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left
% 0.32/0.50 thf(fact_680_add__diff__cancel__left,axiom,
% 0.32/0.50 ! [C: int,A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.50 = ( minus_minus_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel_left
% 0.32/0.50 thf(fact_681_diff__add__cancel,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_cancel
% 0.32/0.50 thf(fact_682_diff__add__cancel,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_cancel
% 0.32/0.50 thf(fact_683_diff__add__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_cancel
% 0.32/0.50 thf(fact_684_min__Null__member,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,X: nat] :
% 0.32/0.50 ( ( vEBT_VEBT_minNull @ T )
% 0.32/0.50 => ~ ( vEBT_vebt_member @ T @ X ) ) ).
% 0.32/0.50
% 0.32/0.50 % min_Null_member
% 0.32/0.50 thf(fact_685_add__left__cancel,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.50 = ( plus_plus_real @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_cancel
% 0.32/0.50 thf(fact_686_add__left__cancel,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.50 = ( plus_plus_rat @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_cancel
% 0.32/0.50 thf(fact_687_add__left__cancel,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ A @ B )
% 0.32/0.50 = ( plus_plus_nat @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_cancel
% 0.32/0.50 thf(fact_688_add__left__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.50 = ( plus_plus_int @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_cancel
% 0.32/0.50 thf(fact_689_add__right__cancel,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ B @ A )
% 0.32/0.50 = ( plus_plus_real @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_cancel
% 0.32/0.50 thf(fact_690_add__right__cancel,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ B @ A )
% 0.32/0.50 = ( plus_plus_rat @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_cancel
% 0.32/0.50 thf(fact_691_add__right__cancel,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ B @ A )
% 0.32/0.50 = ( plus_plus_nat @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_cancel
% 0.32/0.50 thf(fact_692_add__right__cancel,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.50 = ( plus_plus_int @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_cancel
% 0.32/0.50 thf(fact_693_add__le__cancel__left,axiom,
% 0.32/0.50 ! [C: real,A: real,B: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.50 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_cancel_left
% 0.32/0.50 thf(fact_694_add__le__cancel__left,axiom,
% 0.32/0.50 ! [C: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.50 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_cancel_left
% 0.32/0.50 thf(fact_695_add__le__cancel__left,axiom,
% 0.32/0.50 ! [C: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.50 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_cancel_left
% 0.32/0.50 thf(fact_696_add__le__cancel__left,axiom,
% 0.32/0.50 ! [C: int,A: int,B: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.50 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_cancel_left
% 0.32/0.50 thf(fact_697_add__le__cancel__right,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_cancel_right
% 0.32/0.50 thf(fact_698_add__le__cancel__right,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_cancel_right
% 0.32/0.50 thf(fact_699_add__le__cancel__right,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.50 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_cancel_right
% 0.32/0.50 thf(fact_700_add__le__cancel__right,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_cancel_right
% 0.32/0.50 thf(fact_701_add__less__cancel__left,axiom,
% 0.32/0.50 ! [C: real,A: real,B: real] :
% 0.32/0.50 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.50 = ( ord_less_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_cancel_left
% 0.32/0.50 thf(fact_702_add__less__cancel__left,axiom,
% 0.32/0.50 ! [C: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.50 = ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_cancel_left
% 0.32/0.50 thf(fact_703_add__less__cancel__left,axiom,
% 0.32/0.50 ! [C: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.50 = ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_cancel_left
% 0.32/0.50 thf(fact_704_add__less__cancel__left,axiom,
% 0.32/0.50 ! [C: int,A: int,B: int] :
% 0.32/0.50 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.50 = ( ord_less_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_cancel_left
% 0.32/0.50 thf(fact_705_add__less__cancel__right,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 = ( ord_less_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_cancel_right
% 0.32/0.50 thf(fact_706_add__less__cancel__right,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 = ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_cancel_right
% 0.32/0.50 thf(fact_707_add__less__cancel__right,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.50 = ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_cancel_right
% 0.32/0.50 thf(fact_708_add__less__cancel__right,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 = ( ord_less_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_cancel_right
% 0.32/0.50 thf(fact_709_mult_Oright__neutral,axiom,
% 0.32/0.50 ! [A: complex] :
% 0.32/0.50 ( ( times_times_complex @ A @ one_one_complex )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.right_neutral
% 0.32/0.50 thf(fact_710_mult_Oright__neutral,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( times_times_real @ A @ one_one_real )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.right_neutral
% 0.32/0.50 thf(fact_711_mult_Oright__neutral,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( times_times_rat @ A @ one_one_rat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.right_neutral
% 0.32/0.50 thf(fact_712_mult_Oright__neutral,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( times_times_nat @ A @ one_one_nat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.right_neutral
% 0.32/0.50 thf(fact_713_mult_Oright__neutral,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( times_times_int @ A @ one_one_int )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.right_neutral
% 0.32/0.50 thf(fact_714_mult__1,axiom,
% 0.32/0.50 ! [A: complex] :
% 0.32/0.50 ( ( times_times_complex @ one_one_complex @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_1
% 0.32/0.50 thf(fact_715_mult__1,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( times_times_real @ one_one_real @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_1
% 0.32/0.50 thf(fact_716_mult__1,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( times_times_rat @ one_one_rat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_1
% 0.32/0.50 thf(fact_717_mult__1,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( times_times_nat @ one_one_nat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_1
% 0.32/0.50 thf(fact_718_mult__1,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( times_times_int @ one_one_int @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_1
% 0.32/0.50 thf(fact_719_add__diff__cancel,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel
% 0.32/0.50 thf(fact_720_add__diff__cancel,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel
% 0.32/0.50 thf(fact_721_add__diff__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_cancel
% 0.32/0.50 thf(fact_722_real__divide__square__eq,axiom,
% 0.32/0.50 ! [R: real,A: real] :
% 0.32/0.50 ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
% 0.32/0.50 = ( divide_divide_real @ A @ R ) ) ).
% 0.32/0.50
% 0.32/0.50 % real_divide_square_eq
% 0.32/0.50 thf(fact_723_add__diff__assoc__enat,axiom,
% 0.32/0.50 ! [Z: extended_enat,Y3: extended_enat,X: extended_enat] :
% 0.32/0.50 ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 0.32/0.50 => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 0.32/0.50 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y3 ) @ Z ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_assoc_enat
% 0.32/0.50 thf(fact_724_enat__less__induct,axiom,
% 0.32/0.50 ! [P: extended_enat > $o,N: extended_enat] :
% 0.32/0.50 ( ! [N4: extended_enat] :
% 0.32/0.50 ( ! [M2: extended_enat] :
% 0.32/0.50 ( ( ord_le72135733267957522d_enat @ M2 @ N4 )
% 0.32/0.50 => ( P @ M2 ) )
% 0.32/0.50 => ( P @ N4 ) )
% 0.32/0.50 => ( P @ N ) ) ).
% 0.32/0.50
% 0.32/0.50 % enat_less_induct
% 0.32/0.50 thf(fact_725_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.50 thf(fact_726_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.50 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.50 thf(fact_727_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.50 thf(fact_728_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.50 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_mult_class.mult_ac(1)
% 0.32/0.50 thf(fact_729_mult_Oassoc,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.assoc
% 0.32/0.50 thf(fact_730_mult_Oassoc,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.50 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.assoc
% 0.32/0.50 thf(fact_731_mult_Oassoc,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.assoc
% 0.32/0.50 thf(fact_732_mult_Oassoc,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.50 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.assoc
% 0.32/0.50 thf(fact_733_mult_Ocommute,axiom,
% 0.32/0.50 ( times_times_real
% 0.32/0.50 = ( ^ [A3: real,B4: real] : ( times_times_real @ B4 @ A3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.commute
% 0.32/0.50 thf(fact_734_mult_Ocommute,axiom,
% 0.32/0.50 ( times_times_rat
% 0.32/0.50 = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ B4 @ A3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.commute
% 0.32/0.50 thf(fact_735_mult_Ocommute,axiom,
% 0.32/0.50 ( times_times_nat
% 0.32/0.50 = ( ^ [A3: nat,B4: nat] : ( times_times_nat @ B4 @ A3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.commute
% 0.32/0.50 thf(fact_736_mult_Ocommute,axiom,
% 0.32/0.50 ( times_times_int
% 0.32/0.50 = ( ^ [A3: int,B4: int] : ( times_times_int @ B4 @ A3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.commute
% 0.32/0.50 thf(fact_737_mult_Oleft__commute,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.32/0.50 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.left_commute
% 0.32/0.50 thf(fact_738_mult_Oleft__commute,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 0.32/0.50 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.left_commute
% 0.32/0.50 thf(fact_739_mult_Oleft__commute,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.32/0.50 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.left_commute
% 0.32/0.50 thf(fact_740_mult_Oleft__commute,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 0.32/0.50 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult.left_commute
% 0.32/0.50 thf(fact_741_one__reorient,axiom,
% 0.32/0.50 ! [X: complex] :
% 0.32/0.50 ( ( one_one_complex = X )
% 0.32/0.50 = ( X = one_one_complex ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_reorient
% 0.32/0.50 thf(fact_742_one__reorient,axiom,
% 0.32/0.50 ! [X: real] :
% 0.32/0.50 ( ( one_one_real = X )
% 0.32/0.50 = ( X = one_one_real ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_reorient
% 0.32/0.50 thf(fact_743_one__reorient,axiom,
% 0.32/0.50 ! [X: rat] :
% 0.32/0.50 ( ( one_one_rat = X )
% 0.32/0.50 = ( X = one_one_rat ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_reorient
% 0.32/0.50 thf(fact_744_one__reorient,axiom,
% 0.32/0.50 ! [X: nat] :
% 0.32/0.50 ( ( one_one_nat = X )
% 0.32/0.50 = ( X = one_one_nat ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_reorient
% 0.32/0.50 thf(fact_745_one__reorient,axiom,
% 0.32/0.50 ! [X: int] :
% 0.32/0.50 ( ( one_one_int = X )
% 0.32/0.50 = ( X = one_one_int ) ) ).
% 0.32/0.50
% 0.32/0.50 % one_reorient
% 0.32/0.50 thf(fact_746_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.50 thf(fact_747_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.50 thf(fact_748_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.50 thf(fact_749_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ab_semigroup_add_class.add_ac(1)
% 0.32/0.50 thf(fact_750_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ( plus_plus_real @ I @ K )
% 0.32/0.50 = ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(4)
% 0.32/0.50 thf(fact_751_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ( plus_plus_rat @ I @ K )
% 0.32/0.50 = ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(4)
% 0.32/0.50 thf(fact_752_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ( plus_plus_nat @ I @ K )
% 0.32/0.50 = ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(4)
% 0.32/0.50 thf(fact_753_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ( plus_plus_int @ I @ K )
% 0.32/0.50 = ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(4)
% 0.32/0.50 thf(fact_754_group__cancel_Oadd1,axiom,
% 0.32/0.50 ! [A2: real,K: real,A: real,B: real] :
% 0.32/0.50 ( ( A2
% 0.32/0.50 = ( plus_plus_real @ K @ A ) )
% 0.32/0.50 => ( ( plus_plus_real @ A2 @ B )
% 0.32/0.50 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add1
% 0.32/0.50 thf(fact_755_group__cancel_Oadd1,axiom,
% 0.32/0.50 ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( A2
% 0.32/0.50 = ( plus_plus_rat @ K @ A ) )
% 0.32/0.50 => ( ( plus_plus_rat @ A2 @ B )
% 0.32/0.50 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add1
% 0.32/0.50 thf(fact_756_group__cancel_Oadd1,axiom,
% 0.32/0.50 ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( A2
% 0.32/0.50 = ( plus_plus_nat @ K @ A ) )
% 0.32/0.50 => ( ( plus_plus_nat @ A2 @ B )
% 0.32/0.50 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add1
% 0.32/0.50 thf(fact_757_group__cancel_Oadd1,axiom,
% 0.32/0.50 ! [A2: int,K: int,A: int,B: int] :
% 0.32/0.50 ( ( A2
% 0.32/0.50 = ( plus_plus_int @ K @ A ) )
% 0.32/0.50 => ( ( plus_plus_int @ A2 @ B )
% 0.32/0.50 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add1
% 0.32/0.50 thf(fact_758_group__cancel_Oadd2,axiom,
% 0.32/0.50 ! [B5: real,K: real,B: real,A: real] :
% 0.32/0.50 ( ( B5
% 0.32/0.50 = ( plus_plus_real @ K @ B ) )
% 0.32/0.50 => ( ( plus_plus_real @ A @ B5 )
% 0.32/0.50 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add2
% 0.32/0.50 thf(fact_759_group__cancel_Oadd2,axiom,
% 0.32/0.50 ! [B5: rat,K: rat,B: rat,A: rat] :
% 0.32/0.50 ( ( B5
% 0.32/0.50 = ( plus_plus_rat @ K @ B ) )
% 0.32/0.50 => ( ( plus_plus_rat @ A @ B5 )
% 0.32/0.50 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add2
% 0.32/0.50 thf(fact_760_group__cancel_Oadd2,axiom,
% 0.32/0.50 ! [B5: nat,K: nat,B: nat,A: nat] :
% 0.32/0.50 ( ( B5
% 0.32/0.50 = ( plus_plus_nat @ K @ B ) )
% 0.32/0.50 => ( ( plus_plus_nat @ A @ B5 )
% 0.32/0.50 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add2
% 0.32/0.50 thf(fact_761_group__cancel_Oadd2,axiom,
% 0.32/0.50 ! [B5: int,K: int,B: int,A: int] :
% 0.32/0.50 ( ( B5
% 0.32/0.50 = ( plus_plus_int @ K @ B ) )
% 0.32/0.50 => ( ( plus_plus_int @ A @ B5 )
% 0.32/0.50 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.add2
% 0.32/0.50 thf(fact_762_add_Oassoc,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.assoc
% 0.32/0.50 thf(fact_763_add_Oassoc,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.assoc
% 0.32/0.50 thf(fact_764_add_Oassoc,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.assoc
% 0.32/0.50 thf(fact_765_add_Oassoc,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.assoc
% 0.32/0.50 thf(fact_766_add_Oleft__cancel,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.50 = ( plus_plus_real @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_cancel
% 0.32/0.50 thf(fact_767_add_Oleft__cancel,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.50 = ( plus_plus_rat @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_cancel
% 0.32/0.50 thf(fact_768_add_Oleft__cancel,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.50 = ( plus_plus_int @ A @ C ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_cancel
% 0.32/0.50 thf(fact_769_add_Oright__cancel,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ B @ A )
% 0.32/0.50 = ( plus_plus_real @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_cancel
% 0.32/0.50 thf(fact_770_add_Oright__cancel,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ B @ A )
% 0.32/0.50 = ( plus_plus_rat @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_cancel
% 0.32/0.50 thf(fact_771_add_Oright__cancel,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.50 = ( plus_plus_int @ C @ A ) )
% 0.32/0.50 = ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.right_cancel
% 0.32/0.50 thf(fact_772_add_Ocommute,axiom,
% 0.32/0.50 ( plus_plus_real
% 0.32/0.50 = ( ^ [A3: real,B4: real] : ( plus_plus_real @ B4 @ A3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.commute
% 0.32/0.50 thf(fact_773_add_Ocommute,axiom,
% 0.32/0.50 ( plus_plus_rat
% 0.32/0.50 = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ B4 @ A3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.commute
% 0.32/0.50 thf(fact_774_add_Ocommute,axiom,
% 0.32/0.50 ( plus_plus_nat
% 0.32/0.50 = ( ^ [A3: nat,B4: nat] : ( plus_plus_nat @ B4 @ A3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.commute
% 0.32/0.50 thf(fact_775_add_Ocommute,axiom,
% 0.32/0.50 ( plus_plus_int
% 0.32/0.50 = ( ^ [A3: int,B4: int] : ( plus_plus_int @ B4 @ A3 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.commute
% 0.32/0.50 thf(fact_776_add_Oleft__commute,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.32/0.50 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_commute
% 0.32/0.50 thf(fact_777_add_Oleft__commute,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 0.32/0.50 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_commute
% 0.32/0.50 thf(fact_778_add_Oleft__commute,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.32/0.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_commute
% 0.32/0.50 thf(fact_779_add_Oleft__commute,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.32/0.50 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add.left_commute
% 0.32/0.50 thf(fact_780_add__left__imp__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.50 = ( plus_plus_real @ A @ C ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_imp_eq
% 0.32/0.50 thf(fact_781_add__left__imp__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.50 = ( plus_plus_rat @ A @ C ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_imp_eq
% 0.32/0.50 thf(fact_782_add__left__imp__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ A @ B )
% 0.32/0.50 = ( plus_plus_nat @ A @ C ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_imp_eq
% 0.32/0.50 thf(fact_783_add__left__imp__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.50 = ( plus_plus_int @ A @ C ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_imp_eq
% 0.32/0.50 thf(fact_784_add__right__imp__eq,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ B @ A )
% 0.32/0.50 = ( plus_plus_real @ C @ A ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_imp_eq
% 0.32/0.50 thf(fact_785_add__right__imp__eq,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ B @ A )
% 0.32/0.50 = ( plus_plus_rat @ C @ A ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_imp_eq
% 0.32/0.50 thf(fact_786_add__right__imp__eq,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ B @ A )
% 0.32/0.50 = ( plus_plus_nat @ C @ A ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_imp_eq
% 0.32/0.50 thf(fact_787_add__right__imp__eq,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.50 = ( plus_plus_int @ C @ A ) )
% 0.32/0.50 => ( B = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_imp_eq
% 0.32/0.50 thf(fact_788_diff__eq__diff__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.50 ( ( ( minus_minus_real @ A @ B )
% 0.32/0.50 = ( minus_minus_real @ C @ D ) )
% 0.32/0.50 => ( ( A = B )
% 0.32/0.50 = ( C = D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_eq
% 0.32/0.50 thf(fact_789_diff__eq__diff__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.50 ( ( ( minus_minus_rat @ A @ B )
% 0.32/0.50 = ( minus_minus_rat @ C @ D ) )
% 0.32/0.50 => ( ( A = B )
% 0.32/0.50 = ( C = D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_eq
% 0.32/0.50 thf(fact_790_diff__eq__diff__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.50 ( ( ( minus_minus_int @ A @ B )
% 0.32/0.50 = ( minus_minus_int @ C @ D ) )
% 0.32/0.50 => ( ( A = B )
% 0.32/0.50 = ( C = D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_eq
% 0.32/0.50 thf(fact_791_diff__right__commute,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.32/0.50 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_right_commute
% 0.32/0.50 thf(fact_792_diff__right__commute,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 0.32/0.50 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_right_commute
% 0.32/0.50 thf(fact_793_diff__right__commute,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.32/0.50 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_right_commute
% 0.32/0.50 thf(fact_794_diff__right__commute,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.32/0.50 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_right_commute
% 0.32/0.50 thf(fact_795_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( ord_less_eq_real @ I @ J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(3)
% 0.32/0.50 thf(fact_796_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( ord_less_eq_rat @ I @ J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(3)
% 0.32/0.50 thf(fact_797_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(3)
% 0.32/0.50 thf(fact_798_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(3)
% 0.32/0.50 thf(fact_799_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( ord_less_eq_real @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(2)
% 0.32/0.50 thf(fact_800_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(2)
% 0.32/0.50 thf(fact_801_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(2)
% 0.32/0.50 thf(fact_802_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( ord_less_eq_int @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(2)
% 0.32/0.50 thf(fact_803_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( ord_less_eq_real @ I @ J )
% 0.32/0.50 & ( ord_less_eq_real @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(1)
% 0.32/0.50 thf(fact_804_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( ord_less_eq_rat @ I @ J )
% 0.32/0.50 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(1)
% 0.32/0.50 thf(fact_805_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.50 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(1)
% 0.32/0.50 thf(fact_806_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.50 & ( ord_less_eq_int @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_semiring(1)
% 0.32/0.50 thf(fact_807_add__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono
% 0.32/0.50 thf(fact_808_add__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.50 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono
% 0.32/0.50 thf(fact_809_add__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono
% 0.32/0.50 thf(fact_810_add__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.50 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono
% 0.32/0.50 thf(fact_811_add__left__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_mono
% 0.32/0.50 thf(fact_812_add__left__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.50 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_mono
% 0.32/0.50 thf(fact_813_add__left__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_mono
% 0.32/0.50 thf(fact_814_add__left__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.50 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_left_mono
% 0.32/0.50 thf(fact_815_less__eqE,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ~ ! [C2: nat] :
% 0.32/0.50 ( B
% 0.32/0.50 != ( plus_plus_nat @ A @ C2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_eqE
% 0.32/0.50 thf(fact_816_add__right__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.50 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_mono
% 0.32/0.50 thf(fact_817_add__right__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.50 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_mono
% 0.32/0.50 thf(fact_818_add__right__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_mono
% 0.32/0.50 thf(fact_819_add__right__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.50 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_right_mono
% 0.32/0.50 thf(fact_820_le__iff__add,axiom,
% 0.32/0.50 ( ord_less_eq_nat
% 0.32/0.50 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.50 ? [C3: nat] :
% 0.32/0.50 ( B4
% 0.32/0.50 = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_iff_add
% 0.32/0.50 thf(fact_821_add__le__imp__le__left,axiom,
% 0.32/0.50 ! [C: real,A: real,B: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.50 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_left
% 0.32/0.50 thf(fact_822_add__le__imp__le__left,axiom,
% 0.32/0.50 ! [C: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.50 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_left
% 0.32/0.50 thf(fact_823_add__le__imp__le__left,axiom,
% 0.32/0.50 ! [C: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.50 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_left
% 0.32/0.50 thf(fact_824_add__le__imp__le__left,axiom,
% 0.32/0.50 ! [C: int,A: int,B: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.50 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_left
% 0.32/0.50 thf(fact_825_add__le__imp__le__right,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_right
% 0.32/0.50 thf(fact_826_add__le__imp__le__right,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_right
% 0.32/0.50 thf(fact_827_add__le__imp__le__right,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.50 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_right
% 0.32/0.50 thf(fact_828_add__le__imp__le__right,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_imp_le_right
% 0.32/0.50 thf(fact_829_diff__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,D: real,C: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_real @ D @ C )
% 0.32/0.50 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_mono
% 0.32/0.50 thf(fact_830_diff__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_rat @ D @ C )
% 0.32/0.50 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_mono
% 0.32/0.50 thf(fact_831_diff__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,D: int,C: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_int @ D @ C )
% 0.32/0.50 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_mono
% 0.32/0.50 thf(fact_832_diff__left__mono,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.50 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_left_mono
% 0.32/0.50 thf(fact_833_diff__left__mono,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.50 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_left_mono
% 0.32/0.50 thf(fact_834_diff__left__mono,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.50 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_left_mono
% 0.32/0.50 thf(fact_835_diff__right__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.50 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_right_mono
% 0.32/0.50 thf(fact_836_diff__right__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.50 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_right_mono
% 0.32/0.50 thf(fact_837_diff__right__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.50 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_right_mono
% 0.32/0.50 thf(fact_838_diff__eq__diff__less__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.50 ( ( ( minus_minus_real @ A @ B )
% 0.32/0.50 = ( minus_minus_real @ C @ D ) )
% 0.32/0.50 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.50 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_less_eq
% 0.32/0.50 thf(fact_839_diff__eq__diff__less__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.50 ( ( ( minus_minus_rat @ A @ B )
% 0.32/0.50 = ( minus_minus_rat @ C @ D ) )
% 0.32/0.50 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.50 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_less_eq
% 0.32/0.50 thf(fact_840_diff__eq__diff__less__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.50 ( ( ( minus_minus_int @ A @ B )
% 0.32/0.50 = ( minus_minus_int @ C @ D ) )
% 0.32/0.50 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.50 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_less_eq
% 0.32/0.50 thf(fact_841_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( ord_less_real @ I @ J )
% 0.32/0.50 & ( ord_less_real @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(5)
% 0.32/0.50 thf(fact_842_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( ord_less_rat @ I @ J )
% 0.32/0.50 & ( ord_less_rat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(5)
% 0.32/0.50 thf(fact_843_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( ord_less_nat @ I @ J )
% 0.32/0.50 & ( ord_less_nat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(5)
% 0.32/0.50 thf(fact_844_add__mono__thms__linordered__field_I5_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( ord_less_int @ I @ J )
% 0.32/0.50 & ( ord_less_int @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(5)
% 0.32/0.50 thf(fact_845_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( ord_less_real @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(2)
% 0.32/0.50 thf(fact_846_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( ord_less_rat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(2)
% 0.32/0.50 thf(fact_847_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( ord_less_nat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(2)
% 0.32/0.50 thf(fact_848_add__mono__thms__linordered__field_I2_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( I = J )
% 0.32/0.50 & ( ord_less_int @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(2)
% 0.32/0.50 thf(fact_849_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( ord_less_real @ I @ J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(1)
% 0.32/0.50 thf(fact_850_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( ord_less_rat @ I @ J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(1)
% 0.32/0.50 thf(fact_851_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( ord_less_nat @ I @ J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(1)
% 0.32/0.50 thf(fact_852_add__mono__thms__linordered__field_I1_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( ord_less_int @ I @ J )
% 0.32/0.50 & ( K = L2 ) )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(1)
% 0.32/0.50 thf(fact_853_add__strict__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ( ord_less_real @ C @ D )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_mono
% 0.32/0.50 thf(fact_854_add__strict__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ( ord_less_rat @ C @ D )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_mono
% 0.32/0.50 thf(fact_855_add__strict__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.50 ( ( ord_less_nat @ A @ B )
% 0.32/0.50 => ( ( ord_less_nat @ C @ D )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_mono
% 0.32/0.50 thf(fact_856_add__strict__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.50 ( ( ord_less_int @ A @ B )
% 0.32/0.50 => ( ( ord_less_int @ C @ D )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_mono
% 0.32/0.50 thf(fact_857_add__strict__left__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_left_mono
% 0.32/0.50 thf(fact_858_add__strict__left__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_left_mono
% 0.32/0.50 thf(fact_859_add__strict__left__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_nat @ A @ B )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_left_mono
% 0.32/0.50 thf(fact_860_add__strict__left__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ord_less_int @ A @ B )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_left_mono
% 0.32/0.50 thf(fact_861_add__strict__right__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_right_mono
% 0.32/0.50 thf(fact_862_add__strict__right__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_right_mono
% 0.32/0.50 thf(fact_863_add__strict__right__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_nat @ A @ B )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_right_mono
% 0.32/0.50 thf(fact_864_add__strict__right__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ord_less_int @ A @ B )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_strict_right_mono
% 0.32/0.50 thf(fact_865_add__less__imp__less__left,axiom,
% 0.32/0.50 ! [C: real,A: real,B: real] :
% 0.32/0.50 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.32/0.50 => ( ord_less_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_imp_less_left
% 0.32/0.50 thf(fact_866_add__less__imp__less__left,axiom,
% 0.32/0.50 ! [C: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.32/0.50 => ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_imp_less_left
% 0.32/0.50 thf(fact_867_add__less__imp__less__left,axiom,
% 0.32/0.50 ! [C: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.32/0.50 => ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_imp_less_left
% 0.32/0.50 thf(fact_868_add__less__imp__less__left,axiom,
% 0.32/0.50 ! [C: int,A: int,B: int] :
% 0.32/0.50 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.32/0.50 => ( ord_less_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_imp_less_left
% 0.32/0.50 thf(fact_869_add__less__imp__less__right,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 => ( ord_less_real @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_imp_less_right
% 0.32/0.50 thf(fact_870_add__less__imp__less__right,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 => ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_imp_less_right
% 0.32/0.50 thf(fact_871_add__less__imp__less__right,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.50 => ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_imp_less_right
% 0.32/0.50 thf(fact_872_add__less__imp__less__right,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 => ( ord_less_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_imp_less_right
% 0.32/0.50 thf(fact_873_diff__strict__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,D: real,C: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ( ord_less_real @ D @ C )
% 0.32/0.50 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_mono
% 0.32/0.50 thf(fact_874_diff__strict__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,D: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ( ord_less_rat @ D @ C )
% 0.32/0.50 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_mono
% 0.32/0.50 thf(fact_875_diff__strict__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,D: int,C: int] :
% 0.32/0.50 ( ( ord_less_int @ A @ B )
% 0.32/0.50 => ( ( ord_less_int @ D @ C )
% 0.32/0.50 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_mono
% 0.32/0.50 thf(fact_876_diff__eq__diff__less,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.50 ( ( ( minus_minus_real @ A @ B )
% 0.32/0.50 = ( minus_minus_real @ C @ D ) )
% 0.32/0.50 => ( ( ord_less_real @ A @ B )
% 0.32/0.50 = ( ord_less_real @ C @ D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_less
% 0.32/0.50 thf(fact_877_diff__eq__diff__less,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.50 ( ( ( minus_minus_rat @ A @ B )
% 0.32/0.50 = ( minus_minus_rat @ C @ D ) )
% 0.32/0.50 => ( ( ord_less_rat @ A @ B )
% 0.32/0.50 = ( ord_less_rat @ C @ D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_less
% 0.32/0.50 thf(fact_878_diff__eq__diff__less,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.50 ( ( ( minus_minus_int @ A @ B )
% 0.32/0.50 = ( minus_minus_int @ C @ D ) )
% 0.32/0.50 => ( ( ord_less_int @ A @ B )
% 0.32/0.50 = ( ord_less_int @ C @ D ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_diff_less
% 0.32/0.50 thf(fact_879_diff__strict__left__mono,axiom,
% 0.32/0.50 ! [B: real,A: real,C: real] :
% 0.32/0.50 ( ( ord_less_real @ B @ A )
% 0.32/0.50 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_left_mono
% 0.32/0.50 thf(fact_880_diff__strict__left__mono,axiom,
% 0.32/0.50 ! [B: rat,A: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_rat @ B @ A )
% 0.32/0.50 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_left_mono
% 0.32/0.50 thf(fact_881_diff__strict__left__mono,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( ord_less_int @ B @ A )
% 0.32/0.50 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_left_mono
% 0.32/0.50 thf(fact_882_diff__strict__right__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_right_mono
% 0.32/0.50 thf(fact_883_diff__strict__right__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_right_mono
% 0.32/0.50 thf(fact_884_diff__strict__right__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ord_less_int @ A @ B )
% 0.32/0.50 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_strict_right_mono
% 0.32/0.50 thf(fact_885_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.50 ! [A: complex] :
% 0.32/0.50 ( ( times_times_complex @ one_one_complex @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_mult_class.mult_1
% 0.32/0.50 thf(fact_886_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( times_times_real @ one_one_real @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_mult_class.mult_1
% 0.32/0.50 thf(fact_887_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( times_times_rat @ one_one_rat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_mult_class.mult_1
% 0.32/0.50 thf(fact_888_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( times_times_nat @ one_one_nat @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_mult_class.mult_1
% 0.32/0.50 thf(fact_889_comm__monoid__mult__class_Omult__1,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( times_times_int @ one_one_int @ A )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % comm_monoid_mult_class.mult_1
% 0.32/0.50 thf(fact_890_mult_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: complex] :
% 0.32/0.50 ( ( times_times_complex @ A @ one_one_complex )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.comm_neutral
% 0.32/0.50 thf(fact_891_mult_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: real] :
% 0.32/0.50 ( ( times_times_real @ A @ one_one_real )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.comm_neutral
% 0.32/0.50 thf(fact_892_mult_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: rat] :
% 0.32/0.50 ( ( times_times_rat @ A @ one_one_rat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.comm_neutral
% 0.32/0.50 thf(fact_893_mult_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: nat] :
% 0.32/0.50 ( ( times_times_nat @ A @ one_one_nat )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.comm_neutral
% 0.32/0.50 thf(fact_894_mult_Ocomm__neutral,axiom,
% 0.32/0.50 ! [A: int] :
% 0.32/0.50 ( ( times_times_int @ A @ one_one_int )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult.comm_neutral
% 0.32/0.50 thf(fact_895_group__cancel_Osub1,axiom,
% 0.32/0.50 ! [A2: real,K: real,A: real,B: real] :
% 0.32/0.50 ( ( A2
% 0.32/0.50 = ( plus_plus_real @ K @ A ) )
% 0.32/0.50 => ( ( minus_minus_real @ A2 @ B )
% 0.32/0.50 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.sub1
% 0.32/0.50 thf(fact_896_group__cancel_Osub1,axiom,
% 0.32/0.50 ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.32/0.50 ( ( A2
% 0.32/0.50 = ( plus_plus_rat @ K @ A ) )
% 0.32/0.50 => ( ( minus_minus_rat @ A2 @ B )
% 0.32/0.50 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.sub1
% 0.32/0.50 thf(fact_897_group__cancel_Osub1,axiom,
% 0.32/0.50 ! [A2: int,K: int,A: int,B: int] :
% 0.32/0.50 ( ( A2
% 0.32/0.50 = ( plus_plus_int @ K @ A ) )
% 0.32/0.50 => ( ( minus_minus_int @ A2 @ B )
% 0.32/0.50 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % group_cancel.sub1
% 0.32/0.50 thf(fact_898_diff__eq__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ( minus_minus_real @ A @ B )
% 0.32/0.50 = C )
% 0.32/0.50 = ( A
% 0.32/0.50 = ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_eq
% 0.32/0.50 thf(fact_899_diff__eq__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ( minus_minus_rat @ A @ B )
% 0.32/0.50 = C )
% 0.32/0.50 = ( A
% 0.32/0.50 = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_eq
% 0.32/0.50 thf(fact_900_diff__eq__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ( minus_minus_int @ A @ B )
% 0.32/0.50 = C )
% 0.32/0.50 = ( A
% 0.32/0.50 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_eq_eq
% 0.32/0.50 thf(fact_901_eq__diff__eq,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( minus_minus_real @ C @ B ) )
% 0.32/0.50 = ( ( plus_plus_real @ A @ B )
% 0.32/0.50 = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_diff_eq
% 0.32/0.50 thf(fact_902_eq__diff__eq,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( minus_minus_rat @ C @ B ) )
% 0.32/0.50 = ( ( plus_plus_rat @ A @ B )
% 0.32/0.50 = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_diff_eq
% 0.32/0.50 thf(fact_903_eq__diff__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( A
% 0.32/0.50 = ( minus_minus_int @ C @ B ) )
% 0.32/0.50 = ( ( plus_plus_int @ A @ B )
% 0.32/0.50 = C ) ) ).
% 0.32/0.50
% 0.32/0.50 % eq_diff_eq
% 0.32/0.50 thf(fact_904_add__diff__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.32/0.50 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_eq
% 0.32/0.50 thf(fact_905_add__diff__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_eq
% 0.32/0.50 thf(fact_906_add__diff__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.32/0.50 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_diff_eq
% 0.32/0.50 thf(fact_907_diff__diff__eq2,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.32/0.50 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_diff_eq2
% 0.32/0.50 thf(fact_908_diff__diff__eq2,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.32/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_diff_eq2
% 0.32/0.50 thf(fact_909_diff__diff__eq2,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.32/0.50 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_diff_eq2
% 0.32/0.50 thf(fact_910_diff__add__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.50 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_eq
% 0.32/0.50 thf(fact_911_diff__add__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_eq
% 0.32/0.50 thf(fact_912_diff__add__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.50 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_eq
% 0.32/0.50 thf(fact_913_diff__add__eq__diff__diff__swap,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.50 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_eq_diff_diff_swap
% 0.32/0.50 thf(fact_914_diff__add__eq__diff__diff__swap,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.50 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_eq_diff_diff_swap
% 0.32/0.50 thf(fact_915_diff__add__eq__diff__diff__swap,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.50 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add_eq_diff_diff_swap
% 0.32/0.50 thf(fact_916_add__implies__diff,axiom,
% 0.32/0.50 ! [C: real,B: real,A: real] :
% 0.32/0.50 ( ( ( plus_plus_real @ C @ B )
% 0.32/0.50 = A )
% 0.32/0.50 => ( C
% 0.32/0.50 = ( minus_minus_real @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_implies_diff
% 0.32/0.50 thf(fact_917_add__implies__diff,axiom,
% 0.32/0.50 ! [C: rat,B: rat,A: rat] :
% 0.32/0.50 ( ( ( plus_plus_rat @ C @ B )
% 0.32/0.50 = A )
% 0.32/0.50 => ( C
% 0.32/0.50 = ( minus_minus_rat @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_implies_diff
% 0.32/0.50 thf(fact_918_add__implies__diff,axiom,
% 0.32/0.50 ! [C: nat,B: nat,A: nat] :
% 0.32/0.50 ( ( ( plus_plus_nat @ C @ B )
% 0.32/0.50 = A )
% 0.32/0.50 => ( C
% 0.32/0.50 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_implies_diff
% 0.32/0.50 thf(fact_919_add__implies__diff,axiom,
% 0.32/0.50 ! [C: int,B: int,A: int] :
% 0.32/0.50 ( ( ( plus_plus_int @ C @ B )
% 0.32/0.50 = A )
% 0.32/0.50 => ( C
% 0.32/0.50 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_implies_diff
% 0.32/0.50 thf(fact_920_diff__diff__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.50 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_diff_eq
% 0.32/0.50 thf(fact_921_diff__diff__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_diff_eq
% 0.32/0.50 thf(fact_922_diff__diff__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_diff_eq
% 0.32/0.50 thf(fact_923_diff__diff__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.50 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_diff_eq
% 0.32/0.50 thf(fact_924_add__less__le__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_le_mono
% 0.32/0.50 thf(fact_925_add__less__le__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_le_mono
% 0.32/0.50 thf(fact_926_add__less__le__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.50 ( ( ord_less_nat @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_le_mono
% 0.32/0.50 thf(fact_927_add__less__le__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.50 ( ( ord_less_int @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_less_le_mono
% 0.32/0.50 thf(fact_928_add__le__less__mono,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.50 => ( ( ord_less_real @ C @ D )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_less_mono
% 0.32/0.50 thf(fact_929_add__le__less__mono,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.50 => ( ( ord_less_rat @ C @ D )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_less_mono
% 0.32/0.50 thf(fact_930_add__le__less__mono,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( ord_less_nat @ C @ D )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_less_mono
% 0.32/0.50 thf(fact_931_add__le__less__mono,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.50 => ( ( ord_less_int @ C @ D )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_le_less_mono
% 0.32/0.50 thf(fact_932_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( ord_less_real @ I @ J )
% 0.32/0.50 & ( ord_less_eq_real @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(3)
% 0.32/0.50 thf(fact_933_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( ord_less_rat @ I @ J )
% 0.32/0.50 & ( ord_less_eq_rat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(3)
% 0.32/0.50 thf(fact_934_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( ord_less_nat @ I @ J )
% 0.32/0.50 & ( ord_less_eq_nat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(3)
% 0.32/0.50 thf(fact_935_add__mono__thms__linordered__field_I3_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( ord_less_int @ I @ J )
% 0.32/0.50 & ( ord_less_eq_int @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(3)
% 0.32/0.50 thf(fact_936_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.50 ! [I: real,J: real,K: real,L2: real] :
% 0.32/0.50 ( ( ( ord_less_eq_real @ I @ J )
% 0.32/0.50 & ( ord_less_real @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(4)
% 0.32/0.50 thf(fact_937_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.50 ! [I: rat,J: rat,K: rat,L2: rat] :
% 0.32/0.50 ( ( ( ord_less_eq_rat @ I @ J )
% 0.32/0.50 & ( ord_less_rat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(4)
% 0.32/0.50 thf(fact_938_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.50 ! [I: nat,J: nat,K: nat,L2: nat] :
% 0.32/0.50 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.50 & ( ord_less_nat @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(4)
% 0.32/0.50 thf(fact_939_add__mono__thms__linordered__field_I4_J,axiom,
% 0.32/0.50 ! [I: int,J: int,K: int,L2: int] :
% 0.32/0.50 ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.50 & ( ord_less_int @ K @ L2 ) )
% 0.32/0.50 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % add_mono_thms_linordered_field(4)
% 0.32/0.50 thf(fact_940_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( ( minus_minus_nat @ B @ A )
% 0.32/0.50 = C )
% 0.32/0.50 = ( B
% 0.32/0.50 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.32/0.50 thf(fact_941_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.32/0.50 = B ) ) ).
% 0.32/0.50
% 0.32/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.32/0.50 thf(fact_942_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.32/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.32/0.50 thf(fact_943_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.32/0.50 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.32/0.50 thf(fact_944_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.32/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.32/0.50 thf(fact_945_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.32/0.50 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.32/0.50 thf(fact_946_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.32/0.50 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.32/0.50 thf(fact_947_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.32/0.50 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.32/0.50 thf(fact_948_le__add__diff,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_add_diff
% 0.32/0.50 thf(fact_949_diff__add,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.50 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.32/0.50 = B ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_add
% 0.32/0.50 thf(fact_950_le__diff__eq,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.32/0.50 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_diff_eq
% 0.32/0.50 thf(fact_951_le__diff__eq,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.32/0.50 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_diff_eq
% 0.32/0.50 thf(fact_952_le__diff__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.32/0.50 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_diff_eq
% 0.32/0.50 thf(fact_953_diff__le__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.50 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_le_eq
% 0.32/0.50 thf(fact_954_diff__le__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_le_eq
% 0.32/0.50 thf(fact_955_diff__le__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.50 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_le_eq
% 0.32/0.50 thf(fact_956_diff__less__eq,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.50 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_less_eq
% 0.32/0.50 thf(fact_957_diff__less__eq,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.50 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_less_eq
% 0.32/0.50 thf(fact_958_diff__less__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.50 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % diff_less_eq
% 0.32/0.50 thf(fact_959_less__diff__eq,axiom,
% 0.32/0.50 ! [A: real,C: real,B: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.32/0.50 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_diff_eq
% 0.32/0.50 thf(fact_960_less__diff__eq,axiom,
% 0.32/0.50 ! [A: rat,C: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.32/0.50 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_diff_eq
% 0.32/0.50 thf(fact_961_less__diff__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.32/0.50 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_diff_eq
% 0.32/0.50 thf(fact_962_times__divide__eq__right,axiom,
% 0.32/0.50 ! [A: complex,B: complex,C: complex] :
% 0.32/0.50 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.50 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % times_divide_eq_right
% 0.32/0.50 thf(fact_963_times__divide__eq__right,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.50 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % times_divide_eq_right
% 0.32/0.50 thf(fact_964_times__divide__eq__right,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.50 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % times_divide_eq_right
% 0.32/0.50 thf(fact_965_divide__divide__eq__right,axiom,
% 0.32/0.50 ! [A: complex,B: complex,C: complex] :
% 0.32/0.50 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.50 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_divide_eq_right
% 0.32/0.50 thf(fact_966_divide__divide__eq__right,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.50 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_divide_eq_right
% 0.32/0.50 thf(fact_967_divide__divide__eq__right,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.50 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_divide_eq_right
% 0.32/0.50 thf(fact_968_divide__divide__eq__left,axiom,
% 0.32/0.50 ! [A: complex,B: complex,C: complex] :
% 0.32/0.50 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.32/0.50 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_divide_eq_left
% 0.32/0.50 thf(fact_969_divide__divide__eq__left,axiom,
% 0.32/0.50 ! [A: real,B: real,C: real] :
% 0.32/0.50 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.32/0.50 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_divide_eq_left
% 0.32/0.50 thf(fact_970_divide__divide__eq__left,axiom,
% 0.32/0.50 ! [A: rat,B: rat,C: rat] :
% 0.32/0.50 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.32/0.50 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % divide_divide_eq_left
% 0.32/0.50 thf(fact_971_times__divide__eq__left,axiom,
% 0.32/0.50 ! [B: complex,C: complex,A: complex] :
% 0.32/0.50 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 0.32/0.50 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % times_divide_eq_left
% 0.32/0.50 thf(fact_972_times__divide__eq__left,axiom,
% 0.32/0.50 ! [B: real,C: real,A: real] :
% 0.32/0.50 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.50 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % times_divide_eq_left
% 0.32/0.50 thf(fact_973_times__divide__eq__left,axiom,
% 0.32/0.50 ! [B: rat,C: rat,A: rat] :
% 0.32/0.50 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.50 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % times_divide_eq_left
% 0.32/0.50 thf(fact_974_valid__insert__both__member__options__pres,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat,Y3: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.50 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.50 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.32/0.50 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % valid_insert_both_member_options_pres
% 0.32/0.50 thf(fact_975_valid__insert__both__member__options__add,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.50 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % valid_insert_both_member_options_add
% 0.32/0.50 thf(fact_976_maxt__corr__help__empty,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( vEBT_vebt_maxt @ T )
% 0.32/0.50 = none_nat )
% 0.32/0.50 => ( ( vEBT_VEBT_set_vebt @ T )
% 0.32/0.50 = bot_bot_set_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % maxt_corr_help_empty
% 0.32/0.50 thf(fact_977_mint__corr__help__empty,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.32/0.50 = none_nat )
% 0.32/0.50 => ( ( vEBT_VEBT_set_vebt @ T )
% 0.32/0.50 = bot_bot_set_nat ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mint_corr_help_empty
% 0.32/0.50 thf(fact_978_less__half__sum,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_half_sum
% 0.32/0.50 thf(fact_979_less__half__sum,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % less_half_sum
% 0.32/0.50 thf(fact_980_gt__half__sum,axiom,
% 0.32/0.50 ! [A: real,B: real] :
% 0.32/0.50 ( ( ord_less_real @ A @ B )
% 0.32/0.50 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % gt_half_sum
% 0.32/0.50 thf(fact_981_gt__half__sum,axiom,
% 0.32/0.50 ! [A: rat,B: rat] :
% 0.32/0.50 ( ( ord_less_rat @ A @ B )
% 0.32/0.50 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % gt_half_sum
% 0.32/0.50 thf(fact_982_low__def,axiom,
% 0.32/0.50 ( vEBT_VEBT_low
% 0.32/0.50 = ( ^ [X2: nat,N2: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % low_def
% 0.32/0.50 thf(fact_983_not__min__Null__member,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT] :
% 0.32/0.50 ( ~ ( vEBT_VEBT_minNull @ T )
% 0.32/0.50 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 0.32/0.50
% 0.32/0.50 % not_min_Null_member
% 0.32/0.50 thf(fact_984_valid__member__both__member__options,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.32/0.50 => ( vEBT_vebt_member @ T @ X ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % valid_member_both_member_options
% 0.32/0.50 thf(fact_985_both__member__options__equiv__member,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.50 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 0.32/0.50 = ( vEBT_vebt_member @ T @ X ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % both_member_options_equiv_member
% 0.32/0.50 thf(fact_986_maxbmo,axiom,
% 0.32/0.50 ! [T: vEBT_VEBT,X: nat] :
% 0.32/0.50 ( ( ( vEBT_vebt_maxt @ T )
% 0.32/0.50 = ( some_nat @ X ) )
% 0.32/0.50 => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% 0.32/0.50
% 0.32/0.50 % maxbmo
% 0.32/0.50 thf(fact_987_mod__mod__trivial,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.32/0.50 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mod_trivial
% 0.32/0.50 thf(fact_988_mod__mod__trivial,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mod_trivial
% 0.32/0.50 thf(fact_989_mod__mod__trivial,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mod_trivial
% 0.32/0.50 thf(fact_990_mod__add__self1,axiom,
% 0.32/0.50 ! [B: nat,A: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.50 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_self1
% 0.32/0.50 thf(fact_991_mod__add__self1,axiom,
% 0.32/0.50 ! [B: int,A: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_self1
% 0.32/0.50 thf(fact_992_mod__add__self1,axiom,
% 0.32/0.50 ! [B: code_integer,A: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_self1
% 0.32/0.50 thf(fact_993_mod__add__self2,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.50 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_self2
% 0.32/0.50 thf(fact_994_mod__add__self2,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_self2
% 0.32/0.50 thf(fact_995_mod__add__self2,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_self2
% 0.32/0.50 thf(fact_996_minus__mod__self2,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_mod_self2
% 0.32/0.50 thf(fact_997_minus__mod__self2,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_mod_self2
% 0.32/0.50 thf(fact_998_mod__less,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ( ord_less_nat @ M @ N )
% 0.32/0.50 => ( ( modulo_modulo_nat @ M @ N )
% 0.32/0.50 = M ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_less
% 0.32/0.50 thf(fact_999_mod__mult__self1,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.32/0.50 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self1
% 0.32/0.50 thf(fact_1000_mod__mult__self1,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self1
% 0.32/0.50 thf(fact_1001_mod__mult__self1,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self1
% 0.32/0.50 thf(fact_1002_mod__mult__self2,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.32/0.50 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self2
% 0.32/0.50 thf(fact_1003_mod__mult__self2,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self2
% 0.32/0.50 thf(fact_1004_mod__mult__self2,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self2
% 0.32/0.50 thf(fact_1005_mod__mult__self3,axiom,
% 0.32/0.50 ! [C: nat,B: nat,A: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.32/0.50 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self3
% 0.32/0.50 thf(fact_1006_mod__mult__self3,axiom,
% 0.32/0.50 ! [C: int,B: int,A: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self3
% 0.32/0.50 thf(fact_1007_mod__mult__self3,axiom,
% 0.32/0.50 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self3
% 0.32/0.50 thf(fact_1008_mod__mult__self4,axiom,
% 0.32/0.50 ! [B: nat,C: nat,A: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.32/0.50 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self4
% 0.32/0.50 thf(fact_1009_mod__mult__self4,axiom,
% 0.32/0.50 ! [B: int,C: int,A: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self4
% 0.32/0.50 thf(fact_1010_mod__mult__self4,axiom,
% 0.32/0.50 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_self4
% 0.32/0.50 thf(fact_1011_one__mod__two__eq__one,axiom,
% 0.32/0.50 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = one_one_nat ) ).
% 0.32/0.50
% 0.32/0.50 % one_mod_two_eq_one
% 0.32/0.50 thf(fact_1012_one__mod__two__eq__one,axiom,
% 0.32/0.50 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.50 = one_one_int ) ).
% 0.32/0.50
% 0.32/0.50 % one_mod_two_eq_one
% 0.32/0.50 thf(fact_1013_one__mod__two__eq__one,axiom,
% 0.32/0.50 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.50 = one_one_Code_integer ) ).
% 0.32/0.50
% 0.32/0.50 % one_mod_two_eq_one
% 0.32/0.50 thf(fact_1014_bits__one__mod__two__eq__one,axiom,
% 0.32/0.50 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.50 = one_one_nat ) ).
% 0.32/0.50
% 0.32/0.50 % bits_one_mod_two_eq_one
% 0.32/0.50 thf(fact_1015_bits__one__mod__two__eq__one,axiom,
% 0.32/0.50 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.50 = one_one_int ) ).
% 0.32/0.50
% 0.32/0.50 % bits_one_mod_two_eq_one
% 0.32/0.50 thf(fact_1016_bits__one__mod__two__eq__one,axiom,
% 0.32/0.50 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.50 = one_one_Code_integer ) ).
% 0.32/0.50
% 0.32/0.50 % bits_one_mod_two_eq_one
% 0.32/0.50 thf(fact_1017_mod__mult__eq,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_eq
% 0.32/0.50 thf(fact_1018_mod__mult__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_eq
% 0.32/0.50 thf(fact_1019_mod__mult__eq,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_eq
% 0.32/0.50 thf(fact_1020_mod__mult__cong,axiom,
% 0.32/0.50 ! [A: nat,C: nat,A4: nat,B: nat,B2: nat] :
% 0.32/0.50 ( ( ( modulo_modulo_nat @ A @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ A4 @ C ) )
% 0.32/0.50 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ B2 @ C ) )
% 0.32/0.50 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B2 ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_cong
% 0.32/0.50 thf(fact_1021_mod__mult__cong,axiom,
% 0.32/0.50 ! [A: int,C: int,A4: int,B: int,B2: int] :
% 0.32/0.50 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.50 = ( modulo_modulo_int @ A4 @ C ) )
% 0.32/0.50 => ( ( ( modulo_modulo_int @ B @ C )
% 0.32/0.50 = ( modulo_modulo_int @ B2 @ C ) )
% 0.32/0.50 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( times_times_int @ A4 @ B2 ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_cong
% 0.32/0.50 thf(fact_1022_mod__mult__cong,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B2: code_integer] :
% 0.32/0.50 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A4 @ C ) )
% 0.32/0.50 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 0.32/0.50 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B2 ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_cong
% 0.32/0.50 thf(fact_1023_mod__mult__mult2,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.32/0.50 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_mult2
% 0.32/0.50 thf(fact_1024_mod__mult__mult2,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.50 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_mult2
% 0.32/0.50 thf(fact_1025_mod__mult__mult2,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.50 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_mult2
% 0.32/0.50 thf(fact_1026_mult__mod__right,axiom,
% 0.32/0.50 ! [C: nat,A: nat,B: nat] :
% 0.32/0.50 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.50 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult_mod_right
% 0.32/0.50 thf(fact_1027_mult__mod__right,axiom,
% 0.32/0.50 ! [C: int,A: int,B: int] :
% 0.32/0.50 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.50 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult_mod_right
% 0.32/0.50 thf(fact_1028_mult__mod__right,axiom,
% 0.32/0.50 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mult_mod_right
% 0.32/0.50 thf(fact_1029_mod__mult__left__eq,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_left_eq
% 0.32/0.50 thf(fact_1030_mod__mult__left__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_left_eq
% 0.32/0.50 thf(fact_1031_mod__mult__left__eq,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_left_eq
% 0.32/0.50 thf(fact_1032_mod__mult__right__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_right_eq
% 0.32/0.50 thf(fact_1033_mod__mult__right__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_right_eq
% 0.32/0.50 thf(fact_1034_mod__mult__right__eq,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_right_eq
% 0.32/0.50 thf(fact_1035_mod__add__eq,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_eq
% 0.32/0.50 thf(fact_1036_mod__add__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_eq
% 0.32/0.50 thf(fact_1037_mod__add__eq,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_eq
% 0.32/0.50 thf(fact_1038_mod__add__cong,axiom,
% 0.32/0.50 ! [A: nat,C: nat,A4: nat,B: nat,B2: nat] :
% 0.32/0.50 ( ( ( modulo_modulo_nat @ A @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ A4 @ C ) )
% 0.32/0.50 => ( ( ( modulo_modulo_nat @ B @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ B2 @ C ) )
% 0.32/0.50 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B2 ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_cong
% 0.32/0.50 thf(fact_1039_mod__add__cong,axiom,
% 0.32/0.50 ! [A: int,C: int,A4: int,B: int,B2: int] :
% 0.32/0.50 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.50 = ( modulo_modulo_int @ A4 @ C ) )
% 0.32/0.50 => ( ( ( modulo_modulo_int @ B @ C )
% 0.32/0.50 = ( modulo_modulo_int @ B2 @ C ) )
% 0.32/0.50 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B2 ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_cong
% 0.32/0.50 thf(fact_1040_mod__add__cong,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B2: code_integer] :
% 0.32/0.50 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A4 @ C ) )
% 0.32/0.50 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 0.32/0.50 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B2 ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_cong
% 0.32/0.50 thf(fact_1041_mod__add__left__eq,axiom,
% 0.32/0.50 ! [A: nat,C: nat,B: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_left_eq
% 0.32/0.50 thf(fact_1042_mod__add__left__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_left_eq
% 0.32/0.50 thf(fact_1043_mod__add__left__eq,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_left_eq
% 0.32/0.50 thf(fact_1044_mod__add__right__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_right_eq
% 0.32/0.50 thf(fact_1045_mod__add__right__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_right_eq
% 0.32/0.50 thf(fact_1046_mod__add__right__eq,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_add_right_eq
% 0.32/0.50 thf(fact_1047_mod__diff__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_diff_eq
% 0.32/0.50 thf(fact_1048_mod__diff__eq,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_diff_eq
% 0.32/0.50 thf(fact_1049_mod__diff__cong,axiom,
% 0.32/0.50 ! [A: int,C: int,A4: int,B: int,B2: int] :
% 0.32/0.50 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.50 = ( modulo_modulo_int @ A4 @ C ) )
% 0.32/0.50 => ( ( ( modulo_modulo_int @ B @ C )
% 0.32/0.50 = ( modulo_modulo_int @ B2 @ C ) )
% 0.32/0.50 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B2 ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_diff_cong
% 0.32/0.50 thf(fact_1050_mod__diff__cong,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B2: code_integer] :
% 0.32/0.50 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A4 @ C ) )
% 0.32/0.50 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 0.32/0.50 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B2 ) @ C ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_diff_cong
% 0.32/0.50 thf(fact_1051_mod__diff__left__eq,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_diff_left_eq
% 0.32/0.50 thf(fact_1052_mod__diff__left__eq,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_diff_left_eq
% 0.32/0.50 thf(fact_1053_mod__diff__right__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_diff_right_eq
% 0.32/0.50 thf(fact_1054_mod__diff__right__eq,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_diff_right_eq
% 0.32/0.50 thf(fact_1055_power__mod,axiom,
% 0.32/0.50 ! [A: nat,B: nat,N: nat] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 0.32/0.50 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % power_mod
% 0.32/0.50 thf(fact_1056_power__mod,axiom,
% 0.32/0.50 ! [A: int,B: int,N: nat] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 0.32/0.50 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % power_mod
% 0.32/0.50 thf(fact_1057_power__mod,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,N: nat] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % power_mod
% 0.32/0.50 thf(fact_1058_mod__less__eq__dividend,axiom,
% 0.32/0.50 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.32/0.50
% 0.32/0.50 % mod_less_eq_dividend
% 0.32/0.50 thf(fact_1059_cong__exp__iff__simps_I9_J,axiom,
% 0.32/0.50 ! [M: num,Q2: num,N: num] :
% 0.32/0.50 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.50 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.32/0.50 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(9)
% 0.32/0.50 thf(fact_1060_cong__exp__iff__simps_I9_J,axiom,
% 0.32/0.50 ! [M: num,Q2: num,N: num] :
% 0.32/0.50 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.50 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.32/0.50 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(9)
% 0.32/0.50 thf(fact_1061_cong__exp__iff__simps_I9_J,axiom,
% 0.32/0.50 ! [M: num,Q2: num,N: num] :
% 0.32/0.50 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.50 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(9)
% 0.32/0.50 thf(fact_1062_cong__exp__iff__simps_I4_J,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 0.32/0.50 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(4)
% 0.32/0.50 thf(fact_1063_cong__exp__iff__simps_I4_J,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 0.32/0.50 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(4)
% 0.32/0.50 thf(fact_1064_cong__exp__iff__simps_I4_J,axiom,
% 0.32/0.50 ! [M: num,N: num] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 0.32/0.50 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(4)
% 0.32/0.50 thf(fact_1065_mod__eqE,axiom,
% 0.32/0.50 ! [A: int,C: int,B: int] :
% 0.32/0.50 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.50 = ( modulo_modulo_int @ B @ C ) )
% 0.32/0.50 => ~ ! [D3: int] :
% 0.32/0.50 ( B
% 0.32/0.50 != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_eqE
% 0.32/0.50 thf(fact_1066_mod__eqE,axiom,
% 0.32/0.50 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.50 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.50 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.32/0.50 => ~ ! [D3: code_integer] :
% 0.32/0.50 ( B
% 0.32/0.50 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_eqE
% 0.32/0.50 thf(fact_1067_div__add1__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.50 = ( 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 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_add1_eq
% 0.32/0.50 thf(fact_1068_div__add1__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.50 = ( 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 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_add1_eq
% 0.32/0.50 thf(fact_1069_div__add1__eq,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.50 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_add1_eq
% 0.32/0.50 thf(fact_1070_mod__if,axiom,
% 0.32/0.50 ( modulo_modulo_nat
% 0.32/0.50 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_if
% 0.32/0.50 thf(fact_1071_mod__geq,axiom,
% 0.32/0.50 ! [M: nat,N: nat] :
% 0.32/0.50 ( ~ ( ord_less_nat @ M @ N )
% 0.32/0.50 => ( ( modulo_modulo_nat @ M @ N )
% 0.32/0.50 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_geq
% 0.32/0.50 thf(fact_1072_le__mod__geq,axiom,
% 0.32/0.50 ! [N: nat,M: nat] :
% 0.32/0.50 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.50 => ( ( modulo_modulo_nat @ M @ N )
% 0.32/0.50 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % le_mod_geq
% 0.32/0.50 thf(fact_1073_nat__mod__eq__iff,axiom,
% 0.32/0.50 ! [X: nat,N: nat,Y3: nat] :
% 0.32/0.50 ( ( ( modulo_modulo_nat @ X @ N )
% 0.32/0.50 = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.32/0.50 = ( ? [Q1: nat,Q22: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
% 0.32/0.50 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % nat_mod_eq_iff
% 0.32/0.50 thf(fact_1074_cong__exp__iff__simps_I8_J,axiom,
% 0.32/0.50 ! [M: num,Q2: num] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(8)
% 0.32/0.50 thf(fact_1075_cong__exp__iff__simps_I8_J,axiom,
% 0.32/0.50 ! [M: num,Q2: num] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(8)
% 0.32/0.50 thf(fact_1076_cong__exp__iff__simps_I8_J,axiom,
% 0.32/0.50 ! [M: num,Q2: num] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(8)
% 0.32/0.50 thf(fact_1077_cong__exp__iff__simps_I6_J,axiom,
% 0.32/0.50 ! [Q2: num,N: num] :
% 0.32/0.50 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(6)
% 0.32/0.50 thf(fact_1078_cong__exp__iff__simps_I6_J,axiom,
% 0.32/0.50 ! [Q2: num,N: num] :
% 0.32/0.50 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(6)
% 0.32/0.50 thf(fact_1079_cong__exp__iff__simps_I6_J,axiom,
% 0.32/0.50 ! [Q2: num,N: num] :
% 0.32/0.50 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.50 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % cong_exp_iff_simps(6)
% 0.32/0.50 thf(fact_1080_div__mult1__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.50 = ( 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 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult1_eq
% 0.32/0.50 thf(fact_1081_div__mult1__eq,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.50 = ( 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 ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult1_eq
% 0.32/0.50 thf(fact_1082_div__mult1__eq,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.50 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult1_eq
% 0.32/0.50 thf(fact_1083_cancel__div__mod__rules_I2_J,axiom,
% 0.32/0.50 ! [B: nat,A: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ A @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % cancel_div_mod_rules(2)
% 0.32/0.50 thf(fact_1084_cancel__div__mod__rules_I2_J,axiom,
% 0.32/0.50 ! [B: int,A: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.32/0.50 = ( plus_plus_int @ A @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % cancel_div_mod_rules(2)
% 0.32/0.50 thf(fact_1085_cancel__div__mod__rules_I2_J,axiom,
% 0.32/0.50 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.50 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.32/0.50 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % cancel_div_mod_rules(2)
% 0.32/0.50 thf(fact_1086_cancel__div__mod__rules_I1_J,axiom,
% 0.32/0.50 ! [A: nat,B: nat,C: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.32/0.50 = ( plus_plus_nat @ A @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % cancel_div_mod_rules(1)
% 0.32/0.50 thf(fact_1087_cancel__div__mod__rules_I1_J,axiom,
% 0.32/0.50 ! [A: int,B: int,C: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.32/0.50 = ( plus_plus_int @ A @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % cancel_div_mod_rules(1)
% 0.32/0.50 thf(fact_1088_cancel__div__mod__rules_I1_J,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.50 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.32/0.50 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.32/0.50
% 0.32/0.50 % cancel_div_mod_rules(1)
% 0.32/0.50 thf(fact_1089_mod__div__decomp,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( A
% 0.32/0.50 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_div_decomp
% 0.32/0.50 thf(fact_1090_mod__div__decomp,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( A
% 0.32/0.50 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_div_decomp
% 0.32/0.50 thf(fact_1091_mod__div__decomp,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( A
% 0.32/0.50 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % mod_div_decomp
% 0.32/0.50 thf(fact_1092_div__mult__mod__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_mod_eq
% 0.32/0.50 thf(fact_1093_div__mult__mod__eq,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_mod_eq
% 0.32/0.50 thf(fact_1094_div__mult__mod__eq,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % div_mult_mod_eq
% 0.32/0.50 thf(fact_1095_mod__div__mult__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mod_div_mult_eq
% 0.32/0.50 thf(fact_1096_mod__div__mult__eq,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mod_div_mult_eq
% 0.32/0.50 thf(fact_1097_mod__div__mult__eq,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mod_div_mult_eq
% 0.32/0.50 thf(fact_1098_mod__mult__div__eq,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_div_eq
% 0.32/0.50 thf(fact_1099_mod__mult__div__eq,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_div_eq
% 0.32/0.50 thf(fact_1100_mod__mult__div__eq,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mod_mult_div_eq
% 0.32/0.50 thf(fact_1101_mult__div__mod__eq,axiom,
% 0.32/0.50 ! [B: nat,A: nat] :
% 0.32/0.50 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_div_mod_eq
% 0.32/0.50 thf(fact_1102_mult__div__mod__eq,axiom,
% 0.32/0.50 ! [B: int,A: int] :
% 0.32/0.50 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_div_mod_eq
% 0.32/0.50 thf(fact_1103_mult__div__mod__eq,axiom,
% 0.32/0.50 ! [B: code_integer,A: code_integer] :
% 0.32/0.50 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.50 = A ) ).
% 0.32/0.50
% 0.32/0.50 % mult_div_mod_eq
% 0.32/0.50 thf(fact_1104_minus__div__mult__eq__mod,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.32/0.50 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_div_mult_eq_mod
% 0.32/0.50 thf(fact_1105_minus__div__mult__eq__mod,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.32/0.50 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_div_mult_eq_mod
% 0.32/0.50 thf(fact_1106_minus__div__mult__eq__mod,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.32/0.50 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_div_mult_eq_mod
% 0.32/0.50 thf(fact_1107_minus__mod__eq__div__mult,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.50 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_mod_eq_div_mult
% 0.32/0.50 thf(fact_1108_minus__mod__eq__div__mult,axiom,
% 0.32/0.50 ! [A: int,B: int] :
% 0.32/0.50 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.50 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_mod_eq_div_mult
% 0.32/0.50 thf(fact_1109_minus__mod__eq__div__mult,axiom,
% 0.32/0.50 ! [A: code_integer,B: code_integer] :
% 0.32/0.50 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.50 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_mod_eq_div_mult
% 0.32/0.50 thf(fact_1110_minus__mod__eq__mult__div,axiom,
% 0.32/0.50 ! [A: nat,B: nat] :
% 0.32/0.50 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.50 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 0.32/0.50
% 0.32/0.50 % minus_mod_eq_mult_div
% 0.32/0.51 thf(fact_1111_minus__mod__eq__mult__div,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.51 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % minus_mod_eq_mult_div
% 0.32/0.51 thf(fact_1112_minus__mod__eq__mult__div,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer] :
% 0.32/0.51 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.51 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % minus_mod_eq_mult_div
% 0.32/0.51 thf(fact_1113_minus__mult__div__eq__mod,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.32/0.51 = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % minus_mult_div_eq_mod
% 0.32/0.51 thf(fact_1114_minus__mult__div__eq__mod,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.32/0.51 = ( modulo_modulo_int @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % minus_mult_div_eq_mod
% 0.32/0.51 thf(fact_1115_minus__mult__div__eq__mod,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer] :
% 0.32/0.51 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.32/0.51 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % minus_mult_div_eq_mod
% 0.32/0.51 thf(fact_1116_mod__eq__nat1E,axiom,
% 0.32/0.51 ! [M: nat,Q2: nat,N: nat] :
% 0.32/0.51 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.32/0.51 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.32/0.51 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.51 => ~ ! [S3: nat] :
% 0.32/0.51 ( M
% 0.32/0.51 != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mod_eq_nat1E
% 0.32/0.51 thf(fact_1117_mod__eq__nat2E,axiom,
% 0.32/0.51 ! [M: nat,Q2: nat,N: nat] :
% 0.32/0.51 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.32/0.51 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.32/0.51 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.51 => ~ ! [S3: nat] :
% 0.32/0.51 ( N
% 0.32/0.51 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mod_eq_nat2E
% 0.32/0.51 thf(fact_1118_nat__mod__eq__lemma,axiom,
% 0.32/0.51 ! [X: nat,N: nat,Y3: nat] :
% 0.32/0.51 ( ( ( modulo_modulo_nat @ X @ N )
% 0.32/0.51 = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.32/0.51 => ( ( ord_less_eq_nat @ Y3 @ X )
% 0.32/0.51 => ? [Q3: nat] :
% 0.32/0.51 ( X
% 0.32/0.51 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_mod_eq_lemma
% 0.32/0.51 thf(fact_1119_div__mod__decomp,axiom,
% 0.32/0.51 ! [A2: nat,N: nat] :
% 0.32/0.51 ( A2
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mod_decomp
% 0.32/0.51 thf(fact_1120_mod__mult2__eq,axiom,
% 0.32/0.51 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 0.32/0.51 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mod_mult2_eq
% 0.32/0.51 thf(fact_1121_modulo__nat__def,axiom,
% 0.32/0.51 ( modulo_modulo_nat
% 0.32/0.51 = ( ^ [M3: nat,N2: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % modulo_nat_def
% 0.32/0.51 thf(fact_1122_linordered__field__no__ub,axiom,
% 0.32/0.51 ! [X5: real] :
% 0.32/0.51 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 0.32/0.51
% 0.32/0.51 % linordered_field_no_ub
% 0.32/0.51 thf(fact_1123_linordered__field__no__ub,axiom,
% 0.32/0.51 ! [X5: rat] :
% 0.32/0.51 ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% 0.32/0.51
% 0.32/0.51 % linordered_field_no_ub
% 0.32/0.51 thf(fact_1124_linordered__field__no__lb,axiom,
% 0.32/0.51 ! [X5: real] :
% 0.32/0.51 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 0.32/0.51
% 0.32/0.51 % linordered_field_no_lb
% 0.32/0.51 thf(fact_1125_linordered__field__no__lb,axiom,
% 0.32/0.51 ! [X5: rat] :
% 0.32/0.51 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% 0.32/0.51
% 0.32/0.51 % linordered_field_no_lb
% 0.32/0.51 thf(fact_1126_div__exp__mod__exp__eq,axiom,
% 0.32/0.51 ! [A: nat,N: nat,M: nat] :
% 0.32/0.51 ( ( 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 ) )
% 0.32/0.51 = ( 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 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_exp_mod_exp_eq
% 0.32/0.51 thf(fact_1127_div__exp__mod__exp__eq,axiom,
% 0.32/0.51 ! [A: int,N: nat,M: nat] :
% 0.32/0.51 ( ( 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 ) )
% 0.32/0.51 = ( 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 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_exp_mod_exp_eq
% 0.32/0.51 thf(fact_1128_div__exp__mod__exp__eq,axiom,
% 0.32/0.51 ! [A: code_integer,N: nat,M: nat] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.51 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_exp_mod_exp_eq
% 0.32/0.51 thf(fact_1129_mult__exp__mod__exp__eq,axiom,
% 0.32/0.51 ! [M: nat,N: nat,A: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.51 => ( ( 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 ) )
% 0.32/0.51 = ( 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 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_exp_mod_exp_eq
% 0.32/0.51 thf(fact_1130_mult__exp__mod__exp__eq,axiom,
% 0.32/0.51 ! [M: nat,N: nat,A: int] :
% 0.32/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.51 => ( ( 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 ) )
% 0.32/0.51 = ( 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 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_exp_mod_exp_eq
% 0.32/0.51 thf(fact_1131_mult__exp__mod__exp__eq,axiom,
% 0.32/0.51 ! [M: nat,N: nat,A: code_integer] :
% 0.32/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.51 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.51 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_exp_mod_exp_eq
% 0.32/0.51 thf(fact_1132_times__divide__times__eq,axiom,
% 0.32/0.51 ! [X: complex,Y3: complex,Z: complex,W: complex] :
% 0.32/0.51 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % times_divide_times_eq
% 0.32/0.51 thf(fact_1133_times__divide__times__eq,axiom,
% 0.32/0.51 ! [X: real,Y3: real,Z: real,W: real] :
% 0.32/0.51 ( ( times_times_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 0.32/0.51 = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % times_divide_times_eq
% 0.32/0.51 thf(fact_1134_times__divide__times__eq,axiom,
% 0.32/0.51 ! [X: rat,Y3: rat,Z: rat,W: rat] :
% 0.32/0.51 ( ( times_times_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.32/0.51 = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % times_divide_times_eq
% 0.32/0.51 thf(fact_1135_divide__divide__times__eq,axiom,
% 0.32/0.51 ! [X: complex,Y3: complex,Z: complex,W: complex] :
% 0.32/0.51 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_divide_times_eq
% 0.32/0.51 thf(fact_1136_divide__divide__times__eq,axiom,
% 0.32/0.51 ! [X: real,Y3: real,Z: real,W: real] :
% 0.32/0.51 ( ( divide_divide_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 0.32/0.51 = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_divide_times_eq
% 0.32/0.51 thf(fact_1137_divide__divide__times__eq,axiom,
% 0.32/0.51 ! [X: rat,Y3: rat,Z: rat,W: rat] :
% 0.32/0.51 ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 0.32/0.51 = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_divide_times_eq
% 0.32/0.51 thf(fact_1138_divide__divide__eq__left_H,axiom,
% 0.32/0.51 ! [A: complex,B: complex,C: complex] :
% 0.32/0.51 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.32/0.51 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_divide_eq_left'
% 0.32/0.51 thf(fact_1139_divide__divide__eq__left_H,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.32/0.51 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_divide_eq_left'
% 0.32/0.51 thf(fact_1140_divide__divide__eq__left_H,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.32/0.51 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_divide_eq_left'
% 0.32/0.51 thf(fact_1141_add__divide__distrib,axiom,
% 0.32/0.51 ! [A: complex,B: complex,C: complex] :
% 0.32/0.51 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_distrib
% 0.32/0.51 thf(fact_1142_add__divide__distrib,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_distrib
% 0.32/0.51 thf(fact_1143_add__divide__distrib,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.32/0.51 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_divide_distrib
% 0.32/0.51 thf(fact_1144_diff__divide__distrib,axiom,
% 0.32/0.51 ! [A: complex,B: complex,C: complex] :
% 0.32/0.51 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_divide_distrib
% 0.32/0.51 thf(fact_1145_diff__divide__distrib,axiom,
% 0.32/0.51 ! [A: real,B: real,C: real] :
% 0.32/0.51 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_divide_distrib
% 0.32/0.51 thf(fact_1146_diff__divide__distrib,axiom,
% 0.32/0.51 ! [A: rat,B: rat,C: rat] :
% 0.32/0.51 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.32/0.51 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_divide_distrib
% 0.32/0.51 thf(fact_1147_buildup__gives__empty,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 0.32/0.51 = bot_bot_set_nat ) ).
% 0.32/0.51
% 0.32/0.51 % buildup_gives_empty
% 0.32/0.51 thf(fact_1148__C4_Ohyps_C_I11_J,axiom,
% 0.32/0.51 ( ( mi != ma )
% 0.32/0.51 => ! [I3: nat] :
% 0.32/0.51 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.32/0.51 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 0.32/0.51 = I3 )
% 0.32/0.51 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 0.32/0.51 & ! [X5: nat] :
% 0.32/0.51 ( ( ( ( vEBT_VEBT_high @ X5 @ na )
% 0.32/0.51 = I3 )
% 0.32/0.51 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
% 0.32/0.51 => ( ( ord_less_nat @ mi @ X5 )
% 0.32/0.51 & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % "4.hyps"(11)
% 0.32/0.51 thf(fact_1149_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.32/0.51 ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 0.32/0.51 ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa @ Xb )
% 0.32/0.51 = Y3 )
% 0.32/0.51 => ( ( ( Xa = none_P5556105721700978146at_nat )
% 0.32/0.51 => ( Y3 != none_P5556105721700978146at_nat ) )
% 0.32/0.51 => ( ( ? [V2: product_prod_nat_nat] :
% 0.32/0.51 ( Xa
% 0.32/0.51 = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.32/0.51 => ( ( Xb = none_P5556105721700978146at_nat )
% 0.32/0.51 => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 0.32/0.51 => ~ ! [A5: product_prod_nat_nat] :
% 0.32/0.51 ( ( Xa
% 0.32/0.51 = ( some_P7363390416028606310at_nat @ A5 ) )
% 0.32/0.51 => ! [B3: product_prod_nat_nat] :
% 0.32/0.51 ( ( Xb
% 0.32/0.51 = ( some_P7363390416028606310at_nat @ B3 ) )
% 0.32/0.51 => ( Y3
% 0.32/0.51 != ( some_P7363390416028606310at_nat @ ( X @ A5 @ B3 ) ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % VEBT_internal.option_shift.elims
% 0.32/0.51 thf(fact_1150_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.32/0.51 ! [X: num > num > num,Xa: option_num,Xb: option_num,Y3: option_num] :
% 0.32/0.51 ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa @ Xb )
% 0.32/0.51 = Y3 )
% 0.32/0.51 => ( ( ( Xa = none_num )
% 0.32/0.51 => ( Y3 != none_num ) )
% 0.32/0.51 => ( ( ? [V2: num] :
% 0.32/0.51 ( Xa
% 0.32/0.51 = ( some_num @ V2 ) )
% 0.32/0.51 => ( ( Xb = none_num )
% 0.32/0.51 => ( Y3 != none_num ) ) )
% 0.32/0.51 => ~ ! [A5: num] :
% 0.32/0.51 ( ( Xa
% 0.32/0.51 = ( some_num @ A5 ) )
% 0.32/0.51 => ! [B3: num] :
% 0.32/0.51 ( ( Xb
% 0.32/0.51 = ( some_num @ B3 ) )
% 0.32/0.51 => ( Y3
% 0.32/0.51 != ( some_num @ ( X @ A5 @ B3 ) ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % VEBT_internal.option_shift.elims
% 0.32/0.51 thf(fact_1151_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.32/0.51 ! [X: nat > nat > nat,Xa: option_nat,Xb: option_nat,Y3: option_nat] :
% 0.32/0.51 ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa @ Xb )
% 0.32/0.51 = Y3 )
% 0.32/0.51 => ( ( ( Xa = none_nat )
% 0.32/0.51 => ( Y3 != none_nat ) )
% 0.32/0.51 => ( ( ? [V2: nat] :
% 0.32/0.51 ( Xa
% 0.32/0.51 = ( some_nat @ V2 ) )
% 0.32/0.51 => ( ( Xb = none_nat )
% 0.32/0.51 => ( Y3 != none_nat ) ) )
% 0.32/0.51 => ~ ! [A5: nat] :
% 0.32/0.51 ( ( Xa
% 0.32/0.51 = ( some_nat @ A5 ) )
% 0.32/0.51 => ! [B3: nat] :
% 0.32/0.51 ( ( Xb
% 0.32/0.51 = ( some_nat @ B3 ) )
% 0.32/0.51 => ( Y3
% 0.32/0.51 != ( some_nat @ ( X @ A5 @ B3 ) ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % VEBT_internal.option_shift.elims
% 0.32/0.51 thf(fact_1152_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.32/0.51 ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 0.32/0.51 ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 0.32/0.51 = none_P5556105721700978146at_nat ) ).
% 0.32/0.51
% 0.32/0.51 % VEBT_internal.option_shift.simps(2)
% 0.32/0.51 thf(fact_1153_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.32/0.51 ! [Uw: num > num > num,V: num] :
% 0.32/0.51 ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 0.32/0.51 = none_num ) ).
% 0.32/0.51
% 0.32/0.51 % VEBT_internal.option_shift.simps(2)
% 0.32/0.51 thf(fact_1154_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.32/0.51 ! [Uw: nat > nat > nat,V: nat] :
% 0.32/0.51 ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 0.32/0.51 = none_nat ) ).
% 0.32/0.51
% 0.32/0.51 % VEBT_internal.option_shift.simps(2)
% 0.32/0.51 thf(fact_1155_obtain__set__succ,axiom,
% 0.32/0.51 ! [X: nat,Z: nat,A2: set_nat,B5: set_nat] :
% 0.32/0.51 ( ( ord_less_nat @ X @ Z )
% 0.32/0.51 => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
% 0.32/0.51 => ( ( finite_finite_nat @ B5 )
% 0.32/0.51 => ( ( A2 = B5 )
% 0.32/0.51 => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % obtain_set_succ
% 0.32/0.51 thf(fact_1156_True,axiom,
% 0.32/0.51 ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 0.32/0.51
% 0.32/0.51 % True
% 0.32/0.51 thf(fact_1157_dbl__simps_I3_J,axiom,
% 0.32/0.51 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 0.32/0.51 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(3)
% 0.32/0.51 thf(fact_1158_dbl__simps_I3_J,axiom,
% 0.32/0.51 ( ( neg_numeral_dbl_real @ one_one_real )
% 0.32/0.51 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(3)
% 0.32/0.51 thf(fact_1159_dbl__simps_I3_J,axiom,
% 0.32/0.51 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 0.32/0.51 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(3)
% 0.32/0.51 thf(fact_1160_dbl__simps_I3_J,axiom,
% 0.32/0.51 ( ( neg_numeral_dbl_int @ one_one_int )
% 0.32/0.51 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(3)
% 0.32/0.51 thf(fact_1161_divmod__digit__1_I1_J,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer] :
% 0.32/0.51 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.51 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.51 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.51 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 0.32/0.51 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divmod_digit_1(1)
% 0.32/0.51 thf(fact_1162_divmod__digit__1_I1_J,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.51 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.51 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.51 => ( ( 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 )
% 0.32/0.51 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divmod_digit_1(1)
% 0.32/0.51 thf(fact_1163_divmod__digit__1_I1_J,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.51 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.51 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.51 => ( ( 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 )
% 0.32/0.51 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divmod_digit_1(1)
% 0.32/0.51 thf(fact_1164_diff__shunt__var,axiom,
% 0.32/0.51 ! [X: set_real,Y3: set_real] :
% 0.32/0.51 ( ( ( minus_minus_set_real @ X @ Y3 )
% 0.32/0.51 = bot_bot_set_real )
% 0.32/0.51 = ( ord_less_eq_set_real @ X @ Y3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_shunt_var
% 0.32/0.51 thf(fact_1165_diff__shunt__var,axiom,
% 0.32/0.51 ! [X: set_nat,Y3: set_nat] :
% 0.32/0.51 ( ( ( minus_minus_set_nat @ X @ Y3 )
% 0.32/0.51 = bot_bot_set_nat )
% 0.32/0.51 = ( ord_less_eq_set_nat @ X @ Y3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_shunt_var
% 0.32/0.51 thf(fact_1166_diff__shunt__var,axiom,
% 0.32/0.51 ! [X: set_int,Y3: set_int] :
% 0.32/0.51 ( ( ( minus_minus_set_int @ X @ Y3 )
% 0.32/0.51 = bot_bot_set_int )
% 0.32/0.51 = ( ord_less_eq_set_int @ X @ Y3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_shunt_var
% 0.32/0.51 thf(fact_1167_valid__0__not,axiom,
% 0.32/0.51 ! [T: vEBT_VEBT] :
% 0.32/0.51 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % valid_0_not
% 0.32/0.51 thf(fact_1168_valid__tree__deg__neq__0,axiom,
% 0.32/0.51 ! [T: vEBT_VEBT] :
% 0.32/0.51 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % valid_tree_deg_neq_0
% 0.32/0.51 thf(fact_1169_deg__not__0,axiom,
% 0.32/0.51 ! [T: vEBT_VEBT,N: nat] :
% 0.32/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.51 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % deg_not_0
% 0.32/0.51 thf(fact_1170_buildup__gives__valid,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % buildup_gives_valid
% 0.32/0.51 thf(fact_1171_set__vebt__finite,axiom,
% 0.32/0.51 ! [T: vEBT_VEBT,N: nat] :
% 0.32/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.51 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % set_vebt_finite
% 0.32/0.51 thf(fact_1172_succ__none__empty,axiom,
% 0.32/0.51 ! [Xs2: set_nat,A: nat] :
% 0.32/0.51 ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_1 )
% 0.32/0.51 => ( ( finite_finite_nat @ Xs2 )
% 0.32/0.51 => ~ ? [X5: nat] :
% 0.32/0.51 ( ( member_nat @ X5 @ Xs2 )
% 0.32/0.51 & ( ord_less_nat @ A @ X5 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % succ_none_empty
% 0.32/0.51 thf(fact_1173_bot__nat__0_Onot__eq__extremum,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( A != zero_zero_nat )
% 0.32/0.51 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % bot_nat_0.not_eq_extremum
% 0.32/0.51 thf(fact_1174_neq0__conv,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( N != zero_zero_nat )
% 0.32/0.51 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % neq0_conv
% 0.32/0.51 thf(fact_1175_less__nat__zero__code,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % less_nat_zero_code
% 0.32/0.51 thf(fact_1176_bot__nat__0_Oextremum,axiom,
% 0.32/0.51 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.32/0.51
% 0.32/0.51 % bot_nat_0.extremum
% 0.32/0.51 thf(fact_1177_le0,axiom,
% 0.32/0.51 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/0.51
% 0.32/0.51 % le0
% 0.32/0.51 thf(fact_1178__C4_Ohyps_C_I4_J,axiom,
% 0.32/0.51 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.32/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.32/0.51
% 0.32/0.51 % "4.hyps"(4)
% 0.32/0.51 thf(fact_1179_add__is__0,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( plus_plus_nat @ M @ N )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 = ( ( M = zero_zero_nat )
% 0.32/0.51 & ( N = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_is_0
% 0.32/0.51 thf(fact_1180_Nat_Oadd__0__right,axiom,
% 0.32/0.51 ! [M: nat] :
% 0.32/0.51 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.32/0.51 = M ) ).
% 0.32/0.51
% 0.32/0.51 % Nat.add_0_right
% 0.32/0.51 thf(fact_1181_diff__0__eq__0,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % diff_0_eq_0
% 0.32/0.51 thf(fact_1182_diff__self__eq__0,axiom,
% 0.32/0.51 ! [M: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ M @ M )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % diff_self_eq_0
% 0.32/0.51 thf(fact_1183_mult__is__0,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( times_times_nat @ M @ N )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 = ( ( M = zero_zero_nat )
% 0.32/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_is_0
% 0.32/0.51 thf(fact_1184_mult__0__right,axiom,
% 0.32/0.51 ! [M: nat] :
% 0.32/0.51 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mult_0_right
% 0.32/0.51 thf(fact_1185_mult__cancel1,axiom,
% 0.32/0.51 ! [K: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( ( times_times_nat @ K @ M )
% 0.32/0.51 = ( times_times_nat @ K @ N ) )
% 0.32/0.51 = ( ( M = N )
% 0.32/0.51 | ( K = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel1
% 0.32/0.51 thf(fact_1186_mult__cancel2,axiom,
% 0.32/0.51 ! [M: nat,K: nat,N: nat] :
% 0.32/0.51 ( ( ( times_times_nat @ M @ K )
% 0.32/0.51 = ( times_times_nat @ N @ K ) )
% 0.32/0.51 = ( ( M = N )
% 0.32/0.51 | ( K = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel2
% 0.32/0.51 thf(fact_1187_div__pos__pos__trivial,axiom,
% 0.32/0.51 ! [K: int,L2: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.51 => ( ( ord_less_int @ K @ L2 )
% 0.32/0.51 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.51 = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_pos_pos_trivial
% 0.32/0.51 thf(fact_1188_div__neg__neg__trivial,axiom,
% 0.32/0.51 ! [K: int,L2: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.32/0.51 => ( ( ord_less_int @ L2 @ K )
% 0.32/0.51 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.51 = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_neg_neg_trivial
% 0.32/0.51 thf(fact_1189_i0__less,axiom,
% 0.32/0.51 ! [N: extended_enat] :
% 0.32/0.51 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.32/0.51 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 0.32/0.51
% 0.32/0.51 % i0_less
% 0.32/0.51 thf(fact_1190_idiff__0__right,axiom,
% 0.32/0.51 ! [N: extended_enat] :
% 0.32/0.51 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.32/0.51 = N ) ).
% 0.32/0.51
% 0.32/0.51 % idiff_0_right
% 0.32/0.51 thf(fact_1191_idiff__0,axiom,
% 0.32/0.51 ! [N: extended_enat] :
% 0.32/0.51 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.32/0.51 = zero_z5237406670263579293d_enat ) ).
% 0.32/0.51
% 0.32/0.51 % idiff_0
% 0.32/0.51 thf(fact_1192_le__zero__eq,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/0.51 = ( N = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_zero_eq
% 0.32/0.51 thf(fact_1193_not__gr__zero,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.32/0.51 = ( N = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % not_gr_zero
% 0.32/0.51 thf(fact_1194_mult__zero__left,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( times_times_complex @ zero_zero_complex @ A )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_left
% 0.32/0.51 thf(fact_1195_mult__zero__left,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( times_times_real @ zero_zero_real @ A )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_left
% 0.32/0.51 thf(fact_1196_mult__zero__left,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( times_times_rat @ zero_zero_rat @ A )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_left
% 0.32/0.51 thf(fact_1197_mult__zero__left,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( times_times_nat @ zero_zero_nat @ A )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_left
% 0.32/0.51 thf(fact_1198_mult__zero__left,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( times_times_int @ zero_zero_int @ A )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_left
% 0.32/0.51 thf(fact_1199_mult__zero__right,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( times_times_complex @ A @ zero_zero_complex )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_right
% 0.32/0.51 thf(fact_1200_mult__zero__right,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( times_times_real @ A @ zero_zero_real )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_right
% 0.32/0.51 thf(fact_1201_mult__zero__right,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( times_times_rat @ A @ zero_zero_rat )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_right
% 0.32/0.51 thf(fact_1202_mult__zero__right,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( times_times_nat @ A @ zero_zero_nat )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_right
% 0.32/0.51 thf(fact_1203_mult__zero__right,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( times_times_int @ A @ zero_zero_int )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % mult_zero_right
% 0.32/0.51 thf(fact_1204_mult__eq__0__iff,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( ( times_times_complex @ A @ B )
% 0.32/0.51 = zero_zero_complex )
% 0.32/0.51 = ( ( A = zero_zero_complex )
% 0.32/0.51 | ( B = zero_zero_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_eq_0_iff
% 0.32/0.51 thf(fact_1205_mult__eq__0__iff,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ( times_times_real @ A @ B )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( ( A = zero_zero_real )
% 0.32/0.51 | ( B = zero_zero_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_eq_0_iff
% 0.32/0.51 thf(fact_1206_mult__eq__0__iff,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ( times_times_rat @ A @ B )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( ( A = zero_zero_rat )
% 0.32/0.51 | ( B = zero_zero_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_eq_0_iff
% 0.32/0.51 thf(fact_1207_mult__eq__0__iff,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ( times_times_nat @ A @ B )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 = ( ( A = zero_zero_nat )
% 0.32/0.51 | ( B = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_eq_0_iff
% 0.32/0.51 thf(fact_1208_mult__eq__0__iff,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ( times_times_int @ A @ B )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 = ( ( A = zero_zero_int )
% 0.32/0.51 | ( B = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_eq_0_iff
% 0.32/0.51 thf(fact_1209_mult__cancel__left,axiom,
% 0.32/0.51 ! [C: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( ( times_times_complex @ C @ A )
% 0.32/0.51 = ( times_times_complex @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_complex )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left
% 0.32/0.51 thf(fact_1210_mult__cancel__left,axiom,
% 0.32/0.51 ! [C: real,A: real,B: real] :
% 0.32/0.51 ( ( ( times_times_real @ C @ A )
% 0.32/0.51 = ( times_times_real @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_real )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left
% 0.32/0.51 thf(fact_1211_mult__cancel__left,axiom,
% 0.32/0.51 ! [C: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( ( times_times_rat @ C @ A )
% 0.32/0.51 = ( times_times_rat @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_rat )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left
% 0.32/0.51 thf(fact_1212_mult__cancel__left,axiom,
% 0.32/0.51 ! [C: nat,A: nat,B: nat] :
% 0.32/0.51 ( ( ( times_times_nat @ C @ A )
% 0.32/0.51 = ( times_times_nat @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_nat )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left
% 0.32/0.51 thf(fact_1213_mult__cancel__left,axiom,
% 0.32/0.51 ! [C: int,A: int,B: int] :
% 0.32/0.51 ( ( ( times_times_int @ C @ A )
% 0.32/0.51 = ( times_times_int @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_int )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left
% 0.32/0.51 thf(fact_1214_mult__cancel__right,axiom,
% 0.32/0.51 ! [A: complex,C: complex,B: complex] :
% 0.32/0.51 ( ( ( times_times_complex @ A @ C )
% 0.32/0.51 = ( times_times_complex @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_complex )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right
% 0.32/0.51 thf(fact_1215_mult__cancel__right,axiom,
% 0.32/0.51 ! [A: real,C: real,B: real] :
% 0.32/0.51 ( ( ( times_times_real @ A @ C )
% 0.32/0.51 = ( times_times_real @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_real )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right
% 0.32/0.51 thf(fact_1216_mult__cancel__right,axiom,
% 0.32/0.51 ! [A: rat,C: rat,B: rat] :
% 0.32/0.51 ( ( ( times_times_rat @ A @ C )
% 0.32/0.51 = ( times_times_rat @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_rat )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right
% 0.32/0.51 thf(fact_1217_mult__cancel__right,axiom,
% 0.32/0.51 ! [A: nat,C: nat,B: nat] :
% 0.32/0.51 ( ( ( times_times_nat @ A @ C )
% 0.32/0.51 = ( times_times_nat @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_nat )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right
% 0.32/0.51 thf(fact_1218_mult__cancel__right,axiom,
% 0.32/0.51 ! [A: int,C: int,B: int] :
% 0.32/0.51 ( ( ( times_times_int @ A @ C )
% 0.32/0.51 = ( times_times_int @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_int )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right
% 0.32/0.51 thf(fact_1219_add_Oright__neutral,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add.right_neutral
% 0.32/0.51 thf(fact_1220_add_Oright__neutral,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add.right_neutral
% 0.32/0.51 thf(fact_1221_add_Oright__neutral,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add.right_neutral
% 0.32/0.51 thf(fact_1222_add_Oright__neutral,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add.right_neutral
% 0.32/0.51 thf(fact_1223_add_Oright__neutral,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add.right_neutral
% 0.32/0.51 thf(fact_1224_double__zero__sym,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( zero_zero_real
% 0.32/0.51 = ( plus_plus_real @ A @ A ) )
% 0.32/0.51 = ( A = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_zero_sym
% 0.32/0.51 thf(fact_1225_double__zero__sym,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( zero_zero_rat
% 0.32/0.51 = ( plus_plus_rat @ A @ A ) )
% 0.32/0.51 = ( A = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_zero_sym
% 0.32/0.51 thf(fact_1226_double__zero__sym,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( zero_zero_int
% 0.32/0.51 = ( plus_plus_int @ A @ A ) )
% 0.32/0.51 = ( A = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_zero_sym
% 0.32/0.51 thf(fact_1227_add__cancel__left__left,axiom,
% 0.32/0.51 ! [B: complex,A: complex] :
% 0.32/0.51 ( ( ( plus_plus_complex @ B @ A )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_left
% 0.32/0.51 thf(fact_1228_add__cancel__left__left,axiom,
% 0.32/0.51 ! [B: real,A: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ B @ A )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_left
% 0.32/0.51 thf(fact_1229_add__cancel__left__left,axiom,
% 0.32/0.51 ! [B: rat,A: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ B @ A )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_left
% 0.32/0.51 thf(fact_1230_add__cancel__left__left,axiom,
% 0.32/0.51 ! [B: nat,A: nat] :
% 0.32/0.51 ( ( ( plus_plus_nat @ B @ A )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_left
% 0.32/0.51 thf(fact_1231_add__cancel__left__left,axiom,
% 0.32/0.51 ! [B: int,A: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ B @ A )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_left
% 0.32/0.51 thf(fact_1232_add__cancel__left__right,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( ( plus_plus_complex @ A @ B )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_right
% 0.32/0.51 thf(fact_1233_add__cancel__left__right,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_right
% 0.32/0.51 thf(fact_1234_add__cancel__left__right,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_right
% 0.32/0.51 thf(fact_1235_add__cancel__left__right,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ( plus_plus_nat @ A @ B )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_right
% 0.32/0.51 thf(fact_1236_add__cancel__left__right,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.51 = A )
% 0.32/0.51 = ( B = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_left_right
% 0.32/0.51 thf(fact_1237_add__cancel__right__left,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_complex @ B @ A ) )
% 0.32/0.51 = ( B = zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_left
% 0.32/0.51 thf(fact_1238_add__cancel__right__left,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_real @ B @ A ) )
% 0.32/0.51 = ( B = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_left
% 0.32/0.51 thf(fact_1239_add__cancel__right__left,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_rat @ B @ A ) )
% 0.32/0.51 = ( B = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_left
% 0.32/0.51 thf(fact_1240_add__cancel__right__left,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_nat @ B @ A ) )
% 0.32/0.51 = ( B = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_left
% 0.32/0.51 thf(fact_1241_add__cancel__right__left,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_int @ B @ A ) )
% 0.32/0.51 = ( B = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_left
% 0.32/0.51 thf(fact_1242_add__cancel__right__right,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_complex @ A @ B ) )
% 0.32/0.51 = ( B = zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_right
% 0.32/0.51 thf(fact_1243_add__cancel__right__right,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_real @ A @ B ) )
% 0.32/0.51 = ( B = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_right
% 0.32/0.51 thf(fact_1244_add__cancel__right__right,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_rat @ A @ B ) )
% 0.32/0.51 = ( B = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_right
% 0.32/0.51 thf(fact_1245_add__cancel__right__right,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_nat @ A @ B ) )
% 0.32/0.51 = ( B = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_right
% 0.32/0.51 thf(fact_1246_add__cancel__right__right,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( plus_plus_int @ A @ B ) )
% 0.32/0.51 = ( B = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_cancel_right_right
% 0.32/0.51 thf(fact_1247_add__eq__0__iff__both__eq__0,axiom,
% 0.32/0.51 ! [X: nat,Y3: nat] :
% 0.32/0.51 ( ( ( plus_plus_nat @ X @ Y3 )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 = ( ( X = zero_zero_nat )
% 0.32/0.51 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_eq_0_iff_both_eq_0
% 0.32/0.51 thf(fact_1248_zero__eq__add__iff__both__eq__0,axiom,
% 0.32/0.51 ! [X: nat,Y3: nat] :
% 0.32/0.51 ( ( zero_zero_nat
% 0.32/0.51 = ( plus_plus_nat @ X @ Y3 ) )
% 0.32/0.51 = ( ( X = zero_zero_nat )
% 0.32/0.51 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_add_iff_both_eq_0
% 0.32/0.51 thf(fact_1249_add__0,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add_0
% 0.32/0.51 thf(fact_1250_add__0,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add_0
% 0.32/0.51 thf(fact_1251_add__0,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add_0
% 0.32/0.51 thf(fact_1252_add__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add_0
% 0.32/0.51 thf(fact_1253_add__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % add_0
% 0.32/0.51 thf(fact_1254_diff__self,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( minus_minus_complex @ A @ A )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % diff_self
% 0.32/0.51 thf(fact_1255_diff__self,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( minus_minus_real @ A @ A )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % diff_self
% 0.32/0.51 thf(fact_1256_diff__self,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ A @ A )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % diff_self
% 0.32/0.51 thf(fact_1257_diff__self,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( minus_minus_int @ A @ A )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % diff_self
% 0.32/0.51 thf(fact_1258_diff__0__right,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_0_right
% 0.32/0.51 thf(fact_1259_diff__0__right,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_0_right
% 0.32/0.51 thf(fact_1260_diff__0__right,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_0_right
% 0.32/0.51 thf(fact_1261_diff__0__right,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_0_right
% 0.32/0.51 thf(fact_1262_zero__diff,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % zero_diff
% 0.32/0.51 thf(fact_1263_diff__zero,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_zero
% 0.32/0.51 thf(fact_1264_diff__zero,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( minus_minus_real @ A @ zero_zero_real )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_zero
% 0.32/0.51 thf(fact_1265_diff__zero,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_zero
% 0.32/0.51 thf(fact_1266_diff__zero,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_zero
% 0.32/0.51 thf(fact_1267_diff__zero,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( minus_minus_int @ A @ zero_zero_int )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % diff_zero
% 0.32/0.51 thf(fact_1268_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( minus_minus_complex @ A @ A )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.51 thf(fact_1269_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( minus_minus_real @ A @ A )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.51 thf(fact_1270_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( minus_minus_rat @ A @ A )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.51 thf(fact_1271_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ A @ A )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.51 thf(fact_1272_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( minus_minus_int @ A @ A )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % cancel_comm_monoid_add_class.diff_cancel
% 0.32/0.51 thf(fact_1273_bits__div__by__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % bits_div_by_0
% 0.32/0.51 thf(fact_1274_bits__div__by__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % bits_div_by_0
% 0.32/0.51 thf(fact_1275_bits__div__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % bits_div_0
% 0.32/0.51 thf(fact_1276_bits__div__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % bits_div_0
% 0.32/0.51 thf(fact_1277_div__by__0,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % div_by_0
% 0.32/0.51 thf(fact_1278_div__by__0,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % div_by_0
% 0.32/0.51 thf(fact_1279_div__by__0,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % div_by_0
% 0.32/0.51 thf(fact_1280_div__by__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % div_by_0
% 0.32/0.51 thf(fact_1281_div__by__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( divide_divide_int @ A @ zero_zero_int )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % div_by_0
% 0.32/0.51 thf(fact_1282_div__0,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % div_0
% 0.32/0.51 thf(fact_1283_div__0,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( divide_divide_real @ zero_zero_real @ A )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % div_0
% 0.32/0.51 thf(fact_1284_div__0,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % div_0
% 0.32/0.51 thf(fact_1285_div__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % div_0
% 0.32/0.51 thf(fact_1286_div__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( divide_divide_int @ zero_zero_int @ A )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % div_0
% 0.32/0.51 thf(fact_1287_division__ring__divide__zero,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % division_ring_divide_zero
% 0.32/0.51 thf(fact_1288_division__ring__divide__zero,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( divide_divide_real @ A @ zero_zero_real )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % division_ring_divide_zero
% 0.32/0.51 thf(fact_1289_division__ring__divide__zero,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % division_ring_divide_zero
% 0.32/0.51 thf(fact_1290_divide__cancel__right,axiom,
% 0.32/0.51 ! [A: complex,C: complex,B: complex] :
% 0.32/0.51 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.32/0.51 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_complex )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_cancel_right
% 0.32/0.51 thf(fact_1291_divide__cancel__right,axiom,
% 0.32/0.51 ! [A: real,C: real,B: real] :
% 0.32/0.51 ( ( ( divide_divide_real @ A @ C )
% 0.32/0.51 = ( divide_divide_real @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_real )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_cancel_right
% 0.32/0.51 thf(fact_1292_divide__cancel__right,axiom,
% 0.32/0.51 ! [A: rat,C: rat,B: rat] :
% 0.32/0.51 ( ( ( divide_divide_rat @ A @ C )
% 0.32/0.51 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_rat )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_cancel_right
% 0.32/0.51 thf(fact_1293_divide__cancel__left,axiom,
% 0.32/0.51 ! [C: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.32/0.51 = ( divide1717551699836669952omplex @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_complex )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_cancel_left
% 0.32/0.51 thf(fact_1294_divide__cancel__left,axiom,
% 0.32/0.51 ! [C: real,A: real,B: real] :
% 0.32/0.51 ( ( ( divide_divide_real @ C @ A )
% 0.32/0.51 = ( divide_divide_real @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_real )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_cancel_left
% 0.32/0.51 thf(fact_1295_divide__cancel__left,axiom,
% 0.32/0.51 ! [C: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( ( divide_divide_rat @ C @ A )
% 0.32/0.51 = ( divide_divide_rat @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_rat )
% 0.32/0.51 | ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_cancel_left
% 0.32/0.51 thf(fact_1296_divide__eq__0__iff,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.51 = zero_zero_complex )
% 0.32/0.51 = ( ( A = zero_zero_complex )
% 0.32/0.51 | ( B = zero_zero_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_0_iff
% 0.32/0.51 thf(fact_1297_divide__eq__0__iff,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ( divide_divide_real @ A @ B )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( ( A = zero_zero_real )
% 0.32/0.51 | ( B = zero_zero_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_0_iff
% 0.32/0.51 thf(fact_1298_divide__eq__0__iff,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( ( A = zero_zero_rat )
% 0.32/0.51 | ( B = zero_zero_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_0_iff
% 0.32/0.51 thf(fact_1299_bits__mod__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_0
% 0.32/0.51 thf(fact_1300_bits__mod__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_0
% 0.32/0.51 thf(fact_1301_bits__mod__0,axiom,
% 0.32/0.51 ! [A: code_integer] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_0
% 0.32/0.51 thf(fact_1302_mod__self,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ A @ A )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mod_self
% 0.32/0.51 thf(fact_1303_mod__self,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( modulo_modulo_int @ A @ A )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % mod_self
% 0.32/0.51 thf(fact_1304_mod__self,axiom,
% 0.32/0.51 ! [A: code_integer] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ A @ A )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % mod_self
% 0.32/0.51 thf(fact_1305_mod__by__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % mod_by_0
% 0.32/0.51 thf(fact_1306_mod__by__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % mod_by_0
% 0.32/0.51 thf(fact_1307_mod__by__0,axiom,
% 0.32/0.51 ! [A: code_integer] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.32/0.51 = A ) ).
% 0.32/0.51
% 0.32/0.51 % mod_by_0
% 0.32/0.51 thf(fact_1308_mod__0,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mod_0
% 0.32/0.51 thf(fact_1309_mod__0,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % mod_0
% 0.32/0.51 thf(fact_1310_mod__0,axiom,
% 0.32/0.51 ! [A: code_integer] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % mod_0
% 0.32/0.51 thf(fact_1311_add__gr__0,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.51 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.51 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_gr_0
% 0.32/0.51 thf(fact_1312_zero__less__diff,axiom,
% 0.32/0.51 ! [N: nat,M: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.51 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_diff
% 0.32/0.51 thf(fact_1313_nat__0__less__mult__iff,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.32/0.51 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.51 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_0_less_mult_iff
% 0.32/0.51 thf(fact_1314_mult__less__cancel2,axiom,
% 0.32/0.51 ! [M: nat,K: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.32/0.51 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.51 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_less_cancel2
% 0.32/0.51 thf(fact_1315_nat__mult__less__cancel__disj,axiom,
% 0.32/0.51 ! [K: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.51 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.51 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_mult_less_cancel_disj
% 0.32/0.51 thf(fact_1316_not__real__square__gt__zero,axiom,
% 0.32/0.51 ! [X: real] :
% 0.32/0.51 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 0.32/0.51 = ( X = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % not_real_square_gt_zero
% 0.32/0.51 thf(fact_1317_diff__is__0__eq_H,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.51 => ( ( minus_minus_nat @ M @ N )
% 0.32/0.51 = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_is_0_eq'
% 0.32/0.51 thf(fact_1318_diff__is__0__eq,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ( minus_minus_nat @ M @ N )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_is_0_eq
% 0.32/0.51 thf(fact_1319_div__less,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ M @ N )
% 0.32/0.51 => ( ( divide_divide_nat @ M @ N )
% 0.32/0.51 = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_less
% 0.32/0.51 thf(fact_1320_less__one,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ N @ one_one_nat )
% 0.32/0.51 = ( N = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_one
% 0.32/0.51 thf(fact_1321_nat__zero__less__power__iff,axiom,
% 0.32/0.51 ! [X: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 0.32/0.51 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.51 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_zero_less_power_iff
% 0.32/0.51 thf(fact_1322_nat__mult__div__cancel__disj,axiom,
% 0.32/0.51 ! [K: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( ( K = zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.51 = zero_zero_nat ) )
% 0.32/0.51 & ( ( K != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.51 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_mult_div_cancel_disj
% 0.32/0.51 thf(fact_1323__092_060open_062length_AtreeList_A_061_A2_A_094_An_092_060close_062,axiom,
% 0.32/0.51 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.32/0.51 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ).
% 0.32/0.51
% 0.32/0.51 % \<open>length treeList = 2 ^ n\<close>
% 0.32/0.51 thf(fact_1324_dbl__simps_I2_J,axiom,
% 0.32/0.51 ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(2)
% 0.32/0.51 thf(fact_1325_dbl__simps_I2_J,axiom,
% 0.32/0.51 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(2)
% 0.32/0.51 thf(fact_1326_dbl__simps_I2_J,axiom,
% 0.32/0.51 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(2)
% 0.32/0.51 thf(fact_1327_dbl__simps_I2_J,axiom,
% 0.32/0.51 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(2)
% 0.32/0.51 thf(fact_1328_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.32/0.51 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_le_double_add_iff_zero_le_single_add
% 0.32/0.51 thf(fact_1329_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.32/0.51 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_le_double_add_iff_zero_le_single_add
% 0.32/0.51 thf(fact_1330_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.32/0.51 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_le_double_add_iff_zero_le_single_add
% 0.32/0.51 thf(fact_1331_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.32/0.51 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_add_le_zero_iff_single_add_le_zero
% 0.32/0.51 thf(fact_1332_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_add_le_zero_iff_single_add_le_zero
% 0.32/0.51 thf(fact_1333_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.32/0.51 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_add_le_zero_iff_single_add_le_zero
% 0.32/0.51 thf(fact_1334_le__add__same__cancel2,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.32/0.51 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_add_same_cancel2
% 0.32/0.51 thf(fact_1335_le__add__same__cancel2,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.32/0.51 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_add_same_cancel2
% 0.32/0.51 thf(fact_1336_le__add__same__cancel2,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.32/0.51 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_add_same_cancel2
% 0.32/0.51 thf(fact_1337_le__add__same__cancel2,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.32/0.51 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_add_same_cancel2
% 0.32/0.51 thf(fact_1338_le__add__same__cancel1,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.51 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_add_same_cancel1
% 0.32/0.51 thf(fact_1339_le__add__same__cancel1,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.51 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_add_same_cancel1
% 0.32/0.51 thf(fact_1340_le__add__same__cancel1,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.51 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_add_same_cancel1
% 0.32/0.51 thf(fact_1341_le__add__same__cancel1,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.51 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_add_same_cancel1
% 0.32/0.51 thf(fact_1342_add__le__same__cancel2,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.32/0.51 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_same_cancel2
% 0.32/0.51 thf(fact_1343_add__le__same__cancel2,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_same_cancel2
% 0.32/0.51 thf(fact_1344_add__le__same__cancel2,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.51 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_same_cancel2
% 0.32/0.51 thf(fact_1345_add__le__same__cancel2,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.51 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_same_cancel2
% 0.32/0.51 thf(fact_1346_add__le__same__cancel1,axiom,
% 0.32/0.51 ! [B: real,A: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.32/0.51 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_same_cancel1
% 0.32/0.51 thf(fact_1347_add__le__same__cancel1,axiom,
% 0.32/0.51 ! [B: rat,A: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_same_cancel1
% 0.32/0.51 thf(fact_1348_add__le__same__cancel1,axiom,
% 0.32/0.51 ! [B: nat,A: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.51 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_same_cancel1
% 0.32/0.51 thf(fact_1349_add__le__same__cancel1,axiom,
% 0.32/0.51 ! [B: int,A: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.51 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_le_same_cancel1
% 0.32/0.51 thf(fact_1350_diff__ge__0__iff__ge,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.32/0.51 = ( ord_less_eq_real @ B @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_ge_0_iff_ge
% 0.32/0.51 thf(fact_1351_diff__ge__0__iff__ge,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.51 = ( ord_less_eq_rat @ B @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_ge_0_iff_ge
% 0.32/0.51 thf(fact_1352_diff__ge__0__iff__ge,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.32/0.51 = ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_ge_0_iff_ge
% 0.32/0.51 thf(fact_1353_add__less__same__cancel1,axiom,
% 0.32/0.51 ! [B: real,A: real] :
% 0.32/0.51 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.32/0.51 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_less_same_cancel1
% 0.32/0.51 thf(fact_1354_add__less__same__cancel1,axiom,
% 0.32/0.51 ! [B: rat,A: rat] :
% 0.32/0.51 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.32/0.51 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_less_same_cancel1
% 0.32/0.51 thf(fact_1355_add__less__same__cancel1,axiom,
% 0.32/0.51 ! [B: nat,A: nat] :
% 0.32/0.51 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.51 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_less_same_cancel1
% 0.32/0.51 thf(fact_1356_add__less__same__cancel1,axiom,
% 0.32/0.51 ! [B: int,A: int] :
% 0.32/0.51 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.51 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_less_same_cancel1
% 0.32/0.51 thf(fact_1357_add__less__same__cancel2,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.32/0.51 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_less_same_cancel2
% 0.32/0.51 thf(fact_1358_add__less__same__cancel2,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.32/0.51 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_less_same_cancel2
% 0.32/0.51 thf(fact_1359_add__less__same__cancel2,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.51 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_less_same_cancel2
% 0.32/0.51 thf(fact_1360_add__less__same__cancel2,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.51 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % add_less_same_cancel2
% 0.32/0.51 thf(fact_1361_less__add__same__cancel1,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.51 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_add_same_cancel1
% 0.32/0.51 thf(fact_1362_less__add__same__cancel1,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.51 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_add_same_cancel1
% 0.32/0.51 thf(fact_1363_less__add__same__cancel1,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.51 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_add_same_cancel1
% 0.32/0.51 thf(fact_1364_less__add__same__cancel1,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.51 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_add_same_cancel1
% 0.32/0.51 thf(fact_1365_less__add__same__cancel2,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.32/0.51 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_add_same_cancel2
% 0.32/0.51 thf(fact_1366_less__add__same__cancel2,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.32/0.51 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_add_same_cancel2
% 0.32/0.51 thf(fact_1367_less__add__same__cancel2,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.32/0.51 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_add_same_cancel2
% 0.32/0.51 thf(fact_1368_less__add__same__cancel2,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.32/0.51 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_add_same_cancel2
% 0.32/0.51 thf(fact_1369_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.32/0.51 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_add_less_zero_iff_single_add_less_zero
% 0.32/0.51 thf(fact_1370_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.32/0.51 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_add_less_zero_iff_single_add_less_zero
% 0.32/0.51 thf(fact_1371_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.32/0.51 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % double_add_less_zero_iff_single_add_less_zero
% 0.32/0.51 thf(fact_1372_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.32/0.51 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_double_add_iff_zero_less_single_add
% 0.32/0.51 thf(fact_1373_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.32/0.51 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_double_add_iff_zero_less_single_add
% 0.32/0.51 thf(fact_1374_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.32/0.51 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_double_add_iff_zero_less_single_add
% 0.32/0.51 thf(fact_1375_diff__gt__0__iff__gt,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.32/0.51 = ( ord_less_real @ B @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_gt_0_iff_gt
% 0.32/0.51 thf(fact_1376_diff__gt__0__iff__gt,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.51 = ( ord_less_rat @ B @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_gt_0_iff_gt
% 0.32/0.51 thf(fact_1377_diff__gt__0__iff__gt,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.32/0.51 = ( ord_less_int @ B @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % diff_gt_0_iff_gt
% 0.32/0.51 thf(fact_1378_mult__cancel__right2,axiom,
% 0.32/0.51 ! [A: complex,C: complex] :
% 0.32/0.51 ( ( ( times_times_complex @ A @ C )
% 0.32/0.51 = C )
% 0.32/0.51 = ( ( C = zero_zero_complex )
% 0.32/0.51 | ( A = one_one_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right2
% 0.32/0.51 thf(fact_1379_mult__cancel__right2,axiom,
% 0.32/0.51 ! [A: real,C: real] :
% 0.32/0.51 ( ( ( times_times_real @ A @ C )
% 0.32/0.51 = C )
% 0.32/0.51 = ( ( C = zero_zero_real )
% 0.32/0.51 | ( A = one_one_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right2
% 0.32/0.51 thf(fact_1380_mult__cancel__right2,axiom,
% 0.32/0.51 ! [A: rat,C: rat] :
% 0.32/0.51 ( ( ( times_times_rat @ A @ C )
% 0.32/0.51 = C )
% 0.32/0.51 = ( ( C = zero_zero_rat )
% 0.32/0.51 | ( A = one_one_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right2
% 0.32/0.51 thf(fact_1381_mult__cancel__right2,axiom,
% 0.32/0.51 ! [A: int,C: int] :
% 0.32/0.51 ( ( ( times_times_int @ A @ C )
% 0.32/0.51 = C )
% 0.32/0.51 = ( ( C = zero_zero_int )
% 0.32/0.51 | ( A = one_one_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right2
% 0.32/0.51 thf(fact_1382_mult__cancel__right1,axiom,
% 0.32/0.51 ! [C: complex,B: complex] :
% 0.32/0.51 ( ( C
% 0.32/0.51 = ( times_times_complex @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_complex )
% 0.32/0.51 | ( B = one_one_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right1
% 0.32/0.51 thf(fact_1383_mult__cancel__right1,axiom,
% 0.32/0.51 ! [C: real,B: real] :
% 0.32/0.51 ( ( C
% 0.32/0.51 = ( times_times_real @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_real )
% 0.32/0.51 | ( B = one_one_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right1
% 0.32/0.51 thf(fact_1384_mult__cancel__right1,axiom,
% 0.32/0.51 ! [C: rat,B: rat] :
% 0.32/0.51 ( ( C
% 0.32/0.51 = ( times_times_rat @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_rat )
% 0.32/0.51 | ( B = one_one_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right1
% 0.32/0.51 thf(fact_1385_mult__cancel__right1,axiom,
% 0.32/0.51 ! [C: int,B: int] :
% 0.32/0.51 ( ( C
% 0.32/0.51 = ( times_times_int @ B @ C ) )
% 0.32/0.51 = ( ( C = zero_zero_int )
% 0.32/0.51 | ( B = one_one_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_right1
% 0.32/0.51 thf(fact_1386_mult__cancel__left2,axiom,
% 0.32/0.51 ! [C: complex,A: complex] :
% 0.32/0.51 ( ( ( times_times_complex @ C @ A )
% 0.32/0.51 = C )
% 0.32/0.51 = ( ( C = zero_zero_complex )
% 0.32/0.51 | ( A = one_one_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left2
% 0.32/0.51 thf(fact_1387_mult__cancel__left2,axiom,
% 0.32/0.51 ! [C: real,A: real] :
% 0.32/0.51 ( ( ( times_times_real @ C @ A )
% 0.32/0.51 = C )
% 0.32/0.51 = ( ( C = zero_zero_real )
% 0.32/0.51 | ( A = one_one_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left2
% 0.32/0.51 thf(fact_1388_mult__cancel__left2,axiom,
% 0.32/0.51 ! [C: rat,A: rat] :
% 0.32/0.51 ( ( ( times_times_rat @ C @ A )
% 0.32/0.51 = C )
% 0.32/0.51 = ( ( C = zero_zero_rat )
% 0.32/0.51 | ( A = one_one_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left2
% 0.32/0.51 thf(fact_1389_mult__cancel__left2,axiom,
% 0.32/0.51 ! [C: int,A: int] :
% 0.32/0.51 ( ( ( times_times_int @ C @ A )
% 0.32/0.51 = C )
% 0.32/0.51 = ( ( C = zero_zero_int )
% 0.32/0.51 | ( A = one_one_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left2
% 0.32/0.51 thf(fact_1390_mult__cancel__left1,axiom,
% 0.32/0.51 ! [C: complex,B: complex] :
% 0.32/0.51 ( ( C
% 0.32/0.51 = ( times_times_complex @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_complex )
% 0.32/0.51 | ( B = one_one_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left1
% 0.32/0.51 thf(fact_1391_mult__cancel__left1,axiom,
% 0.32/0.51 ! [C: real,B: real] :
% 0.32/0.51 ( ( C
% 0.32/0.51 = ( times_times_real @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_real )
% 0.32/0.51 | ( B = one_one_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left1
% 0.32/0.51 thf(fact_1392_mult__cancel__left1,axiom,
% 0.32/0.51 ! [C: rat,B: rat] :
% 0.32/0.51 ( ( C
% 0.32/0.51 = ( times_times_rat @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_rat )
% 0.32/0.51 | ( B = one_one_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left1
% 0.32/0.51 thf(fact_1393_mult__cancel__left1,axiom,
% 0.32/0.51 ! [C: int,B: int] :
% 0.32/0.51 ( ( C
% 0.32/0.51 = ( times_times_int @ C @ B ) )
% 0.32/0.51 = ( ( C = zero_zero_int )
% 0.32/0.51 | ( B = one_one_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_cancel_left1
% 0.32/0.51 thf(fact_1394_sum__squares__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: real,Y3: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( ( X = zero_zero_real )
% 0.32/0.51 & ( Y3 = zero_zero_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_squares_eq_zero_iff
% 0.32/0.51 thf(fact_1395_sum__squares__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: rat,Y3: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( ( X = zero_zero_rat )
% 0.32/0.51 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_squares_eq_zero_iff
% 0.32/0.51 thf(fact_1396_sum__squares__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: int,Y3: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 = ( ( X = zero_zero_int )
% 0.32/0.51 & ( Y3 = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_squares_eq_zero_iff
% 0.32/0.51 thf(fact_1397_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( A != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 0.32/0.51 = B ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_left
% 0.32/0.51 thf(fact_1398_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( A != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 0.32/0.51 = B ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_left
% 0.32/0.51 thf(fact_1399_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( A != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 0.32/0.51 = B ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_left
% 0.32/0.51 thf(fact_1400_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( A != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.32/0.51 = B ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_left
% 0.32/0.51 thf(fact_1401_nonzero__mult__div__cancel__left,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( A != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 0.32/0.51 = B ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_left
% 0.32/0.51 thf(fact_1402_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.51 ! [B: complex,A: complex] :
% 0.32/0.51 ( ( B != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 0.32/0.51 = A ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_right
% 0.32/0.51 thf(fact_1403_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.51 ! [B: real,A: real] :
% 0.32/0.51 ( ( B != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 0.32/0.51 = A ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_right
% 0.32/0.51 thf(fact_1404_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.51 ! [B: rat,A: rat] :
% 0.32/0.51 ( ( B != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 0.32/0.51 = A ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_right
% 0.32/0.51 thf(fact_1405_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.51 ! [B: nat,A: nat] :
% 0.32/0.51 ( ( B != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.32/0.51 = A ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_right
% 0.32/0.51 thf(fact_1406_nonzero__mult__div__cancel__right,axiom,
% 0.32/0.51 ! [B: int,A: int] :
% 0.32/0.51 ( ( B != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 0.32/0.51 = A ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_div_cancel_right
% 0.32/0.51 thf(fact_1407_div__mult__mult1,axiom,
% 0.32/0.51 ! [C: nat,A: nat,B: nat] :
% 0.32/0.51 ( ( C != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.51 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_mult1
% 0.32/0.51 thf(fact_1408_div__mult__mult1,axiom,
% 0.32/0.51 ! [C: int,A: int,B: int] :
% 0.32/0.51 ( ( C != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.51 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_mult1
% 0.32/0.51 thf(fact_1409_div__mult__mult2,axiom,
% 0.32/0.51 ! [C: nat,A: nat,B: nat] :
% 0.32/0.51 ( ( C != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.32/0.51 = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_mult2
% 0.32/0.51 thf(fact_1410_div__mult__mult2,axiom,
% 0.32/0.51 ! [C: int,A: int,B: int] :
% 0.32/0.51 ( ( C != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.51 = ( divide_divide_int @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_mult2
% 0.32/0.51 thf(fact_1411_div__mult__mult1__if,axiom,
% 0.32/0.51 ! [C: nat,A: nat,B: nat] :
% 0.32/0.51 ( ( ( C = zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.51 = zero_zero_nat ) )
% 0.32/0.51 & ( ( C != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.51 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_mult1_if
% 0.32/0.51 thf(fact_1412_div__mult__mult1__if,axiom,
% 0.32/0.51 ! [C: int,A: int,B: int] :
% 0.32/0.51 ( ( ( C = zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.51 = zero_zero_int ) )
% 0.32/0.51 & ( ( C != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.51 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_mult1_if
% 0.32/0.51 thf(fact_1413_mult__divide__mult__cancel__left__if,axiom,
% 0.32/0.51 ! [C: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( ( C = zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.51 = zero_zero_complex ) )
% 0.32/0.51 & ( ( C != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_divide_mult_cancel_left_if
% 0.32/0.51 thf(fact_1414_mult__divide__mult__cancel__left__if,axiom,
% 0.32/0.51 ! [C: real,A: real,B: real] :
% 0.32/0.51 ( ( ( C = zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.51 = zero_zero_real ) )
% 0.32/0.51 & ( ( C != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.51 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_divide_mult_cancel_left_if
% 0.32/0.51 thf(fact_1415_mult__divide__mult__cancel__left__if,axiom,
% 0.32/0.51 ! [C: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( ( C = zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.51 = zero_zero_rat ) )
% 0.32/0.51 & ( ( C != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.51 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_divide_mult_cancel_left_if
% 0.32/0.51 thf(fact_1416_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.32/0.51 ! [C: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( C != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_left
% 0.32/0.51 thf(fact_1417_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.32/0.51 ! [C: real,A: real,B: real] :
% 0.32/0.51 ( ( C != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.51 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_left
% 0.32/0.51 thf(fact_1418_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.32/0.51 ! [C: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( C != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.51 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_left
% 0.32/0.51 thf(fact_1419_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.32/0.51 ! [C: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( C != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_left2
% 0.32/0.51 thf(fact_1420_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.32/0.51 ! [C: real,A: real,B: real] :
% 0.32/0.51 ( ( C != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 0.32/0.51 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_left2
% 0.32/0.51 thf(fact_1421_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.32/0.51 ! [C: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( C != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.51 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_left2
% 0.32/0.51 thf(fact_1422_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.32/0.51 ! [C: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( C != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_right
% 0.32/0.51 thf(fact_1423_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.32/0.51 ! [C: real,A: real,B: real] :
% 0.32/0.51 ( ( C != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.51 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_right
% 0.32/0.51 thf(fact_1424_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.32/0.51 ! [C: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( C != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.51 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_right
% 0.32/0.51 thf(fact_1425_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.32/0.51 ! [C: complex,A: complex,B: complex] :
% 0.32/0.51 ( ( C != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_right2
% 0.32/0.51 thf(fact_1426_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.32/0.51 ! [C: real,A: real,B: real] :
% 0.32/0.51 ( ( C != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 0.32/0.51 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_right2
% 0.32/0.51 thf(fact_1427_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.32/0.51 ! [C: rat,A: rat,B: rat] :
% 0.32/0.51 ( ( C != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.51 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_mult_divide_mult_cancel_right2
% 0.32/0.51 thf(fact_1428_diff__numeral__special_I9_J,axiom,
% 0.32/0.51 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % diff_numeral_special(9)
% 0.32/0.51 thf(fact_1429_diff__numeral__special_I9_J,axiom,
% 0.32/0.51 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % diff_numeral_special(9)
% 0.32/0.51 thf(fact_1430_diff__numeral__special_I9_J,axiom,
% 0.32/0.51 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % diff_numeral_special(9)
% 0.32/0.51 thf(fact_1431_diff__numeral__special_I9_J,axiom,
% 0.32/0.51 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % diff_numeral_special(9)
% 0.32/0.51 thf(fact_1432_diff__add__zero,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % diff_add_zero
% 0.32/0.51 thf(fact_1433_div__self,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( A != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.32/0.51 = one_one_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_self
% 0.32/0.51 thf(fact_1434_div__self,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( A != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ A @ A )
% 0.32/0.51 = one_one_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_self
% 0.32/0.51 thf(fact_1435_div__self,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( A != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ A @ A )
% 0.32/0.51 = one_one_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_self
% 0.32/0.51 thf(fact_1436_div__self,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( A != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ A @ A )
% 0.32/0.51 = one_one_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_self
% 0.32/0.51 thf(fact_1437_div__self,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( A != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ A @ A )
% 0.32/0.51 = one_one_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_self
% 0.32/0.51 thf(fact_1438_zero__eq__1__divide__iff,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( zero_zero_real
% 0.32/0.51 = ( divide_divide_real @ one_one_real @ A ) )
% 0.32/0.51 = ( A = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_1_divide_iff
% 0.32/0.51 thf(fact_1439_zero__eq__1__divide__iff,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( zero_zero_rat
% 0.32/0.51 = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.32/0.51 = ( A = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_1_divide_iff
% 0.32/0.51 thf(fact_1440_one__divide__eq__0__iff,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ( divide_divide_real @ one_one_real @ A )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( A = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_divide_eq_0_iff
% 0.32/0.51 thf(fact_1441_one__divide__eq__0__iff,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( A = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_divide_eq_0_iff
% 0.32/0.51 thf(fact_1442_eq__divide__eq__1,axiom,
% 0.32/0.51 ! [B: real,A: real] :
% 0.32/0.51 ( ( one_one_real
% 0.32/0.51 = ( divide_divide_real @ B @ A ) )
% 0.32/0.51 = ( ( A != zero_zero_real )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_divide_eq_1
% 0.32/0.51 thf(fact_1443_eq__divide__eq__1,axiom,
% 0.32/0.51 ! [B: rat,A: rat] :
% 0.32/0.51 ( ( one_one_rat
% 0.32/0.51 = ( divide_divide_rat @ B @ A ) )
% 0.32/0.51 = ( ( A != zero_zero_rat )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_divide_eq_1
% 0.32/0.51 thf(fact_1444_divide__eq__eq__1,axiom,
% 0.32/0.51 ! [B: real,A: real] :
% 0.32/0.51 ( ( ( divide_divide_real @ B @ A )
% 0.32/0.51 = one_one_real )
% 0.32/0.51 = ( ( A != zero_zero_real )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_eq_1
% 0.32/0.51 thf(fact_1445_divide__eq__eq__1,axiom,
% 0.32/0.51 ! [B: rat,A: rat] :
% 0.32/0.51 ( ( ( divide_divide_rat @ B @ A )
% 0.32/0.51 = one_one_rat )
% 0.32/0.51 = ( ( A != zero_zero_rat )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_eq_1
% 0.32/0.51 thf(fact_1446_divide__self__if,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( ( A = zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.32/0.51 = zero_zero_complex ) )
% 0.32/0.51 & ( ( A != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.32/0.51 = one_one_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_self_if
% 0.32/0.51 thf(fact_1447_divide__self__if,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ( A = zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ A @ A )
% 0.32/0.51 = zero_zero_real ) )
% 0.32/0.51 & ( ( A != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ A @ A )
% 0.32/0.51 = one_one_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_self_if
% 0.32/0.51 thf(fact_1448_divide__self__if,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ( A = zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ A @ A )
% 0.32/0.51 = zero_zero_rat ) )
% 0.32/0.51 & ( ( A != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ A @ A )
% 0.32/0.51 = one_one_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_self_if
% 0.32/0.51 thf(fact_1449_divide__self,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( A != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ A @ A )
% 0.32/0.51 = one_one_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_self
% 0.32/0.51 thf(fact_1450_divide__self,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( A != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ A @ A )
% 0.32/0.51 = one_one_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_self
% 0.32/0.51 thf(fact_1451_divide__self,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( A != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ A @ A )
% 0.32/0.51 = one_one_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_self
% 0.32/0.51 thf(fact_1452_one__eq__divide__iff,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( one_one_complex
% 0.32/0.51 = ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.51 = ( ( B != zero_zero_complex )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_eq_divide_iff
% 0.32/0.51 thf(fact_1453_one__eq__divide__iff,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( one_one_real
% 0.32/0.51 = ( divide_divide_real @ A @ B ) )
% 0.32/0.51 = ( ( B != zero_zero_real )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_eq_divide_iff
% 0.32/0.51 thf(fact_1454_one__eq__divide__iff,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( one_one_rat
% 0.32/0.51 = ( divide_divide_rat @ A @ B ) )
% 0.32/0.51 = ( ( B != zero_zero_rat )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % one_eq_divide_iff
% 0.32/0.51 thf(fact_1455_divide__eq__1__iff,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.51 = one_one_complex )
% 0.32/0.51 = ( ( B != zero_zero_complex )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_1_iff
% 0.32/0.51 thf(fact_1456_divide__eq__1__iff,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ( divide_divide_real @ A @ B )
% 0.32/0.51 = one_one_real )
% 0.32/0.51 = ( ( B != zero_zero_real )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_1_iff
% 0.32/0.51 thf(fact_1457_divide__eq__1__iff,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.51 = one_one_rat )
% 0.32/0.51 = ( ( B != zero_zero_rat )
% 0.32/0.51 & ( A = B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_1_iff
% 0.32/0.51 thf(fact_1458_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1459_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1460_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1461_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1462_power__zero__numeral,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.32/0.51 = zero_zero_complex ) ).
% 0.32/0.51
% 0.32/0.51 % power_zero_numeral
% 0.32/0.51 thf(fact_1463_power__eq__0__iff,axiom,
% 0.32/0.51 ! [A: rat,N: nat] :
% 0.32/0.51 ( ( ( power_power_rat @ A @ N )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( ( A = zero_zero_rat )
% 0.32/0.51 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_0_iff
% 0.32/0.51 thf(fact_1464_power__eq__0__iff,axiom,
% 0.32/0.51 ! [A: nat,N: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ A @ N )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 = ( ( A = zero_zero_nat )
% 0.32/0.51 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_0_iff
% 0.32/0.51 thf(fact_1465_power__eq__0__iff,axiom,
% 0.32/0.51 ! [A: real,N: nat] :
% 0.32/0.51 ( ( ( power_power_real @ A @ N )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( ( A = zero_zero_real )
% 0.32/0.51 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_0_iff
% 0.32/0.51 thf(fact_1466_power__eq__0__iff,axiom,
% 0.32/0.51 ! [A: int,N: nat] :
% 0.32/0.51 ( ( ( power_power_int @ A @ N )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 = ( ( A = zero_zero_int )
% 0.32/0.51 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_0_iff
% 0.32/0.51 thf(fact_1467_power__eq__0__iff,axiom,
% 0.32/0.51 ! [A: complex,N: nat] :
% 0.32/0.51 ( ( ( power_power_complex @ A @ N )
% 0.32/0.51 = zero_zero_complex )
% 0.32/0.51 = ( ( A = zero_zero_complex )
% 0.32/0.51 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_eq_0_iff
% 0.32/0.51 thf(fact_1468_mod__mult__self2__is__0,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mod_mult_self2_is_0
% 0.32/0.51 thf(fact_1469_mod__mult__self2__is__0,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % mod_mult_self2_is_0
% 0.32/0.51 thf(fact_1470_mod__mult__self2__is__0,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % mod_mult_self2_is_0
% 0.32/0.51 thf(fact_1471_mod__mult__self1__is__0,axiom,
% 0.32/0.51 ! [B: nat,A: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mod_mult_self1_is_0
% 0.32/0.51 thf(fact_1472_mod__mult__self1__is__0,axiom,
% 0.32/0.51 ! [B: int,A: int] :
% 0.32/0.51 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % mod_mult_self1_is_0
% 0.32/0.51 thf(fact_1473_mod__mult__self1__is__0,axiom,
% 0.32/0.51 ! [B: code_integer,A: code_integer] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % mod_mult_self1_is_0
% 0.32/0.51 thf(fact_1474_bits__mod__by__1,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_by_1
% 0.32/0.51 thf(fact_1475_bits__mod__by__1,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_by_1
% 0.32/0.51 thf(fact_1476_bits__mod__by__1,axiom,
% 0.32/0.51 ! [A: code_integer] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_by_1
% 0.32/0.51 thf(fact_1477_mod__by__1,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mod_by_1
% 0.32/0.51 thf(fact_1478_mod__by__1,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( modulo_modulo_int @ A @ one_one_int )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % mod_by_1
% 0.32/0.51 thf(fact_1479_mod__by__1,axiom,
% 0.32/0.51 ! [A: code_integer] :
% 0.32/0.51 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % mod_by_1
% 0.32/0.51 thf(fact_1480_bits__mod__div__trivial,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_div_trivial
% 0.32/0.51 thf(fact_1481_bits__mod__div__trivial,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_div_trivial
% 0.32/0.51 thf(fact_1482_bits__mod__div__trivial,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer] :
% 0.32/0.51 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % bits_mod_div_trivial
% 0.32/0.51 thf(fact_1483_mod__div__trivial,axiom,
% 0.32/0.51 ! [A: nat,B: nat] :
% 0.32/0.51 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % mod_div_trivial
% 0.32/0.51 thf(fact_1484_mod__div__trivial,axiom,
% 0.32/0.51 ! [A: int,B: int] :
% 0.32/0.51 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % mod_div_trivial
% 0.32/0.51 thf(fact_1485_mod__div__trivial,axiom,
% 0.32/0.51 ! [A: code_integer,B: code_integer] :
% 0.32/0.51 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.32/0.51 = zero_z3403309356797280102nteger ) ).
% 0.32/0.51
% 0.32/0.51 % mod_div_trivial
% 0.32/0.51 thf(fact_1486_zmod__numeral__Bit0,axiom,
% 0.32/0.51 ! [V: num,W: num] :
% 0.32/0.51 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.32/0.51 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % zmod_numeral_Bit0
% 0.32/0.51 thf(fact_1487_mult__le__cancel2,axiom,
% 0.32/0.51 ! [M: nat,K: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.32/0.51 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.51 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % mult_le_cancel2
% 0.32/0.51 thf(fact_1488_nat__mult__le__cancel__disj,axiom,
% 0.32/0.51 ! [K: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.51 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.51 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nat_mult_le_cancel_disj
% 0.32/0.51 thf(fact_1489_div__mult__self1__is__m,axiom,
% 0.32/0.51 ! [N: nat,M: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.32/0.51 = M ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self1_is_m
% 0.32/0.51 thf(fact_1490_div__mult__self__is__m,axiom,
% 0.32/0.51 ! [N: nat,M: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.32/0.51 = M ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self_is_m
% 0.32/0.51 thf(fact_1491_dbl__simps_I5_J,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.32/0.51 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(5)
% 0.32/0.51 thf(fact_1492_dbl__simps_I5_J,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 0.32/0.51 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(5)
% 0.32/0.51 thf(fact_1493_dbl__simps_I5_J,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 0.32/0.51 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(5)
% 0.32/0.51 thf(fact_1494_dbl__simps_I5_J,axiom,
% 0.32/0.51 ! [K: num] :
% 0.32/0.51 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 0.32/0.51 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % dbl_simps(5)
% 0.32/0.51 thf(fact_1495__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
% 0.32/0.51 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% 0.32/0.51
% 0.32/0.51 % \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
% 0.32/0.51 thf(fact_1496__C33_C,axiom,
% 0.32/0.51 ~ ? [U2: nat] :
% 0.32/0.51 ( ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U2 )
% 0.32/0.51 & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ U2 ) ) ).
% 0.32/0.51
% 0.32/0.51 % "33"
% 0.32/0.51 thf(fact_1497_divide__le__0__1__iff,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.32/0.51 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_le_0_1_iff
% 0.32/0.51 thf(fact_1498_divide__le__0__1__iff,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_le_0_1_iff
% 0.32/0.51 thf(fact_1499_zero__le__divide__1__iff,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.32/0.51 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_le_divide_1_iff
% 0.32/0.51 thf(fact_1500_zero__le__divide__1__iff,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.32/0.51 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_le_divide_1_iff
% 0.32/0.51 thf(fact_1501_zero__less__divide__1__iff,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.32/0.51 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_divide_1_iff
% 0.32/0.51 thf(fact_1502_zero__less__divide__1__iff,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.32/0.51 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_divide_1_iff
% 0.32/0.51 thf(fact_1503_less__divide__eq__1__pos,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.51 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.51 = ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_divide_eq_1_pos
% 0.32/0.51 thf(fact_1504_less__divide__eq__1__pos,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.51 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.51 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_divide_eq_1_pos
% 0.32/0.51 thf(fact_1505_less__divide__eq__1__neg,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.51 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.51 = ( ord_less_real @ B @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_divide_eq_1_neg
% 0.32/0.51 thf(fact_1506_less__divide__eq__1__neg,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.51 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.51 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % less_divide_eq_1_neg
% 0.32/0.51 thf(fact_1507_divide__less__eq__1__pos,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.51 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.51 = ( ord_less_real @ B @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_less_eq_1_pos
% 0.32/0.51 thf(fact_1508_divide__less__eq__1__pos,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.51 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.51 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_less_eq_1_pos
% 0.32/0.51 thf(fact_1509_divide__less__eq__1__neg,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.51 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.51 = ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_less_eq_1_neg
% 0.32/0.51 thf(fact_1510_divide__less__eq__1__neg,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.51 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.51 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_less_eq_1_neg
% 0.32/0.51 thf(fact_1511_divide__less__0__1__iff,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.32/0.51 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_less_0_1_iff
% 0.32/0.51 thf(fact_1512_divide__less__0__1__iff,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.32/0.51 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_less_0_1_iff
% 0.32/0.51 thf(fact_1513_eq__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.51 ! [A: complex,B: complex,W: num] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.51 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.51 != zero_zero_complex )
% 0.32/0.51 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.51 = B ) )
% 0.32/0.51 & ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.51 = zero_zero_complex )
% 0.32/0.51 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_divide_eq_numeral1(1)
% 0.32/0.51 thf(fact_1514_eq__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.51 ! [A: real,B: real,W: num] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.51 = ( ( ( ( numeral_numeral_real @ W )
% 0.32/0.51 != zero_zero_real )
% 0.32/0.51 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 0.32/0.51 = B ) )
% 0.32/0.51 & ( ( ( numeral_numeral_real @ W )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_divide_eq_numeral1(1)
% 0.32/0.51 thf(fact_1515_eq__divide__eq__numeral1_I1_J,axiom,
% 0.32/0.51 ! [A: rat,B: rat,W: num] :
% 0.32/0.51 ( ( A
% 0.32/0.51 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.51 = ( ( ( ( numeral_numeral_rat @ W )
% 0.32/0.51 != zero_zero_rat )
% 0.32/0.51 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 0.32/0.51 = B ) )
% 0.32/0.51 & ( ( ( numeral_numeral_rat @ W )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % eq_divide_eq_numeral1(1)
% 0.32/0.51 thf(fact_1516_divide__eq__eq__numeral1_I1_J,axiom,
% 0.32/0.51 ! [B: complex,W: num,A: complex] :
% 0.32/0.51 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.51 = A )
% 0.32/0.51 = ( ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.51 != zero_zero_complex )
% 0.32/0.51 => ( B
% 0.32/0.51 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.32/0.51 & ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.51 = zero_zero_complex )
% 0.32/0.51 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_eq_numeral1(1)
% 0.32/0.51 thf(fact_1517_divide__eq__eq__numeral1_I1_J,axiom,
% 0.32/0.51 ! [B: real,W: num,A: real] :
% 0.32/0.51 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 0.32/0.51 = A )
% 0.32/0.51 = ( ( ( ( numeral_numeral_real @ W )
% 0.32/0.51 != zero_zero_real )
% 0.32/0.51 => ( B
% 0.32/0.51 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 0.32/0.51 & ( ( ( numeral_numeral_real @ W )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_eq_numeral1(1)
% 0.32/0.51 thf(fact_1518_divide__eq__eq__numeral1_I1_J,axiom,
% 0.32/0.51 ! [B: rat,W: num,A: rat] :
% 0.32/0.51 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 0.32/0.51 = A )
% 0.32/0.51 = ( ( ( ( numeral_numeral_rat @ W )
% 0.32/0.51 != zero_zero_rat )
% 0.32/0.51 => ( B
% 0.32/0.51 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 0.32/0.51 & ( ( ( numeral_numeral_rat @ W )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_eq_eq_numeral1(1)
% 0.32/0.51 thf(fact_1519_nonzero__divide__mult__cancel__right,axiom,
% 0.32/0.51 ! [B: complex,A: complex] :
% 0.32/0.51 ( ( B != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_divide_mult_cancel_right
% 0.32/0.51 thf(fact_1520_nonzero__divide__mult__cancel__right,axiom,
% 0.32/0.51 ! [B: real,A: real] :
% 0.32/0.51 ( ( B != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 0.32/0.51 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_divide_mult_cancel_right
% 0.32/0.51 thf(fact_1521_nonzero__divide__mult__cancel__right,axiom,
% 0.32/0.51 ! [B: rat,A: rat] :
% 0.32/0.51 ( ( B != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 0.32/0.51 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_divide_mult_cancel_right
% 0.32/0.51 thf(fact_1522_nonzero__divide__mult__cancel__left,axiom,
% 0.32/0.51 ! [A: complex,B: complex] :
% 0.32/0.51 ( ( A != zero_zero_complex )
% 0.32/0.51 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 0.32/0.51 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_divide_mult_cancel_left
% 0.32/0.51 thf(fact_1523_nonzero__divide__mult__cancel__left,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( A != zero_zero_real )
% 0.32/0.51 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 0.32/0.51 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_divide_mult_cancel_left
% 0.32/0.51 thf(fact_1524_nonzero__divide__mult__cancel__left,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( A != zero_zero_rat )
% 0.32/0.51 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 0.32/0.51 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % nonzero_divide_mult_cancel_left
% 0.32/0.51 thf(fact_1525_div__mult__self4,axiom,
% 0.32/0.51 ! [B: nat,C: nat,A: nat] :
% 0.32/0.51 ( ( B != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.32/0.51 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self4
% 0.32/0.51 thf(fact_1526_div__mult__self4,axiom,
% 0.32/0.51 ! [B: int,C: int,A: int] :
% 0.32/0.51 ( ( B != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.32/0.51 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self4
% 0.32/0.51 thf(fact_1527_div__mult__self3,axiom,
% 0.32/0.51 ! [B: nat,C: nat,A: nat] :
% 0.32/0.51 ( ( B != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.32/0.51 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self3
% 0.32/0.51 thf(fact_1528_div__mult__self3,axiom,
% 0.32/0.51 ! [B: int,C: int,A: int] :
% 0.32/0.51 ( ( B != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.32/0.51 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self3
% 0.32/0.51 thf(fact_1529_div__mult__self2,axiom,
% 0.32/0.51 ! [B: nat,A: nat,C: nat] :
% 0.32/0.51 ( ( B != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.32/0.51 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self2
% 0.32/0.51 thf(fact_1530_div__mult__self2,axiom,
% 0.32/0.51 ! [B: int,A: int,C: int] :
% 0.32/0.51 ( ( B != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.32/0.51 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self2
% 0.32/0.51 thf(fact_1531_div__mult__self1,axiom,
% 0.32/0.51 ! [B: nat,A: nat,C: nat] :
% 0.32/0.51 ( ( B != zero_zero_nat )
% 0.32/0.51 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.32/0.51 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self1
% 0.32/0.51 thf(fact_1532_div__mult__self1,axiom,
% 0.32/0.51 ! [B: int,A: int,C: int] :
% 0.32/0.51 ( ( B != zero_zero_int )
% 0.32/0.51 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.32/0.51 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mult_self1
% 0.32/0.51 thf(fact_1533_power__mono__iff,axiom,
% 0.32/0.51 ! [A: real,B: real,N: nat] :
% 0.32/0.51 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.32/0.51 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mono_iff
% 0.32/0.51 thf(fact_1534_power__mono__iff,axiom,
% 0.32/0.51 ! [A: rat,B: rat,N: nat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.51 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mono_iff
% 0.32/0.51 thf(fact_1535_power__mono__iff,axiom,
% 0.32/0.51 ! [A: nat,B: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.51 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mono_iff
% 0.32/0.51 thf(fact_1536_power__mono__iff,axiom,
% 0.32/0.51 ! [A: int,B: int,N: nat] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.51 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.51 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.32/0.51 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_mono_iff
% 0.32/0.51 thf(fact_1537_half__negative__int__iff,axiom,
% 0.32/0.51 ! [K: int] :
% 0.32/0.51 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.32/0.51 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % half_negative_int_iff
% 0.32/0.51 thf(fact_1538_half__nonnegative__int__iff,axiom,
% 0.32/0.51 ! [K: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.51
% 0.32/0.51 % half_nonnegative_int_iff
% 0.32/0.51 thf(fact_1539_i1,axiom,
% 0.32/0.51 ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.51 = none_nat )
% 0.32/0.51 | ~ ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % i1
% 0.32/0.51 thf(fact_1540_divide__le__eq__1__neg,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.51 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.51 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_le_eq_1_neg
% 0.32/0.51 thf(fact_1541_divide__le__eq__1__neg,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.51 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_le_eq_1_neg
% 0.32/0.51 thf(fact_1542_divide__le__eq__1__pos,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.51 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.51 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_le_eq_1_pos
% 0.32/0.51 thf(fact_1543_divide__le__eq__1__pos,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.51 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.51 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % divide_le_eq_1_pos
% 0.32/0.51 thf(fact_1544_le__divide__eq__1__neg,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.51 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.51 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_divide_eq_1_neg
% 0.32/0.51 thf(fact_1545_le__divide__eq__1__neg,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.51 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.51 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_divide_eq_1_neg
% 0.32/0.51 thf(fact_1546_le__divide__eq__1__pos,axiom,
% 0.32/0.51 ! [A: real,B: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.51 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.51 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_divide_eq_1_pos
% 0.32/0.51 thf(fact_1547_le__divide__eq__1__pos,axiom,
% 0.32/0.51 ! [A: rat,B: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.51 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.51 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % le_divide_eq_1_pos
% 0.32/0.51 thf(fact_1548_power__strict__decreasing__iff,axiom,
% 0.32/0.51 ! [B: real,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.51 => ( ( ord_less_real @ B @ one_one_real )
% 0.32/0.51 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.32/0.51 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_strict_decreasing_iff
% 0.32/0.51 thf(fact_1549_power__strict__decreasing__iff,axiom,
% 0.32/0.51 ! [B: rat,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.51 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.32/0.51 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.32/0.51 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_strict_decreasing_iff
% 0.32/0.51 thf(fact_1550_power__strict__decreasing__iff,axiom,
% 0.32/0.51 ! [B: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.51 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.32/0.51 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.51 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_strict_decreasing_iff
% 0.32/0.51 thf(fact_1551_power__strict__decreasing__iff,axiom,
% 0.32/0.51 ! [B: int,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.51 => ( ( ord_less_int @ B @ one_one_int )
% 0.32/0.51 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.32/0.51 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_strict_decreasing_iff
% 0.32/0.51 thf(fact_1552_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( A = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1553_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: nat] :
% 0.32/0.51 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_nat )
% 0.32/0.51 = ( A = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1554_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( A = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1555_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 = ( A = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1556_zero__eq__power2,axiom,
% 0.32/0.51 ! [A: complex] :
% 0.32/0.51 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_complex )
% 0.32/0.51 = ( A = zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_eq_power2
% 0.32/0.51 thf(fact_1557_add__self__mod__2,axiom,
% 0.32/0.51 ! [M: nat] :
% 0.32/0.51 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % add_self_mod_2
% 0.32/0.51 thf(fact_1558_bits__1__div__2,axiom,
% 0.32/0.51 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % bits_1_div_2
% 0.32/0.51 thf(fact_1559_bits__1__div__2,axiom,
% 0.32/0.51 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % bits_1_div_2
% 0.32/0.51 thf(fact_1560_one__div__two__eq__zero,axiom,
% 0.32/0.51 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % one_div_two_eq_zero
% 0.32/0.51 thf(fact_1561_one__div__two__eq__zero,axiom,
% 0.32/0.51 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.51 = zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % one_div_two_eq_zero
% 0.32/0.51 thf(fact_1562_power__decreasing__iff,axiom,
% 0.32/0.51 ! [B: real,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.51 => ( ( ord_less_real @ B @ one_one_real )
% 0.32/0.51 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 0.32/0.51 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_decreasing_iff
% 0.32/0.51 thf(fact_1563_power__decreasing__iff,axiom,
% 0.32/0.51 ! [B: rat,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.51 => ( ( ord_less_rat @ B @ one_one_rat )
% 0.32/0.51 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 0.32/0.51 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_decreasing_iff
% 0.32/0.51 thf(fact_1564_power__decreasing__iff,axiom,
% 0.32/0.51 ! [B: nat,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.51 => ( ( ord_less_nat @ B @ one_one_nat )
% 0.32/0.51 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.51 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_decreasing_iff
% 0.32/0.51 thf(fact_1565_power__decreasing__iff,axiom,
% 0.32/0.51 ! [B: int,M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.51 => ( ( ord_less_int @ B @ one_one_int )
% 0.32/0.51 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 0.32/0.51 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_decreasing_iff
% 0.32/0.51 thf(fact_1566_power2__eq__iff__nonneg,axiom,
% 0.32/0.51 ! [X: real,Y3: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.51 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.51 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( X = Y3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_iff_nonneg
% 0.32/0.51 thf(fact_1567_power2__eq__iff__nonneg,axiom,
% 0.32/0.51 ! [X: rat,Y3: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.51 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.51 => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( X = Y3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_iff_nonneg
% 0.32/0.51 thf(fact_1568_power2__eq__iff__nonneg,axiom,
% 0.32/0.51 ! [X: nat,Y3: nat] :
% 0.32/0.51 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.51 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.32/0.51 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( X = Y3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_iff_nonneg
% 0.32/0.51 thf(fact_1569_power2__eq__iff__nonneg,axiom,
% 0.32/0.51 ! [X: int,Y3: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.51 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.51 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( X = Y3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_eq_iff_nonneg
% 0.32/0.51 thf(fact_1570_power2__less__eq__zero__iff,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.32/0.51 = ( A = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_less_eq_zero_iff
% 0.32/0.51 thf(fact_1571_power2__less__eq__zero__iff,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.32/0.51 = ( A = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_less_eq_zero_iff
% 0.32/0.51 thf(fact_1572_power2__less__eq__zero__iff,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.32/0.51 = ( A = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % power2_less_eq_zero_iff
% 0.32/0.51 thf(fact_1573_zero__less__power2,axiom,
% 0.32/0.51 ! [A: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( A != zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_power2
% 0.32/0.51 thf(fact_1574_zero__less__power2,axiom,
% 0.32/0.51 ! [A: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( A != zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_power2
% 0.32/0.51 thf(fact_1575_zero__less__power2,axiom,
% 0.32/0.51 ! [A: int] :
% 0.32/0.51 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( A != zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_power2
% 0.32/0.51 thf(fact_1576_sum__power2__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: rat,Y3: rat] :
% 0.32/0.51 ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = zero_zero_rat )
% 0.32/0.51 = ( ( X = zero_zero_rat )
% 0.32/0.51 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_power2_eq_zero_iff
% 0.32/0.51 thf(fact_1577_sum__power2__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: real,Y3: real] :
% 0.32/0.51 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = zero_zero_real )
% 0.32/0.51 = ( ( X = zero_zero_real )
% 0.32/0.51 & ( Y3 = zero_zero_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_power2_eq_zero_iff
% 0.32/0.51 thf(fact_1578_sum__power2__eq__zero__iff,axiom,
% 0.32/0.51 ! [X: int,Y3: int] :
% 0.32/0.51 ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 = ( ( X = zero_zero_int )
% 0.32/0.51 & ( Y3 = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % sum_power2_eq_zero_iff
% 0.32/0.51 thf(fact_1579_mod2__gr__0,axiom,
% 0.32/0.51 ! [M: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.51 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.51 = one_one_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % mod2_gr_0
% 0.32/0.51 thf(fact_1580__C4_Ohyps_C_I1_J,axiom,
% 0.32/0.51 ! [X5: vEBT_VEBT] :
% 0.32/0.51 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.32/0.51 => ( ( vEBT_invar_vebt @ X5 @ na )
% 0.32/0.51 & ! [Xa2: nat,Xb2: nat] :
% 0.32/0.51 ( ( ( vEBT_vebt_succ @ X5 @ Xa2 )
% 0.32/0.51 = ( some_nat @ Xb2 ) )
% 0.32/0.51 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ Xa2 @ Xb2 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % "4.hyps"(1)
% 0.32/0.51 thf(fact_1581__C4_Ohyps_C_I7_J,axiom,
% 0.32/0.51 ! [I3: nat] :
% 0.32/0.51 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.32/0.51 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ X6 ) )
% 0.32/0.51 = ( vEBT_V8194947554948674370ptions @ summary @ I3 ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % "4.hyps"(7)
% 0.32/0.51 thf(fact_1582_zero__reorient,axiom,
% 0.32/0.51 ! [X: complex] :
% 0.32/0.51 ( ( zero_zero_complex = X )
% 0.32/0.51 = ( X = zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_reorient
% 0.32/0.51 thf(fact_1583_zero__reorient,axiom,
% 0.32/0.51 ! [X: real] :
% 0.32/0.51 ( ( zero_zero_real = X )
% 0.32/0.51 = ( X = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_reorient
% 0.32/0.51 thf(fact_1584_zero__reorient,axiom,
% 0.32/0.51 ! [X: rat] :
% 0.32/0.51 ( ( zero_zero_rat = X )
% 0.32/0.51 = ( X = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_reorient
% 0.32/0.51 thf(fact_1585_zero__reorient,axiom,
% 0.32/0.51 ! [X: nat] :
% 0.32/0.51 ( ( zero_zero_nat = X )
% 0.32/0.51 = ( X = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_reorient
% 0.32/0.51 thf(fact_1586_zero__reorient,axiom,
% 0.32/0.51 ! [X: int] :
% 0.32/0.51 ( ( zero_zero_int = X )
% 0.32/0.51 = ( X = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_reorient
% 0.32/0.51 thf(fact_1587_zdiv__mono__strict,axiom,
% 0.32/0.51 ! [A2: int,B5: int,N: int] :
% 0.32/0.51 ( ( ord_less_int @ A2 @ B5 )
% 0.32/0.51 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.32/0.51 => ( ( ( modulo_modulo_int @ A2 @ N )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 => ( ( ( modulo_modulo_int @ B5 @ N )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % zdiv_mono_strict
% 0.32/0.51 thf(fact_1588_verit__le__mono__div__int,axiom,
% 0.32/0.51 ! [A2: int,B5: int,N: int] :
% 0.32/0.51 ( ( ord_less_int @ A2 @ B5 )
% 0.32/0.51 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.32/0.51 => ( ord_less_eq_int
% 0.32/0.51 @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 0.32/0.51 @ ( if_int
% 0.32/0.51 @ ( ( modulo_modulo_int @ B5 @ N )
% 0.32/0.51 = zero_zero_int )
% 0.32/0.51 @ one_one_int
% 0.32/0.51 @ zero_zero_int ) )
% 0.32/0.51 @ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % verit_le_mono_div_int
% 0.32/0.51 thf(fact_1589_zmod__zmult2__eq,axiom,
% 0.32/0.51 ! [C: int,A: int,B: int] :
% 0.32/0.51 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.51 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % zmod_zmult2_eq
% 0.32/0.51 thf(fact_1590_split__pos__lemma,axiom,
% 0.32/0.51 ! [K: int,P: int > int > $o,N: int] :
% 0.32/0.51 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.51 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.32/0.51 = ( ! [I4: int,J3: int] :
% 0.32/0.51 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.32/0.51 & ( ord_less_int @ J3 @ K )
% 0.32/0.51 & ( N
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.32/0.51 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % split_pos_lemma
% 0.32/0.51 thf(fact_1591_split__neg__lemma,axiom,
% 0.32/0.51 ! [K: int,P: int > int > $o,N: int] :
% 0.32/0.51 ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.51 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.32/0.51 = ( ! [I4: int,J3: int] :
% 0.32/0.51 ( ( ( ord_less_int @ K @ J3 )
% 0.32/0.51 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.32/0.51 & ( N
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.32/0.51 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % split_neg_lemma
% 0.32/0.51 thf(fact_1592_bounded__nat__set__is__finite,axiom,
% 0.32/0.51 ! [N3: set_nat,N: nat] :
% 0.32/0.51 ( ! [X3: nat] :
% 0.32/0.51 ( ( member_nat @ X3 @ N3 )
% 0.32/0.51 => ( ord_less_nat @ X3 @ N ) )
% 0.32/0.51 => ( finite_finite_nat @ N3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % bounded_nat_set_is_finite
% 0.32/0.51 thf(fact_1593_finite__nat__set__iff__bounded,axiom,
% 0.32/0.51 ( finite_finite_nat
% 0.32/0.51 = ( ^ [N5: set_nat] :
% 0.32/0.51 ? [M3: nat] :
% 0.32/0.51 ! [X2: nat] :
% 0.32/0.51 ( ( member_nat @ X2 @ N5 )
% 0.32/0.51 => ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % finite_nat_set_iff_bounded
% 0.32/0.51 thf(fact_1594_finite__nat__set__iff__bounded__le,axiom,
% 0.32/0.51 ( finite_finite_nat
% 0.32/0.51 = ( ^ [N5: set_nat] :
% 0.32/0.51 ? [M3: nat] :
% 0.32/0.51 ! [X2: nat] :
% 0.32/0.51 ( ( member_nat @ X2 @ N5 )
% 0.32/0.51 => ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % finite_nat_set_iff_bounded_le
% 0.32/0.51 thf(fact_1595_size__neq__size__imp__neq,axiom,
% 0.32/0.51 ! [X: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 0.32/0.51 ( ( ( size_s6755466524823107622T_VEBT @ X )
% 0.32/0.51 != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 0.32/0.51 => ( X != Y3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % size_neq_size_imp_neq
% 0.32/0.51 thf(fact_1596_size__neq__size__imp__neq,axiom,
% 0.32/0.51 ! [X: list_o,Y3: list_o] :
% 0.32/0.51 ( ( ( size_size_list_o @ X )
% 0.32/0.51 != ( size_size_list_o @ Y3 ) )
% 0.32/0.51 => ( X != Y3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % size_neq_size_imp_neq
% 0.32/0.51 thf(fact_1597_size__neq__size__imp__neq,axiom,
% 0.32/0.51 ! [X: list_nat,Y3: list_nat] :
% 0.32/0.51 ( ( ( size_size_list_nat @ X )
% 0.32/0.51 != ( size_size_list_nat @ Y3 ) )
% 0.32/0.51 => ( X != Y3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % size_neq_size_imp_neq
% 0.32/0.51 thf(fact_1598_size__neq__size__imp__neq,axiom,
% 0.32/0.51 ! [X: list_int,Y3: list_int] :
% 0.32/0.51 ( ( ( size_size_list_int @ X )
% 0.32/0.51 != ( size_size_list_int @ Y3 ) )
% 0.32/0.51 => ( X != Y3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % size_neq_size_imp_neq
% 0.32/0.51 thf(fact_1599_size__neq__size__imp__neq,axiom,
% 0.32/0.51 ! [X: num,Y3: num] :
% 0.32/0.51 ( ( ( size_size_num @ X )
% 0.32/0.51 != ( size_size_num @ Y3 ) )
% 0.32/0.51 => ( X != Y3 ) ) ).
% 0.32/0.51
% 0.32/0.51 % size_neq_size_imp_neq
% 0.32/0.51 thf(fact_1600_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.32/0.51 = one_one_rat ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.32/0.51 = zero_zero_rat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1601_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.32/0.51 = one_one_nat ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.32/0.51 = zero_zero_nat ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1602_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_real @ zero_zero_real @ N )
% 0.32/0.51 = one_one_real ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_real @ zero_zero_real @ N )
% 0.32/0.51 = zero_zero_real ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1603_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_int @ zero_zero_int @ N )
% 0.32/0.51 = one_one_int ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_int @ zero_zero_int @ N )
% 0.32/0.51 = zero_zero_int ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1604_power__0__left,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ( N = zero_zero_nat )
% 0.32/0.51 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.32/0.51 = one_one_complex ) )
% 0.32/0.51 & ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.32/0.51 = zero_zero_complex ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % power_0_left
% 0.32/0.51 thf(fact_1605_zero__power,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.32/0.51 = zero_zero_rat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power
% 0.32/0.51 thf(fact_1606_zero__power,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.32/0.51 = zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power
% 0.32/0.51 thf(fact_1607_zero__power,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( power_power_real @ zero_zero_real @ N )
% 0.32/0.51 = zero_zero_real ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power
% 0.32/0.51 thf(fact_1608_zero__power,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( power_power_int @ zero_zero_int @ N )
% 0.32/0.51 = zero_zero_int ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power
% 0.32/0.51 thf(fact_1609_zero__power,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.32/0.51 = zero_zero_complex ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_power
% 0.32/0.51 thf(fact_1610_div__mod__decomp__int,axiom,
% 0.32/0.51 ! [A2: int,N: int] :
% 0.32/0.51 ( A2
% 0.32/0.51 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % div_mod_decomp_int
% 0.32/0.51 thf(fact_1611_zero__le,axiom,
% 0.32/0.51 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 0.32/0.51
% 0.32/0.51 % zero_le
% 0.32/0.51 thf(fact_1612_le__numeral__extra_I3_J,axiom,
% 0.32/0.51 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.32/0.51
% 0.32/0.51 % le_numeral_extra(3)
% 0.32/0.51 thf(fact_1613_le__numeral__extra_I3_J,axiom,
% 0.32/0.51 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.32/0.51
% 0.32/0.51 % le_numeral_extra(3)
% 0.32/0.51 thf(fact_1614_le__numeral__extra_I3_J,axiom,
% 0.32/0.51 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.32/0.51
% 0.32/0.51 % le_numeral_extra(3)
% 0.32/0.51 thf(fact_1615_le__numeral__extra_I3_J,axiom,
% 0.32/0.51 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.32/0.51
% 0.32/0.51 % le_numeral_extra(3)
% 0.32/0.51 thf(fact_1616_gr__zeroI,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( N != zero_zero_nat )
% 0.32/0.51 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % gr_zeroI
% 0.32/0.51 thf(fact_1617_not__less__zero,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % not_less_zero
% 0.32/0.51 thf(fact_1618_gr__implies__not__zero,axiom,
% 0.32/0.51 ! [M: nat,N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ M @ N )
% 0.32/0.51 => ( N != zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % gr_implies_not_zero
% 0.32/0.51 thf(fact_1619_zero__less__iff__neq__zero,axiom,
% 0.32/0.51 ! [N: nat] :
% 0.32/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.51 = ( N != zero_zero_nat ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_less_iff_neq_zero
% 0.32/0.51 thf(fact_1620_field__lbound__gt__zero,axiom,
% 0.32/0.51 ! [D1: real,D22: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.32/0.51 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 0.32/0.51 => ? [E2: real] :
% 0.32/0.51 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.32/0.51 & ( ord_less_real @ E2 @ D1 )
% 0.32/0.51 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % field_lbound_gt_zero
% 0.32/0.51 thf(fact_1621_field__lbound__gt__zero,axiom,
% 0.32/0.51 ! [D1: rat,D22: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.32/0.51 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 0.32/0.51 => ? [E2: rat] :
% 0.32/0.51 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.32/0.51 & ( ord_less_rat @ E2 @ D1 )
% 0.32/0.51 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 0.32/0.51
% 0.32/0.51 % field_lbound_gt_zero
% 0.32/0.51 thf(fact_1622_less__numeral__extra_I3_J,axiom,
% 0.32/0.51 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.32/0.51
% 0.32/0.51 % less_numeral_extra(3)
% 0.32/0.51 thf(fact_1623_less__numeral__extra_I3_J,axiom,
% 0.32/0.51 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.32/0.51
% 0.32/0.51 % less_numeral_extra(3)
% 0.32/0.51 thf(fact_1624_less__numeral__extra_I3_J,axiom,
% 0.32/0.51 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.32/0.51
% 0.32/0.51 % less_numeral_extra(3)
% 0.32/0.51 thf(fact_1625_less__numeral__extra_I3_J,axiom,
% 0.32/0.51 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.32/0.51
% 0.32/0.51 % less_numeral_extra(3)
% 0.32/0.51 thf(fact_1626_zero__neq__numeral,axiom,
% 0.32/0.51 ! [N: num] :
% 0.32/0.51 ( zero_zero_complex
% 0.32/0.51 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.32/0.51
% 0.32/0.51 % zero_neq_numeral
% 0.32/0.51 thf(fact_1627_zero__neq__numeral,axiom,
% 0.32/0.51 ! [N: num] :
% 0.32/0.52 ( zero_zero_real
% 0.32/0.52 != ( numeral_numeral_real @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_numeral
% 0.32/0.52 thf(fact_1628_zero__neq__numeral,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( zero_zero_rat
% 0.32/0.52 != ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_numeral
% 0.32/0.52 thf(fact_1629_zero__neq__numeral,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( zero_zero_nat
% 0.32/0.52 != ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_numeral
% 0.32/0.52 thf(fact_1630_zero__neq__numeral,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ( zero_zero_int
% 0.32/0.52 != ( numeral_numeral_int @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_numeral
% 0.32/0.52 thf(fact_1631_mult__not__zero,axiom,
% 0.32/0.52 ! [A: complex,B: complex] :
% 0.32/0.52 ( ( ( times_times_complex @ A @ B )
% 0.32/0.52 != zero_zero_complex )
% 0.32/0.52 => ( ( A != zero_zero_complex )
% 0.32/0.52 & ( B != zero_zero_complex ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_not_zero
% 0.32/0.52 thf(fact_1632_mult__not__zero,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ( times_times_real @ A @ B )
% 0.32/0.52 != zero_zero_real )
% 0.32/0.52 => ( ( A != zero_zero_real )
% 0.32/0.52 & ( B != zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_not_zero
% 0.32/0.52 thf(fact_1633_mult__not__zero,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ( times_times_rat @ A @ B )
% 0.32/0.52 != zero_zero_rat )
% 0.32/0.52 => ( ( A != zero_zero_rat )
% 0.32/0.52 & ( B != zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_not_zero
% 0.32/0.52 thf(fact_1634_mult__not__zero,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ( times_times_nat @ A @ B )
% 0.32/0.52 != zero_zero_nat )
% 0.32/0.52 => ( ( A != zero_zero_nat )
% 0.32/0.52 & ( B != zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_not_zero
% 0.32/0.52 thf(fact_1635_mult__not__zero,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ( times_times_int @ A @ B )
% 0.32/0.52 != zero_zero_int )
% 0.32/0.52 => ( ( A != zero_zero_int )
% 0.32/0.52 & ( B != zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_not_zero
% 0.32/0.52 thf(fact_1636_divisors__zero,axiom,
% 0.32/0.52 ! [A: complex,B: complex] :
% 0.32/0.52 ( ( ( times_times_complex @ A @ B )
% 0.32/0.52 = zero_zero_complex )
% 0.32/0.52 => ( ( A = zero_zero_complex )
% 0.32/0.52 | ( B = zero_zero_complex ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divisors_zero
% 0.32/0.52 thf(fact_1637_divisors__zero,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ( times_times_real @ A @ B )
% 0.32/0.52 = zero_zero_real )
% 0.32/0.52 => ( ( A = zero_zero_real )
% 0.32/0.52 | ( B = zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divisors_zero
% 0.32/0.52 thf(fact_1638_divisors__zero,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ( times_times_rat @ A @ B )
% 0.32/0.52 = zero_zero_rat )
% 0.32/0.52 => ( ( A = zero_zero_rat )
% 0.32/0.52 | ( B = zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divisors_zero
% 0.32/0.52 thf(fact_1639_divisors__zero,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ( times_times_nat @ A @ B )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 => ( ( A = zero_zero_nat )
% 0.32/0.52 | ( B = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divisors_zero
% 0.32/0.52 thf(fact_1640_divisors__zero,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ( times_times_int @ A @ B )
% 0.32/0.52 = zero_zero_int )
% 0.32/0.52 => ( ( A = zero_zero_int )
% 0.32/0.52 | ( B = zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divisors_zero
% 0.32/0.52 thf(fact_1641_no__zero__divisors,axiom,
% 0.32/0.52 ! [A: complex,B: complex] :
% 0.32/0.52 ( ( A != zero_zero_complex )
% 0.32/0.52 => ( ( B != zero_zero_complex )
% 0.32/0.52 => ( ( times_times_complex @ A @ B )
% 0.32/0.52 != zero_zero_complex ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % no_zero_divisors
% 0.32/0.52 thf(fact_1642_no__zero__divisors,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( A != zero_zero_real )
% 0.32/0.52 => ( ( B != zero_zero_real )
% 0.32/0.52 => ( ( times_times_real @ A @ B )
% 0.32/0.52 != zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % no_zero_divisors
% 0.32/0.52 thf(fact_1643_no__zero__divisors,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( A != zero_zero_rat )
% 0.32/0.52 => ( ( B != zero_zero_rat )
% 0.32/0.52 => ( ( times_times_rat @ A @ B )
% 0.32/0.52 != zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % no_zero_divisors
% 0.32/0.52 thf(fact_1644_no__zero__divisors,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( A != zero_zero_nat )
% 0.32/0.52 => ( ( B != zero_zero_nat )
% 0.32/0.52 => ( ( times_times_nat @ A @ B )
% 0.32/0.52 != zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % no_zero_divisors
% 0.32/0.52 thf(fact_1645_no__zero__divisors,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( A != zero_zero_int )
% 0.32/0.52 => ( ( B != zero_zero_int )
% 0.32/0.52 => ( ( times_times_int @ A @ B )
% 0.32/0.52 != zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % no_zero_divisors
% 0.32/0.52 thf(fact_1646_mult__left__cancel,axiom,
% 0.32/0.52 ! [C: complex,A: complex,B: complex] :
% 0.32/0.52 ( ( C != zero_zero_complex )
% 0.32/0.52 => ( ( ( times_times_complex @ C @ A )
% 0.32/0.52 = ( times_times_complex @ C @ B ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_cancel
% 0.32/0.52 thf(fact_1647_mult__left__cancel,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( C != zero_zero_real )
% 0.32/0.52 => ( ( ( times_times_real @ C @ A )
% 0.32/0.52 = ( times_times_real @ C @ B ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_cancel
% 0.32/0.52 thf(fact_1648_mult__left__cancel,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( C != zero_zero_rat )
% 0.32/0.52 => ( ( ( times_times_rat @ C @ A )
% 0.32/0.52 = ( times_times_rat @ C @ B ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_cancel
% 0.32/0.52 thf(fact_1649_mult__left__cancel,axiom,
% 0.32/0.52 ! [C: nat,A: nat,B: nat] :
% 0.32/0.52 ( ( C != zero_zero_nat )
% 0.32/0.52 => ( ( ( times_times_nat @ C @ A )
% 0.32/0.52 = ( times_times_nat @ C @ B ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_cancel
% 0.32/0.52 thf(fact_1650_mult__left__cancel,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( C != zero_zero_int )
% 0.32/0.52 => ( ( ( times_times_int @ C @ A )
% 0.32/0.52 = ( times_times_int @ C @ B ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_cancel
% 0.32/0.52 thf(fact_1651_mult__right__cancel,axiom,
% 0.32/0.52 ! [C: complex,A: complex,B: complex] :
% 0.32/0.52 ( ( C != zero_zero_complex )
% 0.32/0.52 => ( ( ( times_times_complex @ A @ C )
% 0.32/0.52 = ( times_times_complex @ B @ C ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_cancel
% 0.32/0.52 thf(fact_1652_mult__right__cancel,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( C != zero_zero_real )
% 0.32/0.52 => ( ( ( times_times_real @ A @ C )
% 0.32/0.52 = ( times_times_real @ B @ C ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_cancel
% 0.32/0.52 thf(fact_1653_mult__right__cancel,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( C != zero_zero_rat )
% 0.32/0.52 => ( ( ( times_times_rat @ A @ C )
% 0.32/0.52 = ( times_times_rat @ B @ C ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_cancel
% 0.32/0.52 thf(fact_1654_mult__right__cancel,axiom,
% 0.32/0.52 ! [C: nat,A: nat,B: nat] :
% 0.32/0.52 ( ( C != zero_zero_nat )
% 0.32/0.52 => ( ( ( times_times_nat @ A @ C )
% 0.32/0.52 = ( times_times_nat @ B @ C ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_cancel
% 0.32/0.52 thf(fact_1655_mult__right__cancel,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( C != zero_zero_int )
% 0.32/0.52 => ( ( ( times_times_int @ A @ C )
% 0.32/0.52 = ( times_times_int @ B @ C ) )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_cancel
% 0.32/0.52 thf(fact_1656_zero__neq__one,axiom,
% 0.32/0.52 zero_zero_complex != one_one_complex ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_one
% 0.32/0.52 thf(fact_1657_zero__neq__one,axiom,
% 0.32/0.52 zero_zero_real != one_one_real ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_one
% 0.32/0.52 thf(fact_1658_zero__neq__one,axiom,
% 0.32/0.52 zero_zero_rat != one_one_rat ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_one
% 0.32/0.52 thf(fact_1659_zero__neq__one,axiom,
% 0.32/0.52 zero_zero_nat != one_one_nat ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_one
% 0.32/0.52 thf(fact_1660_zero__neq__one,axiom,
% 0.32/0.52 zero_zero_int != one_one_int ).
% 0.32/0.52
% 0.32/0.52 % zero_neq_one
% 0.32/0.52 thf(fact_1661_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.52 ! [A: complex] :
% 0.32/0.52 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % comm_monoid_add_class.add_0
% 0.32/0.52 thf(fact_1662_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % comm_monoid_add_class.add_0
% 0.32/0.52 thf(fact_1663_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % comm_monoid_add_class.add_0
% 0.32/0.52 thf(fact_1664_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % comm_monoid_add_class.add_0
% 0.32/0.52 thf(fact_1665_comm__monoid__add__class_Oadd__0,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % comm_monoid_add_class.add_0
% 0.32/0.52 thf(fact_1666_add_Ocomm__neutral,axiom,
% 0.32/0.52 ! [A: complex] :
% 0.32/0.52 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.comm_neutral
% 0.32/0.52 thf(fact_1667_add_Ocomm__neutral,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.comm_neutral
% 0.32/0.52 thf(fact_1668_add_Ocomm__neutral,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.comm_neutral
% 0.32/0.52 thf(fact_1669_add_Ocomm__neutral,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.comm_neutral
% 0.32/0.52 thf(fact_1670_add_Ocomm__neutral,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.comm_neutral
% 0.32/0.52 thf(fact_1671_add_Ogroup__left__neutral,axiom,
% 0.32/0.52 ! [A: complex] :
% 0.32/0.52 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.group_left_neutral
% 0.32/0.52 thf(fact_1672_add_Ogroup__left__neutral,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( plus_plus_real @ zero_zero_real @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.group_left_neutral
% 0.32/0.52 thf(fact_1673_add_Ogroup__left__neutral,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.group_left_neutral
% 0.32/0.52 thf(fact_1674_add_Ogroup__left__neutral,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( plus_plus_int @ zero_zero_int @ A )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % add.group_left_neutral
% 0.32/0.52 thf(fact_1675_verit__sum__simplify,axiom,
% 0.32/0.52 ! [A: complex] :
% 0.32/0.52 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_sum_simplify
% 0.32/0.52 thf(fact_1676_verit__sum__simplify,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( plus_plus_real @ A @ zero_zero_real )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_sum_simplify
% 0.32/0.52 thf(fact_1677_verit__sum__simplify,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_sum_simplify
% 0.32/0.52 thf(fact_1678_verit__sum__simplify,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_sum_simplify
% 0.32/0.52 thf(fact_1679_verit__sum__simplify,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( plus_plus_int @ A @ zero_zero_int )
% 0.32/0.52 = A ) ).
% 0.32/0.52
% 0.32/0.52 % verit_sum_simplify
% 0.32/0.52 thf(fact_1680_eq__iff__diff__eq__0,axiom,
% 0.32/0.52 ( ( ^ [Y6: complex,Z4: complex] : ( Y6 = Z4 ) )
% 0.32/0.52 = ( ^ [A3: complex,B4: complex] :
% 0.32/0.52 ( ( minus_minus_complex @ A3 @ B4 )
% 0.32/0.52 = zero_zero_complex ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_iff_diff_eq_0
% 0.32/0.52 thf(fact_1681_eq__iff__diff__eq__0,axiom,
% 0.32/0.52 ( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
% 0.32/0.52 = ( ^ [A3: real,B4: real] :
% 0.32/0.52 ( ( minus_minus_real @ A3 @ B4 )
% 0.32/0.52 = zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_iff_diff_eq_0
% 0.32/0.52 thf(fact_1682_eq__iff__diff__eq__0,axiom,
% 0.32/0.52 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 0.32/0.52 = ( ^ [A3: rat,B4: rat] :
% 0.32/0.52 ( ( minus_minus_rat @ A3 @ B4 )
% 0.32/0.52 = zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_iff_diff_eq_0
% 0.32/0.52 thf(fact_1683_eq__iff__diff__eq__0,axiom,
% 0.32/0.52 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.52 = ( ^ [A3: int,B4: int] :
% 0.32/0.52 ( ( minus_minus_int @ A3 @ B4 )
% 0.32/0.52 = zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_iff_diff_eq_0
% 0.32/0.52 thf(fact_1684_power__not__zero,axiom,
% 0.32/0.52 ! [A: rat,N: nat] :
% 0.32/0.52 ( ( A != zero_zero_rat )
% 0.32/0.52 => ( ( power_power_rat @ A @ N )
% 0.32/0.52 != zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_not_zero
% 0.32/0.52 thf(fact_1685_power__not__zero,axiom,
% 0.32/0.52 ! [A: nat,N: nat] :
% 0.32/0.52 ( ( A != zero_zero_nat )
% 0.32/0.52 => ( ( power_power_nat @ A @ N )
% 0.32/0.52 != zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_not_zero
% 0.32/0.52 thf(fact_1686_power__not__zero,axiom,
% 0.32/0.52 ! [A: real,N: nat] :
% 0.32/0.52 ( ( A != zero_zero_real )
% 0.32/0.52 => ( ( power_power_real @ A @ N )
% 0.32/0.52 != zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_not_zero
% 0.32/0.52 thf(fact_1687_power__not__zero,axiom,
% 0.32/0.52 ! [A: int,N: nat] :
% 0.32/0.52 ( ( A != zero_zero_int )
% 0.32/0.52 => ( ( power_power_int @ A @ N )
% 0.32/0.52 != zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_not_zero
% 0.32/0.52 thf(fact_1688_power__not__zero,axiom,
% 0.32/0.52 ! [A: complex,N: nat] :
% 0.32/0.52 ( ( A != zero_zero_complex )
% 0.32/0.52 => ( ( power_power_complex @ A @ N )
% 0.32/0.52 != zero_zero_complex ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_not_zero
% 0.32/0.52 thf(fact_1689_bot__nat__0_Oextremum__strict,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % bot_nat_0.extremum_strict
% 0.32/0.52 thf(fact_1690_gr0I,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( N != zero_zero_nat )
% 0.32/0.52 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % gr0I
% 0.32/0.52 thf(fact_1691_not__gr0,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.32/0.52 = ( N = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % not_gr0
% 0.32/0.52 thf(fact_1692_not__less0,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % not_less0
% 0.32/0.52 thf(fact_1693_less__zeroE,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % less_zeroE
% 0.32/0.52 thf(fact_1694_gr__implies__not0,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ M @ N )
% 0.32/0.52 => ( N != zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % gr_implies_not0
% 0.32/0.52 thf(fact_1695_infinite__descent0,axiom,
% 0.32/0.52 ! [P: nat > $o,N: nat] :
% 0.32/0.52 ( ( P @ zero_zero_nat )
% 0.32/0.52 => ( ! [N4: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.52 => ( ~ ( P @ N4 )
% 0.32/0.52 => ? [M2: nat] :
% 0.32/0.52 ( ( ord_less_nat @ M2 @ N4 )
% 0.32/0.52 & ~ ( P @ M2 ) ) ) )
% 0.32/0.52 => ( P @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % infinite_descent0
% 0.32/0.52 thf(fact_1696_less__eq__nat_Osimps_I1_J,axiom,
% 0.32/0.52 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.32/0.52
% 0.32/0.52 % less_eq_nat.simps(1)
% 0.32/0.52 thf(fact_1697_bot__nat__0_Oextremum__unique,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.52 = ( A = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % bot_nat_0.extremum_unique
% 0.32/0.52 thf(fact_1698_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.52 => ( A = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % bot_nat_0.extremum_uniqueI
% 0.32/0.52 thf(fact_1699_le__0__eq,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.32/0.52 = ( N = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_0_eq
% 0.32/0.52 thf(fact_1700_plus__nat_Oadd__0,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.32/0.52 = N ) ).
% 0.32/0.52
% 0.32/0.52 % plus_nat.add_0
% 0.32/0.52 thf(fact_1701_add__eq__self__zero,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ( plus_plus_nat @ M @ N )
% 0.32/0.52 = M )
% 0.32/0.52 => ( N = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_eq_self_zero
% 0.32/0.52 thf(fact_1702_ex__min__if__finite,axiom,
% 0.32/0.52 ! [S2: set_real] :
% 0.32/0.52 ( ( finite_finite_real @ S2 )
% 0.32/0.52 => ( ( S2 != bot_bot_set_real )
% 0.32/0.52 => ? [X3: real] :
% 0.32/0.52 ( ( member_real @ X3 @ S2 )
% 0.32/0.52 & ~ ? [Xa2: real] :
% 0.32/0.52 ( ( member_real @ Xa2 @ S2 )
% 0.32/0.52 & ( ord_less_real @ Xa2 @ X3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ex_min_if_finite
% 0.32/0.52 thf(fact_1703_ex__min__if__finite,axiom,
% 0.32/0.52 ! [S2: set_rat] :
% 0.32/0.52 ( ( finite_finite_rat @ S2 )
% 0.32/0.52 => ( ( S2 != bot_bot_set_rat )
% 0.32/0.52 => ? [X3: rat] :
% 0.32/0.52 ( ( member_rat @ X3 @ S2 )
% 0.32/0.52 & ~ ? [Xa2: rat] :
% 0.32/0.52 ( ( member_rat @ Xa2 @ S2 )
% 0.32/0.52 & ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ex_min_if_finite
% 0.32/0.52 thf(fact_1704_ex__min__if__finite,axiom,
% 0.32/0.52 ! [S2: set_num] :
% 0.32/0.52 ( ( finite_finite_num @ S2 )
% 0.32/0.52 => ( ( S2 != bot_bot_set_num )
% 0.32/0.52 => ? [X3: num] :
% 0.32/0.52 ( ( member_num @ X3 @ S2 )
% 0.32/0.52 & ~ ? [Xa2: num] :
% 0.32/0.52 ( ( member_num @ Xa2 @ S2 )
% 0.32/0.52 & ( ord_less_num @ Xa2 @ X3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ex_min_if_finite
% 0.32/0.52 thf(fact_1705_ex__min__if__finite,axiom,
% 0.32/0.52 ! [S2: set_nat] :
% 0.32/0.52 ( ( finite_finite_nat @ S2 )
% 0.32/0.52 => ( ( S2 != bot_bot_set_nat )
% 0.32/0.52 => ? [X3: nat] :
% 0.32/0.52 ( ( member_nat @ X3 @ S2 )
% 0.32/0.52 & ~ ? [Xa2: nat] :
% 0.32/0.52 ( ( member_nat @ Xa2 @ S2 )
% 0.32/0.52 & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ex_min_if_finite
% 0.32/0.52 thf(fact_1706_ex__min__if__finite,axiom,
% 0.32/0.52 ! [S2: set_int] :
% 0.32/0.52 ( ( finite_finite_int @ S2 )
% 0.32/0.52 => ( ( S2 != bot_bot_set_int )
% 0.32/0.52 => ? [X3: int] :
% 0.32/0.52 ( ( member_int @ X3 @ S2 )
% 0.32/0.52 & ~ ? [Xa2: int] :
% 0.32/0.52 ( ( member_int @ Xa2 @ S2 )
% 0.32/0.52 & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ex_min_if_finite
% 0.32/0.52 thf(fact_1707_infinite__growing,axiom,
% 0.32/0.52 ! [X7: set_real] :
% 0.32/0.52 ( ( X7 != bot_bot_set_real )
% 0.32/0.52 => ( ! [X3: real] :
% 0.32/0.52 ( ( member_real @ X3 @ X7 )
% 0.32/0.52 => ? [Xa2: real] :
% 0.32/0.52 ( ( member_real @ Xa2 @ X7 )
% 0.32/0.52 & ( ord_less_real @ X3 @ Xa2 ) ) )
% 0.32/0.52 => ~ ( finite_finite_real @ X7 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % infinite_growing
% 0.32/0.52 thf(fact_1708_infinite__growing,axiom,
% 0.32/0.52 ! [X7: set_rat] :
% 0.32/0.52 ( ( X7 != bot_bot_set_rat )
% 0.32/0.52 => ( ! [X3: rat] :
% 0.32/0.52 ( ( member_rat @ X3 @ X7 )
% 0.32/0.52 => ? [Xa2: rat] :
% 0.32/0.52 ( ( member_rat @ Xa2 @ X7 )
% 0.32/0.52 & ( ord_less_rat @ X3 @ Xa2 ) ) )
% 0.32/0.52 => ~ ( finite_finite_rat @ X7 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % infinite_growing
% 0.32/0.52 thf(fact_1709_infinite__growing,axiom,
% 0.32/0.52 ! [X7: set_num] :
% 0.32/0.52 ( ( X7 != bot_bot_set_num )
% 0.32/0.52 => ( ! [X3: num] :
% 0.32/0.52 ( ( member_num @ X3 @ X7 )
% 0.32/0.52 => ? [Xa2: num] :
% 0.32/0.52 ( ( member_num @ Xa2 @ X7 )
% 0.32/0.52 & ( ord_less_num @ X3 @ Xa2 ) ) )
% 0.32/0.52 => ~ ( finite_finite_num @ X7 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % infinite_growing
% 0.32/0.52 thf(fact_1710_infinite__growing,axiom,
% 0.32/0.52 ! [X7: set_nat] :
% 0.32/0.52 ( ( X7 != bot_bot_set_nat )
% 0.32/0.52 => ( ! [X3: nat] :
% 0.32/0.52 ( ( member_nat @ X3 @ X7 )
% 0.32/0.52 => ? [Xa2: nat] :
% 0.32/0.52 ( ( member_nat @ Xa2 @ X7 )
% 0.32/0.52 & ( ord_less_nat @ X3 @ Xa2 ) ) )
% 0.32/0.52 => ~ ( finite_finite_nat @ X7 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % infinite_growing
% 0.32/0.52 thf(fact_1711_infinite__growing,axiom,
% 0.32/0.52 ! [X7: set_int] :
% 0.32/0.52 ( ( X7 != bot_bot_set_int )
% 0.32/0.52 => ( ! [X3: int] :
% 0.32/0.52 ( ( member_int @ X3 @ X7 )
% 0.32/0.52 => ? [Xa2: int] :
% 0.32/0.52 ( ( member_int @ Xa2 @ X7 )
% 0.32/0.52 & ( ord_less_int @ X3 @ Xa2 ) ) )
% 0.32/0.52 => ~ ( finite_finite_int @ X7 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % infinite_growing
% 0.32/0.52 thf(fact_1712_minus__nat_Odiff__0,axiom,
% 0.32/0.52 ! [M: nat] :
% 0.32/0.52 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.32/0.52 = M ) ).
% 0.32/0.52
% 0.32/0.52 % minus_nat.diff_0
% 0.32/0.52 thf(fact_1713_diffs0__imp__equal,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ( minus_minus_nat @ M @ N )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 => ( ( ( minus_minus_nat @ N @ M )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 => ( M = N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % diffs0_imp_equal
% 0.32/0.52 thf(fact_1714_mult__0,axiom,
% 0.32/0.52 ! [N: nat] :
% 0.32/0.52 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.32/0.52 = zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % mult_0
% 0.32/0.52 thf(fact_1715_nat__mult__eq__cancel__disj,axiom,
% 0.32/0.52 ! [K: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ( times_times_nat @ K @ M )
% 0.32/0.52 = ( times_times_nat @ K @ N ) )
% 0.32/0.52 = ( ( K = zero_zero_nat )
% 0.32/0.52 | ( M = N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_mult_eq_cancel_disj
% 0.32/0.52 thf(fact_1716_div__pos__geq,axiom,
% 0.32/0.52 ! [L2: int,K: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.52 => ( ( ord_less_eq_int @ L2 @ K )
% 0.32/0.52 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.52 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_pos_geq
% 0.32/0.52 thf(fact_1717_int__div__less__self,axiom,
% 0.32/0.52 ! [X: int,K: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ X )
% 0.32/0.52 => ( ( ord_less_int @ one_one_int @ K )
% 0.32/0.52 => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % int_div_less_self
% 0.32/0.52 thf(fact_1718_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.52 = ( ( ord_less_eq_int @ B @ A )
% 0.32/0.52 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nonneg1_imp_zdiv_pos_iff
% 0.32/0.52 thf(fact_1719_pos__imp__zdiv__nonneg__iff,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pos_imp_zdiv_nonneg_iff
% 0.32/0.52 thf(fact_1720_neg__imp__zdiv__nonneg__iff,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.32/0.52 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % neg_imp_zdiv_nonneg_iff
% 0.32/0.52 thf(fact_1721_pos__imp__zdiv__pos__iff,axiom,
% 0.32/0.52 ! [K: int,I: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.32/0.52 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pos_imp_zdiv_pos_iff
% 0.32/0.52 thf(fact_1722_pos__imp__zdiv__neg__iff,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.32/0.52 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pos_imp_zdiv_neg_iff
% 0.32/0.52 thf(fact_1723_neg__imp__zdiv__neg__iff,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.32/0.52 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % neg_imp_zdiv_neg_iff
% 0.32/0.52 thf(fact_1724_div__nonpos__pos__le0,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_nonpos_pos_le0
% 0.32/0.52 thf(fact_1725_div__nonneg__neg__le0,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_nonneg_neg_le0
% 0.32/0.52 thf(fact_1726_div__neg__pos__less0,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_neg_pos_less0
% 0.32/0.52 thf(fact_1727_div__positive__int,axiom,
% 0.32/0.52 ! [L2: int,K: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ L2 @ K )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.52 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_positive_int
% 0.32/0.52 thf(fact_1728_div__int__pos__iff,axiom,
% 0.32/0.52 ! [K: int,L2: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
% 0.32/0.52 = ( ( K = zero_zero_int )
% 0.32/0.52 | ( L2 = zero_zero_int )
% 0.32/0.52 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.52 & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
% 0.32/0.52 | ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.52 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % div_int_pos_iff
% 0.32/0.52 thf(fact_1729_zdiv__mono2__neg,axiom,
% 0.32/0.52 ! [A: int,B2: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.32/0.52 => ( ( ord_less_eq_int @ B2 @ B )
% 0.32/0.52 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zdiv_mono2_neg
% 0.32/0.52 thf(fact_1730_zdiv__mono1__neg,axiom,
% 0.32/0.52 ! [A: int,A4: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ A4 )
% 0.32/0.52 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zdiv_mono1_neg
% 0.32/0.52 thf(fact_1731_zdiv__eq__0__iff,axiom,
% 0.32/0.52 ! [I: int,K: int] :
% 0.32/0.52 ( ( ( divide_divide_int @ I @ K )
% 0.32/0.52 = zero_zero_int )
% 0.32/0.52 = ( ( K = zero_zero_int )
% 0.32/0.52 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.32/0.52 & ( ord_less_int @ I @ K ) )
% 0.32/0.52 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.32/0.52 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zdiv_eq_0_iff
% 0.32/0.52 thf(fact_1732_zdiv__mono2,axiom,
% 0.32/0.52 ! [A: int,B2: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.32/0.52 => ( ( ord_less_eq_int @ B2 @ B )
% 0.32/0.52 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zdiv_mono2
% 0.32/0.52 thf(fact_1733_zdiv__mono1,axiom,
% 0.32/0.52 ! [A: int,A4: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ A4 )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zdiv_mono1
% 0.32/0.52 thf(fact_1734_split__zdiv,axiom,
% 0.32/0.52 ! [P: int > $o,N: int,K: int] :
% 0.32/0.52 ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.32/0.52 = ( ( ( K = zero_zero_int )
% 0.32/0.52 => ( P @ zero_zero_int ) )
% 0.32/0.52 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.52 => ! [I4: int,J3: int] :
% 0.32/0.52 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.32/0.52 & ( ord_less_int @ J3 @ K )
% 0.32/0.52 & ( N
% 0.32/0.52 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.32/0.52 => ( P @ I4 ) ) )
% 0.32/0.52 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.52 => ! [I4: int,J3: int] :
% 0.32/0.52 ( ( ( ord_less_int @ K @ J3 )
% 0.32/0.52 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.32/0.52 & ( N
% 0.32/0.52 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.32/0.52 => ( P @ I4 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % split_zdiv
% 0.32/0.52 thf(fact_1735_int__div__neg__eq,axiom,
% 0.32/0.52 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.52 ( ( A
% 0.32/0.52 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.52 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ B @ R )
% 0.32/0.52 => ( ( divide_divide_int @ A @ B )
% 0.32/0.52 = Q2 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % int_div_neg_eq
% 0.32/0.52 thf(fact_1736_int__div__pos__eq,axiom,
% 0.32/0.52 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.52 ( ( A
% 0.32/0.52 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.52 => ( ( ord_less_int @ R @ B )
% 0.32/0.52 => ( ( divide_divide_int @ A @ B )
% 0.32/0.52 = Q2 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % int_div_pos_eq
% 0.32/0.52 thf(fact_1737_zdiv__zmult2__eq,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.52 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zdiv_zmult2_eq
% 0.32/0.52 thf(fact_1738_neg__zmod__mult__2,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( 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 ) )
% 0.32/0.52 = ( 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 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % neg_zmod_mult_2
% 0.32/0.52 thf(fact_1739_pos__zmod__mult__2,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( 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 ) )
% 0.32/0.52 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pos_zmod_mult_2
% 0.32/0.52 thf(fact_1740_enat__0__less__mult__iff,axiom,
% 0.32/0.52 ! [M: extended_enat,N: extended_enat] :
% 0.32/0.52 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.32/0.52 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.32/0.52 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % enat_0_less_mult_iff
% 0.32/0.52 thf(fact_1741_not__iless0,axiom,
% 0.32/0.52 ! [N: extended_enat] :
% 0.32/0.52 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 0.32/0.52
% 0.32/0.52 % not_iless0
% 0.32/0.52 thf(fact_1742_iadd__is__0,axiom,
% 0.32/0.52 ! [M: extended_enat,N: extended_enat] :
% 0.32/0.52 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 0.32/0.52 = zero_z5237406670263579293d_enat )
% 0.32/0.52 = ( ( M = zero_z5237406670263579293d_enat )
% 0.32/0.52 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % iadd_is_0
% 0.32/0.52 thf(fact_1743_ile0__eq,axiom,
% 0.32/0.52 ! [N: extended_enat] :
% 0.32/0.52 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.32/0.52 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.32/0.52
% 0.32/0.52 % ile0_eq
% 0.32/0.52 thf(fact_1744_i0__lb,axiom,
% 0.32/0.52 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 0.32/0.52
% 0.32/0.52 % i0_lb
% 0.32/0.52 thf(fact_1745_power__eq__iff__eq__base,axiom,
% 0.32/0.52 ! [N: nat,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ( ( power_power_real @ A @ N )
% 0.32/0.52 = ( power_power_real @ B @ N ) )
% 0.32/0.52 = ( A = B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_iff_eq_base
% 0.32/0.52 thf(fact_1746_power__eq__iff__eq__base,axiom,
% 0.32/0.52 ! [N: nat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ( ( power_power_rat @ A @ N )
% 0.32/0.52 = ( power_power_rat @ B @ N ) )
% 0.32/0.52 = ( A = B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_iff_eq_base
% 0.32/0.52 thf(fact_1747_power__eq__iff__eq__base,axiom,
% 0.32/0.52 ! [N: nat,A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ( ( power_power_nat @ A @ N )
% 0.32/0.52 = ( power_power_nat @ B @ N ) )
% 0.32/0.52 = ( A = B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_iff_eq_base
% 0.32/0.52 thf(fact_1748_power__eq__iff__eq__base,axiom,
% 0.32/0.52 ! [N: nat,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ( ( power_power_int @ A @ N )
% 0.32/0.52 = ( power_power_int @ B @ N ) )
% 0.32/0.52 = ( A = B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_iff_eq_base
% 0.32/0.52 thf(fact_1749_power__eq__imp__eq__base,axiom,
% 0.32/0.52 ! [A: real,N: nat,B: real] :
% 0.32/0.52 ( ( ( power_power_real @ A @ N )
% 0.32/0.52 = ( power_power_real @ B @ N ) )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( A = B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_imp_eq_base
% 0.32/0.52 thf(fact_1750_power__eq__imp__eq__base,axiom,
% 0.32/0.52 ! [A: rat,N: nat,B: rat] :
% 0.32/0.52 ( ( ( power_power_rat @ A @ N )
% 0.32/0.52 = ( power_power_rat @ B @ N ) )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( A = B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_imp_eq_base
% 0.32/0.52 thf(fact_1751_power__eq__imp__eq__base,axiom,
% 0.32/0.52 ! [A: nat,N: nat,B: nat] :
% 0.32/0.52 ( ( ( power_power_nat @ A @ N )
% 0.32/0.52 = ( power_power_nat @ B @ N ) )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( A = B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_imp_eq_base
% 0.32/0.52 thf(fact_1752_power__eq__imp__eq__base,axiom,
% 0.32/0.52 ! [A: int,N: nat,B: int] :
% 0.32/0.52 ( ( ( power_power_int @ A @ N )
% 0.32/0.52 = ( power_power_int @ B @ N ) )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( A = B ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_eq_imp_eq_base
% 0.32/0.52 thf(fact_1753_realpow__pos__nth__unique,axiom,
% 0.32/0.52 ! [N: nat,A: real] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ? [X3: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.32/0.52 & ( ( power_power_real @ X3 @ N )
% 0.32/0.52 = A )
% 0.32/0.52 & ! [Y5: real] :
% 0.32/0.52 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 0.32/0.52 & ( ( power_power_real @ Y5 @ N )
% 0.32/0.52 = A ) )
% 0.32/0.52 => ( Y5 = X3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % realpow_pos_nth_unique
% 0.32/0.52 thf(fact_1754_realpow__pos__nth,axiom,
% 0.32/0.52 ! [N: nat,A: real] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ? [R2: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.32/0.52 & ( ( power_power_real @ R2 @ N )
% 0.32/0.52 = A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % realpow_pos_nth
% 0.32/0.52 thf(fact_1755_power__strict__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_mono
% 0.32/0.52 thf(fact_1756_power__strict__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_mono
% 0.32/0.52 thf(fact_1757_power__strict__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_mono
% 0.32/0.52 thf(fact_1758_power__strict__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_strict_mono
% 0.32/0.52 thf(fact_1759_dbl__def,axiom,
% 0.32/0.52 ( neg_numeral_dbl_real
% 0.32/0.52 = ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % dbl_def
% 0.32/0.52 thf(fact_1760_dbl__def,axiom,
% 0.32/0.52 ( neg_numeral_dbl_rat
% 0.32/0.52 = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % dbl_def
% 0.32/0.52 thf(fact_1761_dbl__def,axiom,
% 0.32/0.52 ( neg_numeral_dbl_int
% 0.32/0.52 = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % dbl_def
% 0.32/0.52 thf(fact_1762_zero__le__numeral,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_numeral
% 0.32/0.52 thf(fact_1763_zero__le__numeral,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_numeral
% 0.32/0.52 thf(fact_1764_zero__le__numeral,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_numeral
% 0.32/0.52 thf(fact_1765_zero__le__numeral,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_numeral
% 0.32/0.52 thf(fact_1766_not__numeral__le__zero,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.32/0.52
% 0.32/0.52 % not_numeral_le_zero
% 0.32/0.52 thf(fact_1767_not__numeral__le__zero,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.32/0.52
% 0.32/0.52 % not_numeral_le_zero
% 0.32/0.52 thf(fact_1768_not__numeral__le__zero,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % not_numeral_le_zero
% 0.32/0.52 thf(fact_1769_not__numeral__le__zero,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.32/0.52
% 0.32/0.52 % not_numeral_le_zero
% 0.32/0.52 thf(fact_1770_mult__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_mono
% 0.32/0.52 thf(fact_1771_mult__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_mono
% 0.32/0.52 thf(fact_1772_mult__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_mono
% 0.32/0.52 thf(fact_1773_mult__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_mono
% 0.32/0.52 thf(fact_1774_mult__mono_H,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_mono'
% 0.32/0.52 thf(fact_1775_mult__mono_H,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_mono'
% 0.32/0.52 thf(fact_1776_mult__mono_H,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_mono'
% 0.32/0.52 thf(fact_1777_mult__mono_H,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_mono'
% 0.32/0.52 thf(fact_1778_zero__le__square,axiom,
% 0.32/0.52 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_square
% 0.32/0.52 thf(fact_1779_zero__le__square,axiom,
% 0.32/0.52 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_square
% 0.32/0.52 thf(fact_1780_zero__le__square,axiom,
% 0.32/0.52 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_square
% 0.32/0.52 thf(fact_1781_split__mult__pos__le,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.32/0.52 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.32/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % split_mult_pos_le
% 0.32/0.52 thf(fact_1782_split__mult__pos__le,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.32/0.52 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.32/0.52 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % split_mult_pos_le
% 0.32/0.52 thf(fact_1783_split__mult__pos__le,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.32/0.52 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.32/0.52 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % split_mult_pos_le
% 0.32/0.52 thf(fact_1784_mult__left__mono__neg,axiom,
% 0.32/0.52 ! [B: real,A: real,C: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_mono_neg
% 0.32/0.52 thf(fact_1785_mult__left__mono__neg,axiom,
% 0.32/0.52 ! [B: rat,A: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_mono_neg
% 0.32/0.52 thf(fact_1786_mult__left__mono__neg,axiom,
% 0.32/0.52 ! [B: int,A: int,C: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_mono_neg
% 0.32/0.52 thf(fact_1787_mult__nonpos__nonpos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonpos_nonpos
% 0.32/0.52 thf(fact_1788_mult__nonpos__nonpos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonpos_nonpos
% 0.32/0.52 thf(fact_1789_mult__nonpos__nonpos,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonpos_nonpos
% 0.32/0.52 thf(fact_1790_mult__left__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_mono
% 0.32/0.52 thf(fact_1791_mult__left__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_mono
% 0.32/0.52 thf(fact_1792_mult__left__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_mono
% 0.32/0.52 thf(fact_1793_mult__left__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_mono
% 0.32/0.52 thf(fact_1794_mult__right__mono__neg,axiom,
% 0.32/0.52 ! [B: real,A: real,C: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_mono_neg
% 0.32/0.52 thf(fact_1795_mult__right__mono__neg,axiom,
% 0.32/0.52 ! [B: rat,A: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_mono_neg
% 0.32/0.52 thf(fact_1796_mult__right__mono__neg,axiom,
% 0.32/0.52 ! [B: int,A: int,C: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_mono_neg
% 0.32/0.52 thf(fact_1797_mult__right__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_mono
% 0.32/0.52 thf(fact_1798_mult__right__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_mono
% 0.32/0.52 thf(fact_1799_mult__right__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_mono
% 0.32/0.52 thf(fact_1800_mult__right__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_mono
% 0.32/0.52 thf(fact_1801_mult__le__0__iff,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.32/0.52 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.32/0.52 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_0_iff
% 0.32/0.52 thf(fact_1802_mult__le__0__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.32/0.52 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.32/0.52 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_0_iff
% 0.32/0.52 thf(fact_1803_mult__le__0__iff,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.32/0.52 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.32/0.52 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_0_iff
% 0.32/0.52 thf(fact_1804_split__mult__neg__le,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.32/0.52 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.32/0.52
% 0.32/0.52 % split_mult_neg_le
% 0.32/0.52 thf(fact_1805_split__mult__neg__le,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.32/0.52 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 0.32/0.52
% 0.32/0.52 % split_mult_neg_le
% 0.32/0.52 thf(fact_1806_split__mult__neg__le,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.32/0.52 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.52 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % split_mult_neg_le
% 0.32/0.52 thf(fact_1807_split__mult__neg__le,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.32/0.52 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % split_mult_neg_le
% 0.32/0.52 thf(fact_1808_mult__nonneg__nonneg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonneg
% 0.32/0.52 thf(fact_1809_mult__nonneg__nonneg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonneg
% 0.32/0.52 thf(fact_1810_mult__nonneg__nonneg,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonneg
% 0.32/0.52 thf(fact_1811_mult__nonneg__nonneg,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonneg
% 0.32/0.52 thf(fact_1812_mult__nonneg__nonpos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonpos
% 0.32/0.52 thf(fact_1813_mult__nonneg__nonpos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonpos
% 0.32/0.52 thf(fact_1814_mult__nonneg__nonpos,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonpos
% 0.32/0.52 thf(fact_1815_mult__nonneg__nonpos,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonpos
% 0.32/0.52 thf(fact_1816_mult__nonpos__nonneg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonpos_nonneg
% 0.32/0.52 thf(fact_1817_mult__nonpos__nonneg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonpos_nonneg
% 0.32/0.52 thf(fact_1818_mult__nonpos__nonneg,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonpos_nonneg
% 0.32/0.52 thf(fact_1819_mult__nonpos__nonneg,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonpos_nonneg
% 0.32/0.52 thf(fact_1820_mult__nonneg__nonpos2,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonpos2
% 0.32/0.52 thf(fact_1821_mult__nonneg__nonpos2,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonpos2
% 0.32/0.52 thf(fact_1822_mult__nonneg__nonpos2,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonpos2
% 0.32/0.52 thf(fact_1823_mult__nonneg__nonpos2,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_nonneg_nonpos2
% 0.32/0.52 thf(fact_1824_zero__le__mult__iff,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.32/0.52 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_mult_iff
% 0.32/0.52 thf(fact_1825_zero__le__mult__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.32/0.52 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_mult_iff
% 0.32/0.52 thf(fact_1826_zero__le__mult__iff,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.32/0.52 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_mult_iff
% 0.32/0.52 thf(fact_1827_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.32/0.52 thf(fact_1828_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.32/0.52 thf(fact_1829_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.32/0.52 thf(fact_1830_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.32/0.52 thf(fact_1831_zero__less__numeral,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_numeral
% 0.32/0.52 thf(fact_1832_zero__less__numeral,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_numeral
% 0.32/0.52 thf(fact_1833_zero__less__numeral,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_numeral
% 0.32/0.52 thf(fact_1834_zero__less__numeral,axiom,
% 0.32/0.52 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_numeral
% 0.32/0.52 thf(fact_1835_not__numeral__less__zero,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.32/0.52
% 0.32/0.52 % not_numeral_less_zero
% 0.32/0.52 thf(fact_1836_not__numeral__less__zero,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.32/0.52
% 0.32/0.52 % not_numeral_less_zero
% 0.32/0.52 thf(fact_1837_not__numeral__less__zero,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % not_numeral_less_zero
% 0.32/0.52 thf(fact_1838_not__numeral__less__zero,axiom,
% 0.32/0.52 ! [N: num] :
% 0.32/0.52 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.32/0.52
% 0.32/0.52 % not_numeral_less_zero
% 0.32/0.52 thf(fact_1839_not__one__le__zero,axiom,
% 0.32/0.52 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.32/0.52
% 0.32/0.52 % not_one_le_zero
% 0.32/0.52 thf(fact_1840_not__one__le__zero,axiom,
% 0.32/0.52 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.32/0.52
% 0.32/0.52 % not_one_le_zero
% 0.32/0.52 thf(fact_1841_not__one__le__zero,axiom,
% 0.32/0.52 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % not_one_le_zero
% 0.32/0.52 thf(fact_1842_not__one__le__zero,axiom,
% 0.32/0.52 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.32/0.52
% 0.32/0.52 % not_one_le_zero
% 0.32/0.52 thf(fact_1843_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.32/0.52 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.32/0.52
% 0.32/0.52 % linordered_nonzero_semiring_class.zero_le_one
% 0.32/0.52 thf(fact_1844_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.32/0.52 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.32/0.52
% 0.32/0.52 % linordered_nonzero_semiring_class.zero_le_one
% 0.32/0.52 thf(fact_1845_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.32/0.52 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.52
% 0.32/0.52 % linordered_nonzero_semiring_class.zero_le_one
% 0.32/0.52 thf(fact_1846_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.32/0.52 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.32/0.52
% 0.32/0.52 % linordered_nonzero_semiring_class.zero_le_one
% 0.32/0.52 thf(fact_1847_zero__less__one__class_Ozero__le__one,axiom,
% 0.32/0.52 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.32/0.52
% 0.32/0.52 % zero_less_one_class.zero_le_one
% 0.32/0.52 thf(fact_1848_zero__less__one__class_Ozero__le__one,axiom,
% 0.32/0.52 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.32/0.52
% 0.32/0.52 % zero_less_one_class.zero_le_one
% 0.32/0.52 thf(fact_1849_zero__less__one__class_Ozero__le__one,axiom,
% 0.32/0.52 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.52
% 0.32/0.52 % zero_less_one_class.zero_le_one
% 0.32/0.52 thf(fact_1850_zero__less__one__class_Ozero__le__one,axiom,
% 0.32/0.52 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.32/0.52
% 0.32/0.52 % zero_less_one_class.zero_le_one
% 0.32/0.52 thf(fact_1851_add__nonpos__eq__0__iff,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.32/0.52 => ( ( ( plus_plus_real @ X @ Y3 )
% 0.32/0.52 = zero_zero_real )
% 0.32/0.52 = ( ( X = zero_zero_real )
% 0.32/0.52 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonpos_eq_0_iff
% 0.32/0.52 thf(fact_1852_add__nonpos__eq__0__iff,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.32/0.52 => ( ( ( plus_plus_rat @ X @ Y3 )
% 0.32/0.52 = zero_zero_rat )
% 0.32/0.52 = ( ( X = zero_zero_rat )
% 0.32/0.52 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonpos_eq_0_iff
% 0.32/0.52 thf(fact_1853_add__nonpos__eq__0__iff,axiom,
% 0.32/0.52 ! [X: nat,Y3: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 0.32/0.52 => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 0.32/0.52 => ( ( ( plus_plus_nat @ X @ Y3 )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 = ( ( X = zero_zero_nat )
% 0.32/0.52 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonpos_eq_0_iff
% 0.32/0.52 thf(fact_1854_add__nonpos__eq__0__iff,axiom,
% 0.32/0.52 ! [X: int,Y3: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.32/0.52 => ( ( ( plus_plus_int @ X @ Y3 )
% 0.32/0.52 = zero_zero_int )
% 0.32/0.52 = ( ( X = zero_zero_int )
% 0.32/0.52 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonpos_eq_0_iff
% 0.32/0.52 thf(fact_1855_add__nonneg__eq__0__iff,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.52 => ( ( ( plus_plus_real @ X @ Y3 )
% 0.32/0.52 = zero_zero_real )
% 0.32/0.52 = ( ( X = zero_zero_real )
% 0.32/0.52 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonneg_eq_0_iff
% 0.32/0.52 thf(fact_1856_add__nonneg__eq__0__iff,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.52 => ( ( ( plus_plus_rat @ X @ Y3 )
% 0.32/0.52 = zero_zero_rat )
% 0.32/0.52 = ( ( X = zero_zero_rat )
% 0.32/0.52 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonneg_eq_0_iff
% 0.32/0.52 thf(fact_1857_add__nonneg__eq__0__iff,axiom,
% 0.32/0.52 ! [X: nat,Y3: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.32/0.52 => ( ( ( plus_plus_nat @ X @ Y3 )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 = ( ( X = zero_zero_nat )
% 0.32/0.52 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonneg_eq_0_iff
% 0.32/0.52 thf(fact_1858_add__nonneg__eq__0__iff,axiom,
% 0.32/0.52 ! [X: int,Y3: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.52 => ( ( ( plus_plus_int @ X @ Y3 )
% 0.32/0.52 = zero_zero_int )
% 0.32/0.52 = ( ( X = zero_zero_int )
% 0.32/0.52 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonneg_eq_0_iff
% 0.32/0.52 thf(fact_1859_add__nonpos__nonpos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonpos_nonpos
% 0.32/0.52 thf(fact_1860_add__nonpos__nonpos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonpos_nonpos
% 0.32/0.52 thf(fact_1861_add__nonpos__nonpos,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.52 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.32/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonpos_nonpos
% 0.32/0.52 thf(fact_1862_add__nonpos__nonpos,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonpos_nonpos
% 0.32/0.52 thf(fact_1863_add__nonneg__nonneg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonneg_nonneg
% 0.32/0.52 thf(fact_1864_add__nonneg__nonneg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonneg_nonneg
% 0.32/0.52 thf(fact_1865_add__nonneg__nonneg,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonneg_nonneg
% 0.32/0.52 thf(fact_1866_add__nonneg__nonneg,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_nonneg_nonneg
% 0.32/0.52 thf(fact_1867_add__increasing2,axiom,
% 0.32/0.52 ! [C: real,B: real,A: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ( ord_less_eq_real @ B @ A )
% 0.32/0.52 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_increasing2
% 0.32/0.52 thf(fact_1868_add__increasing2,axiom,
% 0.32/0.52 ! [C: rat,B: rat,A: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.52 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_increasing2
% 0.32/0.52 thf(fact_1869_add__increasing2,axiom,
% 0.32/0.52 ! [C: nat,B: nat,A: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.52 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_increasing2
% 0.32/0.52 thf(fact_1870_add__increasing2,axiom,
% 0.32/0.52 ! [C: int,B: int,A: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ( ord_less_eq_int @ B @ A )
% 0.32/0.52 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_increasing2
% 0.32/0.52 thf(fact_1871_add__decreasing2,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_decreasing2
% 0.32/0.52 thf(fact_1872_add__decreasing2,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_decreasing2
% 0.32/0.52 thf(fact_1873_add__decreasing2,axiom,
% 0.32/0.52 ! [C: nat,A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.32/0.52 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_decreasing2
% 0.32/0.52 thf(fact_1874_add__decreasing2,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_decreasing2
% 0.32/0.52 thf(fact_1875_add__increasing,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.52 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_increasing
% 0.32/0.52 thf(fact_1876_add__increasing,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_increasing
% 0.32/0.52 thf(fact_1877_add__increasing,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.52 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_increasing
% 0.32/0.52 thf(fact_1878_add__increasing,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.52 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_increasing
% 0.32/0.52 thf(fact_1879_add__decreasing,axiom,
% 0.32/0.52 ! [A: real,C: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ C @ B )
% 0.32/0.52 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_decreasing
% 0.32/0.52 thf(fact_1880_add__decreasing,axiom,
% 0.32/0.52 ! [A: rat,C: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ C @ B )
% 0.32/0.52 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_decreasing
% 0.32/0.52 thf(fact_1881_add__decreasing,axiom,
% 0.32/0.52 ! [A: nat,C: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.52 => ( ( ord_less_eq_nat @ C @ B )
% 0.32/0.52 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_decreasing
% 0.32/0.52 thf(fact_1882_add__decreasing,axiom,
% 0.32/0.52 ! [A: int,C: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ C @ B )
% 0.32/0.52 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_decreasing
% 0.32/0.52 thf(fact_1883_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.32/0.52 thf(fact_1884_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.32/0.52 thf(fact_1885_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.32/0.52 thf(fact_1886_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.32/0.52 thf(fact_1887_mult__less__cancel__right__disj,axiom,
% 0.32/0.52 ! [A: real,C: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.52 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 & ( ord_less_real @ A @ B ) )
% 0.32/0.52 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.52 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_right_disj
% 0.32/0.52 thf(fact_1888_mult__less__cancel__right__disj,axiom,
% 0.32/0.52 ! [A: rat,C: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.52 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 & ( ord_less_rat @ A @ B ) )
% 0.32/0.52 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_right_disj
% 0.32/0.52 thf(fact_1889_mult__less__cancel__right__disj,axiom,
% 0.32/0.52 ! [A: int,C: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.52 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 & ( ord_less_int @ A @ B ) )
% 0.32/0.52 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.52 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_right_disj
% 0.32/0.52 thf(fact_1890_mult__strict__right__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_right_mono
% 0.32/0.52 thf(fact_1891_mult__strict__right__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_right_mono
% 0.32/0.52 thf(fact_1892_mult__strict__right__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_right_mono
% 0.32/0.52 thf(fact_1893_mult__strict__right__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_right_mono
% 0.32/0.52 thf(fact_1894_mult__strict__right__mono__neg,axiom,
% 0.32/0.52 ! [B: real,A: real,C: real] :
% 0.32/0.52 ( ( ord_less_real @ B @ A )
% 0.32/0.52 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_right_mono_neg
% 0.32/0.52 thf(fact_1895_mult__strict__right__mono__neg,axiom,
% 0.32/0.52 ! [B: rat,A: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_rat @ B @ A )
% 0.32/0.52 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_right_mono_neg
% 0.32/0.52 thf(fact_1896_mult__strict__right__mono__neg,axiom,
% 0.32/0.52 ! [B: int,A: int,C: int] :
% 0.32/0.52 ( ( ord_less_int @ B @ A )
% 0.32/0.52 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_right_mono_neg
% 0.32/0.52 thf(fact_1897_mult__less__cancel__left__disj,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.52 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 & ( ord_less_real @ A @ B ) )
% 0.32/0.52 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.52 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_disj
% 0.32/0.52 thf(fact_1898_mult__less__cancel__left__disj,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.52 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 & ( ord_less_rat @ A @ B ) )
% 0.32/0.52 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_disj
% 0.32/0.52 thf(fact_1899_mult__less__cancel__left__disj,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.52 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 & ( ord_less_int @ A @ B ) )
% 0.32/0.52 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.52 & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_disj
% 0.32/0.52 thf(fact_1900_mult__strict__left__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_left_mono
% 0.32/0.52 thf(fact_1901_mult__strict__left__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_left_mono
% 0.32/0.52 thf(fact_1902_mult__strict__left__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_left_mono
% 0.32/0.52 thf(fact_1903_mult__strict__left__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_left_mono
% 0.32/0.52 thf(fact_1904_mult__strict__left__mono__neg,axiom,
% 0.32/0.52 ! [B: real,A: real,C: real] :
% 0.32/0.52 ( ( ord_less_real @ B @ A )
% 0.32/0.52 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_left_mono_neg
% 0.32/0.52 thf(fact_1905_mult__strict__left__mono__neg,axiom,
% 0.32/0.52 ! [B: rat,A: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_rat @ B @ A )
% 0.32/0.52 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_left_mono_neg
% 0.32/0.52 thf(fact_1906_mult__strict__left__mono__neg,axiom,
% 0.32/0.52 ! [B: int,A: int,C: int] :
% 0.32/0.52 ( ( ord_less_int @ B @ A )
% 0.32/0.52 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_left_mono_neg
% 0.32/0.52 thf(fact_1907_mult__less__cancel__left__pos,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.52 = ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_pos
% 0.32/0.52 thf(fact_1908_mult__less__cancel__left__pos,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.52 = ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_pos
% 0.32/0.52 thf(fact_1909_mult__less__cancel__left__pos,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.52 = ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_pos
% 0.32/0.52 thf(fact_1910_mult__less__cancel__left__neg,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.52 = ( ord_less_real @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_neg
% 0.32/0.52 thf(fact_1911_mult__less__cancel__left__neg,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.52 = ( ord_less_rat @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_neg
% 0.32/0.52 thf(fact_1912_mult__less__cancel__left__neg,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.52 = ( ord_less_int @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_left_neg
% 0.32/0.52 thf(fact_1913_zero__less__mult__pos2,axiom,
% 0.32/0.52 ! [B: real,A: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_pos2
% 0.32/0.52 thf(fact_1914_zero__less__mult__pos2,axiom,
% 0.32/0.52 ! [B: rat,A: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_pos2
% 0.32/0.52 thf(fact_1915_zero__less__mult__pos2,axiom,
% 0.32/0.52 ! [B: nat,A: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_pos2
% 0.32/0.52 thf(fact_1916_zero__less__mult__pos2,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_pos2
% 0.32/0.52 thf(fact_1917_zero__less__mult__pos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_pos
% 0.32/0.52 thf(fact_1918_zero__less__mult__pos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_pos
% 0.32/0.52 thf(fact_1919_zero__less__mult__pos,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_pos
% 0.32/0.52 thf(fact_1920_zero__less__mult__pos,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_pos
% 0.32/0.52 thf(fact_1921_zero__less__mult__iff,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.32/0.52 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_iff
% 0.32/0.52 thf(fact_1922_zero__less__mult__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.32/0.52 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_iff
% 0.32/0.52 thf(fact_1923_zero__less__mult__iff,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 & ( ord_less_int @ zero_zero_int @ B ) )
% 0.32/0.52 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.52 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_mult_iff
% 0.32/0.52 thf(fact_1924_mult__pos__neg2,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_neg2
% 0.32/0.52 thf(fact_1925_mult__pos__neg2,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_neg2
% 0.32/0.52 thf(fact_1926_mult__pos__neg2,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_neg2
% 0.32/0.52 thf(fact_1927_mult__pos__neg2,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_neg2
% 0.32/0.52 thf(fact_1928_mult__pos__pos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_pos
% 0.32/0.52 thf(fact_1929_mult__pos__pos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_pos
% 0.32/0.52 thf(fact_1930_mult__pos__pos,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_pos
% 0.32/0.52 thf(fact_1931_mult__pos__pos,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_pos
% 0.32/0.52 thf(fact_1932_mult__pos__neg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_neg
% 0.32/0.52 thf(fact_1933_mult__pos__neg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_neg
% 0.32/0.52 thf(fact_1934_mult__pos__neg,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_neg
% 0.32/0.52 thf(fact_1935_mult__pos__neg,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_pos_neg
% 0.32/0.52 thf(fact_1936_mult__neg__pos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_neg_pos
% 0.32/0.52 thf(fact_1937_mult__neg__pos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_neg_pos
% 0.32/0.52 thf(fact_1938_mult__neg__pos,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_neg_pos
% 0.32/0.52 thf(fact_1939_mult__neg__pos,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_neg_pos
% 0.32/0.52 thf(fact_1940_mult__less__0__iff,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.32/0.52 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.32/0.52 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_0_iff
% 0.32/0.52 thf(fact_1941_mult__less__0__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.32/0.52 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.32/0.52 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_0_iff
% 0.32/0.52 thf(fact_1942_mult__less__0__iff,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.32/0.52 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 & ( ord_less_int @ B @ zero_zero_int ) )
% 0.32/0.52 | ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.52 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_0_iff
% 0.32/0.52 thf(fact_1943_not__square__less__zero,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.32/0.52
% 0.32/0.52 % not_square_less_zero
% 0.32/0.52 thf(fact_1944_not__square__less__zero,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.32/0.52
% 0.32/0.52 % not_square_less_zero
% 0.32/0.52 thf(fact_1945_not__square__less__zero,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.32/0.52
% 0.32/0.52 % not_square_less_zero
% 0.32/0.52 thf(fact_1946_mult__neg__neg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_neg_neg
% 0.32/0.52 thf(fact_1947_mult__neg__neg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_neg_neg
% 0.32/0.52 thf(fact_1948_mult__neg__neg,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_neg_neg
% 0.32/0.52 thf(fact_1949_le__iff__diff__le__0,axiom,
% 0.32/0.52 ( ord_less_eq_real
% 0.32/0.52 = ( ^ [A3: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_iff_diff_le_0
% 0.32/0.52 thf(fact_1950_le__iff__diff__le__0,axiom,
% 0.32/0.52 ( ord_less_eq_rat
% 0.32/0.52 = ( ^ [A3: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_iff_diff_le_0
% 0.32/0.52 thf(fact_1951_le__iff__diff__le__0,axiom,
% 0.32/0.52 ( ord_less_eq_int
% 0.32/0.52 = ( ^ [A3: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % le_iff_diff_le_0
% 0.32/0.52 thf(fact_1952_zero__less__one,axiom,
% 0.32/0.52 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.32/0.52
% 0.32/0.52 % zero_less_one
% 0.32/0.52 thf(fact_1953_zero__less__one,axiom,
% 0.32/0.52 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.32/0.52
% 0.32/0.52 % zero_less_one
% 0.32/0.52 thf(fact_1954_zero__less__one,axiom,
% 0.32/0.52 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.52
% 0.32/0.52 % zero_less_one
% 0.32/0.52 thf(fact_1955_zero__less__one,axiom,
% 0.32/0.52 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.32/0.52
% 0.32/0.52 % zero_less_one
% 0.32/0.52 thf(fact_1956_not__one__less__zero,axiom,
% 0.32/0.52 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.32/0.52
% 0.32/0.52 % not_one_less_zero
% 0.32/0.52 thf(fact_1957_not__one__less__zero,axiom,
% 0.32/0.52 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.32/0.52
% 0.32/0.52 % not_one_less_zero
% 0.32/0.52 thf(fact_1958_not__one__less__zero,axiom,
% 0.32/0.52 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % not_one_less_zero
% 0.32/0.52 thf(fact_1959_not__one__less__zero,axiom,
% 0.32/0.52 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.32/0.52
% 0.32/0.52 % not_one_less_zero
% 0.32/0.52 thf(fact_1960_less__numeral__extra_I1_J,axiom,
% 0.32/0.52 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.32/0.52
% 0.32/0.52 % less_numeral_extra(1)
% 0.32/0.52 thf(fact_1961_less__numeral__extra_I1_J,axiom,
% 0.32/0.52 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.32/0.52
% 0.32/0.52 % less_numeral_extra(1)
% 0.32/0.52 thf(fact_1962_less__numeral__extra_I1_J,axiom,
% 0.32/0.52 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.32/0.52
% 0.32/0.52 % less_numeral_extra(1)
% 0.32/0.52 thf(fact_1963_less__numeral__extra_I1_J,axiom,
% 0.32/0.52 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.32/0.52
% 0.32/0.52 % less_numeral_extra(1)
% 0.32/0.52 thf(fact_1964_add__neg__neg,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_neg_neg
% 0.32/0.52 thf(fact_1965_add__neg__neg,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_neg_neg
% 0.32/0.52 thf(fact_1966_add__neg__neg,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.32/0.52 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.32/0.52 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_neg_neg
% 0.32/0.52 thf(fact_1967_add__neg__neg,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.52 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_neg_neg
% 0.32/0.52 thf(fact_1968_add__pos__pos,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.52 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_pos_pos
% 0.32/0.52 thf(fact_1969_add__pos__pos,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.52 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_pos_pos
% 0.32/0.52 thf(fact_1970_add__pos__pos,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_pos_pos
% 0.32/0.52 thf(fact_1971_add__pos__pos,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_pos_pos
% 0.32/0.52 thf(fact_1972_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ B )
% 0.32/0.52 => ~ ! [C2: nat] :
% 0.32/0.52 ( ( B
% 0.32/0.52 = ( plus_plus_nat @ A @ C2 ) )
% 0.32/0.52 => ( C2 = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % canonically_ordered_monoid_add_class.lessE
% 0.32/0.52 thf(fact_1973_pos__add__strict,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_real @ B @ C )
% 0.32/0.52 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pos_add_strict
% 0.32/0.52 thf(fact_1974_pos__add__strict,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_rat @ B @ C )
% 0.32/0.52 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pos_add_strict
% 0.32/0.52 thf(fact_1975_pos__add__strict,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_nat @ B @ C )
% 0.32/0.52 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pos_add_strict
% 0.32/0.52 thf(fact_1976_pos__add__strict,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ B @ C )
% 0.32/0.52 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % pos_add_strict
% 0.32/0.52 thf(fact_1977_add__less__zeroD,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.52 | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_zeroD
% 0.32/0.52 thf(fact_1978_add__less__zeroD,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y3 ) @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.32/0.52 | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_zeroD
% 0.32/0.52 thf(fact_1979_add__less__zeroD,axiom,
% 0.32/0.52 ! [X: int,Y3: int] :
% 0.32/0.52 ( ( ord_less_int @ ( plus_plus_int @ X @ Y3 ) @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_int @ X @ zero_zero_int )
% 0.32/0.52 | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % add_less_zeroD
% 0.32/0.52 thf(fact_1980_divide__le__0__iff,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.32/0.52 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.32/0.52 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_0_iff
% 0.32/0.52 thf(fact_1981_divide__le__0__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.32/0.52 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.32/0.52 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_le_0_iff
% 0.32/0.52 thf(fact_1982_divide__right__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_right_mono
% 0.32/0.52 thf(fact_1983_divide__right__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_right_mono
% 0.32/0.52 thf(fact_1984_zero__le__divide__iff,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.32/0.52 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_divide_iff
% 0.32/0.52 thf(fact_1985_zero__le__divide__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.32/0.52 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_divide_iff
% 0.32/0.52 thf(fact_1986_divide__nonneg__nonneg,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_nonneg_nonneg
% 0.32/0.52 thf(fact_1987_divide__nonneg__nonneg,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.52 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_nonneg_nonneg
% 0.32/0.52 thf(fact_1988_divide__nonneg__nonpos,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.52 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_nonneg_nonpos
% 0.32/0.52 thf(fact_1989_divide__nonneg__nonpos,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.52 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_nonneg_nonpos
% 0.32/0.52 thf(fact_1990_divide__nonpos__nonneg,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.52 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_nonpos_nonneg
% 0.32/0.52 thf(fact_1991_divide__nonpos__nonneg,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.52 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_nonpos_nonneg
% 0.32/0.52 thf(fact_1992_divide__nonpos__nonpos,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_nonpos_nonpos
% 0.32/0.52 thf(fact_1993_divide__nonpos__nonpos,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_nonpos_nonpos
% 0.32/0.52 thf(fact_1994_divide__right__mono__neg,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_right_mono_neg
% 0.32/0.52 thf(fact_1995_divide__right__mono__neg,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_right_mono_neg
% 0.32/0.52 thf(fact_1996_less__iff__diff__less__0,axiom,
% 0.32/0.52 ( ord_less_real
% 0.32/0.52 = ( ^ [A3: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_iff_diff_less_0
% 0.32/0.52 thf(fact_1997_less__iff__diff__less__0,axiom,
% 0.32/0.52 ( ord_less_rat
% 0.32/0.52 = ( ^ [A3: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_iff_diff_less_0
% 0.32/0.52 thf(fact_1998_less__iff__diff__less__0,axiom,
% 0.32/0.52 ( ord_less_int
% 0.32/0.52 = ( ^ [A3: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_iff_diff_less_0
% 0.32/0.52 thf(fact_1999_divide__strict__right__mono__neg,axiom,
% 0.32/0.52 ! [B: real,A: real,C: real] :
% 0.32/0.52 ( ( ord_less_real @ B @ A )
% 0.32/0.52 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_strict_right_mono_neg
% 0.32/0.52 thf(fact_2000_divide__strict__right__mono__neg,axiom,
% 0.32/0.52 ! [B: rat,A: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_rat @ B @ A )
% 0.32/0.52 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_strict_right_mono_neg
% 0.32/0.52 thf(fact_2001_divide__strict__right__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_strict_right_mono
% 0.32/0.52 thf(fact_2002_divide__strict__right__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_strict_right_mono
% 0.32/0.52 thf(fact_2003_zero__less__divide__iff,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_real @ zero_zero_real @ B ) )
% 0.32/0.52 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_divide_iff
% 0.32/0.52 thf(fact_2004_zero__less__divide__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.52 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.32/0.52 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_divide_iff
% 0.32/0.52 thf(fact_2005_divide__less__cancel,axiom,
% 0.32/0.52 ! [A: real,C: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.52 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ A @ B ) )
% 0.32/0.52 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ B @ A ) )
% 0.32/0.52 & ( C != zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_cancel
% 0.32/0.52 thf(fact_2006_divide__less__cancel,axiom,
% 0.32/0.52 ! [A: rat,C: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.52 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ A @ B ) )
% 0.32/0.52 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ B @ A ) )
% 0.32/0.52 & ( C != zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_cancel
% 0.32/0.52 thf(fact_2007_divide__less__0__iff,axiom,
% 0.32/0.52 ! [A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.32/0.52 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 & ( ord_less_real @ B @ zero_zero_real ) )
% 0.32/0.52 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.52 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_0_iff
% 0.32/0.52 thf(fact_2008_divide__less__0__iff,axiom,
% 0.32/0.52 ! [A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.32/0.52 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.32/0.52 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.52 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_less_0_iff
% 0.32/0.52 thf(fact_2009_divide__pos__pos,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.52 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_pos_pos
% 0.32/0.52 thf(fact_2010_divide__pos__pos,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.52 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_pos_pos
% 0.32/0.52 thf(fact_2011_divide__pos__neg,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.52 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_pos_neg
% 0.32/0.52 thf(fact_2012_divide__pos__neg,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.52 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_pos_neg
% 0.32/0.52 thf(fact_2013_divide__neg__pos,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.52 => ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_neg_pos
% 0.32/0.52 thf(fact_2014_divide__neg__pos,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.52 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_neg_pos
% 0.32/0.52 thf(fact_2015_divide__neg__neg,axiom,
% 0.32/0.52 ! [X: real,Y3: real] :
% 0.32/0.52 ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_neg_neg
% 0.32/0.52 thf(fact_2016_divide__neg__neg,axiom,
% 0.32/0.52 ! [X: rat,Y3: rat] :
% 0.32/0.52 ( ( ord_less_rat @ X @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_neg_neg
% 0.32/0.52 thf(fact_2017_power__mono,axiom,
% 0.32/0.52 ! [A: real,B: real,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_mono
% 0.32/0.52 thf(fact_2018_power__mono,axiom,
% 0.32/0.52 ! [A: rat,B: rat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_mono
% 0.32/0.52 thf(fact_2019_power__mono,axiom,
% 0.32/0.52 ! [A: nat,B: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_mono
% 0.32/0.52 thf(fact_2020_power__mono,axiom,
% 0.32/0.52 ! [A: int,B: int,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % power_mono
% 0.32/0.52 thf(fact_2021_zero__le__power,axiom,
% 0.32/0.52 ! [A: real,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_power
% 0.32/0.52 thf(fact_2022_zero__le__power,axiom,
% 0.32/0.52 ! [A: rat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_power
% 0.32/0.52 thf(fact_2023_zero__le__power,axiom,
% 0.32/0.52 ! [A: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_power
% 0.32/0.52 thf(fact_2024_zero__le__power,axiom,
% 0.32/0.52 ! [A: int,N: nat] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_le_power
% 0.32/0.52 thf(fact_2025_zero__less__power,axiom,
% 0.32/0.52 ! [A: real,N: nat] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_power
% 0.32/0.52 thf(fact_2026_zero__less__power,axiom,
% 0.32/0.52 ! [A: rat,N: nat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_power
% 0.32/0.52 thf(fact_2027_zero__less__power,axiom,
% 0.32/0.52 ! [A: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_power
% 0.32/0.52 thf(fact_2028_zero__less__power,axiom,
% 0.32/0.52 ! [A: int,N: nat] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % zero_less_power
% 0.32/0.52 thf(fact_2029_frac__eq__eq,axiom,
% 0.32/0.52 ! [Y3: complex,Z: complex,X: complex,W: complex] :
% 0.32/0.52 ( ( Y3 != zero_zero_complex )
% 0.32/0.52 => ( ( Z != zero_zero_complex )
% 0.32/0.52 => ( ( ( divide1717551699836669952omplex @ X @ Y3 )
% 0.32/0.52 = ( divide1717551699836669952omplex @ W @ Z ) )
% 0.32/0.52 = ( ( times_times_complex @ X @ Z )
% 0.32/0.52 = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % frac_eq_eq
% 0.32/0.52 thf(fact_2030_frac__eq__eq,axiom,
% 0.32/0.52 ! [Y3: real,Z: real,X: real,W: real] :
% 0.32/0.52 ( ( Y3 != zero_zero_real )
% 0.32/0.52 => ( ( Z != zero_zero_real )
% 0.32/0.52 => ( ( ( divide_divide_real @ X @ Y3 )
% 0.32/0.52 = ( divide_divide_real @ W @ Z ) )
% 0.32/0.52 = ( ( times_times_real @ X @ Z )
% 0.32/0.52 = ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % frac_eq_eq
% 0.32/0.52 thf(fact_2031_frac__eq__eq,axiom,
% 0.32/0.52 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.52 ( ( Y3 != zero_zero_rat )
% 0.32/0.52 => ( ( Z != zero_zero_rat )
% 0.32/0.52 => ( ( ( divide_divide_rat @ X @ Y3 )
% 0.32/0.52 = ( divide_divide_rat @ W @ Z ) )
% 0.32/0.52 = ( ( times_times_rat @ X @ Z )
% 0.32/0.52 = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % frac_eq_eq
% 0.32/0.52 thf(fact_2032_divide__eq__eq,axiom,
% 0.32/0.52 ! [B: complex,C: complex,A: complex] :
% 0.32/0.52 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.52 = A )
% 0.32/0.52 = ( ( ( C != zero_zero_complex )
% 0.32/0.52 => ( B
% 0.32/0.52 = ( times_times_complex @ A @ C ) ) )
% 0.32/0.52 & ( ( C = zero_zero_complex )
% 0.32/0.52 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_eq_eq
% 0.32/0.52 thf(fact_2033_divide__eq__eq,axiom,
% 0.32/0.52 ! [B: real,C: real,A: real] :
% 0.32/0.52 ( ( ( divide_divide_real @ B @ C )
% 0.32/0.52 = A )
% 0.32/0.52 = ( ( ( C != zero_zero_real )
% 0.32/0.52 => ( B
% 0.32/0.52 = ( times_times_real @ A @ C ) ) )
% 0.32/0.52 & ( ( C = zero_zero_real )
% 0.32/0.52 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_eq_eq
% 0.32/0.52 thf(fact_2034_divide__eq__eq,axiom,
% 0.32/0.52 ! [B: rat,C: rat,A: rat] :
% 0.32/0.52 ( ( ( divide_divide_rat @ B @ C )
% 0.32/0.52 = A )
% 0.32/0.52 = ( ( ( C != zero_zero_rat )
% 0.32/0.52 => ( B
% 0.32/0.52 = ( times_times_rat @ A @ C ) ) )
% 0.32/0.52 & ( ( C = zero_zero_rat )
% 0.32/0.52 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_eq_eq
% 0.32/0.52 thf(fact_2035_eq__divide__eq,axiom,
% 0.32/0.52 ! [A: complex,B: complex,C: complex] :
% 0.32/0.52 ( ( A
% 0.32/0.52 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.52 = ( ( ( C != zero_zero_complex )
% 0.32/0.52 => ( ( times_times_complex @ A @ C )
% 0.32/0.52 = B ) )
% 0.32/0.52 & ( ( C = zero_zero_complex )
% 0.32/0.52 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_divide_eq
% 0.32/0.52 thf(fact_2036_eq__divide__eq,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real] :
% 0.32/0.52 ( ( A
% 0.32/0.52 = ( divide_divide_real @ B @ C ) )
% 0.32/0.52 = ( ( ( C != zero_zero_real )
% 0.32/0.52 => ( ( times_times_real @ A @ C )
% 0.32/0.52 = B ) )
% 0.32/0.52 & ( ( C = zero_zero_real )
% 0.32/0.52 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_divide_eq
% 0.32/0.52 thf(fact_2037_eq__divide__eq,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat] :
% 0.32/0.52 ( ( A
% 0.32/0.52 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.52 = ( ( ( C != zero_zero_rat )
% 0.32/0.52 => ( ( times_times_rat @ A @ C )
% 0.32/0.52 = B ) )
% 0.32/0.52 & ( ( C = zero_zero_rat )
% 0.32/0.52 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_divide_eq
% 0.32/0.52 thf(fact_2038_divide__eq__imp,axiom,
% 0.32/0.52 ! [C: complex,B: complex,A: complex] :
% 0.32/0.52 ( ( C != zero_zero_complex )
% 0.32/0.52 => ( ( B
% 0.32/0.52 = ( times_times_complex @ A @ C ) )
% 0.32/0.52 => ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.52 = A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_eq_imp
% 0.32/0.52 thf(fact_2039_divide__eq__imp,axiom,
% 0.32/0.52 ! [C: real,B: real,A: real] :
% 0.32/0.52 ( ( C != zero_zero_real )
% 0.32/0.52 => ( ( B
% 0.32/0.52 = ( times_times_real @ A @ C ) )
% 0.32/0.52 => ( ( divide_divide_real @ B @ C )
% 0.32/0.52 = A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_eq_imp
% 0.32/0.52 thf(fact_2040_divide__eq__imp,axiom,
% 0.32/0.52 ! [C: rat,B: rat,A: rat] :
% 0.32/0.52 ( ( C != zero_zero_rat )
% 0.32/0.52 => ( ( B
% 0.32/0.52 = ( times_times_rat @ A @ C ) )
% 0.32/0.52 => ( ( divide_divide_rat @ B @ C )
% 0.32/0.52 = A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % divide_eq_imp
% 0.32/0.52 thf(fact_2041_eq__divide__imp,axiom,
% 0.32/0.52 ! [C: complex,A: complex,B: complex] :
% 0.32/0.52 ( ( C != zero_zero_complex )
% 0.32/0.52 => ( ( ( times_times_complex @ A @ C )
% 0.32/0.52 = B )
% 0.32/0.52 => ( A
% 0.32/0.52 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_divide_imp
% 0.32/0.52 thf(fact_2042_eq__divide__imp,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( C != zero_zero_real )
% 0.32/0.52 => ( ( ( times_times_real @ A @ C )
% 0.32/0.52 = B )
% 0.32/0.52 => ( A
% 0.32/0.52 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_divide_imp
% 0.32/0.52 thf(fact_2043_eq__divide__imp,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( C != zero_zero_rat )
% 0.32/0.52 => ( ( ( times_times_rat @ A @ C )
% 0.32/0.52 = B )
% 0.32/0.52 => ( A
% 0.32/0.52 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % eq_divide_imp
% 0.32/0.52 thf(fact_2044_nonzero__divide__eq__eq,axiom,
% 0.32/0.52 ! [C: complex,B: complex,A: complex] :
% 0.32/0.52 ( ( C != zero_zero_complex )
% 0.32/0.52 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.52 = A )
% 0.32/0.52 = ( B
% 0.32/0.52 = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nonzero_divide_eq_eq
% 0.32/0.52 thf(fact_2045_nonzero__divide__eq__eq,axiom,
% 0.32/0.52 ! [C: real,B: real,A: real] :
% 0.32/0.52 ( ( C != zero_zero_real )
% 0.32/0.52 => ( ( ( divide_divide_real @ B @ C )
% 0.32/0.52 = A )
% 0.32/0.52 = ( B
% 0.32/0.52 = ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nonzero_divide_eq_eq
% 0.32/0.52 thf(fact_2046_nonzero__divide__eq__eq,axiom,
% 0.32/0.52 ! [C: rat,B: rat,A: rat] :
% 0.32/0.52 ( ( C != zero_zero_rat )
% 0.32/0.52 => ( ( ( divide_divide_rat @ B @ C )
% 0.32/0.52 = A )
% 0.32/0.52 = ( B
% 0.32/0.52 = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nonzero_divide_eq_eq
% 0.32/0.52 thf(fact_2047_nonzero__eq__divide__eq,axiom,
% 0.32/0.52 ! [C: complex,A: complex,B: complex] :
% 0.32/0.52 ( ( C != zero_zero_complex )
% 0.32/0.52 => ( ( A
% 0.32/0.52 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.52 = ( ( times_times_complex @ A @ C )
% 0.32/0.52 = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nonzero_eq_divide_eq
% 0.32/0.52 thf(fact_2048_nonzero__eq__divide__eq,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( C != zero_zero_real )
% 0.32/0.52 => ( ( A
% 0.32/0.52 = ( divide_divide_real @ B @ C ) )
% 0.32/0.52 = ( ( times_times_real @ A @ C )
% 0.32/0.52 = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nonzero_eq_divide_eq
% 0.32/0.52 thf(fact_2049_nonzero__eq__divide__eq,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( C != zero_zero_rat )
% 0.32/0.52 => ( ( A
% 0.32/0.52 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.52 = ( ( times_times_rat @ A @ C )
% 0.32/0.52 = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nonzero_eq_divide_eq
% 0.32/0.52 thf(fact_2050_right__inverse__eq,axiom,
% 0.32/0.52 ! [B: complex,A: complex] :
% 0.32/0.52 ( ( B != zero_zero_complex )
% 0.32/0.52 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.52 = one_one_complex )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % right_inverse_eq
% 0.32/0.52 thf(fact_2051_right__inverse__eq,axiom,
% 0.32/0.52 ! [B: real,A: real] :
% 0.32/0.52 ( ( B != zero_zero_real )
% 0.32/0.52 => ( ( ( divide_divide_real @ A @ B )
% 0.32/0.52 = one_one_real )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % right_inverse_eq
% 0.32/0.52 thf(fact_2052_right__inverse__eq,axiom,
% 0.32/0.52 ! [B: rat,A: rat] :
% 0.32/0.52 ( ( B != zero_zero_rat )
% 0.32/0.52 => ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.52 = one_one_rat )
% 0.32/0.52 = ( A = B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % right_inverse_eq
% 0.32/0.52 thf(fact_2053_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.32/0.52 ! [A: code_integer,B: code_integer] :
% 0.32/0.52 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.52 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.32/0.52 thf(fact_2054_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.32/0.52 thf(fact_2055_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 0.32/0.52
% 0.32/0.52 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.32/0.52 thf(fact_2056_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.32/0.52 ! [B: nat,A: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.52 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.32/0.52 thf(fact_2057_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.32/0.52 ! [B: int,A: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.52 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.32/0.52 thf(fact_2058_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.32/0.52 ! [B: code_integer,A: code_integer] :
% 0.32/0.52 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.52 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 0.32/0.52
% 0.32/0.52 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.32/0.52 thf(fact_2059_power__0,axiom,
% 0.32/0.52 ! [A: rat] :
% 0.32/0.52 ( ( power_power_rat @ A @ zero_zero_nat )
% 0.32/0.52 = one_one_rat ) ).
% 0.32/0.52
% 0.32/0.52 % power_0
% 0.32/0.52 thf(fact_2060_power__0,axiom,
% 0.32/0.52 ! [A: nat] :
% 0.32/0.52 ( ( power_power_nat @ A @ zero_zero_nat )
% 0.32/0.52 = one_one_nat ) ).
% 0.32/0.52
% 0.32/0.52 % power_0
% 0.32/0.52 thf(fact_2061_power__0,axiom,
% 0.32/0.52 ! [A: real] :
% 0.32/0.52 ( ( power_power_real @ A @ zero_zero_nat )
% 0.32/0.52 = one_one_real ) ).
% 0.32/0.52
% 0.32/0.52 % power_0
% 0.32/0.52 thf(fact_2062_power__0,axiom,
% 0.32/0.52 ! [A: int] :
% 0.32/0.52 ( ( power_power_int @ A @ zero_zero_nat )
% 0.32/0.52 = one_one_int ) ).
% 0.32/0.52
% 0.32/0.52 % power_0
% 0.32/0.52 thf(fact_2063_power__0,axiom,
% 0.32/0.52 ! [A: complex] :
% 0.32/0.52 ( ( power_power_complex @ A @ zero_zero_nat )
% 0.32/0.52 = one_one_complex ) ).
% 0.32/0.52
% 0.32/0.52 % power_0
% 0.32/0.52 thf(fact_2064_mod__eq__self__iff__div__eq__0,axiom,
% 0.32/0.52 ! [A: nat,B: nat] :
% 0.32/0.52 ( ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.52 = A )
% 0.32/0.52 = ( ( divide_divide_nat @ A @ B )
% 0.32/0.52 = zero_zero_nat ) ) ).
% 0.32/0.52
% 0.32/0.52 % mod_eq_self_iff_div_eq_0
% 0.32/0.52 thf(fact_2065_mod__eq__self__iff__div__eq__0,axiom,
% 0.32/0.52 ! [A: int,B: int] :
% 0.32/0.52 ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.52 = A )
% 0.32/0.52 = ( ( divide_divide_int @ A @ B )
% 0.32/0.52 = zero_zero_int ) ) ).
% 0.32/0.52
% 0.32/0.52 % mod_eq_self_iff_div_eq_0
% 0.32/0.52 thf(fact_2066_mod__eq__self__iff__div__eq__0,axiom,
% 0.32/0.52 ! [A: code_integer,B: code_integer] :
% 0.32/0.52 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.52 = A )
% 0.32/0.52 = ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.52 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.52
% 0.32/0.52 % mod_eq_self_iff_div_eq_0
% 0.32/0.52 thf(fact_2067_ex__least__nat__le,axiom,
% 0.32/0.52 ! [P: nat > $o,N: nat] :
% 0.32/0.52 ( ( P @ N )
% 0.32/0.52 => ( ~ ( P @ zero_zero_nat )
% 0.32/0.52 => ? [K3: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ K3 @ N )
% 0.32/0.52 & ! [I3: nat] :
% 0.32/0.52 ( ( ord_less_nat @ I3 @ K3 )
% 0.32/0.52 => ~ ( P @ I3 ) )
% 0.32/0.52 & ( P @ K3 ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % ex_least_nat_le
% 0.32/0.52 thf(fact_2068_less__imp__add__positive,axiom,
% 0.32/0.52 ! [I: nat,J: nat] :
% 0.32/0.52 ( ( ord_less_nat @ I @ J )
% 0.32/0.52 => ? [K3: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 0.32/0.52 & ( ( plus_plus_nat @ I @ K3 )
% 0.32/0.52 = J ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % less_imp_add_positive
% 0.32/0.52 thf(fact_2069_diff__less,axiom,
% 0.32/0.52 ! [N: nat,M: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.52 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % diff_less
% 0.32/0.52 thf(fact_2070_mult__less__mono2,axiom,
% 0.32/0.52 ! [I: nat,J: nat,K: nat] :
% 0.32/0.52 ( ( ord_less_nat @ I @ J )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_mono2
% 0.32/0.52 thf(fact_2071_mult__less__mono1,axiom,
% 0.32/0.52 ! [I: nat,J: nat,K: nat] :
% 0.32/0.52 ( ( ord_less_nat @ I @ J )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_mono1
% 0.32/0.52 thf(fact_2072_nat__mult__less__cancel1,axiom,
% 0.32/0.52 ! [K: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.52 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.52 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_mult_less_cancel1
% 0.32/0.52 thf(fact_2073_nat__mult__eq__cancel1,axiom,
% 0.32/0.52 ! [K: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.52 => ( ( ( times_times_nat @ K @ M )
% 0.32/0.52 = ( times_times_nat @ K @ N ) )
% 0.32/0.52 = ( M = N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_mult_eq_cancel1
% 0.32/0.52 thf(fact_2074_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( ( divide_divide_nat @ M @ N )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 = ( ( ord_less_nat @ M @ N )
% 0.32/0.52 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % Euclidean_Division.div_eq_0_iff
% 0.32/0.52 thf(fact_2075_diff__add__0,axiom,
% 0.32/0.52 ! [N: nat,M: nat] :
% 0.32/0.52 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.52 = zero_zero_nat ) ).
% 0.32/0.52
% 0.32/0.52 % diff_add_0
% 0.32/0.52 thf(fact_2076_nat__power__less__imp__less,axiom,
% 0.32/0.52 ! [I: nat,M: nat,N: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.32/0.52 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.32/0.52 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % nat_power_less_imp_less
% 0.32/0.52 thf(fact_2077_mult__eq__self__implies__10,axiom,
% 0.32/0.52 ! [M: nat,N: nat] :
% 0.32/0.52 ( ( M
% 0.32/0.52 = ( times_times_nat @ M @ N ) )
% 0.32/0.52 => ( ( N = one_one_nat )
% 0.32/0.52 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_eq_self_implies_10
% 0.32/0.52 thf(fact_2078_mod__less__divisor,axiom,
% 0.32/0.52 ! [N: nat,M: nat] :
% 0.32/0.52 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.52 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.32/0.52
% 0.32/0.52 % mod_less_divisor
% 0.32/0.52 thf(fact_2079_mod__eq__0D,axiom,
% 0.32/0.52 ! [M: nat,D: nat] :
% 0.32/0.52 ( ( ( modulo_modulo_nat @ M @ D )
% 0.32/0.52 = zero_zero_nat )
% 0.32/0.52 => ? [Q3: nat] :
% 0.32/0.52 ( M
% 0.32/0.52 = ( times_times_nat @ D @ Q3 ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mod_eq_0D
% 0.32/0.52 thf(fact_2080_mult__less__le__imp__less,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_real @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_le_imp_less
% 0.32/0.52 thf(fact_2081_mult__less__le__imp__less,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_rat @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_le_imp_less
% 0.32/0.52 thf(fact_2082_mult__less__le__imp__less,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_nat @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_le_imp_less
% 0.32/0.52 thf(fact_2083_mult__less__le__imp__less,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.52 ( ( ord_less_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_eq_int @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_le_imp_less
% 0.32/0.52 thf(fact_2084_mult__le__less__imp__less,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_real @ C @ D )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_less_imp_less
% 0.32/0.52 thf(fact_2085_mult__le__less__imp__less,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_rat @ C @ D )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_less_imp_less
% 0.32/0.52 thf(fact_2086_mult__le__less__imp__less,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_nat @ C @ D )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_less_imp_less
% 0.32/0.52 thf(fact_2087_mult__le__less__imp__less,axiom,
% 0.32/0.52 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.52 => ( ( ord_less_int @ C @ D )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.52 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_less_imp_less
% 0.32/0.52 thf(fact_2088_mult__right__le__imp__le,axiom,
% 0.32/0.52 ! [A: real,C: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_le_imp_le
% 0.32/0.52 thf(fact_2089_mult__right__le__imp__le,axiom,
% 0.32/0.52 ! [A: rat,C: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_le_imp_le
% 0.32/0.52 thf(fact_2090_mult__right__le__imp__le,axiom,
% 0.32/0.52 ! [A: nat,C: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_le_imp_le
% 0.32/0.52 thf(fact_2091_mult__right__le__imp__le,axiom,
% 0.32/0.52 ! [A: int,C: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_right_le_imp_le
% 0.32/0.52 thf(fact_2092_mult__left__le__imp__le,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.52 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_le_imp_le
% 0.32/0.52 thf(fact_2093_mult__left__le__imp__le,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.52 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_le_imp_le
% 0.32/0.52 thf(fact_2094_mult__left__le__imp__le,axiom,
% 0.32/0.52 ! [C: nat,A: nat,B: nat] :
% 0.32/0.52 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.52 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_le_imp_le
% 0.32/0.52 thf(fact_2095_mult__left__le__imp__le,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.52 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_left_le_imp_le
% 0.32/0.52 thf(fact_2096_mult__le__cancel__left__pos,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.52 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_cancel_left_pos
% 0.32/0.52 thf(fact_2097_mult__le__cancel__left__pos,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.52 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_cancel_left_pos
% 0.32/0.52 thf(fact_2098_mult__le__cancel__left__pos,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.52 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_cancel_left_pos
% 0.32/0.52 thf(fact_2099_mult__le__cancel__left__neg,axiom,
% 0.32/0.52 ! [C: real,A: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.52 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_cancel_left_neg
% 0.32/0.52 thf(fact_2100_mult__le__cancel__left__neg,axiom,
% 0.32/0.52 ! [C: rat,A: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.52 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_cancel_left_neg
% 0.32/0.52 thf(fact_2101_mult__le__cancel__left__neg,axiom,
% 0.32/0.52 ! [C: int,A: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.52 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.52 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_le_cancel_left_neg
% 0.32/0.52 thf(fact_2102_mult__less__cancel__right,axiom,
% 0.32/0.52 ! [A: real,C: real,B: real] :
% 0.32/0.52 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.52 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ A @ B ) )
% 0.32/0.52 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.52 => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_right
% 0.32/0.52 thf(fact_2103_mult__less__cancel__right,axiom,
% 0.32/0.52 ! [A: rat,C: rat,B: rat] :
% 0.32/0.52 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.52 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ A @ B ) )
% 0.32/0.52 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.52 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_right
% 0.32/0.52 thf(fact_2104_mult__less__cancel__right,axiom,
% 0.32/0.52 ! [A: int,C: int,B: int] :
% 0.32/0.52 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.52 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.52 => ( ord_less_int @ A @ B ) )
% 0.32/0.52 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.52 => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_less_cancel_right
% 0.32/0.52 thf(fact_2105_mult__strict__mono_H,axiom,
% 0.32/0.52 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.52 ( ( ord_less_real @ A @ B )
% 0.32/0.52 => ( ( ord_less_real @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.52 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.52 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_mono'
% 0.32/0.52 thf(fact_2106_mult__strict__mono_H,axiom,
% 0.32/0.52 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.52 ( ( ord_less_rat @ A @ B )
% 0.32/0.52 => ( ( ord_less_rat @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.52 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.52 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.52
% 0.32/0.52 % mult_strict_mono'
% 0.32/0.52 thf(fact_2107_mult__strict__mono_H,axiom,
% 0.32/0.52 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.52 ( ( ord_less_nat @ A @ B )
% 0.32/0.52 => ( ( ord_less_nat @ C @ D )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.52 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.52 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono'
% 0.32/0.53 thf(fact_2108_mult__strict__mono_H,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_int @ C @ D )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono'
% 0.32/0.53 thf(fact_2109_mult__right__less__imp__less,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_less_imp_less
% 0.32/0.53 thf(fact_2110_mult__right__less__imp__less,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_less_imp_less
% 0.32/0.53 thf(fact_2111_mult__right__less__imp__less,axiom,
% 0.32/0.53 ! [A: nat,C: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_less_imp_less
% 0.32/0.53 thf(fact_2112_mult__right__less__imp__less,axiom,
% 0.32/0.53 ! [A: int,C: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_less_imp_less
% 0.32/0.53 thf(fact_2113_mult__less__cancel__left,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left
% 0.32/0.53 thf(fact_2114_mult__less__cancel__left,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left
% 0.32/0.53 thf(fact_2115_mult__less__cancel__left,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left
% 0.32/0.53 thf(fact_2116_mult__strict__mono,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_real @ C @ D )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono
% 0.32/0.53 thf(fact_2117_mult__strict__mono,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_rat @ C @ D )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono
% 0.32/0.53 thf(fact_2118_mult__strict__mono,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( ord_less_nat @ C @ D )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono
% 0.32/0.53 thf(fact_2119_mult__strict__mono,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( ord_less_int @ C @ D )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_strict_mono
% 0.32/0.53 thf(fact_2120_mult__left__less__imp__less,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_less_imp_less
% 0.32/0.53 thf(fact_2121_mult__left__less__imp__less,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_less_imp_less
% 0.32/0.53 thf(fact_2122_mult__left__less__imp__less,axiom,
% 0.32/0.53 ! [C: nat,A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_less_imp_less
% 0.32/0.53 thf(fact_2123_mult__left__less__imp__less,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_less_imp_less
% 0.32/0.53 thf(fact_2124_mult__le__cancel__right,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right
% 0.32/0.53 thf(fact_2125_mult__le__cancel__right,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right
% 0.32/0.53 thf(fact_2126_mult__le__cancel__right,axiom,
% 0.32/0.53 ! [A: int,C: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right
% 0.32/0.53 thf(fact_2127_mult__le__cancel__left,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left
% 0.32/0.53 thf(fact_2128_mult__le__cancel__left,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left
% 0.32/0.53 thf(fact_2129_mult__le__cancel__left,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left
% 0.32/0.53 thf(fact_2130_add__strict__increasing2,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_real @ B @ C )
% 0.32/0.53 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing2
% 0.32/0.53 thf(fact_2131_add__strict__increasing2,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_rat @ B @ C )
% 0.32/0.53 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing2
% 0.32/0.53 thf(fact_2132_add__strict__increasing2,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ B @ C )
% 0.32/0.53 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing2
% 0.32/0.53 thf(fact_2133_add__strict__increasing2,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ B @ C )
% 0.32/0.53 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing2
% 0.32/0.53 thf(fact_2134_add__strict__increasing,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.53 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing
% 0.32/0.53 thf(fact_2135_add__strict__increasing,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.53 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing
% 0.32/0.53 thf(fact_2136_add__strict__increasing,axiom,
% 0.32/0.53 ! [A: nat,B: nat,C: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.53 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing
% 0.32/0.53 thf(fact_2137_add__strict__increasing,axiom,
% 0.32/0.53 ! [A: int,B: int,C: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.53 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_strict_increasing
% 0.32/0.53 thf(fact_2138_add__pos__nonneg,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_pos_nonneg
% 0.32/0.53 thf(fact_2139_add__pos__nonneg,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_pos_nonneg
% 0.32/0.53 thf(fact_2140_add__pos__nonneg,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_pos_nonneg
% 0.32/0.53 thf(fact_2141_add__pos__nonneg,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_pos_nonneg
% 0.32/0.53 thf(fact_2142_add__nonpos__neg,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ B @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonpos_neg
% 0.32/0.53 thf(fact_2143_add__nonpos__neg,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonpos_neg
% 0.32/0.53 thf(fact_2144_add__nonpos__neg,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.32/0.53 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonpos_neg
% 0.32/0.53 thf(fact_2145_add__nonpos__neg,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.53 => ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonpos_neg
% 0.32/0.53 thf(fact_2146_add__nonneg__pos,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonneg_pos
% 0.32/0.53 thf(fact_2147_add__nonneg__pos,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonneg_pos
% 0.32/0.53 thf(fact_2148_add__nonneg__pos,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonneg_pos
% 0.32/0.53 thf(fact_2149_add__nonneg__pos,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_nonneg_pos
% 0.32/0.53 thf(fact_2150_add__neg__nonpos,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_neg_nonpos
% 0.32/0.53 thf(fact_2151_add__neg__nonpos,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_neg_nonpos
% 0.32/0.53 thf(fact_2152_add__neg__nonpos,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.32/0.53 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.32/0.53 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_neg_nonpos
% 0.32/0.53 thf(fact_2153_add__neg__nonpos,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.53 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_neg_nonpos
% 0.32/0.53 thf(fact_2154_field__le__epsilon,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ! [E2: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.32/0.53 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y3 @ E2 ) ) )
% 0.32/0.53 => ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.53
% 0.32/0.53 % field_le_epsilon
% 0.32/0.53 thf(fact_2155_field__le__epsilon,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ! [E2: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.32/0.53 => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y3 @ E2 ) ) )
% 0.32/0.53 => ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 0.32/0.53
% 0.32/0.53 % field_le_epsilon
% 0.32/0.53 thf(fact_2156_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( divide_divide_nat @ A @ B )
% 0.32/0.53 = zero_zero_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.div_less
% 0.32/0.53 thf(fact_2157_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( divide_divide_int @ A @ B )
% 0.32/0.53 = zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.div_less
% 0.32/0.53 thf(fact_2158_div__positive,axiom,
% 0.32/0.53 ! [B: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.53 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_positive
% 0.32/0.53 thf(fact_2159_div__positive,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ B @ A )
% 0.32/0.53 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_positive
% 0.32/0.53 thf(fact_2160_frac__le,axiom,
% 0.32/0.53 ! [Y3: real,X: real,W: real,Z: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.32/0.53 => ( ( ord_less_eq_real @ W @ Z )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_le
% 0.32/0.53 thf(fact_2161_frac__le,axiom,
% 0.32/0.53 ! [Y3: rat,X: rat,W: rat,Z: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.32/0.53 => ( ( ord_less_eq_rat @ W @ Z )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_le
% 0.32/0.53 thf(fact_2162_frac__less,axiom,
% 0.32/0.53 ! [X: real,Y3: real,W: real,Z: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_real @ X @ Y3 )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.32/0.53 => ( ( ord_less_eq_real @ W @ Z )
% 0.32/0.53 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less
% 0.32/0.53 thf(fact_2163_frac__less,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat,W: rat,Z: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.32/0.53 => ( ( ord_less_eq_rat @ W @ Z )
% 0.32/0.53 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less
% 0.32/0.53 thf(fact_2164_frac__less2,axiom,
% 0.32/0.53 ! [X: real,Y3: real,W: real,Z: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ W )
% 0.32/0.53 => ( ( ord_less_real @ W @ Z )
% 0.32/0.53 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less2
% 0.32/0.53 thf(fact_2165_frac__less2,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat,W: rat,Z: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 0.32/0.53 => ( ( ord_less_rat @ W @ Z )
% 0.32/0.53 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less2
% 0.32/0.53 thf(fact_2166_divide__le__cancel,axiom,
% 0.32/0.53 ! [A: real,C: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_cancel
% 0.32/0.53 thf(fact_2167_divide__le__cancel,axiom,
% 0.32/0.53 ! [A: rat,C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_cancel
% 0.32/0.53 thf(fact_2168_divide__nonneg__neg,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonneg_neg
% 0.32/0.53 thf(fact_2169_divide__nonneg__neg,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonneg_neg
% 0.32/0.53 thf(fact_2170_divide__nonneg__pos,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonneg_pos
% 0.32/0.53 thf(fact_2171_divide__nonneg__pos,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonneg_pos
% 0.32/0.53 thf(fact_2172_divide__nonpos__neg,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonpos_neg
% 0.32/0.53 thf(fact_2173_divide__nonpos__neg,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonpos_neg
% 0.32/0.53 thf(fact_2174_divide__nonpos__pos,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonpos_pos
% 0.32/0.53 thf(fact_2175_divide__nonpos__pos,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_nonpos_pos
% 0.32/0.53 thf(fact_2176_mult__left__le,axiom,
% 0.32/0.53 ! [C: real,A: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ C @ one_one_real )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le
% 0.32/0.53 thf(fact_2177_mult__left__le,axiom,
% 0.32/0.53 ! [C: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le
% 0.32/0.53 thf(fact_2178_mult__left__le,axiom,
% 0.32/0.53 ! [C: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le
% 0.32/0.53 thf(fact_2179_mult__left__le,axiom,
% 0.32/0.53 ! [C: int,A: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ C @ one_one_int )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le
% 0.32/0.53 thf(fact_2180_mult__le__one,axiom,
% 0.32/0.53 ! [A: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_one
% 0.32/0.53 thf(fact_2181_mult__le__one,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_one
% 0.32/0.53 thf(fact_2182_mult__le__one,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.32/0.53 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_one
% 0.32/0.53 thf(fact_2183_mult__le__one,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.32/0.53 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_one
% 0.32/0.53 thf(fact_2184_mult__right__le__one__le,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ X @ Y3 ) @ X ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_le_one_le
% 0.32/0.53 thf(fact_2185_mult__right__le__one__le,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y3 ) @ X ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_le_one_le
% 0.32/0.53 thf(fact_2186_mult__right__le__one__le,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.32/0.53 => ( ord_less_eq_int @ ( times_times_int @ X @ Y3 ) @ X ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_right_le_one_le
% 0.32/0.53 thf(fact_2187_mult__left__le__one__le,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X ) @ X ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le_one_le
% 0.32/0.53 thf(fact_2188_mult__left__le__one__le,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X ) @ X ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le_one_le
% 0.32/0.53 thf(fact_2189_mult__left__le__one__le,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.32/0.53 => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X ) @ X ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_left_le_one_le
% 0.32/0.53 thf(fact_2190_sum__squares__ge__zero,axiom,
% 0.32/0.53 ! [X: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_ge_zero
% 0.32/0.53 thf(fact_2191_sum__squares__ge__zero,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_ge_zero
% 0.32/0.53 thf(fact_2192_sum__squares__ge__zero,axiom,
% 0.32/0.53 ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_ge_zero
% 0.32/0.53 thf(fact_2193_sum__squares__le__zero__iff,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 0.32/0.53 = ( ( X = zero_zero_real )
% 0.32/0.53 & ( Y3 = zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_le_zero_iff
% 0.32/0.53 thf(fact_2194_sum__squares__le__zero__iff,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 0.32/0.53 = ( ( X = zero_zero_rat )
% 0.32/0.53 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_le_zero_iff
% 0.32/0.53 thf(fact_2195_sum__squares__le__zero__iff,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 0.32/0.53 = ( ( X = zero_zero_int )
% 0.32/0.53 & ( Y3 = zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_le_zero_iff
% 0.32/0.53 thf(fact_2196_power__less__imp__less__base,axiom,
% 0.32/0.53 ! [A: real,N: nat,B: real] :
% 0.32/0.53 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.53 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_base
% 0.32/0.53 thf(fact_2197_power__less__imp__less__base,axiom,
% 0.32/0.53 ! [A: rat,N: nat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.53 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_base
% 0.32/0.53 thf(fact_2198_power__less__imp__less__base,axiom,
% 0.32/0.53 ! [A: nat,N: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_base
% 0.32/0.53 thf(fact_2199_power__less__imp__less__base,axiom,
% 0.32/0.53 ! [A: int,N: nat,B: int] :
% 0.32/0.53 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_less_imp_less_base
% 0.32/0.53 thf(fact_2200_not__sum__squares__lt__zero,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 0.32/0.53
% 0.32/0.53 % not_sum_squares_lt_zero
% 0.32/0.53 thf(fact_2201_not__sum__squares__lt__zero,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 0.32/0.53
% 0.32/0.53 % not_sum_squares_lt_zero
% 0.32/0.53 thf(fact_2202_not__sum__squares__lt__zero,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 0.32/0.53
% 0.32/0.53 % not_sum_squares_lt_zero
% 0.32/0.53 thf(fact_2203_sum__squares__gt__zero__iff,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 0.32/0.53 = ( ( X != zero_zero_real )
% 0.32/0.53 | ( Y3 != zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_gt_zero_iff
% 0.32/0.53 thf(fact_2204_sum__squares__gt__zero__iff,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 0.32/0.53 = ( ( X != zero_zero_rat )
% 0.32/0.53 | ( Y3 != zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_gt_zero_iff
% 0.32/0.53 thf(fact_2205_sum__squares__gt__zero__iff,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 0.32/0.53 = ( ( X != zero_zero_int )
% 0.32/0.53 | ( Y3 != zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_squares_gt_zero_iff
% 0.32/0.53 thf(fact_2206_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.32/0.53 ! [C: nat,A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.53 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.32/0.53 thf(fact_2207_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.53 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.32/0.53 thf(fact_2208_zero__less__two,axiom,
% 0.32/0.53 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.32/0.53
% 0.32/0.53 % zero_less_two
% 0.32/0.53 thf(fact_2209_zero__less__two,axiom,
% 0.32/0.53 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.32/0.53
% 0.32/0.53 % zero_less_two
% 0.32/0.53 thf(fact_2210_zero__less__two,axiom,
% 0.32/0.53 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.32/0.53
% 0.32/0.53 % zero_less_two
% 0.32/0.53 thf(fact_2211_zero__less__two,axiom,
% 0.32/0.53 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.32/0.53
% 0.32/0.53 % zero_less_two
% 0.32/0.53 thf(fact_2212_divide__strict__left__mono__neg,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.53 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_strict_left_mono_neg
% 0.32/0.53 thf(fact_2213_divide__strict__left__mono__neg,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.53 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_strict_left_mono_neg
% 0.32/0.53 thf(fact_2214_divide__strict__left__mono,axiom,
% 0.32/0.53 ! [B: real,A: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ B @ A )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.53 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_strict_left_mono
% 0.32/0.53 thf(fact_2215_divide__strict__left__mono,axiom,
% 0.32/0.53 ! [B: rat,A: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ B @ A )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.53 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_strict_left_mono
% 0.32/0.53 thf(fact_2216_mult__imp__less__div__pos,axiom,
% 0.32/0.53 ! [Y3: real,Z: real,X: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X )
% 0.32/0.53 => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_imp_less_div_pos
% 0.32/0.53 thf(fact_2217_mult__imp__less__div__pos,axiom,
% 0.32/0.53 ! [Y3: rat,Z: rat,X: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X )
% 0.32/0.53 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_imp_less_div_pos
% 0.32/0.53 thf(fact_2218_mult__imp__div__pos__less,axiom,
% 0.32/0.53 ! [Y3: real,X: real,Z: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y3 ) )
% 0.32/0.53 => ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_imp_div_pos_less
% 0.32/0.53 thf(fact_2219_mult__imp__div__pos__less,axiom,
% 0.32/0.53 ! [Y3: rat,X: rat,Z: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y3 ) )
% 0.32/0.53 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_imp_div_pos_less
% 0.32/0.53 thf(fact_2220_pos__less__divide__eq,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_less_divide_eq
% 0.32/0.53 thf(fact_2221_pos__less__divide__eq,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_less_divide_eq
% 0.32/0.53 thf(fact_2222_pos__divide__less__eq,axiom,
% 0.32/0.53 ! [C: real,B: real,A: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.53 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_divide_less_eq
% 0.32/0.53 thf(fact_2223_pos__divide__less__eq,axiom,
% 0.32/0.53 ! [C: rat,B: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.53 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_divide_less_eq
% 0.32/0.53 thf(fact_2224_neg__less__divide__eq,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_less_divide_eq
% 0.32/0.53 thf(fact_2225_neg__less__divide__eq,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_less_divide_eq
% 0.32/0.53 thf(fact_2226_neg__divide__less__eq,axiom,
% 0.32/0.53 ! [C: real,B: real,A: real] :
% 0.32/0.53 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.53 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_divide_less_eq
% 0.32/0.53 thf(fact_2227_neg__divide__less__eq,axiom,
% 0.32/0.53 ! [C: rat,B: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.53 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_divide_less_eq
% 0.32/0.53 thf(fact_2228_less__divide__eq,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_divide_eq
% 0.32/0.53 thf(fact_2229_less__divide__eq,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_divide_eq
% 0.32/0.53 thf(fact_2230_divide__less__eq,axiom,
% 0.32/0.53 ! [B: real,C: real,A: real] :
% 0.32/0.53 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_less_eq
% 0.32/0.53 thf(fact_2231_divide__less__eq,axiom,
% 0.32/0.53 ! [B: rat,C: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_less_eq
% 0.32/0.53 thf(fact_2232_less__divide__eq__1,axiom,
% 0.32/0.53 ! [B: real,A: real] :
% 0.32/0.53 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 & ( ord_less_real @ A @ B ) )
% 0.32/0.53 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.53 & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_divide_eq_1
% 0.32/0.53 thf(fact_2233_less__divide__eq__1,axiom,
% 0.32/0.53 ! [B: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 & ( ord_less_rat @ A @ B ) )
% 0.32/0.53 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.53 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_divide_eq_1
% 0.32/0.53 thf(fact_2234_divide__less__eq__1,axiom,
% 0.32/0.53 ! [B: real,A: real] :
% 0.32/0.53 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 & ( ord_less_real @ B @ A ) )
% 0.32/0.53 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.53 & ( ord_less_real @ A @ B ) )
% 0.32/0.53 | ( A = zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_less_eq_1
% 0.32/0.53 thf(fact_2235_divide__less__eq__1,axiom,
% 0.32/0.53 ! [B: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 & ( ord_less_rat @ B @ A ) )
% 0.32/0.53 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.53 & ( ord_less_rat @ A @ B ) )
% 0.32/0.53 | ( A = zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_less_eq_1
% 0.32/0.53 thf(fact_2236_power__le__one,axiom,
% 0.32/0.53 ! [A: real,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_le_one
% 0.32/0.53 thf(fact_2237_power__le__one,axiom,
% 0.32/0.53 ! [A: rat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_le_one
% 0.32/0.53 thf(fact_2238_power__le__one,axiom,
% 0.32/0.53 ! [A: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.53 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_le_one
% 0.32/0.53 thf(fact_2239_power__le__one,axiom,
% 0.32/0.53 ! [A: int,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.53 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_le_one
% 0.32/0.53 thf(fact_2240_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [W: num,B: complex,C: complex] :
% 0.32/0.53 ( ( ( numera6690914467698888265omplex @ W )
% 0.32/0.53 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.53 = ( ( ( C != zero_zero_complex )
% 0.32/0.53 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 0.32/0.53 = B ) )
% 0.32/0.53 & ( ( C = zero_zero_complex )
% 0.32/0.53 => ( ( numera6690914467698888265omplex @ W )
% 0.32/0.53 = zero_zero_complex ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % eq_divide_eq_numeral(1)
% 0.32/0.53 thf(fact_2241_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [W: num,B: real,C: real] :
% 0.32/0.53 ( ( ( numeral_numeral_real @ W )
% 0.32/0.53 = ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ( ( C != zero_zero_real )
% 0.32/0.53 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 0.32/0.53 = B ) )
% 0.32/0.53 & ( ( C = zero_zero_real )
% 0.32/0.53 => ( ( numeral_numeral_real @ W )
% 0.32/0.53 = zero_zero_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % eq_divide_eq_numeral(1)
% 0.32/0.53 thf(fact_2242_eq__divide__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [W: num,B: rat,C: rat] :
% 0.32/0.53 ( ( ( numeral_numeral_rat @ W )
% 0.32/0.53 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( C != zero_zero_rat )
% 0.32/0.53 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 0.32/0.53 = B ) )
% 0.32/0.53 & ( ( C = zero_zero_rat )
% 0.32/0.53 => ( ( numeral_numeral_rat @ W )
% 0.32/0.53 = zero_zero_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % eq_divide_eq_numeral(1)
% 0.32/0.53 thf(fact_2243_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [B: complex,C: complex,W: num] :
% 0.32/0.53 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.53 = ( numera6690914467698888265omplex @ W ) )
% 0.32/0.53 = ( ( ( C != zero_zero_complex )
% 0.32/0.53 => ( B
% 0.32/0.53 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 0.32/0.53 & ( ( C = zero_zero_complex )
% 0.32/0.53 => ( ( numera6690914467698888265omplex @ W )
% 0.32/0.53 = zero_zero_complex ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_eq_eq_numeral(1)
% 0.32/0.53 thf(fact_2244_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [B: real,C: real,W: num] :
% 0.32/0.53 ( ( ( divide_divide_real @ B @ C )
% 0.32/0.53 = ( numeral_numeral_real @ W ) )
% 0.32/0.53 = ( ( ( C != zero_zero_real )
% 0.32/0.53 => ( B
% 0.32/0.53 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.53 & ( ( C = zero_zero_real )
% 0.32/0.53 => ( ( numeral_numeral_real @ W )
% 0.32/0.53 = zero_zero_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_eq_eq_numeral(1)
% 0.32/0.53 thf(fact_2245_divide__eq__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [B: rat,C: rat,W: num] :
% 0.32/0.53 ( ( ( divide_divide_rat @ B @ C )
% 0.32/0.53 = ( numeral_numeral_rat @ W ) )
% 0.32/0.53 = ( ( ( C != zero_zero_rat )
% 0.32/0.53 => ( B
% 0.32/0.53 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.53 & ( ( C = zero_zero_rat )
% 0.32/0.53 => ( ( numeral_numeral_rat @ W )
% 0.32/0.53 = zero_zero_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_eq_eq_numeral(1)
% 0.32/0.53 thf(fact_2246_divide__add__eq__iff,axiom,
% 0.32/0.53 ! [Z: complex,X: complex,Y3: complex] :
% 0.32/0.53 ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y3 )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_add_eq_iff
% 0.32/0.53 thf(fact_2247_divide__add__eq__iff,axiom,
% 0.32/0.53 ! [Z: real,X: real,Y3: real] :
% 0.32/0.53 ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y3 )
% 0.32/0.53 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_add_eq_iff
% 0.32/0.53 thf(fact_2248_divide__add__eq__iff,axiom,
% 0.32/0.53 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.53 ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y3 )
% 0.32/0.53 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_add_eq_iff
% 0.32/0.53 thf(fact_2249_add__divide__eq__iff,axiom,
% 0.32/0.53 ! [Z: complex,X: complex,Y3: complex] :
% 0.32/0.53 ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_iff
% 0.32/0.53 thf(fact_2250_add__divide__eq__iff,axiom,
% 0.32/0.53 ! [Z: real,X: real,Y3: real] :
% 0.32/0.53 ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y3 @ Z ) )
% 0.32/0.53 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_iff
% 0.32/0.53 thf(fact_2251_add__divide__eq__iff,axiom,
% 0.32/0.53 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.53 ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.32/0.53 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_iff
% 0.32/0.53 thf(fact_2252_add__num__frac,axiom,
% 0.32/0.53 ! [Y3: complex,Z: complex,X: complex] :
% 0.32/0.53 ( ( Y3 != zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y3 ) )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_num_frac
% 0.32/0.53 thf(fact_2253_add__num__frac,axiom,
% 0.32/0.53 ! [Y3: real,Z: real,X: real] :
% 0.32/0.53 ( ( Y3 != zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y3 ) )
% 0.32/0.53 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_num_frac
% 0.32/0.53 thf(fact_2254_add__num__frac,axiom,
% 0.32/0.53 ! [Y3: rat,Z: rat,X: rat] :
% 0.32/0.53 ( ( Y3 != zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y3 ) )
% 0.32/0.53 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_num_frac
% 0.32/0.53 thf(fact_2255_add__frac__num,axiom,
% 0.32/0.53 ! [Y3: complex,X: complex,Z: complex] :
% 0.32/0.53 ( ( Y3 != zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ Z )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_frac_num
% 0.32/0.53 thf(fact_2256_add__frac__num,axiom,
% 0.32/0.53 ! [Y3: real,X: real,Z: real] :
% 0.32/0.53 ( ( Y3 != zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y3 ) @ Z )
% 0.32/0.53 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_frac_num
% 0.32/0.53 thf(fact_2257_add__frac__num,axiom,
% 0.32/0.53 ! [Y3: rat,X: rat,Z: rat] :
% 0.32/0.53 ( ( Y3 != zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z )
% 0.32/0.53 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_frac_num
% 0.32/0.53 thf(fact_2258_add__frac__eq,axiom,
% 0.32/0.53 ! [Y3: complex,Z: complex,X: complex,W: complex] :
% 0.32/0.53 ( ( Y3 != zero_zero_complex )
% 0.32/0.53 => ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_frac_eq
% 0.32/0.53 thf(fact_2259_add__frac__eq,axiom,
% 0.32/0.53 ! [Y3: real,Z: real,X: real,W: real] :
% 0.32/0.53 ( ( Y3 != zero_zero_real )
% 0.32/0.53 => ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.32/0.53 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_frac_eq
% 0.32/0.53 thf(fact_2260_add__frac__eq,axiom,
% 0.32/0.53 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.53 ( ( Y3 != zero_zero_rat )
% 0.32/0.53 => ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.32/0.53 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_frac_eq
% 0.32/0.53 thf(fact_2261_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.53 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.53 ( ( ( Z = zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.32/0.53 = A ) )
% 0.32/0.53 & ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(1)
% 0.32/0.53 thf(fact_2262_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.53 ! [Z: real,A: real,B: real] :
% 0.32/0.53 ( ( ( Z = zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.32/0.53 = A ) )
% 0.32/0.53 & ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.32/0.53 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(1)
% 0.32/0.53 thf(fact_2263_add__divide__eq__if__simps_I1_J,axiom,
% 0.32/0.53 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ( Z = zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.32/0.53 = A ) )
% 0.32/0.53 & ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.32/0.53 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(1)
% 0.32/0.53 thf(fact_2264_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.53 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.53 ( ( ( Z = zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.32/0.53 = B ) )
% 0.32/0.53 & ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(2)
% 0.32/0.53 thf(fact_2265_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.53 ! [Z: real,A: real,B: real] :
% 0.32/0.53 ( ( ( Z = zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.32/0.53 = B ) )
% 0.32/0.53 & ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.32/0.53 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(2)
% 0.32/0.53 thf(fact_2266_add__divide__eq__if__simps_I2_J,axiom,
% 0.32/0.53 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ( Z = zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.32/0.53 = B ) )
% 0.32/0.53 & ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.32/0.53 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(2)
% 0.32/0.53 thf(fact_2267_div__add__self2,axiom,
% 0.32/0.53 ! [B: nat,A: nat] :
% 0.32/0.53 ( ( B != zero_zero_nat )
% 0.32/0.53 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.32/0.53 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_add_self2
% 0.32/0.53 thf(fact_2268_div__add__self2,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( B != zero_zero_int )
% 0.32/0.53 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.32/0.53 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_add_self2
% 0.32/0.53 thf(fact_2269_div__add__self1,axiom,
% 0.32/0.53 ! [B: nat,A: nat] :
% 0.32/0.53 ( ( B != zero_zero_nat )
% 0.32/0.53 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.32/0.53 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_add_self1
% 0.32/0.53 thf(fact_2270_div__add__self1,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( B != zero_zero_int )
% 0.32/0.53 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.32/0.53 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_add_self1
% 0.32/0.53 thf(fact_2271_divide__diff__eq__iff,axiom,
% 0.32/0.53 ! [Z: complex,X: complex,Y3: complex] :
% 0.32/0.53 ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y3 )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_diff_eq_iff
% 0.32/0.53 thf(fact_2272_divide__diff__eq__iff,axiom,
% 0.32/0.53 ! [Z: real,X: real,Y3: real] :
% 0.32/0.53 ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y3 )
% 0.32/0.53 = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_diff_eq_iff
% 0.32/0.53 thf(fact_2273_divide__diff__eq__iff,axiom,
% 0.32/0.53 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.53 ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y3 )
% 0.32/0.53 = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_diff_eq_iff
% 0.32/0.53 thf(fact_2274_diff__divide__eq__iff,axiom,
% 0.32/0.53 ! [Z: complex,X: complex,Y3: complex] :
% 0.32/0.53 ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_divide_eq_iff
% 0.32/0.53 thf(fact_2275_diff__divide__eq__iff,axiom,
% 0.32/0.53 ! [Z: real,X: real,Y3: real] :
% 0.32/0.53 ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y3 @ Z ) )
% 0.32/0.53 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_divide_eq_iff
% 0.32/0.53 thf(fact_2276_diff__divide__eq__iff,axiom,
% 0.32/0.53 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.53 ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.32/0.53 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_divide_eq_iff
% 0.32/0.53 thf(fact_2277_diff__frac__eq,axiom,
% 0.32/0.53 ! [Y3: complex,Z: complex,X: complex,W: complex] :
% 0.32/0.53 ( ( Y3 != zero_zero_complex )
% 0.32/0.53 => ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_frac_eq
% 0.32/0.53 thf(fact_2278_diff__frac__eq,axiom,
% 0.32/0.53 ! [Y3: real,Z: real,X: real,W: real] :
% 0.32/0.53 ( ( Y3 != zero_zero_real )
% 0.32/0.53 => ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.32/0.53 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_frac_eq
% 0.32/0.53 thf(fact_2279_diff__frac__eq,axiom,
% 0.32/0.53 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.53 ( ( Y3 != zero_zero_rat )
% 0.32/0.53 => ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.32/0.53 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % diff_frac_eq
% 0.32/0.53 thf(fact_2280_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.53 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.53 ( ( ( Z = zero_zero_complex )
% 0.32/0.53 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.32/0.53 = A ) )
% 0.32/0.53 & ( ( Z != zero_zero_complex )
% 0.32/0.53 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(4)
% 0.32/0.53 thf(fact_2281_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.53 ! [Z: real,A: real,B: real] :
% 0.32/0.53 ( ( ( Z = zero_zero_real )
% 0.32/0.53 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.32/0.53 = A ) )
% 0.32/0.53 & ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 0.32/0.53 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(4)
% 0.32/0.53 thf(fact_2282_add__divide__eq__if__simps_I4_J,axiom,
% 0.32/0.53 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ( Z = zero_zero_rat )
% 0.32/0.53 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.32/0.53 = A ) )
% 0.32/0.53 & ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 0.32/0.53 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % add_divide_eq_if_simps(4)
% 0.32/0.53 thf(fact_2283_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.32/0.53 ! [B: code_integer,A: code_integer] :
% 0.32/0.53 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.53 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.32/0.53 thf(fact_2284_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.32/0.53 ! [B: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.32/0.53 thf(fact_2285_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.32/0.53 thf(fact_2286_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.32/0.53 ! [A: code_integer,B: code_integer] :
% 0.32/0.53 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.53 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.32/0.53 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.53 = A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.mod_less
% 0.32/0.53 thf(fact_2287_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.53 = A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.mod_less
% 0.32/0.53 thf(fact_2288_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ A @ B )
% 0.32/0.53 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.53 = A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.mod_less
% 0.32/0.53 thf(fact_2289_cong__exp__iff__simps_I2_J,axiom,
% 0.32/0.53 ! [N: num,Q2: num] :
% 0.32/0.53 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.53 = zero_zero_nat )
% 0.32/0.53 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.32/0.53 = zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % cong_exp_iff_simps(2)
% 0.32/0.53 thf(fact_2290_cong__exp__iff__simps_I2_J,axiom,
% 0.32/0.53 ! [N: num,Q2: num] :
% 0.32/0.53 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.53 = zero_zero_int )
% 0.32/0.53 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.32/0.53 = zero_zero_int ) ) ).
% 0.32/0.53
% 0.32/0.53 % cong_exp_iff_simps(2)
% 0.32/0.53 thf(fact_2291_cong__exp__iff__simps_I2_J,axiom,
% 0.32/0.53 ! [N: num,Q2: num] :
% 0.32/0.53 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.53 = zero_z3403309356797280102nteger )
% 0.32/0.53 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.32/0.53 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.53
% 0.32/0.53 % cong_exp_iff_simps(2)
% 0.32/0.53 thf(fact_2292_cong__exp__iff__simps_I1_J,axiom,
% 0.32/0.53 ! [N: num] :
% 0.32/0.53 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.32/0.53 = zero_zero_nat ) ).
% 0.32/0.53
% 0.32/0.53 % cong_exp_iff_simps(1)
% 0.32/0.53 thf(fact_2293_cong__exp__iff__simps_I1_J,axiom,
% 0.32/0.53 ! [N: num] :
% 0.32/0.53 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.32/0.53 = zero_zero_int ) ).
% 0.32/0.53
% 0.32/0.53 % cong_exp_iff_simps(1)
% 0.32/0.53 thf(fact_2294_cong__exp__iff__simps_I1_J,axiom,
% 0.32/0.53 ! [N: num] :
% 0.32/0.53 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.32/0.53 = zero_z3403309356797280102nteger ) ).
% 0.32/0.53
% 0.32/0.53 % cong_exp_iff_simps(1)
% 0.32/0.53 thf(fact_2295_bounded__Max__nat,axiom,
% 0.32/0.53 ! [P: nat > $o,X: nat,M5: nat] :
% 0.32/0.53 ( ( P @ X )
% 0.32/0.53 => ( ! [X3: nat] :
% 0.32/0.53 ( ( P @ X3 )
% 0.32/0.53 => ( ord_less_eq_nat @ X3 @ M5 ) )
% 0.32/0.53 => ~ ! [M4: nat] :
% 0.32/0.53 ( ( P @ M4 )
% 0.32/0.53 => ~ ! [X5: nat] :
% 0.32/0.53 ( ( P @ X5 )
% 0.32/0.53 => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % bounded_Max_nat
% 0.32/0.53 thf(fact_2296_not__exp__less__eq__0__int,axiom,
% 0.32/0.53 ! [N: nat] :
% 0.32/0.53 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 0.32/0.53
% 0.32/0.53 % not_exp_less_eq_0_int
% 0.32/0.53 thf(fact_2297_nat__mult__le__cancel1,axiom,
% 0.32/0.53 ! [K: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.53 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.53 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_mult_le_cancel1
% 0.32/0.53 thf(fact_2298_nat__diff__split,axiom,
% 0.32/0.53 ! [P: nat > $o,A: nat,B: nat] :
% 0.32/0.53 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.53 = ( ( ( ord_less_nat @ A @ B )
% 0.32/0.53 => ( P @ zero_zero_nat ) )
% 0.32/0.53 & ! [D2: nat] :
% 0.32/0.53 ( ( A
% 0.32/0.53 = ( plus_plus_nat @ B @ D2 ) )
% 0.32/0.53 => ( P @ D2 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_diff_split
% 0.32/0.53 thf(fact_2299_nat__diff__split__asm,axiom,
% 0.32/0.53 ! [P: nat > $o,A: nat,B: nat] :
% 0.32/0.53 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.53 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.32/0.53 & ~ ( P @ zero_zero_nat ) )
% 0.32/0.53 | ? [D2: nat] :
% 0.32/0.53 ( ( A
% 0.32/0.53 = ( plus_plus_nat @ B @ D2 ) )
% 0.32/0.53 & ~ ( P @ D2 ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_diff_split_asm
% 0.32/0.53 thf(fact_2300_div__le__mono2,axiom,
% 0.32/0.53 ! [M: nat,N: nat,K: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.53 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_le_mono2
% 0.32/0.53 thf(fact_2301_div__greater__zero__iff,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.53 = ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_greater_zero_iff
% 0.32/0.53 thf(fact_2302_real__arch__pow__inv,axiom,
% 0.32/0.53 ! [Y3: real,X: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.53 => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % real_arch_pow_inv
% 0.32/0.53 thf(fact_2303_div__less__iff__less__mult,axiom,
% 0.32/0.53 ! [Q2: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.32/0.53 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 0.32/0.53 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_less_iff_less_mult
% 0.32/0.53 thf(fact_2304_nat__mult__div__cancel1,axiom,
% 0.32/0.53 ! [K: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.53 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.53 = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_mult_div_cancel1
% 0.32/0.53 thf(fact_2305_div__eq__dividend__iff,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.53 => ( ( ( divide_divide_nat @ M @ N )
% 0.32/0.53 = M )
% 0.32/0.53 = ( N = one_one_nat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_eq_dividend_iff
% 0.32/0.53 thf(fact_2306_div__less__dividend,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ one_one_nat @ N )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.53 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_less_dividend
% 0.32/0.53 thf(fact_2307_mod__le__divisor,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.32/0.53
% 0.32/0.53 % mod_le_divisor
% 0.32/0.53 thf(fact_2308_div__less__mono,axiom,
% 0.32/0.53 ! [A2: nat,B5: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A2 @ B5 )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( ( modulo_modulo_nat @ A2 @ N )
% 0.32/0.53 = zero_zero_nat )
% 0.32/0.53 => ( ( ( modulo_modulo_nat @ B5 @ N )
% 0.32/0.53 = zero_zero_nat )
% 0.32/0.53 => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % div_less_mono
% 0.32/0.53 thf(fact_2309_mult__less__cancel__right2,axiom,
% 0.32/0.53 ! [A: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.32/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ one_one_real ) )
% 0.32/0.53 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right2
% 0.32/0.53 thf(fact_2310_mult__less__cancel__right2,axiom,
% 0.32/0.53 ! [A: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.32/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.32/0.53 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right2
% 0.32/0.53 thf(fact_2311_mult__less__cancel__right2,axiom,
% 0.32/0.53 ! [A: int,C: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.32/0.53 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ one_one_int ) )
% 0.32/0.53 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right2
% 0.32/0.53 thf(fact_2312_mult__less__cancel__right1,axiom,
% 0.32/0.53 ! [C: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ one_one_real @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right1
% 0.32/0.53 thf(fact_2313_mult__less__cancel__right1,axiom,
% 0.32/0.53 ! [C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right1
% 0.32/0.53 thf(fact_2314_mult__less__cancel__right1,axiom,
% 0.32/0.53 ! [C: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ one_one_int @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_right1
% 0.32/0.53 thf(fact_2315_mult__less__cancel__left2,axiom,
% 0.32/0.53 ! [C: real,A: real] :
% 0.32/0.53 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.32/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ A @ one_one_real ) )
% 0.32/0.53 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left2
% 0.32/0.53 thf(fact_2316_mult__less__cancel__left2,axiom,
% 0.32/0.53 ! [C: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.32/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ A @ one_one_rat ) )
% 0.32/0.53 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left2
% 0.32/0.53 thf(fact_2317_mult__less__cancel__left2,axiom,
% 0.32/0.53 ! [C: int,A: int] :
% 0.32/0.53 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.32/0.53 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ A @ one_one_int ) )
% 0.32/0.53 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left2
% 0.32/0.53 thf(fact_2318_mult__less__cancel__left1,axiom,
% 0.32/0.53 ! [C: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ one_one_real @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left1
% 0.32/0.53 thf(fact_2319_mult__less__cancel__left1,axiom,
% 0.32/0.53 ! [C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ one_one_rat @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left1
% 0.32/0.53 thf(fact_2320_mult__less__cancel__left1,axiom,
% 0.32/0.53 ! [C: int,B: int] :
% 0.32/0.53 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_int @ one_one_int @ B ) )
% 0.32/0.53 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_less_cancel_left1
% 0.32/0.53 thf(fact_2321_mult__le__cancel__right2,axiom,
% 0.32/0.53 ! [A: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right2
% 0.32/0.53 thf(fact_2322_mult__le__cancel__right2,axiom,
% 0.32/0.53 ! [A: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right2
% 0.32/0.53 thf(fact_2323_mult__le__cancel__right2,axiom,
% 0.32/0.53 ! [A: int,C: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.32/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.32/0.53 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right2
% 0.32/0.53 thf(fact_2324_mult__le__cancel__right1,axiom,
% 0.32/0.53 ! [C: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right1
% 0.32/0.53 thf(fact_2325_mult__le__cancel__right1,axiom,
% 0.32/0.53 ! [C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right1
% 0.32/0.53 thf(fact_2326_mult__le__cancel__right1,axiom,
% 0.32/0.53 ! [C: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.32/0.53 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_right1
% 0.32/0.53 thf(fact_2327_mult__le__cancel__left2,axiom,
% 0.32/0.53 ! [C: real,A: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left2
% 0.32/0.53 thf(fact_2328_mult__le__cancel__left2,axiom,
% 0.32/0.53 ! [C: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left2
% 0.32/0.53 thf(fact_2329_mult__le__cancel__left2,axiom,
% 0.32/0.53 ! [C: int,A: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.32/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.32/0.53 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left2
% 0.32/0.53 thf(fact_2330_mult__le__cancel__left1,axiom,
% 0.32/0.53 ! [C: real,B: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.32/0.53 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left1
% 0.32/0.53 thf(fact_2331_mult__le__cancel__left1,axiom,
% 0.32/0.53 ! [C: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.32/0.53 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left1
% 0.32/0.53 thf(fact_2332_mult__le__cancel__left1,axiom,
% 0.32/0.53 ! [C: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.32/0.53 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.32/0.53 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.32/0.53 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_le_cancel_left1
% 0.32/0.53 thf(fact_2333_field__le__mult__one__interval,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ! [Z5: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ Z5 )
% 0.32/0.53 => ( ( ord_less_real @ Z5 @ one_one_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ Z5 @ X ) @ Y3 ) ) )
% 0.32/0.53 => ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.53
% 0.32/0.53 % field_le_mult_one_interval
% 0.32/0.53 thf(fact_2334_field__le__mult__one__interval,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ! [Z5: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ Z5 )
% 0.32/0.53 => ( ( ord_less_rat @ Z5 @ one_one_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ Z5 @ X ) @ Y3 ) ) )
% 0.32/0.53 => ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 0.32/0.53
% 0.32/0.53 % field_le_mult_one_interval
% 0.32/0.53 thf(fact_2335_divide__le__eq,axiom,
% 0.32/0.53 ! [B: real,C: real,A: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_eq
% 0.32/0.53 thf(fact_2336_divide__le__eq,axiom,
% 0.32/0.53 ! [B: rat,C: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_eq
% 0.32/0.53 thf(fact_2337_le__divide__eq,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_divide_eq
% 0.32/0.53 thf(fact_2338_le__divide__eq,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_divide_eq
% 0.32/0.53 thf(fact_2339_divide__left__mono,axiom,
% 0.32/0.53 ! [B: real,A: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_left_mono
% 0.32/0.53 thf(fact_2340_divide__left__mono,axiom,
% 0.32/0.53 ! [B: rat,A: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_left_mono
% 0.32/0.53 thf(fact_2341_neg__divide__le__eq,axiom,
% 0.32/0.53 ! [C: real,B: real,A: real] :
% 0.32/0.53 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.53 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_divide_le_eq
% 0.32/0.53 thf(fact_2342_neg__divide__le__eq,axiom,
% 0.32/0.53 ! [C: rat,B: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.53 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_divide_le_eq
% 0.32/0.53 thf(fact_2343_neg__le__divide__eq,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_le_divide_eq
% 0.32/0.53 thf(fact_2344_neg__le__divide__eq,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_le_divide_eq
% 0.32/0.53 thf(fact_2345_pos__divide__le__eq,axiom,
% 0.32/0.53 ! [C: real,B: real,A: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.32/0.53 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_divide_le_eq
% 0.32/0.53 thf(fact_2346_pos__divide__le__eq,axiom,
% 0.32/0.53 ! [C: rat,B: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.32/0.53 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_divide_le_eq
% 0.32/0.53 thf(fact_2347_pos__le__divide__eq,axiom,
% 0.32/0.53 ! [C: real,A: real,B: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_le_divide_eq
% 0.32/0.53 thf(fact_2348_pos__le__divide__eq,axiom,
% 0.32/0.53 ! [C: rat,A: rat,B: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_le_divide_eq
% 0.32/0.53 thf(fact_2349_mult__imp__div__pos__le,axiom,
% 0.32/0.53 ! [Y3: real,X: real,Z: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y3 ) )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_imp_div_pos_le
% 0.32/0.53 thf(fact_2350_mult__imp__div__pos__le,axiom,
% 0.32/0.53 ! [Y3: rat,X: rat,Z: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y3 ) )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_imp_div_pos_le
% 0.32/0.53 thf(fact_2351_mult__imp__le__div__pos,axiom,
% 0.32/0.53 ! [Y3: real,Z: real,X: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X )
% 0.32/0.53 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_imp_le_div_pos
% 0.32/0.53 thf(fact_2352_mult__imp__le__div__pos,axiom,
% 0.32/0.53 ! [Y3: rat,Z: rat,X: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X )
% 0.32/0.53 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_imp_le_div_pos
% 0.32/0.53 thf(fact_2353_divide__left__mono__neg,axiom,
% 0.32/0.53 ! [A: real,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.32/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_left_mono_neg
% 0.32/0.53 thf(fact_2354_divide__left__mono__neg,axiom,
% 0.32/0.53 ! [A: rat,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.53 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_left_mono_neg
% 0.32/0.53 thf(fact_2355_divide__le__eq__1,axiom,
% 0.32/0.53 ! [B: real,A: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 & ( ord_less_eq_real @ B @ A ) )
% 0.32/0.53 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.53 & ( ord_less_eq_real @ A @ B ) )
% 0.32/0.53 | ( A = zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_eq_1
% 0.32/0.53 thf(fact_2356_divide__le__eq__1,axiom,
% 0.32/0.53 ! [B: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 & ( ord_less_eq_rat @ B @ A ) )
% 0.32/0.53 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.53 & ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.53 | ( A = zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_eq_1
% 0.32/0.53 thf(fact_2357_le__divide__eq__1,axiom,
% 0.32/0.53 ! [B: real,A: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 & ( ord_less_eq_real @ A @ B ) )
% 0.32/0.53 | ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.53 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_divide_eq_1
% 0.32/0.53 thf(fact_2358_le__divide__eq__1,axiom,
% 0.32/0.53 ! [B: rat,A: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 & ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.53 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.53 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_divide_eq_1
% 0.32/0.53 thf(fact_2359_convex__bound__le,axiom,
% 0.32/0.53 ! [X: real,A: real,Y3: real,U: real,V: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ X @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ Y3 @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.32/0.53 => ( ( ( plus_plus_real @ U @ V )
% 0.32/0.53 = one_one_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % convex_bound_le
% 0.32/0.53 thf(fact_2360_convex__bound__le,axiom,
% 0.32/0.53 ! [X: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ X @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ Y3 @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.32/0.53 => ( ( ( plus_plus_rat @ U @ V )
% 0.32/0.53 = one_one_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % convex_bound_le
% 0.32/0.53 thf(fact_2361_convex__bound__le,axiom,
% 0.32/0.53 ! [X: int,A: int,Y3: int,U: int,V: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ X @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ Y3 @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.32/0.53 => ( ( ( plus_plus_int @ U @ V )
% 0.32/0.53 = one_one_int )
% 0.32/0.53 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % convex_bound_le
% 0.32/0.53 thf(fact_2362_less__divide__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [W: num,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_divide_eq_numeral(1)
% 0.32/0.53 thf(fact_2363_less__divide__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [W: num,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_divide_eq_numeral(1)
% 0.32/0.53 thf(fact_2364_divide__less__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [B: real,C: real,W: num] :
% 0.32/0.53 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_less_eq_numeral(1)
% 0.32/0.53 thf(fact_2365_divide__less__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [B: rat,C: rat,W: num] :
% 0.32/0.53 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_less_eq_numeral(1)
% 0.32/0.53 thf(fact_2366_frac__le__eq,axiom,
% 0.32/0.53 ! [Y3: real,Z: real,X: real,W: real] :
% 0.32/0.53 ( ( Y3 != zero_zero_real )
% 0.32/0.53 => ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.32/0.53 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_le_eq
% 0.32/0.53 thf(fact_2367_frac__le__eq,axiom,
% 0.32/0.53 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.53 ( ( Y3 != zero_zero_rat )
% 0.32/0.53 => ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.32/0.53 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_le_eq
% 0.32/0.53 thf(fact_2368_frac__less__eq,axiom,
% 0.32/0.53 ! [Y3: real,Z: real,X: real,W: real] :
% 0.32/0.53 ( ( Y3 != zero_zero_real )
% 0.32/0.53 => ( ( Z != zero_zero_real )
% 0.32/0.53 => ( ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 0.32/0.53 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less_eq
% 0.32/0.53 thf(fact_2369_frac__less__eq,axiom,
% 0.32/0.53 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 0.32/0.53 ( ( Y3 != zero_zero_rat )
% 0.32/0.53 => ( ( Z != zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 0.32/0.53 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % frac_less_eq
% 0.32/0.53 thf(fact_2370_power__Suc__less,axiom,
% 0.32/0.53 ! [A: real,N: nat] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.53 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_Suc_less
% 0.32/0.53 thf(fact_2371_power__Suc__less,axiom,
% 0.32/0.53 ! [A: rat,N: nat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.53 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_Suc_less
% 0.32/0.53 thf(fact_2372_power__Suc__less,axiom,
% 0.32/0.53 ! [A: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.53 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_Suc_less
% 0.32/0.53 thf(fact_2373_power__Suc__less,axiom,
% 0.32/0.53 ! [A: int,N: nat] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.53 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_Suc_less
% 0.32/0.53 thf(fact_2374_power__strict__decreasing,axiom,
% 0.32/0.53 ! [N: nat,N3: nat,A: real] :
% 0.32/0.53 ( ( ord_less_nat @ N @ N3 )
% 0.32/0.53 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.53 => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_decreasing
% 0.32/0.53 thf(fact_2375_power__strict__decreasing,axiom,
% 0.32/0.53 ! [N: nat,N3: nat,A: rat] :
% 0.32/0.53 ( ( ord_less_nat @ N @ N3 )
% 0.32/0.53 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.53 => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_decreasing
% 0.32/0.53 thf(fact_2376_power__strict__decreasing,axiom,
% 0.32/0.53 ! [N: nat,N3: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_nat @ N @ N3 )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.53 => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_decreasing
% 0.32/0.53 thf(fact_2377_power__strict__decreasing,axiom,
% 0.32/0.53 ! [N: nat,N3: nat,A: int] :
% 0.32/0.53 ( ( ord_less_nat @ N @ N3 )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.53 => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_strict_decreasing
% 0.32/0.53 thf(fact_2378_power__decreasing,axiom,
% 0.32/0.53 ! [N: nat,N3: nat,A: real] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_decreasing
% 0.32/0.53 thf(fact_2379_power__decreasing,axiom,
% 0.32/0.53 ! [N: nat,N3: nat,A: rat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_decreasing
% 0.32/0.53 thf(fact_2380_power__decreasing,axiom,
% 0.32/0.53 ! [N: nat,N3: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.53 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_decreasing
% 0.32/0.53 thf(fact_2381_power__decreasing,axiom,
% 0.32/0.53 ! [N: nat,N3: nat,A: int] :
% 0.32/0.53 ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.53 => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_decreasing
% 0.32/0.53 thf(fact_2382_zero__power2,axiom,
% 0.32/0.53 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = zero_zero_rat ) ).
% 0.32/0.53
% 0.32/0.53 % zero_power2
% 0.32/0.53 thf(fact_2383_zero__power2,axiom,
% 0.32/0.53 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = zero_zero_nat ) ).
% 0.32/0.53
% 0.32/0.53 % zero_power2
% 0.32/0.53 thf(fact_2384_zero__power2,axiom,
% 0.32/0.53 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = zero_zero_real ) ).
% 0.32/0.53
% 0.32/0.53 % zero_power2
% 0.32/0.53 thf(fact_2385_zero__power2,axiom,
% 0.32/0.53 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = zero_zero_int ) ).
% 0.32/0.53
% 0.32/0.53 % zero_power2
% 0.32/0.53 thf(fact_2386_zero__power2,axiom,
% 0.32/0.53 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = zero_zero_complex ) ).
% 0.32/0.53
% 0.32/0.53 % zero_power2
% 0.32/0.53 thf(fact_2387_self__le__power,axiom,
% 0.32/0.53 ! [A: real,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_real @ one_one_real @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % self_le_power
% 0.32/0.53 thf(fact_2388_self__le__power,axiom,
% 0.32/0.53 ! [A: rat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % self_le_power
% 0.32/0.53 thf(fact_2389_self__le__power,axiom,
% 0.32/0.53 ! [A: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % self_le_power
% 0.32/0.53 thf(fact_2390_self__le__power,axiom,
% 0.32/0.53 ! [A: int,N: nat] :
% 0.32/0.53 ( ( ord_less_eq_int @ one_one_int @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % self_le_power
% 0.32/0.53 thf(fact_2391_one__less__power,axiom,
% 0.32/0.53 ! [A: real,N: nat] :
% 0.32/0.53 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % one_less_power
% 0.32/0.53 thf(fact_2392_one__less__power,axiom,
% 0.32/0.53 ! [A: rat,N: nat] :
% 0.32/0.53 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % one_less_power
% 0.32/0.53 thf(fact_2393_one__less__power,axiom,
% 0.32/0.53 ! [A: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % one_less_power
% 0.32/0.53 thf(fact_2394_one__less__power,axiom,
% 0.32/0.53 ! [A: int,N: nat] :
% 0.32/0.53 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % one_less_power
% 0.32/0.53 thf(fact_2395_pos2,axiom,
% 0.32/0.53 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos2
% 0.32/0.53 thf(fact_2396_power__diff,axiom,
% 0.32/0.53 ! [A: complex,N: nat,M: nat] :
% 0.32/0.53 ( ( A != zero_zero_complex )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.53 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_diff
% 0.32/0.53 thf(fact_2397_power__diff,axiom,
% 0.32/0.53 ! [A: real,N: nat,M: nat] :
% 0.32/0.53 ( ( A != zero_zero_real )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.53 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_diff
% 0.32/0.53 thf(fact_2398_power__diff,axiom,
% 0.32/0.53 ! [A: rat,N: nat,M: nat] :
% 0.32/0.53 ( ( A != zero_zero_rat )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.53 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_diff
% 0.32/0.53 thf(fact_2399_power__diff,axiom,
% 0.32/0.53 ! [A: nat,N: nat,M: nat] :
% 0.32/0.53 ( ( A != zero_zero_nat )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.53 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_diff
% 0.32/0.53 thf(fact_2400_power__diff,axiom,
% 0.32/0.53 ! [A: int,N: nat,M: nat] :
% 0.32/0.53 ( ( A != zero_zero_int )
% 0.32/0.53 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.53 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_diff
% 0.32/0.53 thf(fact_2401_pos__zdiv__mult__2,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( 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 ) )
% 0.32/0.53 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % pos_zdiv_mult_2
% 0.32/0.53 thf(fact_2402_neg__zdiv__mult__2,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.53 => ( ( 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 ) )
% 0.32/0.53 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % neg_zdiv_mult_2
% 0.32/0.53 thf(fact_2403_less__eq__div__iff__mult__less__eq,axiom,
% 0.32/0.53 ! [Q2: nat,M: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 0.32/0.53 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 0.32/0.53 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % less_eq_div_iff_mult_less_eq
% 0.32/0.53 thf(fact_2404_dividend__less__times__div,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % dividend_less_times_div
% 0.32/0.53 thf(fact_2405_dividend__less__div__times,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % dividend_less_div_times
% 0.32/0.53 thf(fact_2406_split__div,axiom,
% 0.32/0.53 ! [P: nat > $o,M: nat,N: nat] :
% 0.32/0.53 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.53 = ( ( ( N = zero_zero_nat )
% 0.32/0.53 => ( P @ zero_zero_nat ) )
% 0.32/0.53 & ( ( N != zero_zero_nat )
% 0.32/0.53 => ! [I4: nat,J3: nat] :
% 0.32/0.53 ( ( ord_less_nat @ J3 @ N )
% 0.32/0.53 => ( ( M
% 0.32/0.53 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.32/0.53 => ( P @ I4 ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % split_div
% 0.32/0.53 thf(fact_2407_mult__eq__if,axiom,
% 0.32/0.53 ( times_times_nat
% 0.32/0.53 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mult_eq_if
% 0.32/0.53 thf(fact_2408_split__mod,axiom,
% 0.32/0.53 ! [P: nat > $o,M: nat,N: nat] :
% 0.32/0.53 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.53 = ( ( ( N = zero_zero_nat )
% 0.32/0.53 => ( P @ M ) )
% 0.32/0.53 & ( ( N != zero_zero_nat )
% 0.32/0.53 => ! [I4: nat,J3: nat] :
% 0.32/0.53 ( ( ord_less_nat @ J3 @ N )
% 0.32/0.53 => ( ( M
% 0.32/0.53 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.32/0.53 => ( P @ J3 ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % split_mod
% 0.32/0.53 thf(fact_2409_convex__bound__lt,axiom,
% 0.32/0.53 ! [X: real,A: real,Y3: real,U: real,V: real] :
% 0.32/0.53 ( ( ord_less_real @ X @ A )
% 0.32/0.53 => ( ( ord_less_real @ Y3 @ A )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.32/0.53 => ( ( ( plus_plus_real @ U @ V )
% 0.32/0.53 = one_one_real )
% 0.32/0.53 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % convex_bound_lt
% 0.32/0.53 thf(fact_2410_convex__bound__lt,axiom,
% 0.32/0.53 ! [X: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.32/0.53 ( ( ord_less_rat @ X @ A )
% 0.32/0.53 => ( ( ord_less_rat @ Y3 @ A )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.32/0.53 => ( ( ( plus_plus_rat @ U @ V )
% 0.32/0.53 = one_one_rat )
% 0.32/0.53 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % convex_bound_lt
% 0.32/0.53 thf(fact_2411_convex__bound__lt,axiom,
% 0.32/0.53 ! [X: int,A: int,Y3: int,U: int,V: int] :
% 0.32/0.53 ( ( ord_less_int @ X @ A )
% 0.32/0.53 => ( ( ord_less_int @ Y3 @ A )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.32/0.53 => ( ( ( plus_plus_int @ U @ V )
% 0.32/0.53 = one_one_int )
% 0.32/0.53 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % convex_bound_lt
% 0.32/0.53 thf(fact_2412_divide__le__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [B: real,C: real,W: num] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_eq_numeral(1)
% 0.32/0.53 thf(fact_2413_divide__le__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [B: rat,C: rat,W: num] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divide_le_eq_numeral(1)
% 0.32/0.53 thf(fact_2414_le__divide__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [W: num,B: real,C: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.53 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.53 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_divide_eq_numeral(1)
% 0.32/0.53 thf(fact_2415_le__divide__eq__numeral_I1_J,axiom,
% 0.32/0.53 ! [W: num,B: rat,C: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.53 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.53 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 0.32/0.53 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.53 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % le_divide_eq_numeral(1)
% 0.32/0.53 thf(fact_2416_half__gt__zero__iff,axiom,
% 0.32/0.53 ! [A: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.53 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.53
% 0.32/0.53 % half_gt_zero_iff
% 0.32/0.53 thf(fact_2417_half__gt__zero__iff,axiom,
% 0.32/0.53 ! [A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.53
% 0.32/0.53 % half_gt_zero_iff
% 0.32/0.53 thf(fact_2418_half__gt__zero,axiom,
% 0.32/0.53 ! [A: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.53 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % half_gt_zero
% 0.32/0.53 thf(fact_2419_half__gt__zero,axiom,
% 0.32/0.53 ! [A: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.53 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % half_gt_zero
% 0.32/0.53 thf(fact_2420_scaling__mono,axiom,
% 0.32/0.53 ! [U: real,V: real,R: real,S: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ U @ V )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.32/0.53 => ( ( ord_less_eq_real @ R @ S )
% 0.32/0.53 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % scaling_mono
% 0.32/0.53 thf(fact_2421_scaling__mono,axiom,
% 0.32/0.53 ! [U: rat,V: rat,R: rat,S: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ U @ V )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 0.32/0.53 => ( ( ord_less_eq_rat @ R @ S )
% 0.32/0.53 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % scaling_mono
% 0.32/0.53 thf(fact_2422_power2__le__imp__le,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_le_imp_le
% 0.32/0.53 thf(fact_2423_power2__le__imp__le,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ord_less_eq_rat @ X @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_le_imp_le
% 0.32/0.53 thf(fact_2424_power2__le__imp__le,axiom,
% 0.32/0.53 ! [X: nat,Y3: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.32/0.53 => ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_le_imp_le
% 0.32/0.53 thf(fact_2425_power2__le__imp__le,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.53 => ( ord_less_eq_int @ X @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_le_imp_le
% 0.32/0.53 thf(fact_2426_power2__eq__imp__eq,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( X = Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_eq_imp_eq
% 0.32/0.53 thf(fact_2427_power2__eq__imp__eq,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( X = Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_eq_imp_eq
% 0.32/0.53 thf(fact_2428_power2__eq__imp__eq,axiom,
% 0.32/0.53 ! [X: nat,Y3: nat] :
% 0.32/0.53 ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.32/0.53 => ( X = Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_eq_imp_eq
% 0.32/0.53 thf(fact_2429_power2__eq__imp__eq,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.53 => ( X = Y3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_eq_imp_eq
% 0.32/0.53 thf(fact_2430_zero__le__power2,axiom,
% 0.32/0.53 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zero_le_power2
% 0.32/0.53 thf(fact_2431_zero__le__power2,axiom,
% 0.32/0.53 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zero_le_power2
% 0.32/0.53 thf(fact_2432_zero__le__power2,axiom,
% 0.32/0.53 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zero_le_power2
% 0.32/0.53 thf(fact_2433_power2__less__0,axiom,
% 0.32/0.53 ! [A: real] :
% 0.32/0.53 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.32/0.53
% 0.32/0.53 % power2_less_0
% 0.32/0.53 thf(fact_2434_power2__less__0,axiom,
% 0.32/0.53 ! [A: rat] :
% 0.32/0.53 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.32/0.53
% 0.32/0.53 % power2_less_0
% 0.32/0.53 thf(fact_2435_power2__less__0,axiom,
% 0.32/0.53 ! [A: int] :
% 0.32/0.53 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.32/0.53
% 0.32/0.53 % power2_less_0
% 0.32/0.53 thf(fact_2436_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.32/0.53 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.53 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.32/0.53 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.53 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.32/0.53 thf(fact_2437_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.32/0.53 ! [C: nat,A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.32/0.53 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.53 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.32/0.53 thf(fact_2438_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.32/0.53 ! [C: int,A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.32/0.53 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.53 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.32/0.53 thf(fact_2439_exp__add__not__zero__imp__right,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.53 != zero_zero_nat )
% 0.32/0.53 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.53 != zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % exp_add_not_zero_imp_right
% 0.32/0.53 thf(fact_2440_exp__add__not__zero__imp__right,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.53 != zero_zero_int )
% 0.32/0.53 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.53 != zero_zero_int ) ) ).
% 0.32/0.53
% 0.32/0.53 % exp_add_not_zero_imp_right
% 0.32/0.53 thf(fact_2441_exp__add__not__zero__imp__left,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.53 != zero_zero_nat )
% 0.32/0.53 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.32/0.53 != zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % exp_add_not_zero_imp_left
% 0.32/0.53 thf(fact_2442_exp__add__not__zero__imp__left,axiom,
% 0.32/0.53 ! [M: nat,N: nat] :
% 0.32/0.53 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.53 != zero_zero_int )
% 0.32/0.53 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.32/0.53 != zero_zero_int ) ) ).
% 0.32/0.53
% 0.32/0.53 % exp_add_not_zero_imp_left
% 0.32/0.53 thf(fact_2443_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.53 != zero_zero_nat )
% 0.32/0.53 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.53 != zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % exp_not_zero_imp_exp_diff_not_zero
% 0.32/0.53 thf(fact_2444_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.32/0.53 ! [N: nat,M: nat] :
% 0.32/0.53 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.53 != zero_zero_int )
% 0.32/0.53 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.53 != zero_zero_int ) ) ).
% 0.32/0.53
% 0.32/0.53 % exp_not_zero_imp_exp_diff_not_zero
% 0.32/0.53 thf(fact_2445_power__diff__power__eq,axiom,
% 0.32/0.53 ! [A: nat,N: nat,M: nat] :
% 0.32/0.53 ( ( A != zero_zero_nat )
% 0.32/0.53 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.53 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.32/0.53 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.53 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_diff_power_eq
% 0.32/0.53 thf(fact_2446_power__diff__power__eq,axiom,
% 0.32/0.53 ! [A: int,N: nat,M: nat] :
% 0.32/0.53 ( ( A != zero_zero_int )
% 0.32/0.53 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.32/0.53 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.32/0.53 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.53 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 0.32/0.53 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_diff_power_eq
% 0.32/0.53 thf(fact_2447_power__eq__if,axiom,
% 0.32/0.53 ( power_power_complex
% 0.32/0.53 = ( ^ [P4: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_eq_if
% 0.32/0.53 thf(fact_2448_power__eq__if,axiom,
% 0.32/0.53 ( power_power_real
% 0.32/0.53 = ( ^ [P4: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_eq_if
% 0.32/0.53 thf(fact_2449_power__eq__if,axiom,
% 0.32/0.53 ( power_power_rat
% 0.32/0.53 = ( ^ [P4: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_eq_if
% 0.32/0.53 thf(fact_2450_power__eq__if,axiom,
% 0.32/0.53 ( power_power_nat
% 0.32/0.53 = ( ^ [P4: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_eq_if
% 0.32/0.53 thf(fact_2451_power__eq__if,axiom,
% 0.32/0.53 ( power_power_int
% 0.32/0.53 = ( ^ [P4: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_eq_if
% 0.32/0.53 thf(fact_2452_power__minus__mult,axiom,
% 0.32/0.53 ! [N: nat,A: complex] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.53 = ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_minus_mult
% 0.32/0.53 thf(fact_2453_power__minus__mult,axiom,
% 0.32/0.53 ! [N: nat,A: real] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.53 = ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_minus_mult
% 0.32/0.53 thf(fact_2454_power__minus__mult,axiom,
% 0.32/0.53 ! [N: nat,A: rat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.53 = ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_minus_mult
% 0.32/0.53 thf(fact_2455_power__minus__mult,axiom,
% 0.32/0.53 ! [N: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.53 = ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_minus_mult
% 0.32/0.53 thf(fact_2456_power__minus__mult,axiom,
% 0.32/0.53 ! [N: nat,A: int] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 0.32/0.53 = ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power_minus_mult
% 0.32/0.53 thf(fact_2457_power2__less__imp__less,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ord_less_real @ X @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_less_imp_less
% 0.32/0.53 thf(fact_2458_power2__less__imp__less,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ord_less_rat @ X @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_less_imp_less
% 0.32/0.53 thf(fact_2459_power2__less__imp__less,axiom,
% 0.32/0.53 ! [X: nat,Y3: nat] :
% 0.32/0.53 ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.32/0.53 => ( ord_less_nat @ X @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_less_imp_less
% 0.32/0.53 thf(fact_2460_power2__less__imp__less,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.53 => ( ord_less_int @ X @ Y3 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % power2_less_imp_less
% 0.32/0.53 thf(fact_2461_sum__power2__le__zero__iff,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.32/0.53 = ( ( X = zero_zero_real )
% 0.32/0.53 & ( Y3 = zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_le_zero_iff
% 0.32/0.53 thf(fact_2462_sum__power2__le__zero__iff,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.32/0.53 = ( ( X = zero_zero_rat )
% 0.32/0.53 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_le_zero_iff
% 0.32/0.53 thf(fact_2463_sum__power2__le__zero__iff,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.32/0.53 = ( ( X = zero_zero_int )
% 0.32/0.53 & ( Y3 = zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_le_zero_iff
% 0.32/0.53 thf(fact_2464_sum__power2__ge__zero,axiom,
% 0.32/0.53 ! [X: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_ge_zero
% 0.32/0.53 thf(fact_2465_sum__power2__ge__zero,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_ge_zero
% 0.32/0.53 thf(fact_2466_sum__power2__ge__zero,axiom,
% 0.32/0.53 ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_ge_zero
% 0.32/0.53 thf(fact_2467_sum__power2__gt__zero__iff,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.53 = ( ( X != zero_zero_real )
% 0.32/0.53 | ( Y3 != zero_zero_real ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_gt_zero_iff
% 0.32/0.53 thf(fact_2468_sum__power2__gt__zero__iff,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.53 = ( ( X != zero_zero_rat )
% 0.32/0.53 | ( Y3 != zero_zero_rat ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_gt_zero_iff
% 0.32/0.53 thf(fact_2469_sum__power2__gt__zero__iff,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.53 = ( ( X != zero_zero_int )
% 0.32/0.53 | ( Y3 != zero_zero_int ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % sum_power2_gt_zero_iff
% 0.32/0.53 thf(fact_2470_not__sum__power2__lt__zero,axiom,
% 0.32/0.53 ! [X: real,Y3: real] :
% 0.32/0.53 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.32/0.53
% 0.32/0.53 % not_sum_power2_lt_zero
% 0.32/0.53 thf(fact_2471_not__sum__power2__lt__zero,axiom,
% 0.32/0.53 ! [X: rat,Y3: rat] :
% 0.32/0.53 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.32/0.53
% 0.32/0.53 % not_sum_power2_lt_zero
% 0.32/0.53 thf(fact_2472_not__sum__power2__lt__zero,axiom,
% 0.32/0.53 ! [X: int,Y3: int] :
% 0.32/0.53 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.32/0.53
% 0.32/0.53 % not_sum_power2_lt_zero
% 0.32/0.53 thf(fact_2473_divmod__digit__0_I2_J,axiom,
% 0.32/0.53 ! [B: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 0.32/0.53 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_0(2)
% 0.32/0.53 thf(fact_2474_divmod__digit__0_I2_J,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 0.32/0.53 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_0(2)
% 0.32/0.53 thf(fact_2475_divmod__digit__0_I2_J,axiom,
% 0.32/0.53 ! [B: code_integer,A: code_integer] :
% 0.32/0.53 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.53 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 0.32/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_0(2)
% 0.32/0.53 thf(fact_2476_bits__stable__imp__add__self,axiom,
% 0.32/0.53 ! [A: nat] :
% 0.32/0.53 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = A )
% 0.32/0.53 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.53 = zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % bits_stable_imp_add_self
% 0.32/0.53 thf(fact_2477_bits__stable__imp__add__self,axiom,
% 0.32/0.53 ! [A: int] :
% 0.32/0.53 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.53 = A )
% 0.32/0.53 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.53 = zero_zero_int ) ) ).
% 0.32/0.53
% 0.32/0.53 % bits_stable_imp_add_self
% 0.32/0.53 thf(fact_2478_bits__stable__imp__add__self,axiom,
% 0.32/0.53 ! [A: code_integer] :
% 0.32/0.53 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.53 = A )
% 0.32/0.53 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.32/0.53 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.53
% 0.32/0.53 % bits_stable_imp_add_self
% 0.32/0.53 thf(fact_2479_zero__le__even__power_H,axiom,
% 0.32/0.53 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zero_le_even_power'
% 0.32/0.53 thf(fact_2480_zero__le__even__power_H,axiom,
% 0.32/0.53 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zero_le_even_power'
% 0.32/0.53 thf(fact_2481_zero__le__even__power_H,axiom,
% 0.32/0.53 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % zero_le_even_power'
% 0.32/0.53 thf(fact_2482_verit__le__mono__div,axiom,
% 0.32/0.53 ! [A2: nat,B5: nat,N: nat] :
% 0.32/0.53 ( ( ord_less_nat @ A2 @ B5 )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ord_less_eq_nat
% 0.32/0.53 @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 0.32/0.53 @ ( if_nat
% 0.32/0.53 @ ( ( modulo_modulo_nat @ B5 @ N )
% 0.32/0.53 = zero_zero_nat )
% 0.32/0.53 @ one_one_nat
% 0.32/0.53 @ zero_zero_nat ) )
% 0.32/0.53 @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % verit_le_mono_div
% 0.32/0.53 thf(fact_2483_divmod__digit__0_I1_J,axiom,
% 0.32/0.53 ! [B: nat,A: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.53 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_0(1)
% 0.32/0.53 thf(fact_2484_divmod__digit__0_I1_J,axiom,
% 0.32/0.53 ! [B: int,A: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.53 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_0(1)
% 0.32/0.53 thf(fact_2485_divmod__digit__0_I1_J,axiom,
% 0.32/0.53 ! [B: code_integer,A: code_integer] :
% 0.32/0.53 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.53 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.53 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_0(1)
% 0.32/0.53 thf(fact_2486_mod__double__modulus,axiom,
% 0.32/0.53 ! [M: code_integer,X: code_integer] :
% 0.32/0.53 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.32/0.53 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.32/0.53 => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.53 = ( modulo364778990260209775nteger @ X @ M ) )
% 0.32/0.53 | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.53 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mod_double_modulus
% 0.32/0.53 thf(fact_2487_mod__double__modulus,axiom,
% 0.32/0.53 ! [M: nat,X: nat] :
% 0.32/0.53 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.53 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 0.32/0.53 => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.53 = ( modulo_modulo_nat @ X @ M ) )
% 0.32/0.53 | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.53 = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mod_double_modulus
% 0.32/0.53 thf(fact_2488_mod__double__modulus,axiom,
% 0.32/0.53 ! [M: int,X: int] :
% 0.32/0.53 ( ( ord_less_int @ zero_zero_int @ M )
% 0.32/0.53 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.53 => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.53 = ( modulo_modulo_int @ X @ M ) )
% 0.32/0.53 | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.53 = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % mod_double_modulus
% 0.32/0.53 thf(fact_2489_divmod__digit__1_I2_J,axiom,
% 0.32/0.53 ! [A: code_integer,B: code_integer] :
% 0.32/0.53 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.53 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.53 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.53 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_1(2)
% 0.32/0.53 thf(fact_2490_divmod__digit__1_I2_J,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.32/0.53 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.53 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_1(2)
% 0.32/0.53 thf(fact_2491_divmod__digit__1_I2_J,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.53 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.53 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.32/0.53 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.32/0.53 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % divmod_digit_1(2)
% 0.32/0.53 thf(fact_2492_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.32/0.53 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 0.32/0.53 ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 0.32/0.53 = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % VEBT_internal.option_shift.simps(3)
% 0.32/0.53 thf(fact_2493_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.32/0.53 ! [F: num > num > num,A: num,B: num] :
% 0.32/0.53 ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 0.32/0.53 = ( some_num @ ( F @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % VEBT_internal.option_shift.simps(3)
% 0.32/0.53 thf(fact_2494_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.32/0.53 ! [F: nat > nat > nat,A: nat,B: nat] :
% 0.32/0.53 ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 0.32/0.53 = ( some_nat @ ( F @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % VEBT_internal.option_shift.simps(3)
% 0.32/0.53 thf(fact_2495_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.32/0.53 ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 0.32/0.53 ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 0.32/0.53 = none_P5556105721700978146at_nat ) ).
% 0.32/0.53
% 0.32/0.53 % VEBT_internal.option_shift.simps(1)
% 0.32/0.53 thf(fact_2496_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.32/0.53 ! [Uu: num > num > num,Uv: option_num] :
% 0.32/0.53 ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 0.32/0.53 = none_num ) ).
% 0.32/0.53
% 0.32/0.53 % VEBT_internal.option_shift.simps(1)
% 0.32/0.53 thf(fact_2497_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.32/0.53 ! [Uu: nat > nat > nat,Uv: option_nat] :
% 0.32/0.53 ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 0.32/0.53 = none_nat ) ).
% 0.32/0.53
% 0.32/0.53 % VEBT_internal.option_shift.simps(1)
% 0.32/0.53 thf(fact_2498_arith__geo__mean,axiom,
% 0.32/0.53 ! [U: real,X: real,Y3: real] :
% 0.32/0.53 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = ( times_times_real @ X @ Y3 ) )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.53 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.53 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % arith_geo_mean
% 0.32/0.53 thf(fact_2499_arith__geo__mean,axiom,
% 0.32/0.53 ! [U: rat,X: rat,Y3: rat] :
% 0.32/0.53 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = ( times_times_rat @ X @ Y3 ) )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.53 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.53 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % arith_geo_mean
% 0.32/0.53 thf(fact_2500_in__children__def,axiom,
% 0.32/0.53 ( vEBT_V5917875025757280293ildren
% 0.32/0.53 = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ N2 ) ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % in_children_def
% 0.32/0.53 thf(fact_2501_not__mod__2__eq__1__eq__0,axiom,
% 0.32/0.53 ! [A: nat] :
% 0.32/0.53 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 != one_one_nat )
% 0.32/0.53 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = zero_zero_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_mod_2_eq_1_eq_0
% 0.32/0.53 thf(fact_2502_not__mod__2__eq__1__eq__0,axiom,
% 0.32/0.53 ! [A: int] :
% 0.32/0.53 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.53 != one_one_int )
% 0.32/0.53 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.53 = zero_zero_int ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_mod_2_eq_1_eq_0
% 0.32/0.53 thf(fact_2503_not__mod__2__eq__1__eq__0,axiom,
% 0.32/0.53 ! [A: code_integer] :
% 0.32/0.53 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.53 != one_one_Code_integer )
% 0.32/0.53 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.53 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_mod_2_eq_1_eq_0
% 0.32/0.53 thf(fact_2504_not__mod__2__eq__0__eq__1,axiom,
% 0.32/0.53 ! [A: nat] :
% 0.32/0.53 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 != zero_zero_nat )
% 0.32/0.53 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.53 = one_one_nat ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_mod_2_eq_0_eq_1
% 0.32/0.53 thf(fact_2505_not__mod__2__eq__0__eq__1,axiom,
% 0.32/0.53 ! [A: int] :
% 0.32/0.53 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.53 != zero_zero_int )
% 0.32/0.53 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.53 = one_one_int ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_mod_2_eq_0_eq_1
% 0.32/0.53 thf(fact_2506_not__mod__2__eq__0__eq__1,axiom,
% 0.32/0.53 ! [A: code_integer] :
% 0.32/0.53 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.53 != zero_z3403309356797280102nteger )
% 0.32/0.53 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.53 = one_one_Code_integer ) ) ).
% 0.32/0.53
% 0.32/0.53 % not_mod_2_eq_0_eq_1
% 0.32/0.53 thf(fact_2507_both__member__options__ding,axiom,
% 0.32/0.53 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.53 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 0.32/0.53 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.32/0.53 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) )
% 0.32/0.53 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % both_member_options_ding
% 0.32/0.53 thf(fact_2508_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.32/0.53 ! [X: nat,N: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.53 => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % VEBT_internal.exp_split_high_low(2)
% 0.32/0.53 thf(fact_2509_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.32/0.53 ! [X: nat,N: nat,M: nat] :
% 0.32/0.53 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.53 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.53 => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % VEBT_internal.exp_split_high_low(1)
% 0.32/0.53 thf(fact_2510_set__n__deg__not__0,axiom,
% 0.32/0.53 ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
% 0.32/0.53 ( ! [X3: vEBT_VEBT] :
% 0.32/0.53 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.53 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.32/0.53 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.32/0.53 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.53 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % set_n_deg_not_0
% 0.32/0.53 thf(fact_2511_inrange,axiom,
% 0.32/0.53 ! [T: vEBT_VEBT,N: nat] :
% 0.32/0.53 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.53 => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inrange
% 0.32/0.53 thf(fact_2512_set__bit__0,axiom,
% 0.32/0.53 ! [A: int] :
% 0.32/0.53 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.32/0.53 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % set_bit_0
% 0.32/0.53 thf(fact_2513_set__bit__0,axiom,
% 0.32/0.53 ! [A: nat] :
% 0.32/0.53 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.32/0.53 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % set_bit_0
% 0.32/0.53 thf(fact_2514_nat__induct2,axiom,
% 0.32/0.53 ! [P: nat > $o,N: nat] :
% 0.32/0.53 ( ( P @ zero_zero_nat )
% 0.32/0.53 => ( ( P @ one_one_nat )
% 0.32/0.53 => ( ! [N4: nat] :
% 0.32/0.53 ( ( P @ N4 )
% 0.32/0.53 => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.53 => ( P @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % nat_induct2
% 0.32/0.53 thf(fact_2515_deg__deg__n,axiom,
% 0.32/0.53 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.32/0.53 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 0.32/0.53 => ( Deg = N ) ) ).
% 0.32/0.53
% 0.32/0.53 % deg_deg_n
% 0.32/0.53 thf(fact_2516_inthall,axiom,
% 0.32/0.53 ! [Xs2: list_real,P: real > $o,N: nat] :
% 0.32/0.53 ( ! [X3: real] :
% 0.32/0.53 ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 0.32/0.53 => ( P @ X3 ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.32/0.53 => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inthall
% 0.32/0.53 thf(fact_2517_inthall,axiom,
% 0.32/0.53 ! [Xs2: list_complex,P: complex > $o,N: nat] :
% 0.32/0.53 ( ! [X3: complex] :
% 0.32/0.53 ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 0.32/0.53 => ( P @ X3 ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.32/0.53 => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inthall
% 0.32/0.53 thf(fact_2518_inthall,axiom,
% 0.32/0.53 ! [Xs2: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,N: nat] :
% 0.32/0.53 ( ! [X3: produc859450856879609959at_nat] :
% 0.32/0.53 ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 0.32/0.53 => ( P @ X3 ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 0.32/0.53 => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inthall
% 0.32/0.53 thf(fact_2519_inthall,axiom,
% 0.32/0.53 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.32/0.53 ( ! [X3: vEBT_VEBT] :
% 0.32/0.53 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.32/0.53 => ( P @ X3 ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.53 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inthall
% 0.32/0.53 thf(fact_2520_inthall,axiom,
% 0.32/0.53 ! [Xs2: list_o,P: $o > $o,N: nat] :
% 0.32/0.53 ( ! [X3: $o] :
% 0.32/0.53 ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 0.32/0.53 => ( P @ X3 ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.53 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inthall
% 0.32/0.53 thf(fact_2521_inthall,axiom,
% 0.32/0.53 ! [Xs2: list_nat,P: nat > $o,N: nat] :
% 0.32/0.53 ( ! [X3: nat] :
% 0.32/0.53 ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 0.32/0.53 => ( P @ X3 ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.53 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inthall
% 0.32/0.53 thf(fact_2522_inthall,axiom,
% 0.32/0.53 ! [Xs2: list_int,P: int > $o,N: nat] :
% 0.32/0.53 ( ! [X3: int] :
% 0.32/0.53 ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 0.32/0.53 => ( P @ X3 ) )
% 0.32/0.53 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.53 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % inthall
% 0.32/0.53 thf(fact_2523__C4_Ohyps_C_I8_J,axiom,
% 0.32/0.53 ( ( mi = ma )
% 0.32/0.53 => ! [X5: vEBT_VEBT] :
% 0.32/0.53 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.32/0.53 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % "4.hyps"(8)
% 0.32/0.53 thf(fact_2524_atLeastAtMost__iff,axiom,
% 0.32/0.53 ! [I: set_int,L2: set_int,U: set_int] :
% 0.32/0.53 ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L2 @ U ) )
% 0.32/0.53 = ( ( ord_less_eq_set_int @ L2 @ I )
% 0.32/0.53 & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastAtMost_iff
% 0.32/0.53 thf(fact_2525_atLeastAtMost__iff,axiom,
% 0.32/0.53 ! [I: rat,L2: rat,U: rat] :
% 0.32/0.53 ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L2 @ U ) )
% 0.32/0.53 = ( ( ord_less_eq_rat @ L2 @ I )
% 0.32/0.53 & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastAtMost_iff
% 0.32/0.53 thf(fact_2526_atLeastAtMost__iff,axiom,
% 0.32/0.53 ! [I: num,L2: num,U: num] :
% 0.32/0.53 ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U ) )
% 0.32/0.53 = ( ( ord_less_eq_num @ L2 @ I )
% 0.32/0.53 & ( ord_less_eq_num @ I @ U ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastAtMost_iff
% 0.32/0.53 thf(fact_2527_atLeastAtMost__iff,axiom,
% 0.32/0.53 ! [I: nat,L2: nat,U: nat] :
% 0.32/0.53 ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 0.32/0.53 = ( ( ord_less_eq_nat @ L2 @ I )
% 0.32/0.53 & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastAtMost_iff
% 0.32/0.53 thf(fact_2528_atLeastAtMost__iff,axiom,
% 0.32/0.53 ! [I: int,L2: int,U: int] :
% 0.32/0.53 ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 0.32/0.53 = ( ( ord_less_eq_int @ L2 @ I )
% 0.32/0.53 & ( ord_less_eq_int @ I @ U ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastAtMost_iff
% 0.32/0.53 thf(fact_2529_atLeastAtMost__iff,axiom,
% 0.32/0.53 ! [I: real,L2: real,U: real] :
% 0.32/0.53 ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U ) )
% 0.32/0.53 = ( ( ord_less_eq_real @ L2 @ I )
% 0.32/0.53 & ( ord_less_eq_real @ I @ U ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastAtMost_iff
% 0.32/0.53 thf(fact_2530_Icc__eq__Icc,axiom,
% 0.32/0.53 ! [L2: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 0.32/0.53 ( ( ( set_or370866239135849197et_int @ L2 @ H2 )
% 0.32/0.53 = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 0.32/0.53 = ( ( ( L2 = L3 )
% 0.32/0.53 & ( H2 = H3 ) )
% 0.32/0.53 | ( ~ ( ord_less_eq_set_int @ L2 @ H2 )
% 0.32/0.53 & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Icc_eq_Icc
% 0.32/0.53 thf(fact_2531_Icc__eq__Icc,axiom,
% 0.32/0.53 ! [L2: rat,H2: rat,L3: rat,H3: rat] :
% 0.32/0.53 ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
% 0.32/0.53 = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 0.32/0.53 = ( ( ( L2 = L3 )
% 0.32/0.53 & ( H2 = H3 ) )
% 0.32/0.53 | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
% 0.32/0.53 & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Icc_eq_Icc
% 0.32/0.53 thf(fact_2532_Icc__eq__Icc,axiom,
% 0.32/0.53 ! [L2: num,H2: num,L3: num,H3: num] :
% 0.32/0.53 ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
% 0.32/0.53 = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 0.32/0.53 = ( ( ( L2 = L3 )
% 0.32/0.53 & ( H2 = H3 ) )
% 0.32/0.53 | ( ~ ( ord_less_eq_num @ L2 @ H2 )
% 0.32/0.53 & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Icc_eq_Icc
% 0.32/0.53 thf(fact_2533_Icc__eq__Icc,axiom,
% 0.32/0.53 ! [L2: nat,H2: nat,L3: nat,H3: nat] :
% 0.32/0.53 ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
% 0.32/0.53 = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 0.32/0.53 = ( ( ( L2 = L3 )
% 0.32/0.53 & ( H2 = H3 ) )
% 0.32/0.53 | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
% 0.32/0.53 & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Icc_eq_Icc
% 0.32/0.53 thf(fact_2534_Icc__eq__Icc,axiom,
% 0.32/0.53 ! [L2: int,H2: int,L3: int,H3: int] :
% 0.32/0.53 ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
% 0.32/0.53 = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 0.32/0.53 = ( ( ( L2 = L3 )
% 0.32/0.53 & ( H2 = H3 ) )
% 0.32/0.53 | ( ~ ( ord_less_eq_int @ L2 @ H2 )
% 0.32/0.53 & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Icc_eq_Icc
% 0.32/0.53 thf(fact_2535_Icc__eq__Icc,axiom,
% 0.32/0.53 ! [L2: real,H2: real,L3: real,H3: real] :
% 0.32/0.53 ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
% 0.32/0.53 = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 0.32/0.53 = ( ( ( L2 = L3 )
% 0.32/0.53 & ( H2 = H3 ) )
% 0.32/0.53 | ( ~ ( ord_less_eq_real @ L2 @ H2 )
% 0.32/0.53 & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % Icc_eq_Icc
% 0.32/0.53 thf(fact_2536_set__bit__nonnegative__int__iff,axiom,
% 0.32/0.53 ! [N: nat,K: int] :
% 0.32/0.53 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 0.32/0.53 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.53
% 0.32/0.53 % set_bit_nonnegative_int_iff
% 0.32/0.53 thf(fact_2537_set__bit__negative__int__iff,axiom,
% 0.32/0.53 ! [N: nat,K: int] :
% 0.32/0.53 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.53 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.53
% 0.32/0.53 % set_bit_negative_int_iff
% 0.32/0.53 thf(fact_2538_atLeastatMost__empty__iff2,axiom,
% 0.32/0.53 ! [A: set_int,B: set_int] :
% 0.32/0.53 ( ( bot_bot_set_set_int
% 0.32/0.53 = ( set_or370866239135849197et_int @ A @ B ) )
% 0.32/0.53 = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastatMost_empty_iff2
% 0.32/0.53 thf(fact_2539_atLeastatMost__empty__iff2,axiom,
% 0.32/0.53 ! [A: rat,B: rat] :
% 0.32/0.53 ( ( bot_bot_set_rat
% 0.32/0.53 = ( set_or633870826150836451st_rat @ A @ B ) )
% 0.32/0.53 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastatMost_empty_iff2
% 0.32/0.53 thf(fact_2540_atLeastatMost__empty__iff2,axiom,
% 0.32/0.53 ! [A: num,B: num] :
% 0.32/0.53 ( ( bot_bot_set_num
% 0.32/0.53 = ( set_or7049704709247886629st_num @ A @ B ) )
% 0.32/0.53 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastatMost_empty_iff2
% 0.32/0.53 thf(fact_2541_atLeastatMost__empty__iff2,axiom,
% 0.32/0.53 ! [A: nat,B: nat] :
% 0.32/0.53 ( ( bot_bot_set_nat
% 0.32/0.53 = ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.32/0.53 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.53
% 0.32/0.53 % atLeastatMost_empty_iff2
% 0.32/0.53 thf(fact_2542_atLeastatMost__empty__iff2,axiom,
% 0.32/0.53 ! [A: int,B: int] :
% 0.32/0.53 ( ( bot_bot_set_int
% 0.32/0.53 = ( set_or1266510415728281911st_int @ A @ B ) )
% 0.32/0.53 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty_iff2
% 0.32/0.54 thf(fact_2543_atLeastatMost__empty__iff2,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( bot_bot_set_real
% 0.32/0.54 = ( set_or1222579329274155063t_real @ A @ B ) )
% 0.32/0.54 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty_iff2
% 0.32/0.54 thf(fact_2544_atLeastatMost__empty__iff,axiom,
% 0.32/0.54 ! [A: set_int,B: set_int] :
% 0.32/0.54 ( ( ( set_or370866239135849197et_int @ A @ B )
% 0.32/0.54 = bot_bot_set_set_int )
% 0.32/0.54 = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty_iff
% 0.32/0.54 thf(fact_2545_atLeastatMost__empty__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ( set_or633870826150836451st_rat @ A @ B )
% 0.32/0.54 = bot_bot_set_rat )
% 0.32/0.54 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty_iff
% 0.32/0.54 thf(fact_2546_atLeastatMost__empty__iff,axiom,
% 0.32/0.54 ! [A: num,B: num] :
% 0.32/0.54 ( ( ( set_or7049704709247886629st_num @ A @ B )
% 0.32/0.54 = bot_bot_set_num )
% 0.32/0.54 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty_iff
% 0.32/0.54 thf(fact_2547_atLeastatMost__empty__iff,axiom,
% 0.32/0.54 ! [A: nat,B: nat] :
% 0.32/0.54 ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.32/0.54 = bot_bot_set_nat )
% 0.32/0.54 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty_iff
% 0.32/0.54 thf(fact_2548_atLeastatMost__empty__iff,axiom,
% 0.32/0.54 ! [A: int,B: int] :
% 0.32/0.54 ( ( ( set_or1266510415728281911st_int @ A @ B )
% 0.32/0.54 = bot_bot_set_int )
% 0.32/0.54 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty_iff
% 0.32/0.54 thf(fact_2549_atLeastatMost__empty__iff,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ( set_or1222579329274155063t_real @ A @ B )
% 0.32/0.54 = bot_bot_set_real )
% 0.32/0.54 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty_iff
% 0.32/0.54 thf(fact_2550_atLeastatMost__subset__iff,axiom,
% 0.32/0.54 ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 0.32/0.54 ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.32/0.54 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_set_int @ C @ A )
% 0.32/0.54 & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_subset_iff
% 0.32/0.54 thf(fact_2551_atLeastatMost__subset__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.54 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.32/0.54 = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_rat @ C @ A )
% 0.32/0.54 & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_subset_iff
% 0.32/0.54 thf(fact_2552_atLeastatMost__subset__iff,axiom,
% 0.32/0.54 ! [A: num,B: num,C: num,D: num] :
% 0.32/0.54 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.32/0.54 = ( ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_num @ C @ A )
% 0.32/0.54 & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_subset_iff
% 0.32/0.54 thf(fact_2553_atLeastatMost__subset__iff,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.54 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.32/0.54 = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.54 & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_subset_iff
% 0.32/0.54 thf(fact_2554_atLeastatMost__subset__iff,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.54 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.32/0.54 = ( ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_int @ C @ A )
% 0.32/0.54 & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_subset_iff
% 0.32/0.54 thf(fact_2555_atLeastatMost__subset__iff,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.54 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.32/0.54 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_real @ C @ A )
% 0.32/0.54 & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_subset_iff
% 0.32/0.54 thf(fact_2556_atLeastatMost__empty,axiom,
% 0.32/0.54 ! [B: rat,A: rat] :
% 0.32/0.54 ( ( ord_less_rat @ B @ A )
% 0.32/0.54 => ( ( set_or633870826150836451st_rat @ A @ B )
% 0.32/0.54 = bot_bot_set_rat ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty
% 0.32/0.54 thf(fact_2557_atLeastatMost__empty,axiom,
% 0.32/0.54 ! [B: num,A: num] :
% 0.32/0.54 ( ( ord_less_num @ B @ A )
% 0.32/0.54 => ( ( set_or7049704709247886629st_num @ A @ B )
% 0.32/0.54 = bot_bot_set_num ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty
% 0.32/0.54 thf(fact_2558_atLeastatMost__empty,axiom,
% 0.32/0.54 ! [B: nat,A: nat] :
% 0.32/0.54 ( ( ord_less_nat @ B @ A )
% 0.32/0.54 => ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.32/0.54 = bot_bot_set_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty
% 0.32/0.54 thf(fact_2559_atLeastatMost__empty,axiom,
% 0.32/0.54 ! [B: int,A: int] :
% 0.32/0.54 ( ( ord_less_int @ B @ A )
% 0.32/0.54 => ( ( set_or1266510415728281911st_int @ A @ B )
% 0.32/0.54 = bot_bot_set_int ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty
% 0.32/0.54 thf(fact_2560_atLeastatMost__empty,axiom,
% 0.32/0.54 ! [B: real,A: real] :
% 0.32/0.54 ( ( ord_less_real @ B @ A )
% 0.32/0.54 => ( ( set_or1222579329274155063t_real @ A @ B )
% 0.32/0.54 = bot_bot_set_real ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_empty
% 0.32/0.54 thf(fact_2561_infinite__Icc__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 0.32/0.54 = ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.54
% 0.32/0.54 % infinite_Icc_iff
% 0.32/0.54 thf(fact_2562_infinite__Icc__iff,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 0.32/0.54 = ( ord_less_real @ A @ B ) ) ).
% 0.32/0.54
% 0.32/0.54 % infinite_Icc_iff
% 0.32/0.54 thf(fact_2563_mod__neg__neg__trivial,axiom,
% 0.32/0.54 ! [K: int,L2: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ L2 @ K )
% 0.32/0.54 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.54 = K ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod_neg_neg_trivial
% 0.32/0.54 thf(fact_2564_mod__pos__pos__trivial,axiom,
% 0.32/0.54 ! [K: int,L2: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.54 => ( ( ord_less_int @ K @ L2 )
% 0.32/0.54 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.54 = K ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod_pos_pos_trivial
% 0.32/0.54 thf(fact_2565_split__zmod,axiom,
% 0.32/0.54 ! [P: int > $o,N: int,K: int] :
% 0.32/0.54 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.32/0.54 = ( ( ( K = zero_zero_int )
% 0.32/0.54 => ( P @ N ) )
% 0.32/0.54 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.54 => ! [I4: int,J3: int] :
% 0.32/0.54 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.32/0.54 & ( ord_less_int @ J3 @ K )
% 0.32/0.54 & ( N
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.32/0.54 => ( P @ J3 ) ) )
% 0.32/0.54 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.54 => ! [I4: int,J3: int] :
% 0.32/0.54 ( ( ( ord_less_int @ K @ J3 )
% 0.32/0.54 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.32/0.54 & ( N
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.32/0.54 => ( P @ J3 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % split_zmod
% 0.32/0.54 thf(fact_2566_mod__pos__geq,axiom,
% 0.32/0.54 ! [L2: int,K: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.54 => ( ( ord_less_eq_int @ L2 @ K )
% 0.32/0.54 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.54 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod_pos_geq
% 0.32/0.54 thf(fact_2567_q__pos__lemma,axiom,
% 0.32/0.54 ! [B2: int,Q4: int,R3: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) )
% 0.32/0.54 => ( ( ord_less_int @ R3 @ B2 )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.32/0.54 => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % q_pos_lemma
% 0.32/0.54 thf(fact_2568_neg__mod__conj,axiom,
% 0.32/0.54 ! [B: int,A: int] :
% 0.32/0.54 ( ( ord_less_int @ B @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 0.32/0.54 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % neg_mod_conj
% 0.32/0.54 thf(fact_2569_pos__mod__conj,axiom,
% 0.32/0.54 ! [B: int,A: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.54 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_mod_conj
% 0.32/0.54 thf(fact_2570_int__mod__neg__eq,axiom,
% 0.32/0.54 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.54 ( ( A
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ B @ R )
% 0.32/0.54 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.54 = R ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_mod_neg_eq
% 0.32/0.54 thf(fact_2571_int__mod__pos__eq,axiom,
% 0.32/0.54 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.54 ( ( A
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.54 => ( ( ord_less_int @ R @ B )
% 0.32/0.54 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.54 = R ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_mod_pos_eq
% 0.32/0.54 thf(fact_2572_zdiv__mono2__lemma,axiom,
% 0.32/0.54 ! [B: int,Q2: int,R: int,B2: int,Q4: int,R3: int] :
% 0.32/0.54 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) )
% 0.32/0.54 => ( ( ord_less_int @ R3 @ B2 )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.32/0.54 => ( ( ord_less_eq_int @ B2 @ B )
% 0.32/0.54 => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zdiv_mono2_lemma
% 0.32/0.54 thf(fact_2573_zmod__trivial__iff,axiom,
% 0.32/0.54 ! [I: int,K: int] :
% 0.32/0.54 ( ( ( modulo_modulo_int @ I @ K )
% 0.32/0.54 = I )
% 0.32/0.54 = ( ( K = zero_zero_int )
% 0.32/0.54 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.32/0.54 & ( ord_less_int @ I @ K ) )
% 0.32/0.54 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.32/0.54 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zmod_trivial_iff
% 0.32/0.54 thf(fact_2574_zdiv__mono2__neg__lemma,axiom,
% 0.32/0.54 ! [B: int,Q2: int,R: int,B2: int,Q4: int,R3: int] :
% 0.32/0.54 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) )
% 0.32/0.54 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ R @ B )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 0.32/0.54 => ( ( ord_less_eq_int @ B2 @ B )
% 0.32/0.54 => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zdiv_mono2_neg_lemma
% 0.32/0.54 thf(fact_2575_unique__quotient__lemma,axiom,
% 0.32/0.54 ! [B: int,Q4: int,R3: int,Q2: int,R: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
% 0.32/0.54 => ( ( ord_less_int @ R3 @ B )
% 0.32/0.54 => ( ( ord_less_int @ R @ B )
% 0.32/0.54 => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % unique_quotient_lemma
% 0.32/0.54 thf(fact_2576_neg__mod__sign,axiom,
% 0.32/0.54 ! [L2: int,K: int] :
% 0.32/0.54 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.54 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).
% 0.32/0.54
% 0.32/0.54 % neg_mod_sign
% 0.32/0.54 thf(fact_2577_Euclidean__Division_Opos__mod__sign,axiom,
% 0.32/0.54 ! [L2: int,K: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.54 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Euclidean_Division.pos_mod_sign
% 0.32/0.54 thf(fact_2578_unique__quotient__lemma__neg,axiom,
% 0.32/0.54 ! [B: int,Q4: int,R3: int,Q2: int,R: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 0.32/0.54 => ( ( ord_less_int @ B @ R )
% 0.32/0.54 => ( ( ord_less_int @ B @ R3 )
% 0.32/0.54 => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % unique_quotient_lemma_neg
% 0.32/0.54 thf(fact_2579_mod__pos__neg__trivial,axiom,
% 0.32/0.54 ! [K: int,L2: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.54 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 0.32/0.54 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.54 = ( plus_plus_int @ K @ L2 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod_pos_neg_trivial
% 0.32/0.54 thf(fact_2580_zmod__eq__0__iff,axiom,
% 0.32/0.54 ! [M: int,D: int] :
% 0.32/0.54 ( ( ( modulo_modulo_int @ M @ D )
% 0.32/0.54 = zero_zero_int )
% 0.32/0.54 = ( ? [Q5: int] :
% 0.32/0.54 ( M
% 0.32/0.54 = ( times_times_int @ D @ Q5 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zmod_eq_0_iff
% 0.32/0.54 thf(fact_2581_zmod__eq__0D,axiom,
% 0.32/0.54 ! [M: int,D: int] :
% 0.32/0.54 ( ( ( modulo_modulo_int @ M @ D )
% 0.32/0.54 = zero_zero_int )
% 0.32/0.54 => ? [Q3: int] :
% 0.32/0.54 ( M
% 0.32/0.54 = ( times_times_int @ D @ Q3 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zmod_eq_0D
% 0.32/0.54 thf(fact_2582_zmod__le__nonneg__dividend,axiom,
% 0.32/0.54 ! [M: int,K: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.32/0.54 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 0.32/0.54
% 0.32/0.54 % zmod_le_nonneg_dividend
% 0.32/0.54 thf(fact_2583_bot__enat__def,axiom,
% 0.32/0.54 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.32/0.54
% 0.32/0.54 % bot_enat_def
% 0.32/0.54 thf(fact_2584_Euclidean__Division_Opos__mod__bound,axiom,
% 0.32/0.54 ! [L2: int,K: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.54 => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).
% 0.32/0.54
% 0.32/0.54 % Euclidean_Division.pos_mod_bound
% 0.32/0.54 thf(fact_2585_neg__mod__bound,axiom,
% 0.32/0.54 ! [L2: int,K: int] :
% 0.32/0.54 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.54 => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % neg_mod_bound
% 0.32/0.54 thf(fact_2586_bot__nat__def,axiom,
% 0.32/0.54 bot_bot_nat = zero_zero_nat ).
% 0.32/0.54
% 0.32/0.54 % bot_nat_def
% 0.32/0.54 thf(fact_2587_imult__is__0,axiom,
% 0.32/0.54 ! [M: extended_enat,N: extended_enat] :
% 0.32/0.54 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 0.32/0.54 = zero_z5237406670263579293d_enat )
% 0.32/0.54 = ( ( M = zero_z5237406670263579293d_enat )
% 0.32/0.54 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % imult_is_0
% 0.32/0.54 thf(fact_2588_zero__one__enat__neq_I1_J,axiom,
% 0.32/0.54 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 0.32/0.54
% 0.32/0.54 % zero_one_enat_neq(1)
% 0.32/0.54 thf(fact_2589_set__bit__greater__eq,axiom,
% 0.32/0.54 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 0.32/0.54
% 0.32/0.54 % set_bit_greater_eq
% 0.32/0.54 thf(fact_2590_verit__la__generic,axiom,
% 0.32/0.54 ! [A: int,X: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ A @ X )
% 0.32/0.54 | ( A = X )
% 0.32/0.54 | ( ord_less_eq_int @ X @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % verit_la_generic
% 0.32/0.54 thf(fact_2591_subset__eq__atLeast0__atMost__finite,axiom,
% 0.32/0.54 ! [N3: set_nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.54 => ( finite_finite_nat @ N3 ) ) ).
% 0.32/0.54
% 0.32/0.54 % subset_eq_atLeast0_atMost_finite
% 0.32/0.54 thf(fact_2592_atLeastatMost__psubset__iff,axiom,
% 0.32/0.54 ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 0.32/0.54 ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.32/0.54 = ( ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_set_int @ C @ A )
% 0.32/0.54 & ( ord_less_eq_set_int @ B @ D )
% 0.32/0.54 & ( ( ord_less_set_int @ C @ A )
% 0.32/0.54 | ( ord_less_set_int @ B @ D ) ) ) )
% 0.32/0.54 & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_psubset_iff
% 0.32/0.54 thf(fact_2593_atLeastatMost__psubset__iff,axiom,
% 0.32/0.54 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.54 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.32/0.54 = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_rat @ C @ A )
% 0.32/0.54 & ( ord_less_eq_rat @ B @ D )
% 0.32/0.54 & ( ( ord_less_rat @ C @ A )
% 0.32/0.54 | ( ord_less_rat @ B @ D ) ) ) )
% 0.32/0.54 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_psubset_iff
% 0.32/0.54 thf(fact_2594_atLeastatMost__psubset__iff,axiom,
% 0.32/0.54 ! [A: num,B: num,C: num,D: num] :
% 0.32/0.54 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.32/0.54 = ( ( ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_num @ C @ A )
% 0.32/0.54 & ( ord_less_eq_num @ B @ D )
% 0.32/0.54 & ( ( ord_less_num @ C @ A )
% 0.32/0.54 | ( ord_less_num @ B @ D ) ) ) )
% 0.32/0.54 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_psubset_iff
% 0.32/0.54 thf(fact_2595_atLeastatMost__psubset__iff,axiom,
% 0.32/0.54 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.54 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.32/0.54 = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.54 & ( ord_less_eq_nat @ B @ D )
% 0.32/0.54 & ( ( ord_less_nat @ C @ A )
% 0.32/0.54 | ( ord_less_nat @ B @ D ) ) ) )
% 0.32/0.54 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_psubset_iff
% 0.32/0.54 thf(fact_2596_atLeastatMost__psubset__iff,axiom,
% 0.32/0.54 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.54 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.32/0.54 = ( ( ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_int @ C @ A )
% 0.32/0.54 & ( ord_less_eq_int @ B @ D )
% 0.32/0.54 & ( ( ord_less_int @ C @ A )
% 0.32/0.54 | ( ord_less_int @ B @ D ) ) ) )
% 0.32/0.54 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_psubset_iff
% 0.32/0.54 thf(fact_2597_atLeastatMost__psubset__iff,axiom,
% 0.32/0.54 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.54 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.32/0.54 = ( ( ~ ( ord_less_eq_real @ A @ B )
% 0.32/0.54 | ( ( ord_less_eq_real @ C @ A )
% 0.32/0.54 & ( ord_less_eq_real @ B @ D )
% 0.32/0.54 & ( ( ord_less_real @ C @ A )
% 0.32/0.54 | ( ord_less_real @ B @ D ) ) ) )
% 0.32/0.54 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % atLeastatMost_psubset_iff
% 0.32/0.54 thf(fact_2598_infinite__Icc,axiom,
% 0.32/0.54 ! [A: rat,B: rat] :
% 0.32/0.54 ( ( ord_less_rat @ A @ B )
% 0.32/0.54 => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % infinite_Icc
% 0.32/0.54 thf(fact_2599_infinite__Icc,axiom,
% 0.32/0.54 ! [A: real,B: real] :
% 0.32/0.54 ( ( ord_less_real @ A @ B )
% 0.32/0.54 => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % infinite_Icc
% 0.32/0.54 thf(fact_2600_num_Osize_I4_J,axiom,
% 0.32/0.54 ( ( size_size_num @ one )
% 0.32/0.54 = zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % num.size(4)
% 0.32/0.54 thf(fact_2601_vebt__succ_Osimps_I3_J,axiom,
% 0.32/0.54 ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 0.32/0.54 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 0.32/0.54 = none_nat ) ).
% 0.32/0.54
% 0.32/0.54 % vebt_succ.simps(3)
% 0.32/0.54 thf(fact_2602_vebt__succ_Osimps_I4_J,axiom,
% 0.32/0.54 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
% 0.32/0.54 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
% 0.32/0.54 = none_nat ) ).
% 0.32/0.54
% 0.32/0.54 % vebt_succ.simps(4)
% 0.32/0.54 thf(fact_2603_vebt__mint_Osimps_I2_J,axiom,
% 0.32/0.54 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.32/0.54 ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.32/0.54 = none_nat ) ).
% 0.32/0.54
% 0.32/0.54 % vebt_mint.simps(2)
% 0.32/0.54 thf(fact_2604_vebt__maxt_Osimps_I2_J,axiom,
% 0.32/0.54 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.32/0.54 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.32/0.54 = none_nat ) ).
% 0.32/0.54
% 0.32/0.54 % vebt_maxt.simps(2)
% 0.32/0.54 thf(fact_2605_invar__vebt_Ointros_I2_J,axiom,
% 0.32/0.54 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.32/0.54 ( ! [X3: vEBT_VEBT] :
% 0.32/0.54 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.54 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.32/0.54 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.32/0.54 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.32/0.54 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.54 => ( ( M = N )
% 0.32/0.54 => ( ( Deg
% 0.32/0.54 = ( plus_plus_nat @ N @ M ) )
% 0.32/0.54 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.32/0.54 => ( ! [X3: vEBT_VEBT] :
% 0.32/0.54 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.54 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 0.32/0.54 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % invar_vebt.intros(2)
% 0.32/0.54 thf(fact_2606_zle__diff1__eq,axiom,
% 0.32/0.54 ! [W: int,Z: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.32/0.54 = ( ord_less_int @ W @ Z ) ) ).
% 0.32/0.54
% 0.32/0.54 % zle_diff1_eq
% 0.32/0.54 thf(fact_2607_zle__add1__eq__le,axiom,
% 0.32/0.54 ! [W: int,Z: int] :
% 0.32/0.54 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.32/0.54 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.32/0.54
% 0.32/0.54 % zle_add1_eq_le
% 0.32/0.54 thf(fact_2608_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.32/0.54 ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.32/0.54 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.32/0.54 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) )
% 0.32/0.54 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % both_member_options_from_chilf_to_complete_tree
% 0.32/0.54 thf(fact_2609_member__inv,axiom,
% 0.32/0.54 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.32/0.54 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.54 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.32/0.54 & ( ( X = Mi )
% 0.32/0.54 | ( X = Ma )
% 0.32/0.54 | ( ( ord_less_nat @ X @ Ma )
% 0.32/0.54 & ( ord_less_nat @ Mi @ X )
% 0.32/0.54 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.54 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % member_inv
% 0.32/0.54 thf(fact_2610_mintlistlength,axiom,
% 0.32/0.54 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.32/0.54 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.32/0.54 => ( ( Mi != Ma )
% 0.32/0.54 => ( ( ord_less_nat @ Mi @ Ma )
% 0.32/0.54 & ? [M4: nat] :
% 0.32/0.54 ( ( ( some_nat @ M4 )
% 0.32/0.54 = ( vEBT_vebt_mint @ Summary ) )
% 0.32/0.54 & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mintlistlength
% 0.32/0.54 thf(fact_2611_both__member__options__from__complete__tree__to__child,axiom,
% 0.32/0.54 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.32/0.54 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.54 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) )
% 0.32/0.54 | ( X = Mi )
% 0.32/0.54 | ( X = Ma ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % both_member_options_from_complete_tree_to_child
% 0.32/0.54 thf(fact_2612_succ__list__to__short,axiom,
% 0.32/0.54 ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.32/0.54 => ( ( ord_less_eq_nat @ Mi @ X )
% 0.32/0.54 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.54 = none_nat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % succ_list_to_short
% 0.32/0.54 thf(fact_2613__C1_C,axiom,
% 0.32/0.54 ( ( ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.54 != none_nat )
% 0.32/0.54 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.32/0.54 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.54 & ( ~ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.54 != none_nat )
% 0.32/0.54 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.54 => ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.54 = none_nat )
% 0.32/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.32/0.54 = none_nat ) )
% 0.32/0.54 & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.54 != none_nat )
% 0.32/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.32/0.54 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % "1"
% 0.32/0.54 thf(fact_2614_succ__min,axiom,
% 0.32/0.54 ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.32/0.54 => ( ( ord_less_nat @ X @ Mi )
% 0.32/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.54 = ( some_nat @ Mi ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % succ_min
% 0.32/0.54 thf(fact_2615_mi__eq__ma__no__ch,axiom,
% 0.32/0.54 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.32/0.54 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.32/0.54 => ( ( Mi = Ma )
% 0.32/0.54 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.54 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.54 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 0.32/0.54 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mi_eq_ma_no_ch
% 0.32/0.54 thf(fact_2616_double__eq__0__iff,axiom,
% 0.32/0.54 ! [A: real] :
% 0.32/0.54 ( ( ( plus_plus_real @ A @ A )
% 0.32/0.54 = zero_zero_real )
% 0.32/0.54 = ( A = zero_zero_real ) ) ).
% 0.32/0.54
% 0.32/0.54 % double_eq_0_iff
% 0.32/0.54 thf(fact_2617_double__eq__0__iff,axiom,
% 0.32/0.54 ! [A: rat] :
% 0.32/0.54 ( ( ( plus_plus_rat @ A @ A )
% 0.32/0.54 = zero_zero_rat )
% 0.32/0.54 = ( A = zero_zero_rat ) ) ).
% 0.32/0.54
% 0.32/0.54 % double_eq_0_iff
% 0.32/0.54 thf(fact_2618_double__eq__0__iff,axiom,
% 0.32/0.54 ! [A: int] :
% 0.32/0.54 ( ( ( plus_plus_int @ A @ A )
% 0.32/0.54 = zero_zero_int )
% 0.32/0.54 = ( A = zero_zero_int ) ) ).
% 0.32/0.54
% 0.32/0.54 % double_eq_0_iff
% 0.32/0.54 thf(fact_2619_insert__simp__mima,axiom,
% 0.32/0.54 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.32/0.54 ( ( ( X = Mi )
% 0.32/0.54 | ( X = Ma ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.32/0.54 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.54 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % insert_simp_mima
% 0.32/0.54 thf(fact_2620_mi__ma__2__deg,axiom,
% 0.32/0.54 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.32/0.54 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.32/0.54 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.32/0.54 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mi_ma_2_deg
% 0.32/0.54 thf(fact_2621_summaxma,axiom,
% 0.32/0.54 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.32/0.54 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.32/0.54 => ( ( Mi != Ma )
% 0.32/0.54 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 0.32/0.54 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % summaxma
% 0.32/0.54 thf(fact_2622_option_Ocollapse,axiom,
% 0.32/0.54 ! [Option: option_nat] :
% 0.32/0.54 ( ( Option != none_nat )
% 0.32/0.54 => ( ( some_nat @ ( the_nat @ Option ) )
% 0.32/0.54 = Option ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.collapse
% 0.32/0.54 thf(fact_2623_option_Ocollapse,axiom,
% 0.32/0.54 ! [Option: option4927543243414619207at_nat] :
% 0.32/0.54 ( ( Option != none_P5556105721700978146at_nat )
% 0.32/0.54 => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 0.32/0.54 = Option ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.collapse
% 0.32/0.54 thf(fact_2624_option_Ocollapse,axiom,
% 0.32/0.54 ! [Option: option_num] :
% 0.32/0.54 ( ( Option != none_num )
% 0.32/0.54 => ( ( some_num @ ( the_num @ Option ) )
% 0.32/0.54 = Option ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.collapse
% 0.32/0.54 thf(fact_2625__092_060open_062vebt__succ_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ASome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
% 0.32/0.54 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.32/0.54 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % \<open>vebt_succ (Node (Some (mi, ma)) deg treeList summary) x = Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))\<close>
% 0.32/0.54 thf(fact_2626__C2_C,axiom,
% 0.32/0.54 ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.54 = none_nat )
% 0.32/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.32/0.54 = none_nat ) )
% 0.32/0.54 & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.54 != none_nat )
% 0.32/0.54 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 0.32/0.54 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % "2"
% 0.32/0.54 thf(fact_2627_option_Osel,axiom,
% 0.32/0.54 ! [X22: nat] :
% 0.32/0.54 ( ( the_nat @ ( some_nat @ X22 ) )
% 0.32/0.54 = X22 ) ).
% 0.32/0.54
% 0.32/0.54 % option.sel
% 0.32/0.54 thf(fact_2628_option_Osel,axiom,
% 0.32/0.54 ! [X22: product_prod_nat_nat] :
% 0.32/0.54 ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.32/0.54 = X22 ) ).
% 0.32/0.54
% 0.32/0.54 % option.sel
% 0.32/0.54 thf(fact_2629_option_Osel,axiom,
% 0.32/0.54 ! [X22: num] :
% 0.32/0.54 ( ( the_num @ ( some_num @ X22 ) )
% 0.32/0.54 = X22 ) ).
% 0.32/0.54
% 0.32/0.54 % option.sel
% 0.32/0.54 thf(fact_2630_option_Oexpand,axiom,
% 0.32/0.54 ! [Option: option_nat,Option2: option_nat] :
% 0.32/0.54 ( ( ( Option = none_nat )
% 0.32/0.54 = ( Option2 = none_nat ) )
% 0.32/0.54 => ( ( ( Option != none_nat )
% 0.32/0.54 => ( ( Option2 != none_nat )
% 0.32/0.54 => ( ( the_nat @ Option )
% 0.32/0.54 = ( the_nat @ Option2 ) ) ) )
% 0.32/0.54 => ( Option = Option2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.expand
% 0.32/0.54 thf(fact_2631_option_Oexpand,axiom,
% 0.32/0.54 ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 0.32/0.54 ( ( ( Option = none_P5556105721700978146at_nat )
% 0.32/0.54 = ( Option2 = none_P5556105721700978146at_nat ) )
% 0.32/0.54 => ( ( ( Option != none_P5556105721700978146at_nat )
% 0.32/0.54 => ( ( Option2 != none_P5556105721700978146at_nat )
% 0.32/0.54 => ( ( the_Pr8591224930841456533at_nat @ Option )
% 0.32/0.54 = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 0.32/0.54 => ( Option = Option2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.expand
% 0.32/0.54 thf(fact_2632_option_Oexpand,axiom,
% 0.32/0.54 ! [Option: option_num,Option2: option_num] :
% 0.32/0.54 ( ( ( Option = none_num )
% 0.32/0.54 = ( Option2 = none_num ) )
% 0.32/0.54 => ( ( ( Option != none_num )
% 0.32/0.54 => ( ( Option2 != none_num )
% 0.32/0.54 => ( ( the_num @ Option )
% 0.32/0.54 = ( the_num @ Option2 ) ) ) )
% 0.32/0.54 => ( Option = Option2 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.expand
% 0.32/0.54 thf(fact_2633_option_Oexhaust__sel,axiom,
% 0.32/0.54 ! [Option: option_nat] :
% 0.32/0.54 ( ( Option != none_nat )
% 0.32/0.54 => ( Option
% 0.32/0.54 = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.exhaust_sel
% 0.32/0.54 thf(fact_2634_option_Oexhaust__sel,axiom,
% 0.32/0.54 ! [Option: option4927543243414619207at_nat] :
% 0.32/0.54 ( ( Option != none_P5556105721700978146at_nat )
% 0.32/0.54 => ( Option
% 0.32/0.54 = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.exhaust_sel
% 0.32/0.54 thf(fact_2635_option_Oexhaust__sel,axiom,
% 0.32/0.54 ! [Option: option_num] :
% 0.32/0.54 ( ( Option != none_num )
% 0.32/0.54 => ( Option
% 0.32/0.54 = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.exhaust_sel
% 0.32/0.54 thf(fact_2636_vebt__mint_Osimps_I3_J,axiom,
% 0.32/0.54 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.32/0.54 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.32/0.54 = ( some_nat @ Mi ) ) ).
% 0.32/0.54
% 0.32/0.54 % vebt_mint.simps(3)
% 0.32/0.54 thf(fact_2637_vebt__maxt_Osimps_I3_J,axiom,
% 0.32/0.54 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.32/0.54 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.32/0.54 = ( some_nat @ Ma ) ) ).
% 0.32/0.54
% 0.32/0.54 % vebt_maxt.simps(3)
% 0.32/0.54 thf(fact_2638_less__eq__int__code_I1_J,axiom,
% 0.32/0.54 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.32/0.54
% 0.32/0.54 % less_eq_int_code(1)
% 0.32/0.54 thf(fact_2639_less__int__code_I1_J,axiom,
% 0.32/0.54 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % less_int_code(1)
% 0.32/0.54 thf(fact_2640_times__int__code_I2_J,axiom,
% 0.32/0.54 ! [L2: int] :
% 0.32/0.54 ( ( times_times_int @ zero_zero_int @ L2 )
% 0.32/0.54 = zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % times_int_code(2)
% 0.32/0.54 thf(fact_2641_times__int__code_I1_J,axiom,
% 0.32/0.54 ! [K: int] :
% 0.32/0.54 ( ( times_times_int @ K @ zero_zero_int )
% 0.32/0.54 = zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % times_int_code(1)
% 0.32/0.54 thf(fact_2642_plus__int__code_I1_J,axiom,
% 0.32/0.54 ! [K: int] :
% 0.32/0.54 ( ( plus_plus_int @ K @ zero_zero_int )
% 0.32/0.54 = K ) ).
% 0.32/0.54
% 0.32/0.54 % plus_int_code(1)
% 0.32/0.54 thf(fact_2643_plus__int__code_I2_J,axiom,
% 0.32/0.54 ! [L2: int] :
% 0.32/0.54 ( ( plus_plus_int @ zero_zero_int @ L2 )
% 0.32/0.54 = L2 ) ).
% 0.32/0.54
% 0.32/0.54 % plus_int_code(2)
% 0.32/0.54 thf(fact_2644_minus__int__code_I1_J,axiom,
% 0.32/0.54 ! [K: int] :
% 0.32/0.54 ( ( minus_minus_int @ K @ zero_zero_int )
% 0.32/0.54 = K ) ).
% 0.32/0.54
% 0.32/0.54 % minus_int_code(1)
% 0.32/0.54 thf(fact_2645_int__distrib_I1_J,axiom,
% 0.32/0.54 ! [Z1: int,Z22: int,W: int] :
% 0.32/0.54 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_distrib(1)
% 0.32/0.54 thf(fact_2646_int__distrib_I2_J,axiom,
% 0.32/0.54 ! [W: int,Z1: int,Z22: int] :
% 0.32/0.54 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.32/0.54 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_distrib(2)
% 0.32/0.54 thf(fact_2647_int__distrib_I3_J,axiom,
% 0.32/0.54 ! [Z1: int,Z22: int,W: int] :
% 0.32/0.54 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 0.32/0.54 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_distrib(3)
% 0.32/0.54 thf(fact_2648_int__distrib_I4_J,axiom,
% 0.32/0.54 ! [W: int,Z1: int,Z22: int] :
% 0.32/0.54 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.32/0.54 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_distrib(4)
% 0.32/0.54 thf(fact_2649_zmult__zless__mono2,axiom,
% 0.32/0.54 ! [I: int,J: int,K: int] :
% 0.32/0.54 ( ( ord_less_int @ I @ J )
% 0.32/0.54 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.54 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zmult_zless_mono2
% 0.32/0.54 thf(fact_2650_odd__nonzero,axiom,
% 0.32/0.54 ! [Z: int] :
% 0.32/0.54 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 0.32/0.54 != zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % odd_nonzero
% 0.32/0.54 thf(fact_2651_int__ge__induct,axiom,
% 0.32/0.54 ! [K: int,I: int,P: int > $o] :
% 0.32/0.54 ( ( ord_less_eq_int @ K @ I )
% 0.32/0.54 => ( ( P @ K )
% 0.32/0.54 => ( ! [I2: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ K @ I2 )
% 0.32/0.54 => ( ( P @ I2 )
% 0.32/0.54 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.54 => ( P @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_ge_induct
% 0.32/0.54 thf(fact_2652_zless__add1__eq,axiom,
% 0.32/0.54 ! [W: int,Z: int] :
% 0.32/0.54 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.32/0.54 = ( ( ord_less_int @ W @ Z )
% 0.32/0.54 | ( W = Z ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zless_add1_eq
% 0.32/0.54 thf(fact_2653_int__gr__induct,axiom,
% 0.32/0.54 ! [K: int,I: int,P: int > $o] :
% 0.32/0.54 ( ( ord_less_int @ K @ I )
% 0.32/0.54 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.32/0.54 => ( ! [I2: int] :
% 0.32/0.54 ( ( ord_less_int @ K @ I2 )
% 0.32/0.54 => ( ( P @ I2 )
% 0.32/0.54 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.54 => ( P @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_gr_induct
% 0.32/0.54 thf(fact_2654_int__le__induct,axiom,
% 0.32/0.54 ! [I: int,K: int,P: int > $o] :
% 0.32/0.54 ( ( ord_less_eq_int @ I @ K )
% 0.32/0.54 => ( ( P @ K )
% 0.32/0.54 => ( ! [I2: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ I2 @ K )
% 0.32/0.54 => ( ( P @ I2 )
% 0.32/0.54 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.54 => ( P @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_le_induct
% 0.32/0.54 thf(fact_2655_int__less__induct,axiom,
% 0.32/0.54 ! [I: int,K: int,P: int > $o] :
% 0.32/0.54 ( ( ord_less_int @ I @ K )
% 0.32/0.54 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.32/0.54 => ( ! [I2: int] :
% 0.32/0.54 ( ( ord_less_int @ I2 @ K )
% 0.32/0.54 => ( ( P @ I2 )
% 0.32/0.54 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.54 => ( P @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_less_induct
% 0.32/0.54 thf(fact_2656_int__one__le__iff__zero__less,axiom,
% 0.32/0.54 ! [Z: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.32/0.54 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_one_le_iff_zero_less
% 0.32/0.54 thf(fact_2657_pos__zmult__eq__1__iff,axiom,
% 0.32/0.54 ! [M: int,N: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ M )
% 0.32/0.54 => ( ( ( times_times_int @ M @ N )
% 0.32/0.54 = one_one_int )
% 0.32/0.54 = ( ( M = one_one_int )
% 0.32/0.54 & ( N = one_one_int ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pos_zmult_eq_1_iff
% 0.32/0.54 thf(fact_2658_odd__less__0__iff,axiom,
% 0.32/0.54 ! [Z: int] :
% 0.32/0.54 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.32/0.54 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.32/0.54
% 0.32/0.54 % odd_less_0_iff
% 0.32/0.54 thf(fact_2659_add1__zle__eq,axiom,
% 0.32/0.54 ! [W: int,Z: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 0.32/0.54 = ( ord_less_int @ W @ Z ) ) ).
% 0.32/0.54
% 0.32/0.54 % add1_zle_eq
% 0.32/0.54 thf(fact_2660_zless__imp__add1__zle,axiom,
% 0.32/0.54 ! [W: int,Z: int] :
% 0.32/0.54 ( ( ord_less_int @ W @ Z )
% 0.32/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 0.32/0.54
% 0.32/0.54 % zless_imp_add1_zle
% 0.32/0.54 thf(fact_2661_int__induct,axiom,
% 0.32/0.54 ! [P: int > $o,K: int,I: int] :
% 0.32/0.54 ( ( P @ K )
% 0.32/0.54 => ( ! [I2: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ K @ I2 )
% 0.32/0.54 => ( ( P @ I2 )
% 0.32/0.54 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.54 => ( ! [I2: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ I2 @ K )
% 0.32/0.54 => ( ( P @ I2 )
% 0.32/0.54 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.32/0.54 => ( P @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % int_induct
% 0.32/0.54 thf(fact_2662_invar__vebt_Ointros_I4_J,axiom,
% 0.32/0.54 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.32/0.54 ( ! [X3: vEBT_VEBT] :
% 0.32/0.54 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.54 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.32/0.54 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.32/0.54 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.32/0.54 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.54 => ( ( M = N )
% 0.32/0.54 => ( ( Deg
% 0.32/0.54 = ( plus_plus_nat @ N @ M ) )
% 0.32/0.54 => ( ! [I2: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.54 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
% 0.32/0.54 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.32/0.54 => ( ( ( Mi = Ma )
% 0.32/0.54 => ! [X3: vEBT_VEBT] :
% 0.32/0.54 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.54 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.32/0.54 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.32/0.54 => ( ( ( Mi != Ma )
% 0.32/0.54 => ! [I2: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.54 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.32/0.54 = I2 )
% 0.32/0.54 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.32/0.54 & ! [X3: nat] :
% 0.32/0.54 ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 0.32/0.54 = I2 )
% 0.32/0.54 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 0.32/0.54 => ( ( ord_less_nat @ Mi @ X3 )
% 0.32/0.54 & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 0.32/0.54 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % invar_vebt.intros(4)
% 0.32/0.54 thf(fact_2663_le__imp__0__less,axiom,
% 0.32/0.54 ! [Z: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.54 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_imp_0_less
% 0.32/0.54 thf(fact_2664_nested__mint,axiom,
% 0.32/0.54 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 0.32/0.54 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.32/0.54 => ( ( N
% 0.32/0.54 = ( suc @ ( suc @ Va ) ) )
% 0.32/0.54 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 0.32/0.54 => ( ( Ma != Mi )
% 0.32/0.54 => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nested_mint
% 0.32/0.54 thf(fact_2665_divmod__step__eq,axiom,
% 0.32/0.54 ! [L2: num,R: nat,Q2: nat] :
% 0.32/0.54 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
% 0.32/0.54 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.32/0.54 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L2 ) ) ) ) )
% 0.32/0.54 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
% 0.32/0.54 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.32/0.54 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divmod_step_eq
% 0.32/0.54 thf(fact_2666_divmod__step__eq,axiom,
% 0.32/0.54 ! [L2: num,R: int,Q2: int] :
% 0.32/0.54 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
% 0.32/0.54 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.54 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L2 ) ) ) ) )
% 0.32/0.54 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
% 0.32/0.54 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.54 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divmod_step_eq
% 0.32/0.54 thf(fact_2667_divmod__step__eq,axiom,
% 0.32/0.54 ! [L2: num,R: code_integer,Q2: code_integer] :
% 0.32/0.54 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
% 0.32/0.54 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 0.32/0.54 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
% 0.32/0.54 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
% 0.32/0.54 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 0.32/0.54 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % divmod_step_eq
% 0.32/0.54 thf(fact_2668_cpmi,axiom,
% 0.32/0.54 ! [D4: int,P: int > $o,P5: int > $o,B5: set_int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.54 => ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ! [X3: int] :
% 0.32/0.54 ( ! [Xa2: int] :
% 0.32/0.54 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 => ! [Xb2: int] :
% 0.32/0.54 ( ( member_int @ Xb2 @ B5 )
% 0.32/0.54 => ( X3
% 0.32/0.54 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.32/0.54 => ( ! [X3: int,K3: int] :
% 0.32/0.54 ( ( P5 @ X3 )
% 0.32/0.54 = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.32/0.54 => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.32/0.54 = ( ? [X2: int] :
% 0.32/0.54 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 & ( P5 @ X2 ) )
% 0.32/0.54 | ? [X2: int] :
% 0.32/0.54 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 & ? [Y: int] :
% 0.32/0.54 ( ( member_int @ Y @ B5 )
% 0.32/0.54 & ( P @ ( plus_plus_int @ Y @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % cpmi
% 0.32/0.54 thf(fact_2669_cppi,axiom,
% 0.32/0.54 ! [D4: int,P: int > $o,P5: int > $o,A2: set_int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.54 => ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ! [X3: int] :
% 0.32/0.54 ( ! [Xa2: int] :
% 0.32/0.54 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 => ! [Xb2: int] :
% 0.32/0.54 ( ( member_int @ Xb2 @ A2 )
% 0.32/0.54 => ( X3
% 0.32/0.54 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.32/0.54 => ( ! [X3: int,K3: int] :
% 0.32/0.54 ( ( P5 @ X3 )
% 0.32/0.54 = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.32/0.54 => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.32/0.54 = ( ? [X2: int] :
% 0.32/0.54 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 & ( P5 @ X2 ) )
% 0.32/0.54 | ? [X2: int] :
% 0.32/0.54 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 & ? [Y: int] :
% 0.32/0.54 ( ( member_int @ Y @ A2 )
% 0.32/0.54 & ( P @ ( minus_minus_int @ Y @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % cppi
% 0.32/0.54 thf(fact_2670_bset_I6_J,axiom,
% 0.32/0.54 ! [D4: int,B5: set_int,T: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.54 => ! [X5: int] :
% 0.32/0.54 ( ! [Xa3: int] :
% 0.32/0.54 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 => ! [Xb3: int] :
% 0.32/0.54 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.54 => ( X5
% 0.32/0.54 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ X5 @ T )
% 0.32/0.54 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % bset(6)
% 0.32/0.54 thf(fact_2671_bset_I8_J,axiom,
% 0.32/0.54 ! [D4: int,T: int,B5: set_int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.54 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 0.32/0.54 => ! [X5: int] :
% 0.32/0.54 ( ! [Xa3: int] :
% 0.32/0.54 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 => ! [Xb3: int] :
% 0.32/0.54 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.54 => ( X5
% 0.32/0.54 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ T @ X5 )
% 0.32/0.54 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % bset(8)
% 0.32/0.54 thf(fact_2672_aset_I6_J,axiom,
% 0.32/0.54 ! [D4: int,T: int,A2: set_int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.54 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.32/0.54 => ! [X5: int] :
% 0.32/0.54 ( ! [Xa3: int] :
% 0.32/0.54 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 => ! [Xb3: int] :
% 0.32/0.54 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.54 => ( X5
% 0.32/0.54 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ X5 @ T )
% 0.32/0.54 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % aset(6)
% 0.32/0.54 thf(fact_2673_aset_I8_J,axiom,
% 0.32/0.54 ! [D4: int,A2: set_int,T: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.54 => ! [X5: int] :
% 0.32/0.54 ( ! [Xa3: int] :
% 0.32/0.54 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 => ! [Xb3: int] :
% 0.32/0.54 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.54 => ( X5
% 0.32/0.54 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ T @ X5 )
% 0.32/0.54 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % aset(8)
% 0.32/0.54 thf(fact_2674_aset_I7_J,axiom,
% 0.32/0.54 ! [D4: int,A2: set_int,T: int] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.54 => ! [X5: int] :
% 0.32/0.54 ( ! [Xa3: int] :
% 0.32/0.54 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.54 => ! [Xb3: int] :
% 0.32/0.54 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.54 => ( X5
% 0.32/0.54 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.54 => ( ( ord_less_int @ T @ X5 )
% 0.32/0.54 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % aset(7)
% 0.32/0.54 thf(fact_2675_even__odd__cases,axiom,
% 0.32/0.54 ! [X: nat] :
% 0.32/0.54 ( ! [N4: nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( plus_plus_nat @ N4 @ N4 ) )
% 0.32/0.54 => ~ ! [N4: nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % even_odd_cases
% 0.32/0.54 thf(fact_2676_deg__SUcn__Node,axiom,
% 0.32/0.54 ! [Tree: vEBT_VEBT,N: nat] :
% 0.32/0.54 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 0.32/0.54 => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.32/0.54 ( Tree
% 0.32/0.54 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % deg_SUcn_Node
% 0.32/0.54 thf(fact_2677_nat_Oinject,axiom,
% 0.32/0.54 ! [X22: nat,Y2: nat] :
% 0.32/0.54 ( ( ( suc @ X22 )
% 0.32/0.54 = ( suc @ Y2 ) )
% 0.32/0.54 = ( X22 = Y2 ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat.inject
% 0.32/0.54 thf(fact_2678_old_Onat_Oinject,axiom,
% 0.32/0.54 ! [Nat: nat,Nat2: nat] :
% 0.32/0.54 ( ( ( suc @ Nat )
% 0.32/0.54 = ( suc @ Nat2 ) )
% 0.32/0.54 = ( Nat = Nat2 ) ) ).
% 0.32/0.54
% 0.32/0.54 % old.nat.inject
% 0.32/0.54 thf(fact_2679_Suc__less__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.54 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_less_eq
% 0.32/0.54 thf(fact_2680_Suc__mono,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ N )
% 0.32/0.54 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_mono
% 0.32/0.54 thf(fact_2681_lessI,axiom,
% 0.32/0.54 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % lessI
% 0.32/0.54 thf(fact_2682_Suc__le__mono,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.32/0.54 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_le_mono
% 0.32/0.54 thf(fact_2683_add__Suc__right,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.32/0.54 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_Suc_right
% 0.32/0.54 thf(fact_2684_Suc__diff__diff,axiom,
% 0.32/0.54 ! [M: nat,N: nat,K: nat] :
% 0.32/0.54 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.32/0.54 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_diff_diff
% 0.32/0.54 thf(fact_2685_diff__Suc__Suc,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.54 = ( minus_minus_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_Suc_Suc
% 0.32/0.54 thf(fact_2686_power__0__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.32/0.54 = zero_zero_rat ) ).
% 0.32/0.54
% 0.32/0.54 % power_0_Suc
% 0.32/0.54 thf(fact_2687_power__0__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.32/0.54 = zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % power_0_Suc
% 0.32/0.54 thf(fact_2688_power__0__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.32/0.54 = zero_zero_real ) ).
% 0.32/0.54
% 0.32/0.54 % power_0_Suc
% 0.32/0.54 thf(fact_2689_power__0__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.32/0.54 = zero_zero_int ) ).
% 0.32/0.54
% 0.32/0.54 % power_0_Suc
% 0.32/0.54 thf(fact_2690_power__0__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.32/0.54 = zero_zero_complex ) ).
% 0.32/0.54
% 0.32/0.54 % power_0_Suc
% 0.32/0.54 thf(fact_2691_power__Suc0__right,axiom,
% 0.32/0.54 ! [A: nat] :
% 0.32/0.54 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.32/0.54 = A ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc0_right
% 0.32/0.54 thf(fact_2692_power__Suc0__right,axiom,
% 0.32/0.54 ! [A: real] :
% 0.32/0.54 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.32/0.54 = A ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc0_right
% 0.32/0.54 thf(fact_2693_power__Suc0__right,axiom,
% 0.32/0.54 ! [A: int] :
% 0.32/0.54 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.32/0.54 = A ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc0_right
% 0.32/0.54 thf(fact_2694_power__Suc0__right,axiom,
% 0.32/0.54 ! [A: complex] :
% 0.32/0.54 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.32/0.54 = A ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc0_right
% 0.32/0.54 thf(fact_2695_less__Suc0,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.54 = ( N = zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_Suc0
% 0.32/0.54 thf(fact_2696_zero__less__Suc,axiom,
% 0.32/0.54 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_less_Suc
% 0.32/0.54 thf(fact_2697_one__eq__mult__iff,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ( suc @ zero_zero_nat )
% 0.32/0.54 = ( times_times_nat @ M @ N ) )
% 0.32/0.54 = ( ( M
% 0.32/0.54 = ( suc @ zero_zero_nat ) )
% 0.32/0.54 & ( N
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % one_eq_mult_iff
% 0.32/0.54 thf(fact_2698_mult__eq__1__iff,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ( times_times_nat @ M @ N )
% 0.32/0.54 = ( suc @ zero_zero_nat ) )
% 0.32/0.54 = ( ( M
% 0.32/0.54 = ( suc @ zero_zero_nat ) )
% 0.32/0.54 & ( N
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_eq_1_iff
% 0.32/0.54 thf(fact_2699_div__by__Suc__0,axiom,
% 0.32/0.54 ! [M: nat] :
% 0.32/0.54 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.32/0.54 = M ) ).
% 0.32/0.54
% 0.32/0.54 % div_by_Suc_0
% 0.32/0.54 thf(fact_2700_mult__Suc__right,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.32/0.54 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_Suc_right
% 0.32/0.54 thf(fact_2701_power__Suc__0,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_0
% 0.32/0.54 thf(fact_2702_nat__power__eq__Suc__0__iff,axiom,
% 0.32/0.54 ! [X: nat,M: nat] :
% 0.32/0.54 ( ( ( power_power_nat @ X @ M )
% 0.32/0.54 = ( suc @ zero_zero_nat ) )
% 0.32/0.54 = ( ( M = zero_zero_nat )
% 0.32/0.54 | ( X
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat_power_eq_Suc_0_iff
% 0.32/0.54 thf(fact_2703_diff__Suc__1,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.32/0.54 = N ) ).
% 0.32/0.54
% 0.32/0.54 % diff_Suc_1
% 0.32/0.54 thf(fact_2704_mod__by__Suc__0,axiom,
% 0.32/0.54 ! [M: nat] :
% 0.32/0.54 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.32/0.54 = zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % mod_by_Suc_0
% 0.32/0.54 thf(fact_2705_Suc__pred,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.32/0.54 = N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_pred
% 0.32/0.54 thf(fact_2706_one__le__mult__iff,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.32/0.54 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.54 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % one_le_mult_iff
% 0.32/0.54 thf(fact_2707_diff__Suc__diff__eq1,axiom,
% 0.32/0.54 ! [K: nat,J: nat,I: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.54 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.32/0.54 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_Suc_diff_eq1
% 0.32/0.54 thf(fact_2708_diff__Suc__diff__eq2,axiom,
% 0.32/0.54 ! [K: nat,J: nat,I: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ K @ J )
% 0.32/0.54 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.32/0.54 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_Suc_diff_eq2
% 0.32/0.54 thf(fact_2709_Suc__numeral,axiom,
% 0.32/0.54 ! [N: num] :
% 0.32/0.54 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 0.32/0.54 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_numeral
% 0.32/0.54 thf(fact_2710_Suc__mod__mult__self4,axiom,
% 0.32/0.54 ! [N: nat,K: nat,M: nat] :
% 0.32/0.54 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 0.32/0.54 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_mod_mult_self4
% 0.32/0.54 thf(fact_2711_Suc__mod__mult__self3,axiom,
% 0.32/0.54 ! [K: nat,N: nat,M: nat] :
% 0.32/0.54 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 0.32/0.54 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_mod_mult_self3
% 0.32/0.54 thf(fact_2712_Suc__mod__mult__self2,axiom,
% 0.32/0.54 ! [M: nat,N: nat,K: nat] :
% 0.32/0.54 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 0.32/0.54 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_mod_mult_self2
% 0.32/0.54 thf(fact_2713_Suc__mod__mult__self1,axiom,
% 0.32/0.54 ! [M: nat,K: nat,N: nat] :
% 0.32/0.54 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 0.32/0.54 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_mod_mult_self1
% 0.32/0.54 thf(fact_2714_add__2__eq__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.54 = ( suc @ ( suc @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_2_eq_Suc
% 0.32/0.54 thf(fact_2715_add__2__eq__Suc_H,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.54 = ( suc @ ( suc @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_2_eq_Suc'
% 0.32/0.54 thf(fact_2716_div2__Suc__Suc,axiom,
% 0.32/0.54 ! [M: nat] :
% 0.32/0.54 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.54 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % div2_Suc_Suc
% 0.32/0.54 thf(fact_2717_Suc__1,axiom,
% 0.32/0.54 ( ( suc @ one_one_nat )
% 0.32/0.54 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_1
% 0.32/0.54 thf(fact_2718_mod2__Suc__Suc,axiom,
% 0.32/0.54 ! [M: nat] :
% 0.32/0.54 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.54 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod2_Suc_Suc
% 0.32/0.54 thf(fact_2719_Suc__diff__1,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.32/0.54 = N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_diff_1
% 0.32/0.54 thf(fact_2720_Suc__times__numeral__mod__eq,axiom,
% 0.32/0.54 ! [K: num,N: nat] :
% 0.32/0.54 ( ( ( numeral_numeral_nat @ K )
% 0.32/0.54 != one_one_nat )
% 0.32/0.54 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.54 = one_one_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_times_numeral_mod_eq
% 0.32/0.54 thf(fact_2721_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.54 != ( suc @ zero_zero_nat ) )
% 0.32/0.54 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.54 = zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % not_mod2_eq_Suc_0_eq_0
% 0.32/0.54 thf(fact_2722_Suc__inject,axiom,
% 0.32/0.54 ! [X: nat,Y3: nat] :
% 0.32/0.54 ( ( ( suc @ X )
% 0.32/0.54 = ( suc @ Y3 ) )
% 0.32/0.54 => ( X = Y3 ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_inject
% 0.32/0.54 thf(fact_2723_n__not__Suc__n,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( N
% 0.32/0.54 != ( suc @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % n_not_Suc_n
% 0.32/0.54 thf(fact_2724_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.32/0.54 ! [X: produc8306885398267862888on_nat] :
% 0.32/0.54 ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.32/0.54 => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.32/0.54 => ~ ! [F2: nat > nat > nat,A5: nat,B3: nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A5 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % VEBT_internal.option_shift.cases
% 0.32/0.54 thf(fact_2725_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.32/0.54 ! [X: produc5542196010084753463at_nat] :
% 0.32/0.54 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.32/0.54 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.32/0.54 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A5: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A5 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % VEBT_internal.option_shift.cases
% 0.32/0.54 thf(fact_2726_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.32/0.54 ! [X: produc1193250871479095198on_num] :
% 0.32/0.54 ( ! [Uu2: num > num > num,Uv2: option_num] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.32/0.54 => ( ! [Uw2: num > num > num,V2: num] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.32/0.54 => ~ ! [F2: num > num > num,A5: num,B3: num] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A5 ) @ ( some_num @ B3 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % VEBT_internal.option_shift.cases
% 0.32/0.54 thf(fact_2727_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.32/0.54 ! [X: produc2233624965454879586on_nat] :
% 0.32/0.54 ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.32/0.54 => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.32/0.54 => ~ ! [F2: nat > nat > $o,X3: nat,Y4: nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % VEBT_internal.option_comp_shift.cases
% 0.32/0.54 thf(fact_2728_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.32/0.54 ! [X: produc5491161045314408544at_nat] :
% 0.32/0.54 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.32/0.54 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.32/0.54 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % VEBT_internal.option_comp_shift.cases
% 0.32/0.54 thf(fact_2729_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.32/0.54 ! [X: produc7036089656553540234on_num] :
% 0.32/0.54 ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.32/0.54 => ( ! [Uw2: num > num > $o,V2: num] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.32/0.54 => ~ ! [F2: num > num > $o,X3: num,Y4: num] :
% 0.32/0.54 ( X
% 0.32/0.54 != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % VEBT_internal.option_comp_shift.cases
% 0.32/0.54 thf(fact_2730_not0__implies__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( N != zero_zero_nat )
% 0.32/0.54 => ? [M4: nat] :
% 0.32/0.54 ( N
% 0.32/0.54 = ( suc @ M4 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % not0_implies_Suc
% 0.32/0.54 thf(fact_2731_Zero__not__Suc,axiom,
% 0.32/0.54 ! [M: nat] :
% 0.32/0.54 ( zero_zero_nat
% 0.32/0.54 != ( suc @ M ) ) ).
% 0.32/0.54
% 0.32/0.54 % Zero_not_Suc
% 0.32/0.54 thf(fact_2732_Zero__neq__Suc,axiom,
% 0.32/0.54 ! [M: nat] :
% 0.32/0.54 ( zero_zero_nat
% 0.32/0.54 != ( suc @ M ) ) ).
% 0.32/0.54
% 0.32/0.54 % Zero_neq_Suc
% 0.32/0.54 thf(fact_2733_Suc__neq__Zero,axiom,
% 0.32/0.54 ! [M: nat] :
% 0.32/0.54 ( ( suc @ M )
% 0.32/0.54 != zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_neq_Zero
% 0.32/0.54 thf(fact_2734_zero__induct,axiom,
% 0.32/0.54 ! [P: nat > $o,K: nat] :
% 0.32/0.54 ( ( P @ K )
% 0.32/0.54 => ( ! [N4: nat] :
% 0.32/0.54 ( ( P @ ( suc @ N4 ) )
% 0.32/0.54 => ( P @ N4 ) )
% 0.32/0.54 => ( P @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_induct
% 0.32/0.54 thf(fact_2735_diff__induct,axiom,
% 0.32/0.54 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.32/0.54 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 0.32/0.54 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.32/0.54 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.54 ( ( P @ X3 @ Y4 )
% 0.32/0.54 => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 0.32/0.54 => ( P @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_induct
% 0.32/0.54 thf(fact_2736_nat__induct,axiom,
% 0.32/0.54 ! [P: nat > $o,N: nat] :
% 0.32/0.54 ( ( P @ zero_zero_nat )
% 0.32/0.54 => ( ! [N4: nat] :
% 0.32/0.54 ( ( P @ N4 )
% 0.32/0.54 => ( P @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( P @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat_induct
% 0.32/0.54 thf(fact_2737_old_Onat_Oexhaust,axiom,
% 0.32/0.54 ! [Y3: nat] :
% 0.32/0.54 ( ( Y3 != zero_zero_nat )
% 0.32/0.54 => ~ ! [Nat3: nat] :
% 0.32/0.54 ( Y3
% 0.32/0.54 != ( suc @ Nat3 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % old.nat.exhaust
% 0.32/0.54 thf(fact_2738_nat_OdiscI,axiom,
% 0.32/0.54 ! [Nat: nat,X22: nat] :
% 0.32/0.54 ( ( Nat
% 0.32/0.54 = ( suc @ X22 ) )
% 0.32/0.54 => ( Nat != zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat.discI
% 0.32/0.54 thf(fact_2739_old_Onat_Odistinct_I1_J,axiom,
% 0.32/0.54 ! [Nat2: nat] :
% 0.32/0.54 ( zero_zero_nat
% 0.32/0.54 != ( suc @ Nat2 ) ) ).
% 0.32/0.54
% 0.32/0.54 % old.nat.distinct(1)
% 0.32/0.54 thf(fact_2740_old_Onat_Odistinct_I2_J,axiom,
% 0.32/0.54 ! [Nat2: nat] :
% 0.32/0.54 ( ( suc @ Nat2 )
% 0.32/0.54 != zero_zero_nat ) ).
% 0.32/0.54
% 0.32/0.54 % old.nat.distinct(2)
% 0.32/0.54 thf(fact_2741_nat_Odistinct_I1_J,axiom,
% 0.32/0.54 ! [X22: nat] :
% 0.32/0.54 ( zero_zero_nat
% 0.32/0.54 != ( suc @ X22 ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat.distinct(1)
% 0.32/0.54 thf(fact_2742_not__less__less__Suc__eq,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ~ ( ord_less_nat @ N @ M )
% 0.32/0.54 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.32/0.54 = ( N = M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % not_less_less_Suc_eq
% 0.32/0.54 thf(fact_2743_strict__inc__induct,axiom,
% 0.32/0.54 ! [I: nat,J: nat,P: nat > $o] :
% 0.32/0.54 ( ( ord_less_nat @ I @ J )
% 0.32/0.54 => ( ! [I2: nat] :
% 0.32/0.54 ( ( J
% 0.32/0.54 = ( suc @ I2 ) )
% 0.32/0.54 => ( P @ I2 ) )
% 0.32/0.54 => ( ! [I2: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I2 @ J )
% 0.32/0.54 => ( ( P @ ( suc @ I2 ) )
% 0.32/0.54 => ( P @ I2 ) ) )
% 0.32/0.54 => ( P @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % strict_inc_induct
% 0.32/0.54 thf(fact_2744_less__Suc__induct,axiom,
% 0.32/0.54 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.32/0.54 ( ( ord_less_nat @ I @ J )
% 0.32/0.54 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.32/0.54 => ( ! [I2: nat,J2: nat,K3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I2 @ J2 )
% 0.32/0.54 => ( ( ord_less_nat @ J2 @ K3 )
% 0.32/0.54 => ( ( P @ I2 @ J2 )
% 0.32/0.54 => ( ( P @ J2 @ K3 )
% 0.32/0.54 => ( P @ I2 @ K3 ) ) ) ) )
% 0.32/0.54 => ( P @ I @ J ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_Suc_induct
% 0.32/0.54 thf(fact_2745_less__trans__Suc,axiom,
% 0.32/0.54 ! [I: nat,J: nat,K: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I @ J )
% 0.32/0.54 => ( ( ord_less_nat @ J @ K )
% 0.32/0.54 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_trans_Suc
% 0.32/0.54 thf(fact_2746_Suc__less__SucD,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.54 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_less_SucD
% 0.32/0.54 thf(fact_2747_less__antisym,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ~ ( ord_less_nat @ N @ M )
% 0.32/0.54 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.32/0.54 => ( M = N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_antisym
% 0.32/0.54 thf(fact_2748_Suc__less__eq2,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.54 = ( ? [M6: nat] :
% 0.32/0.54 ( ( M
% 0.32/0.54 = ( suc @ M6 ) )
% 0.32/0.54 & ( ord_less_nat @ N @ M6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_less_eq2
% 0.32/0.54 thf(fact_2749_All__less__Suc,axiom,
% 0.32/0.54 ! [N: nat,P: nat > $o] :
% 0.32/0.54 ( ( ! [I4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.32/0.54 => ( P @ I4 ) ) )
% 0.32/0.54 = ( ( P @ N )
% 0.32/0.54 & ! [I4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I4 @ N )
% 0.32/0.54 => ( P @ I4 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % All_less_Suc
% 0.32/0.54 thf(fact_2750_not__less__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.32/0.54 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % not_less_eq
% 0.32/0.54 thf(fact_2751_less__Suc__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.32/0.54 = ( ( ord_less_nat @ M @ N )
% 0.32/0.54 | ( M = N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_Suc_eq
% 0.32/0.54 thf(fact_2752_Ex__less__Suc,axiom,
% 0.32/0.54 ! [N: nat,P: nat > $o] :
% 0.32/0.54 ( ( ? [I4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.32/0.54 & ( P @ I4 ) ) )
% 0.32/0.54 = ( ( P @ N )
% 0.32/0.54 | ? [I4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I4 @ N )
% 0.32/0.54 & ( P @ I4 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Ex_less_Suc
% 0.32/0.54 thf(fact_2753_less__SucI,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ N )
% 0.32/0.54 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_SucI
% 0.32/0.54 thf(fact_2754_less__SucE,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.32/0.54 => ( ~ ( ord_less_nat @ M @ N )
% 0.32/0.54 => ( M = N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_SucE
% 0.32/0.54 thf(fact_2755_Suc__lessI,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ N )
% 0.32/0.54 => ( ( ( suc @ M )
% 0.32/0.54 != N )
% 0.32/0.54 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_lessI
% 0.32/0.54 thf(fact_2756_Suc__lessE,axiom,
% 0.32/0.54 ! [I: nat,K: nat] :
% 0.32/0.54 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.32/0.54 => ~ ! [J2: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I @ J2 )
% 0.32/0.54 => ( K
% 0.32/0.54 != ( suc @ J2 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_lessE
% 0.32/0.54 thf(fact_2757_Suc__lessD,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.32/0.54 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_lessD
% 0.32/0.54 thf(fact_2758_Nat_OlessE,axiom,
% 0.32/0.54 ! [I: nat,K: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I @ K )
% 0.32/0.54 => ( ( K
% 0.32/0.54 != ( suc @ I ) )
% 0.32/0.54 => ~ ! [J2: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I @ J2 )
% 0.32/0.54 => ( K
% 0.32/0.54 != ( suc @ J2 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Nat.lessE
% 0.32/0.54 thf(fact_2759_transitive__stepwise__le,axiom,
% 0.32/0.54 ! [M: nat,N: nat,R4: nat > nat > $o] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( ! [X3: nat] : ( R4 @ X3 @ X3 )
% 0.32/0.54 => ( ! [X3: nat,Y4: nat,Z5: nat] :
% 0.32/0.54 ( ( R4 @ X3 @ Y4 )
% 0.32/0.54 => ( ( R4 @ Y4 @ Z5 )
% 0.32/0.54 => ( R4 @ X3 @ Z5 ) ) )
% 0.32/0.54 => ( ! [N4: nat] : ( R4 @ N4 @ ( suc @ N4 ) )
% 0.32/0.54 => ( R4 @ M @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % transitive_stepwise_le
% 0.32/0.54 thf(fact_2760_nat__induct__at__least,axiom,
% 0.32/0.54 ! [M: nat,N: nat,P: nat > $o] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( ( P @ M )
% 0.32/0.54 => ( ! [N4: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N4 )
% 0.32/0.54 => ( ( P @ N4 )
% 0.32/0.54 => ( P @ ( suc @ N4 ) ) ) )
% 0.32/0.54 => ( P @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat_induct_at_least
% 0.32/0.54 thf(fact_2761_full__nat__induct,axiom,
% 0.32/0.54 ! [P: nat > $o,N: nat] :
% 0.32/0.54 ( ! [N4: nat] :
% 0.32/0.54 ( ! [M2: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N4 )
% 0.32/0.54 => ( P @ M2 ) )
% 0.32/0.54 => ( P @ N4 ) )
% 0.32/0.54 => ( P @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % full_nat_induct
% 0.32/0.54 thf(fact_2762_not__less__eq__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.32/0.54 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.32/0.54
% 0.32/0.54 % not_less_eq_eq
% 0.32/0.54 thf(fact_2763_Suc__n__not__le__n,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_n_not_le_n
% 0.32/0.54 thf(fact_2764_le__Suc__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.54 = ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 | ( M
% 0.32/0.54 = ( suc @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_Suc_eq
% 0.32/0.54 thf(fact_2765_Suc__le__D,axiom,
% 0.32/0.54 ! [N: nat,M7: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
% 0.32/0.54 => ? [M4: nat] :
% 0.32/0.54 ( M7
% 0.32/0.54 = ( suc @ M4 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_le_D
% 0.32/0.54 thf(fact_2766_le__SucI,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_SucI
% 0.32/0.54 thf(fact_2767_le__SucE,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.54 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( M
% 0.32/0.54 = ( suc @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_SucE
% 0.32/0.54 thf(fact_2768_Suc__leD,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.32/0.54 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_leD
% 0.32/0.54 thf(fact_2769_add__Suc__shift,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.32/0.54 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_Suc_shift
% 0.32/0.54 thf(fact_2770_add__Suc,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.32/0.54 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_Suc
% 0.32/0.54 thf(fact_2771_nat__arith_Osuc1,axiom,
% 0.32/0.54 ! [A2: nat,K: nat,A: nat] :
% 0.32/0.54 ( ( A2
% 0.32/0.54 = ( plus_plus_nat @ K @ A ) )
% 0.32/0.54 => ( ( suc @ A2 )
% 0.32/0.54 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat_arith.suc1
% 0.32/0.54 thf(fact_2772_zero__induct__lemma,axiom,
% 0.32/0.54 ! [P: nat > $o,K: nat,I: nat] :
% 0.32/0.54 ( ( P @ K )
% 0.32/0.54 => ( ! [N4: nat] :
% 0.32/0.54 ( ( P @ ( suc @ N4 ) )
% 0.32/0.54 => ( P @ N4 ) )
% 0.32/0.54 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % zero_induct_lemma
% 0.32/0.54 thf(fact_2773_Suc__mult__cancel1,axiom,
% 0.32/0.54 ! [K: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.32/0.54 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.32/0.54 = ( M = N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_mult_cancel1
% 0.32/0.54 thf(fact_2774_mod__Suc__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 0.32/0.54 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod_Suc_eq
% 0.32/0.54 thf(fact_2775_mod__Suc__Suc__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 0.32/0.54 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod_Suc_Suc_eq
% 0.32/0.54 thf(fact_2776_lift__Suc__mono__less__iff,axiom,
% 0.32/0.54 ! [F: nat > real,N: nat,M: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.54 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less_iff
% 0.32/0.54 thf(fact_2777_lift__Suc__mono__less__iff,axiom,
% 0.32/0.54 ! [F: nat > rat,N: nat,M: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.54 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less_iff
% 0.32/0.54 thf(fact_2778_lift__Suc__mono__less__iff,axiom,
% 0.32/0.54 ! [F: nat > num,N: nat,M: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.54 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less_iff
% 0.32/0.54 thf(fact_2779_lift__Suc__mono__less__iff,axiom,
% 0.32/0.54 ! [F: nat > nat,N: nat,M: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.54 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less_iff
% 0.32/0.54 thf(fact_2780_lift__Suc__mono__less__iff,axiom,
% 0.32/0.54 ! [F: nat > int,N: nat,M: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.32/0.54 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less_iff
% 0.32/0.54 thf(fact_2781_lift__Suc__mono__less,axiom,
% 0.32/0.54 ! [F: nat > real,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_real @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less
% 0.32/0.54 thf(fact_2782_lift__Suc__mono__less,axiom,
% 0.32/0.54 ! [F: nat > rat,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_rat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less
% 0.32/0.54 thf(fact_2783_lift__Suc__mono__less,axiom,
% 0.32/0.54 ! [F: nat > num,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_num @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less
% 0.32/0.54 thf(fact_2784_lift__Suc__mono__less,axiom,
% 0.32/0.54 ! [F: nat > nat,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less
% 0.32/0.54 thf(fact_2785_lift__Suc__mono__less,axiom,
% 0.32/0.54 ! [F: nat > int,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_less
% 0.32/0.54 thf(fact_2786_power__Suc,axiom,
% 0.32/0.54 ! [A: complex,N: nat] :
% 0.32/0.54 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc
% 0.32/0.54 thf(fact_2787_power__Suc,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc
% 0.32/0.54 thf(fact_2788_power__Suc,axiom,
% 0.32/0.54 ! [A: rat,N: nat] :
% 0.32/0.54 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc
% 0.32/0.54 thf(fact_2789_power__Suc,axiom,
% 0.32/0.54 ! [A: nat,N: nat] :
% 0.32/0.54 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc
% 0.32/0.54 thf(fact_2790_power__Suc,axiom,
% 0.32/0.54 ! [A: int,N: nat] :
% 0.32/0.54 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc
% 0.32/0.54 thf(fact_2791_power__Suc2,axiom,
% 0.32/0.54 ! [A: complex,N: nat] :
% 0.32/0.54 ( ( power_power_complex @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc2
% 0.32/0.54 thf(fact_2792_power__Suc2,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( power_power_real @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc2
% 0.32/0.54 thf(fact_2793_power__Suc2,axiom,
% 0.32/0.54 ! [A: rat,N: nat] :
% 0.32/0.54 ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc2
% 0.32/0.54 thf(fact_2794_power__Suc2,axiom,
% 0.32/0.54 ! [A: nat,N: nat] :
% 0.32/0.54 ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc2
% 0.32/0.54 thf(fact_2795_power__Suc2,axiom,
% 0.32/0.54 ! [A: int,N: nat] :
% 0.32/0.54 ( ( power_power_int @ A @ ( suc @ N ) )
% 0.32/0.54 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc2
% 0.32/0.54 thf(fact_2796_lift__Suc__antimono__le,axiom,
% 0.32/0.54 ! [F: nat > set_int,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_set_int @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_antimono_le
% 0.32/0.54 thf(fact_2797_lift__Suc__antimono__le,axiom,
% 0.32/0.54 ! [F: nat > rat,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_rat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_antimono_le
% 0.32/0.54 thf(fact_2798_lift__Suc__antimono__le,axiom,
% 0.32/0.54 ! [F: nat > num,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_num @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_antimono_le
% 0.32/0.54 thf(fact_2799_lift__Suc__antimono__le,axiom,
% 0.32/0.54 ! [F: nat > nat,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_antimono_le
% 0.32/0.54 thf(fact_2800_lift__Suc__antimono__le,axiom,
% 0.32/0.54 ! [F: nat > int,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_int @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_antimono_le
% 0.32/0.54 thf(fact_2801_lift__Suc__mono__le,axiom,
% 0.32/0.54 ! [F: nat > set_int,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_le
% 0.32/0.54 thf(fact_2802_lift__Suc__mono__le,axiom,
% 0.32/0.54 ! [F: nat > rat,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_le
% 0.32/0.54 thf(fact_2803_lift__Suc__mono__le,axiom,
% 0.32/0.54 ! [F: nat > num,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_le
% 0.32/0.54 thf(fact_2804_lift__Suc__mono__le,axiom,
% 0.32/0.54 ! [F: nat > nat,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_le
% 0.32/0.54 thf(fact_2805_lift__Suc__mono__le,axiom,
% 0.32/0.54 ! [F: nat > int,N: nat,N6: nat] :
% 0.32/0.54 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.32/0.54 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % lift_Suc_mono_le
% 0.32/0.54 thf(fact_2806_Ex__less__Suc2,axiom,
% 0.32/0.54 ! [N: nat,P: nat > $o] :
% 0.32/0.54 ( ( ? [I4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.32/0.54 & ( P @ I4 ) ) )
% 0.32/0.54 = ( ( P @ zero_zero_nat )
% 0.32/0.54 | ? [I4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I4 @ N )
% 0.32/0.54 & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Ex_less_Suc2
% 0.32/0.54 thf(fact_2807_gr0__conv__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 = ( ? [M3: nat] :
% 0.32/0.54 ( N
% 0.32/0.54 = ( suc @ M3 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % gr0_conv_Suc
% 0.32/0.54 thf(fact_2808_All__less__Suc2,axiom,
% 0.32/0.54 ! [N: nat,P: nat > $o] :
% 0.32/0.54 ( ( ! [I4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.32/0.54 => ( P @ I4 ) ) )
% 0.32/0.54 = ( ( P @ zero_zero_nat )
% 0.32/0.54 & ! [I4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ I4 @ N )
% 0.32/0.54 => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % All_less_Suc2
% 0.32/0.54 thf(fact_2809_gr0__implies__Suc,axiom,
% 0.32/0.54 ! [N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ? [M4: nat] :
% 0.32/0.54 ( N
% 0.32/0.54 = ( suc @ M4 ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % gr0_implies_Suc
% 0.32/0.54 thf(fact_2810_less__Suc__eq__0__disj,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.32/0.54 = ( ( M = zero_zero_nat )
% 0.32/0.54 | ? [J3: nat] :
% 0.32/0.54 ( ( M
% 0.32/0.54 = ( suc @ J3 ) )
% 0.32/0.54 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_Suc_eq_0_disj
% 0.32/0.54 thf(fact_2811_Suc__leI,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ N )
% 0.32/0.54 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_leI
% 0.32/0.54 thf(fact_2812_Suc__le__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.32/0.54 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_le_eq
% 0.32/0.54 thf(fact_2813_dec__induct,axiom,
% 0.32/0.54 ! [I: nat,J: nat,P: nat > $o] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.54 => ( ( P @ I )
% 0.32/0.54 => ( ! [N4: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I @ N4 )
% 0.32/0.54 => ( ( ord_less_nat @ N4 @ J )
% 0.32/0.54 => ( ( P @ N4 )
% 0.32/0.54 => ( P @ ( suc @ N4 ) ) ) ) )
% 0.32/0.54 => ( P @ J ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % dec_induct
% 0.32/0.54 thf(fact_2814_inc__induct,axiom,
% 0.32/0.54 ! [I: nat,J: nat,P: nat > $o] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.54 => ( ( P @ J )
% 0.32/0.54 => ( ! [N4: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I @ N4 )
% 0.32/0.54 => ( ( ord_less_nat @ N4 @ J )
% 0.32/0.54 => ( ( P @ ( suc @ N4 ) )
% 0.32/0.54 => ( P @ N4 ) ) ) )
% 0.32/0.54 => ( P @ I ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % inc_induct
% 0.32/0.54 thf(fact_2815_Suc__le__lessD,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.32/0.54 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_le_lessD
% 0.32/0.54 thf(fact_2816_le__less__Suc__eq,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.32/0.54 = ( N = M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_less_Suc_eq
% 0.32/0.54 thf(fact_2817_less__Suc__eq__le,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.32/0.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_Suc_eq_le
% 0.32/0.54 thf(fact_2818_less__eq__Suc__le,axiom,
% 0.32/0.54 ( ord_less_nat
% 0.32/0.54 = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_eq_Suc_le
% 0.32/0.54 thf(fact_2819_le__imp__less__Suc,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.54 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % le_imp_less_Suc
% 0.32/0.54 thf(fact_2820_add__is__1,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ( plus_plus_nat @ M @ N )
% 0.32/0.54 = ( suc @ zero_zero_nat ) )
% 0.32/0.54 = ( ( ( M
% 0.32/0.54 = ( suc @ zero_zero_nat ) )
% 0.32/0.54 & ( N = zero_zero_nat ) )
% 0.32/0.54 | ( ( M = zero_zero_nat )
% 0.32/0.54 & ( N
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % add_is_1
% 0.32/0.54 thf(fact_2821_one__is__add,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ( suc @ zero_zero_nat )
% 0.32/0.54 = ( plus_plus_nat @ M @ N ) )
% 0.32/0.54 = ( ( ( M
% 0.32/0.54 = ( suc @ zero_zero_nat ) )
% 0.32/0.54 & ( N = zero_zero_nat ) )
% 0.32/0.54 | ( ( M = zero_zero_nat )
% 0.32/0.54 & ( N
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % one_is_add
% 0.32/0.54 thf(fact_2822_less__imp__Suc__add,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ N )
% 0.32/0.54 => ? [K3: nat] :
% 0.32/0.54 ( N
% 0.32/0.54 = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_imp_Suc_add
% 0.32/0.54 thf(fact_2823_less__iff__Suc__add,axiom,
% 0.32/0.54 ( ord_less_nat
% 0.32/0.54 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.54 ? [K2: nat] :
% 0.32/0.54 ( N2
% 0.32/0.54 = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_iff_Suc_add
% 0.32/0.54 thf(fact_2824_less__add__Suc2,axiom,
% 0.32/0.54 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_add_Suc2
% 0.32/0.54 thf(fact_2825_less__add__Suc1,axiom,
% 0.32/0.54 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_add_Suc1
% 0.32/0.54 thf(fact_2826_less__natE,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ M @ N )
% 0.32/0.54 => ~ ! [Q3: nat] :
% 0.32/0.54 ( N
% 0.32/0.54 != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % less_natE
% 0.32/0.54 thf(fact_2827_diff__less__Suc,axiom,
% 0.32/0.54 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_less_Suc
% 0.32/0.54 thf(fact_2828_Suc__diff__Suc,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_nat @ N @ M )
% 0.32/0.54 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.32/0.54 = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_diff_Suc
% 0.32/0.54 thf(fact_2829_Suc__mult__less__cancel1,axiom,
% 0.32/0.54 ! [K: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.32/0.54 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_mult_less_cancel1
% 0.32/0.54 thf(fact_2830_Suc__diff__le,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.54 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.32/0.54 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_diff_le
% 0.32/0.54 thf(fact_2831_One__nat__def,axiom,
% 0.32/0.54 ( one_one_nat
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % One_nat_def
% 0.32/0.54 thf(fact_2832_Suc__mult__le__cancel1,axiom,
% 0.32/0.54 ! [K: nat,M: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.32/0.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_mult_le_cancel1
% 0.32/0.54 thf(fact_2833_Suc__div__le__mono,axiom,
% 0.32/0.54 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_div_le_mono
% 0.32/0.54 thf(fact_2834_mult__Suc,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.32/0.54 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mult_Suc
% 0.32/0.54 thf(fact_2835_Suc__eq__plus1__left,axiom,
% 0.32/0.54 ( suc
% 0.32/0.54 = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_eq_plus1_left
% 0.32/0.54 thf(fact_2836_plus__1__eq__Suc,axiom,
% 0.32/0.54 ( ( plus_plus_nat @ one_one_nat )
% 0.32/0.54 = suc ) ).
% 0.32/0.54
% 0.32/0.54 % plus_1_eq_Suc
% 0.32/0.54 thf(fact_2837_Suc__eq__plus1,axiom,
% 0.32/0.54 ( suc
% 0.32/0.54 = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_eq_plus1
% 0.32/0.54 thf(fact_2838_diff__Suc__eq__diff__pred,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.32/0.54 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_Suc_eq_diff_pred
% 0.32/0.54 thf(fact_2839_mod__Suc,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.54 = N )
% 0.32/0.54 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.32/0.54 = zero_zero_nat ) )
% 0.32/0.54 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.54 != N )
% 0.32/0.54 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.32/0.54 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod_Suc
% 0.32/0.54 thf(fact_2840_mod__induct,axiom,
% 0.32/0.54 ! [P: nat > $o,N: nat,P6: nat,M: nat] :
% 0.32/0.54 ( ( P @ N )
% 0.32/0.54 => ( ( ord_less_nat @ N @ P6 )
% 0.32/0.54 => ( ( ord_less_nat @ M @ P6 )
% 0.32/0.54 => ( ! [N4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ N4 @ P6 )
% 0.32/0.54 => ( ( P @ N4 )
% 0.32/0.54 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P6 ) ) ) )
% 0.32/0.54 => ( P @ M ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % mod_induct
% 0.32/0.54 thf(fact_2841_mod__Suc__le__divisor,axiom,
% 0.32/0.54 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.32/0.54
% 0.32/0.54 % mod_Suc_le_divisor
% 0.32/0.54 thf(fact_2842_power__le__imp__le__base,axiom,
% 0.32/0.54 ! [A: real,N: nat,B: real] :
% 0.32/0.54 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.54 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_le_imp_le_base
% 0.32/0.54 thf(fact_2843_power__le__imp__le__base,axiom,
% 0.32/0.54 ! [A: rat,N: nat,B: rat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_le_imp_le_base
% 0.32/0.54 thf(fact_2844_power__le__imp__le__base,axiom,
% 0.32/0.54 ! [A: nat,N: nat,B: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_le_imp_le_base
% 0.32/0.54 thf(fact_2845_power__le__imp__le__base,axiom,
% 0.32/0.54 ! [A: int,N: nat,B: int] :
% 0.32/0.54 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.54 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_le_imp_le_base
% 0.32/0.54 thf(fact_2846_power__inject__base,axiom,
% 0.32/0.54 ! [A: real,N: nat,B: real] :
% 0.32/0.54 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 0.32/0.54 = ( power_power_real @ B @ ( suc @ N ) ) )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.54 => ( A = B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_inject_base
% 0.32/0.54 thf(fact_2847_power__inject__base,axiom,
% 0.32/0.54 ! [A: rat,N: nat,B: rat] :
% 0.32/0.54 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 0.32/0.54 = ( power_power_rat @ B @ ( suc @ N ) ) )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.54 => ( A = B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_inject_base
% 0.32/0.54 thf(fact_2848_power__inject__base,axiom,
% 0.32/0.54 ! [A: nat,N: nat,B: nat] :
% 0.32/0.54 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 0.32/0.54 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.32/0.54 => ( A = B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_inject_base
% 0.32/0.54 thf(fact_2849_power__inject__base,axiom,
% 0.32/0.54 ! [A: int,N: nat,B: int] :
% 0.32/0.54 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 0.32/0.54 = ( power_power_int @ B @ ( suc @ N ) ) )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.54 => ( A = B ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_inject_base
% 0.32/0.54 thf(fact_2850_power__gt1,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.54 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_gt1
% 0.32/0.54 thf(fact_2851_power__gt1,axiom,
% 0.32/0.54 ! [A: rat,N: nat] :
% 0.32/0.54 ( ( ord_less_rat @ one_one_rat @ A )
% 0.32/0.54 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_gt1
% 0.32/0.54 thf(fact_2852_power__gt1,axiom,
% 0.32/0.54 ! [A: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ one_one_nat @ A )
% 0.32/0.54 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_gt1
% 0.32/0.54 thf(fact_2853_power__gt1,axiom,
% 0.32/0.54 ! [A: int,N: nat] :
% 0.32/0.54 ( ( ord_less_int @ one_one_int @ A )
% 0.32/0.54 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_gt1
% 0.32/0.54 thf(fact_2854_numeral__1__eq__Suc__0,axiom,
% 0.32/0.54 ( ( numeral_numeral_nat @ one )
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % numeral_1_eq_Suc_0
% 0.32/0.54 thf(fact_2855_ex__least__nat__less,axiom,
% 0.32/0.54 ! [P: nat > $o,N: nat] :
% 0.32/0.54 ( ( P @ N )
% 0.32/0.54 => ( ~ ( P @ zero_zero_nat )
% 0.32/0.54 => ? [K3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ K3 @ N )
% 0.32/0.54 & ! [I3: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ I3 @ K3 )
% 0.32/0.54 => ~ ( P @ I3 ) )
% 0.32/0.54 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % ex_least_nat_less
% 0.32/0.54 thf(fact_2856_diff__Suc__less,axiom,
% 0.32/0.54 ! [N: nat,I: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % diff_Suc_less
% 0.32/0.54 thf(fact_2857_one__less__mult,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.54 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % one_less_mult
% 0.32/0.54 thf(fact_2858_n__less__m__mult__n,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.54 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % n_less_m_mult_n
% 0.32/0.54 thf(fact_2859_n__less__n__mult__m,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.54 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % n_less_n_mult_m
% 0.32/0.54 thf(fact_2860_pinf_I1_J,axiom,
% 0.32/0.54 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 0.32/0.54 ( ? [Z3: real] :
% 0.32/0.54 ! [X3: real] :
% 0.32/0.54 ( ( ord_less_real @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: real] :
% 0.32/0.54 ! [X3: real] :
% 0.32/0.54 ( ( ord_less_real @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(1)
% 0.32/0.54 thf(fact_2861_pinf_I1_J,axiom,
% 0.32/0.54 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.32/0.54 ( ? [Z3: rat] :
% 0.32/0.54 ! [X3: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: rat] :
% 0.32/0.54 ! [X3: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(1)
% 0.32/0.54 thf(fact_2862_pinf_I1_J,axiom,
% 0.32/0.54 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 0.32/0.54 ( ? [Z3: num] :
% 0.32/0.54 ! [X3: num] :
% 0.32/0.54 ( ( ord_less_num @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: num] :
% 0.32/0.54 ! [X3: num] :
% 0.32/0.54 ( ( ord_less_num @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(1)
% 0.32/0.54 thf(fact_2863_pinf_I1_J,axiom,
% 0.32/0.54 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.32/0.54 ( ? [Z3: nat] :
% 0.32/0.54 ! [X3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: nat] :
% 0.32/0.54 ! [X3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(1)
% 0.32/0.54 thf(fact_2864_pinf_I1_J,axiom,
% 0.32/0.54 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 0.32/0.54 ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(1)
% 0.32/0.54 thf(fact_2865_pinf_I2_J,axiom,
% 0.32/0.54 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 0.32/0.54 ( ? [Z3: real] :
% 0.32/0.54 ! [X3: real] :
% 0.32/0.54 ( ( ord_less_real @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: real] :
% 0.32/0.54 ! [X3: real] :
% 0.32/0.54 ( ( ord_less_real @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(2)
% 0.32/0.54 thf(fact_2866_pinf_I2_J,axiom,
% 0.32/0.54 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.32/0.54 ( ? [Z3: rat] :
% 0.32/0.54 ! [X3: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: rat] :
% 0.32/0.54 ! [X3: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(2)
% 0.32/0.54 thf(fact_2867_pinf_I2_J,axiom,
% 0.32/0.54 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 0.32/0.54 ( ? [Z3: num] :
% 0.32/0.54 ! [X3: num] :
% 0.32/0.54 ( ( ord_less_num @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: num] :
% 0.32/0.54 ! [X3: num] :
% 0.32/0.54 ( ( ord_less_num @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(2)
% 0.32/0.54 thf(fact_2868_pinf_I2_J,axiom,
% 0.32/0.54 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.32/0.54 ( ? [Z3: nat] :
% 0.32/0.54 ! [X3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: nat] :
% 0.32/0.54 ! [X3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(2)
% 0.32/0.54 thf(fact_2869_pinf_I2_J,axiom,
% 0.32/0.54 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 0.32/0.54 ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ Z3 @ X3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ Z3 @ X3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(2)
% 0.32/0.54 thf(fact_2870_pinf_I3_J,axiom,
% 0.32/0.54 ! [T: real] :
% 0.32/0.54 ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(3)
% 0.32/0.54 thf(fact_2871_pinf_I3_J,axiom,
% 0.32/0.54 ! [T: rat] :
% 0.32/0.54 ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(3)
% 0.32/0.54 thf(fact_2872_pinf_I3_J,axiom,
% 0.32/0.54 ! [T: num] :
% 0.32/0.54 ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(3)
% 0.32/0.54 thf(fact_2873_pinf_I3_J,axiom,
% 0.32/0.54 ! [T: nat] :
% 0.32/0.54 ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(3)
% 0.32/0.54 thf(fact_2874_pinf_I3_J,axiom,
% 0.32/0.54 ! [T: int] :
% 0.32/0.54 ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(3)
% 0.32/0.54 thf(fact_2875_pinf_I4_J,axiom,
% 0.32/0.54 ! [T: real] :
% 0.32/0.54 ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(4)
% 0.32/0.54 thf(fact_2876_pinf_I4_J,axiom,
% 0.32/0.54 ! [T: rat] :
% 0.32/0.54 ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(4)
% 0.32/0.54 thf(fact_2877_pinf_I4_J,axiom,
% 0.32/0.54 ! [T: num] :
% 0.32/0.54 ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(4)
% 0.32/0.54 thf(fact_2878_pinf_I4_J,axiom,
% 0.32/0.54 ! [T: nat] :
% 0.32/0.54 ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(4)
% 0.32/0.54 thf(fact_2879_pinf_I4_J,axiom,
% 0.32/0.54 ! [T: int] :
% 0.32/0.54 ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(4)
% 0.32/0.54 thf(fact_2880_pinf_I5_J,axiom,
% 0.32/0.54 ! [T: real] :
% 0.32/0.54 ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.54 => ~ ( ord_less_real @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(5)
% 0.32/0.54 thf(fact_2881_pinf_I5_J,axiom,
% 0.32/0.54 ! [T: rat] :
% 0.32/0.54 ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.54 => ~ ( ord_less_rat @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(5)
% 0.32/0.54 thf(fact_2882_pinf_I5_J,axiom,
% 0.32/0.54 ! [T: num] :
% 0.32/0.54 ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.54 => ~ ( ord_less_num @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(5)
% 0.32/0.54 thf(fact_2883_pinf_I5_J,axiom,
% 0.32/0.54 ! [T: nat] :
% 0.32/0.54 ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.54 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(5)
% 0.32/0.54 thf(fact_2884_pinf_I5_J,axiom,
% 0.32/0.54 ! [T: int] :
% 0.32/0.54 ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.54 => ~ ( ord_less_int @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(5)
% 0.32/0.54 thf(fact_2885_pinf_I7_J,axiom,
% 0.32/0.54 ! [T: real] :
% 0.32/0.54 ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.54 => ( ord_less_real @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(7)
% 0.32/0.54 thf(fact_2886_pinf_I7_J,axiom,
% 0.32/0.54 ! [T: rat] :
% 0.32/0.54 ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.54 => ( ord_less_rat @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(7)
% 0.32/0.54 thf(fact_2887_pinf_I7_J,axiom,
% 0.32/0.54 ! [T: num] :
% 0.32/0.54 ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.54 => ( ord_less_num @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(7)
% 0.32/0.54 thf(fact_2888_pinf_I7_J,axiom,
% 0.32/0.54 ! [T: nat] :
% 0.32/0.54 ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.54 => ( ord_less_nat @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(7)
% 0.32/0.54 thf(fact_2889_pinf_I7_J,axiom,
% 0.32/0.54 ! [T: int] :
% 0.32/0.54 ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.54 => ( ord_less_int @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % pinf(7)
% 0.32/0.54 thf(fact_2890_minf_I1_J,axiom,
% 0.32/0.54 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 0.32/0.54 ( ? [Z3: real] :
% 0.32/0.54 ! [X3: real] :
% 0.32/0.54 ( ( ord_less_real @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: real] :
% 0.32/0.54 ! [X3: real] :
% 0.32/0.54 ( ( ord_less_real @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(1)
% 0.32/0.54 thf(fact_2891_minf_I1_J,axiom,
% 0.32/0.54 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.32/0.54 ( ? [Z3: rat] :
% 0.32/0.54 ! [X3: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: rat] :
% 0.32/0.54 ! [X3: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(1)
% 0.32/0.54 thf(fact_2892_minf_I1_J,axiom,
% 0.32/0.54 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 0.32/0.54 ( ? [Z3: num] :
% 0.32/0.54 ! [X3: num] :
% 0.32/0.54 ( ( ord_less_num @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: num] :
% 0.32/0.54 ! [X3: num] :
% 0.32/0.54 ( ( ord_less_num @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(1)
% 0.32/0.54 thf(fact_2893_minf_I1_J,axiom,
% 0.32/0.54 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.32/0.54 ( ? [Z3: nat] :
% 0.32/0.54 ! [X3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: nat] :
% 0.32/0.54 ! [X3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(1)
% 0.32/0.54 thf(fact_2894_minf_I1_J,axiom,
% 0.32/0.54 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 0.32/0.54 ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 & ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(1)
% 0.32/0.54 thf(fact_2895_minf_I2_J,axiom,
% 0.32/0.54 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 0.32/0.54 ( ? [Z3: real] :
% 0.32/0.54 ! [X3: real] :
% 0.32/0.54 ( ( ord_less_real @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: real] :
% 0.32/0.54 ! [X3: real] :
% 0.32/0.54 ( ( ord_less_real @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(2)
% 0.32/0.54 thf(fact_2896_minf_I2_J,axiom,
% 0.32/0.54 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 0.32/0.54 ( ? [Z3: rat] :
% 0.32/0.54 ! [X3: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: rat] :
% 0.32/0.54 ! [X3: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(2)
% 0.32/0.54 thf(fact_2897_minf_I2_J,axiom,
% 0.32/0.54 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 0.32/0.54 ( ? [Z3: num] :
% 0.32/0.54 ! [X3: num] :
% 0.32/0.54 ( ( ord_less_num @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: num] :
% 0.32/0.54 ! [X3: num] :
% 0.32/0.54 ( ( ord_less_num @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(2)
% 0.32/0.54 thf(fact_2898_minf_I2_J,axiom,
% 0.32/0.54 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 0.32/0.54 ( ? [Z3: nat] :
% 0.32/0.54 ! [X3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: nat] :
% 0.32/0.54 ! [X3: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(2)
% 0.32/0.54 thf(fact_2899_minf_I2_J,axiom,
% 0.32/0.54 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 0.32/0.54 ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ X3 @ Z3 )
% 0.32/0.54 => ( ( P @ X3 )
% 0.32/0.54 = ( P5 @ X3 ) ) )
% 0.32/0.54 => ( ? [Z3: int] :
% 0.32/0.54 ! [X3: int] :
% 0.32/0.54 ( ( ord_less_int @ X3 @ Z3 )
% 0.32/0.54 => ( ( Q @ X3 )
% 0.32/0.54 = ( Q6 @ X3 ) ) )
% 0.32/0.54 => ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.54 => ( ( ( P @ X5 )
% 0.32/0.54 | ( Q @ X5 ) )
% 0.32/0.54 = ( ( P5 @ X5 )
% 0.32/0.54 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(2)
% 0.32/0.54 thf(fact_2900_minf_I3_J,axiom,
% 0.32/0.54 ! [T: real] :
% 0.32/0.54 ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(3)
% 0.32/0.54 thf(fact_2901_minf_I3_J,axiom,
% 0.32/0.54 ! [T: rat] :
% 0.32/0.54 ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(3)
% 0.32/0.54 thf(fact_2902_minf_I3_J,axiom,
% 0.32/0.54 ! [T: num] :
% 0.32/0.54 ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(3)
% 0.32/0.54 thf(fact_2903_minf_I3_J,axiom,
% 0.32/0.54 ! [T: nat] :
% 0.32/0.54 ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(3)
% 0.32/0.54 thf(fact_2904_minf_I3_J,axiom,
% 0.32/0.54 ! [T: int] :
% 0.32/0.54 ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(3)
% 0.32/0.54 thf(fact_2905_minf_I4_J,axiom,
% 0.32/0.54 ! [T: real] :
% 0.32/0.54 ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(4)
% 0.32/0.54 thf(fact_2906_minf_I4_J,axiom,
% 0.32/0.54 ! [T: rat] :
% 0.32/0.54 ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(4)
% 0.32/0.54 thf(fact_2907_minf_I4_J,axiom,
% 0.32/0.54 ! [T: num] :
% 0.32/0.54 ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(4)
% 0.32/0.54 thf(fact_2908_minf_I4_J,axiom,
% 0.32/0.54 ! [T: nat] :
% 0.32/0.54 ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(4)
% 0.32/0.54 thf(fact_2909_minf_I4_J,axiom,
% 0.32/0.54 ! [T: int] :
% 0.32/0.54 ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.54 => ( X5 != T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(4)
% 0.32/0.54 thf(fact_2910_minf_I5_J,axiom,
% 0.32/0.54 ! [T: real] :
% 0.32/0.54 ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.54 => ( ord_less_real @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(5)
% 0.32/0.54 thf(fact_2911_minf_I5_J,axiom,
% 0.32/0.54 ! [T: rat] :
% 0.32/0.54 ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.54 => ( ord_less_rat @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(5)
% 0.32/0.54 thf(fact_2912_minf_I5_J,axiom,
% 0.32/0.54 ! [T: num] :
% 0.32/0.54 ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.54 => ( ord_less_num @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(5)
% 0.32/0.54 thf(fact_2913_minf_I5_J,axiom,
% 0.32/0.54 ! [T: nat] :
% 0.32/0.54 ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.54 => ( ord_less_nat @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(5)
% 0.32/0.54 thf(fact_2914_minf_I5_J,axiom,
% 0.32/0.54 ! [T: int] :
% 0.32/0.54 ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.54 => ( ord_less_int @ X5 @ T ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(5)
% 0.32/0.54 thf(fact_2915_minf_I7_J,axiom,
% 0.32/0.54 ! [T: real] :
% 0.32/0.54 ? [Z5: real] :
% 0.32/0.54 ! [X5: real] :
% 0.32/0.54 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.54 => ~ ( ord_less_real @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(7)
% 0.32/0.54 thf(fact_2916_minf_I7_J,axiom,
% 0.32/0.54 ! [T: rat] :
% 0.32/0.54 ? [Z5: rat] :
% 0.32/0.54 ! [X5: rat] :
% 0.32/0.54 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.54 => ~ ( ord_less_rat @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(7)
% 0.32/0.54 thf(fact_2917_minf_I7_J,axiom,
% 0.32/0.54 ! [T: num] :
% 0.32/0.54 ? [Z5: num] :
% 0.32/0.54 ! [X5: num] :
% 0.32/0.54 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.54 => ~ ( ord_less_num @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(7)
% 0.32/0.54 thf(fact_2918_minf_I7_J,axiom,
% 0.32/0.54 ! [T: nat] :
% 0.32/0.54 ? [Z5: nat] :
% 0.32/0.54 ! [X5: nat] :
% 0.32/0.54 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.54 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(7)
% 0.32/0.54 thf(fact_2919_minf_I7_J,axiom,
% 0.32/0.54 ! [T: int] :
% 0.32/0.54 ? [Z5: int] :
% 0.32/0.54 ! [X5: int] :
% 0.32/0.54 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.54 => ~ ( ord_less_int @ T @ X5 ) ) ).
% 0.32/0.54
% 0.32/0.54 % minf(7)
% 0.32/0.54 thf(fact_2920_nat__induct__non__zero,axiom,
% 0.32/0.54 ! [N: nat,P: nat > $o] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ( P @ one_one_nat )
% 0.32/0.54 => ( ! [N4: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.54 => ( ( P @ N4 )
% 0.32/0.54 => ( P @ ( suc @ N4 ) ) ) )
% 0.32/0.54 => ( P @ N ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat_induct_non_zero
% 0.32/0.54 thf(fact_2921_realpow__pos__nth2,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ? [R2: real] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.32/0.54 & ( ( power_power_real @ R2 @ ( suc @ N ) )
% 0.32/0.54 = A ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % realpow_pos_nth2
% 0.32/0.54 thf(fact_2922_power__gt__expt,axiom,
% 0.32/0.54 ! [N: nat,K: nat] :
% 0.32/0.54 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.54 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_gt_expt
% 0.32/0.54 thf(fact_2923_nat__one__le__power,axiom,
% 0.32/0.54 ! [I: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.32/0.54 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % nat_one_le_power
% 0.32/0.54 thf(fact_2924_power__Suc__le__self,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.32/0.54 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_le_self
% 0.32/0.54 thf(fact_2925_power__Suc__le__self,axiom,
% 0.32/0.54 ! [A: rat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.32/0.54 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_le_self
% 0.32/0.54 thf(fact_2926_power__Suc__le__self,axiom,
% 0.32/0.54 ! [A: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.32/0.54 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_le_self
% 0.32/0.54 thf(fact_2927_power__Suc__le__self,axiom,
% 0.32/0.54 ! [A: int,N: nat] :
% 0.32/0.54 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.32/0.54 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_le_self
% 0.32/0.54 thf(fact_2928_power__Suc__less__one,axiom,
% 0.32/0.54 ! [A: real,N: nat] :
% 0.32/0.54 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.54 => ( ( ord_less_real @ A @ one_one_real )
% 0.32/0.54 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_less_one
% 0.32/0.54 thf(fact_2929_power__Suc__less__one,axiom,
% 0.32/0.54 ! [A: rat,N: nat] :
% 0.32/0.54 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.54 => ( ( ord_less_rat @ A @ one_one_rat )
% 0.32/0.54 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_less_one
% 0.32/0.54 thf(fact_2930_power__Suc__less__one,axiom,
% 0.32/0.54 ! [A: nat,N: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.32/0.54 => ( ( ord_less_nat @ A @ one_one_nat )
% 0.32/0.54 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_less_one
% 0.32/0.54 thf(fact_2931_power__Suc__less__one,axiom,
% 0.32/0.54 ! [A: int,N: nat] :
% 0.32/0.54 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.54 => ( ( ord_less_int @ A @ one_one_int )
% 0.32/0.54 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % power_Suc_less_one
% 0.32/0.54 thf(fact_2932_option_Osize_I4_J,axiom,
% 0.32/0.54 ! [X22: nat] :
% 0.32/0.54 ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.size(4)
% 0.32/0.54 thf(fact_2933_option_Osize_I4_J,axiom,
% 0.32/0.54 ! [X22: product_prod_nat_nat] :
% 0.32/0.54 ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.size(4)
% 0.32/0.54 thf(fact_2934_option_Osize_I4_J,axiom,
% 0.32/0.54 ! [X22: num] :
% 0.32/0.54 ( ( size_size_option_num @ ( some_num @ X22 ) )
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.size(4)
% 0.32/0.54 thf(fact_2935_option_Osize_I3_J,axiom,
% 0.32/0.54 ( ( size_size_option_nat @ none_nat )
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.size(3)
% 0.32/0.54 thf(fact_2936_option_Osize_I3_J,axiom,
% 0.32/0.54 ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.size(3)
% 0.32/0.54 thf(fact_2937_option_Osize_I3_J,axiom,
% 0.32/0.54 ( ( size_size_option_num @ none_num )
% 0.32/0.54 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.54
% 0.32/0.54 % option.size(3)
% 0.32/0.54 thf(fact_2938_numeral__2__eq__2,axiom,
% 0.32/0.54 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.32/0.54 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % numeral_2_eq_2
% 0.32/0.54 thf(fact_2939_Suc__double__not__eq__double,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.54 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.54
% 0.32/0.54 % Suc_double_not_eq_double
% 0.32/0.54 thf(fact_2940_double__not__eq__Suc__double,axiom,
% 0.32/0.54 ! [M: nat,N: nat] :
% 0.32/0.54 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.32/0.54 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % double_not_eq_Suc_double
% 0.32/0.54 thf(fact_2941_div__geq,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.54 => ( ~ ( ord_less_nat @ M @ N )
% 0.32/0.54 => ( ( divide_divide_nat @ M @ N )
% 0.32/0.54 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % div_geq
% 0.32/0.54 thf(fact_2942_div__if,axiom,
% 0.32/0.54 ( divide_divide_nat
% 0.32/0.54 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.54 ( if_nat
% 0.32/0.54 @ ( ( ord_less_nat @ M3 @ N2 )
% 0.32/0.54 | ( N2 = zero_zero_nat ) )
% 0.32/0.54 @ zero_zero_nat
% 0.32/0.54 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.32/0.54
% 0.32/0.54 % div_if
% 0.32/0.54 thf(fact_2943_Suc__diff__eq__diff__pred,axiom,
% 0.32/0.54 ! [N: nat,M: nat] :
% 0.32/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.32/0.55 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_diff_eq_diff_pred
% 0.32/0.55 thf(fact_2944_Suc__pred_H,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( N
% 0.32/0.55 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_pred'
% 0.32/0.55 thf(fact_2945_div__nat__eqI,axiom,
% 0.32/0.55 ! [N: nat,Q2: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 0.32/0.55 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 0.32/0.55 => ( ( divide_divide_nat @ M @ N )
% 0.32/0.55 = Q2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_nat_eqI
% 0.32/0.55 thf(fact_2946_add__eq__if,axiom,
% 0.32/0.55 ( plus_plus_nat
% 0.32/0.55 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % add_eq_if
% 0.32/0.55 thf(fact_2947_Suc__nat__number__of__add,axiom,
% 0.32/0.55 ! [V: num,N: nat] :
% 0.32/0.55 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 0.32/0.55 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_nat_number_of_add
% 0.32/0.55 thf(fact_2948_num_Osize_I5_J,axiom,
% 0.32/0.55 ! [X22: num] :
% 0.32/0.55 ( ( size_size_num @ ( bit0 @ X22 ) )
% 0.32/0.55 = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % num.size(5)
% 0.32/0.55 thf(fact_2949_less__2__cases,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.55 => ( ( N = zero_zero_nat )
% 0.32/0.55 | ( N
% 0.32/0.55 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_2_cases
% 0.32/0.55 thf(fact_2950_less__2__cases__iff,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.55 = ( ( N = zero_zero_nat )
% 0.32/0.55 | ( N
% 0.32/0.55 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_2_cases_iff
% 0.32/0.55 thf(fact_2951_le__div__geq,axiom,
% 0.32/0.55 ! [N: nat,M: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.55 => ( ( divide_divide_nat @ M @ N )
% 0.32/0.55 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % le_div_geq
% 0.32/0.55 thf(fact_2952_split__div_H,axiom,
% 0.32/0.55 ! [P: nat > $o,M: nat,N: nat] :
% 0.32/0.55 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.55 = ( ( ( N = zero_zero_nat )
% 0.32/0.55 & ( P @ zero_zero_nat ) )
% 0.32/0.55 | ? [Q5: nat] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
% 0.32/0.55 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
% 0.32/0.55 & ( P @ Q5 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % split_div'
% 0.32/0.55 thf(fact_2953_Suc__times__mod__eq,axiom,
% 0.32/0.55 ! [M: nat,N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.32/0.55 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.32/0.55 = one_one_nat ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_times_mod_eq
% 0.32/0.55 thf(fact_2954_vebt__succ_Osimps_I5_J,axiom,
% 0.32/0.55 ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 0.32/0.55 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
% 0.32/0.55 = none_nat ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_succ.simps(5)
% 0.32/0.55 thf(fact_2955_power__odd__eq,axiom,
% 0.32/0.55 ! [A: complex,N: nat] :
% 0.32/0.55 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.55 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_odd_eq
% 0.32/0.55 thf(fact_2956_power__odd__eq,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.55 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_odd_eq
% 0.32/0.55 thf(fact_2957_power__odd__eq,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.55 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_odd_eq
% 0.32/0.55 thf(fact_2958_power__odd__eq,axiom,
% 0.32/0.55 ! [A: nat,N: nat] :
% 0.32/0.55 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.55 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_odd_eq
% 0.32/0.55 thf(fact_2959_power__odd__eq,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.55 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_odd_eq
% 0.32/0.55 thf(fact_2960_nat__bit__induct,axiom,
% 0.32/0.55 ! [P: nat > $o,N: nat] :
% 0.32/0.55 ( ( P @ zero_zero_nat )
% 0.32/0.55 => ( ! [N4: nat] :
% 0.32/0.55 ( ( P @ N4 )
% 0.32/0.55 => ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.55 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 0.32/0.55 => ( ! [N4: nat] :
% 0.32/0.55 ( ( P @ N4 )
% 0.32/0.55 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 0.32/0.55 => ( P @ N ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % nat_bit_induct
% 0.32/0.55 thf(fact_2961_div__2__gt__zero,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.55 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % div_2_gt_zero
% 0.32/0.55 thf(fact_2962_Suc__n__div__2__gt__zero,axiom,
% 0.32/0.55 ! [N: nat] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.55 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Suc_n_div_2_gt_zero
% 0.32/0.55 thf(fact_2963_pinf_I6_J,axiom,
% 0.32/0.55 ! [T: real] :
% 0.32/0.55 ? [Z5: real] :
% 0.32/0.55 ! [X5: real] :
% 0.32/0.55 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.55 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(6)
% 0.32/0.55 thf(fact_2964_pinf_I6_J,axiom,
% 0.32/0.55 ! [T: rat] :
% 0.32/0.55 ? [Z5: rat] :
% 0.32/0.55 ! [X5: rat] :
% 0.32/0.55 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.55 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(6)
% 0.32/0.55 thf(fact_2965_pinf_I6_J,axiom,
% 0.32/0.55 ! [T: num] :
% 0.32/0.55 ? [Z5: num] :
% 0.32/0.55 ! [X5: num] :
% 0.32/0.55 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.55 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(6)
% 0.32/0.55 thf(fact_2966_pinf_I6_J,axiom,
% 0.32/0.55 ! [T: nat] :
% 0.32/0.55 ? [Z5: nat] :
% 0.32/0.55 ! [X5: nat] :
% 0.32/0.55 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.55 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(6)
% 0.32/0.55 thf(fact_2967_pinf_I6_J,axiom,
% 0.32/0.55 ! [T: int] :
% 0.32/0.55 ? [Z5: int] :
% 0.32/0.55 ! [X5: int] :
% 0.32/0.55 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.55 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(6)
% 0.32/0.55 thf(fact_2968_pinf_I8_J,axiom,
% 0.32/0.55 ! [T: real] :
% 0.32/0.55 ? [Z5: real] :
% 0.32/0.55 ! [X5: real] :
% 0.32/0.55 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.55 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(8)
% 0.32/0.55 thf(fact_2969_pinf_I8_J,axiom,
% 0.32/0.55 ! [T: rat] :
% 0.32/0.55 ? [Z5: rat] :
% 0.32/0.55 ! [X5: rat] :
% 0.32/0.55 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.55 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(8)
% 0.32/0.55 thf(fact_2970_pinf_I8_J,axiom,
% 0.32/0.55 ! [T: num] :
% 0.32/0.55 ? [Z5: num] :
% 0.32/0.55 ! [X5: num] :
% 0.32/0.55 ( ( ord_less_num @ Z5 @ X5 )
% 0.32/0.55 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(8)
% 0.32/0.55 thf(fact_2971_pinf_I8_J,axiom,
% 0.32/0.55 ! [T: nat] :
% 0.32/0.55 ? [Z5: nat] :
% 0.32/0.55 ! [X5: nat] :
% 0.32/0.55 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.55 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(8)
% 0.32/0.55 thf(fact_2972_pinf_I8_J,axiom,
% 0.32/0.55 ! [T: int] :
% 0.32/0.55 ? [Z5: int] :
% 0.32/0.55 ! [X5: int] :
% 0.32/0.55 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.55 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % pinf(8)
% 0.32/0.55 thf(fact_2973_minf_I6_J,axiom,
% 0.32/0.55 ! [T: real] :
% 0.32/0.55 ? [Z5: real] :
% 0.32/0.55 ! [X5: real] :
% 0.32/0.55 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.55 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(6)
% 0.32/0.55 thf(fact_2974_minf_I6_J,axiom,
% 0.32/0.55 ! [T: rat] :
% 0.32/0.55 ? [Z5: rat] :
% 0.32/0.55 ! [X5: rat] :
% 0.32/0.55 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.55 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(6)
% 0.32/0.55 thf(fact_2975_minf_I6_J,axiom,
% 0.32/0.55 ! [T: num] :
% 0.32/0.55 ? [Z5: num] :
% 0.32/0.55 ! [X5: num] :
% 0.32/0.55 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.55 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(6)
% 0.32/0.55 thf(fact_2976_minf_I6_J,axiom,
% 0.32/0.55 ! [T: nat] :
% 0.32/0.55 ? [Z5: nat] :
% 0.32/0.55 ! [X5: nat] :
% 0.32/0.55 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.55 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(6)
% 0.32/0.55 thf(fact_2977_minf_I6_J,axiom,
% 0.32/0.55 ! [T: int] :
% 0.32/0.55 ? [Z5: int] :
% 0.32/0.55 ! [X5: int] :
% 0.32/0.55 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.55 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(6)
% 0.32/0.55 thf(fact_2978_minf_I8_J,axiom,
% 0.32/0.55 ! [T: real] :
% 0.32/0.55 ? [Z5: real] :
% 0.32/0.55 ! [X5: real] :
% 0.32/0.55 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.55 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(8)
% 0.32/0.55 thf(fact_2979_minf_I8_J,axiom,
% 0.32/0.55 ! [T: rat] :
% 0.32/0.55 ? [Z5: rat] :
% 0.32/0.55 ! [X5: rat] :
% 0.32/0.55 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.55 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(8)
% 0.32/0.55 thf(fact_2980_minf_I8_J,axiom,
% 0.32/0.55 ! [T: num] :
% 0.32/0.55 ? [Z5: num] :
% 0.32/0.55 ! [X5: num] :
% 0.32/0.55 ( ( ord_less_num @ X5 @ Z5 )
% 0.32/0.55 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(8)
% 0.32/0.55 thf(fact_2981_minf_I8_J,axiom,
% 0.32/0.55 ! [T: nat] :
% 0.32/0.55 ? [Z5: nat] :
% 0.32/0.55 ! [X5: nat] :
% 0.32/0.55 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.55 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(8)
% 0.32/0.55 thf(fact_2982_minf_I8_J,axiom,
% 0.32/0.55 ! [T: int] :
% 0.32/0.55 ? [Z5: int] :
% 0.32/0.55 ! [X5: int] :
% 0.32/0.55 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.55 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % minf(8)
% 0.32/0.55 thf(fact_2983_inf__period_I1_J,axiom,
% 0.32/0.55 ! [P: real > $o,D4: real,Q: real > $o] :
% 0.32/0.55 ( ! [X3: real,K3: real] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: real,K3: real] :
% 0.32/0.55 ( ( Q @ X3 )
% 0.32/0.55 = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: real,K4: real] :
% 0.32/0.55 ( ( ( P @ X5 )
% 0.32/0.55 & ( Q @ X5 ) )
% 0.32/0.55 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
% 0.32/0.55 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % inf_period(1)
% 0.32/0.55 thf(fact_2984_inf__period_I1_J,axiom,
% 0.32/0.55 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 0.32/0.55 ( ! [X3: rat,K3: rat] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: rat,K3: rat] :
% 0.32/0.55 ( ( Q @ X3 )
% 0.32/0.55 = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: rat,K4: rat] :
% 0.32/0.55 ( ( ( P @ X5 )
% 0.32/0.55 & ( Q @ X5 ) )
% 0.32/0.55 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
% 0.32/0.55 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % inf_period(1)
% 0.32/0.55 thf(fact_2985_inf__period_I1_J,axiom,
% 0.32/0.55 ! [P: int > $o,D4: int,Q: int > $o] :
% 0.32/0.55 ( ! [X3: int,K3: int] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: int,K3: int] :
% 0.32/0.55 ( ( Q @ X3 )
% 0.32/0.55 = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: int,K4: int] :
% 0.32/0.55 ( ( ( P @ X5 )
% 0.32/0.55 & ( Q @ X5 ) )
% 0.32/0.55 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
% 0.32/0.55 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % inf_period(1)
% 0.32/0.55 thf(fact_2986_inf__period_I2_J,axiom,
% 0.32/0.55 ! [P: real > $o,D4: real,Q: real > $o] :
% 0.32/0.55 ( ! [X3: real,K3: real] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: real,K3: real] :
% 0.32/0.55 ( ( Q @ X3 )
% 0.32/0.55 = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: real,K4: real] :
% 0.32/0.55 ( ( ( P @ X5 )
% 0.32/0.55 | ( Q @ X5 ) )
% 0.32/0.55 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
% 0.32/0.55 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % inf_period(2)
% 0.32/0.55 thf(fact_2987_inf__period_I2_J,axiom,
% 0.32/0.55 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 0.32/0.55 ( ! [X3: rat,K3: rat] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: rat,K3: rat] :
% 0.32/0.55 ( ( Q @ X3 )
% 0.32/0.55 = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: rat,K4: rat] :
% 0.32/0.55 ( ( ( P @ X5 )
% 0.32/0.55 | ( Q @ X5 ) )
% 0.32/0.55 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
% 0.32/0.55 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % inf_period(2)
% 0.32/0.55 thf(fact_2988_inf__period_I2_J,axiom,
% 0.32/0.55 ! [P: int > $o,D4: int,Q: int > $o] :
% 0.32/0.55 ( ! [X3: int,K3: int] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: int,K3: int] :
% 0.32/0.55 ( ( Q @ X3 )
% 0.32/0.55 = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: int,K4: int] :
% 0.32/0.55 ( ( ( P @ X5 )
% 0.32/0.55 | ( Q @ X5 ) )
% 0.32/0.55 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
% 0.32/0.55 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % inf_period(2)
% 0.32/0.55 thf(fact_2989_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.55 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.55
% 0.32/0.55 % odd_0_le_power_imp_0_le
% 0.32/0.55 thf(fact_2990_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.55
% 0.32/0.55 % odd_0_le_power_imp_0_le
% 0.32/0.55 thf(fact_2991_odd__0__le__power__imp__0__le,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.55 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.55
% 0.32/0.55 % odd_0_le_power_imp_0_le
% 0.32/0.55 thf(fact_2992_odd__power__less__zero,axiom,
% 0.32/0.55 ! [A: real,N: nat] :
% 0.32/0.55 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.55 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.32/0.55
% 0.32/0.55 % odd_power_less_zero
% 0.32/0.55 thf(fact_2993_odd__power__less__zero,axiom,
% 0.32/0.55 ! [A: rat,N: nat] :
% 0.32/0.55 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.55 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.32/0.55
% 0.32/0.55 % odd_power_less_zero
% 0.32/0.55 thf(fact_2994_odd__power__less__zero,axiom,
% 0.32/0.55 ! [A: int,N: nat] :
% 0.32/0.55 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.55 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.32/0.55
% 0.32/0.55 % odd_power_less_zero
% 0.32/0.55 thf(fact_2995_imp__le__cong,axiom,
% 0.32/0.55 ! [X: int,X8: int,P: $o,P5: $o] :
% 0.32/0.55 ( ( X = X8 )
% 0.32/0.55 => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 0.32/0.55 => ( P = P5 ) )
% 0.32/0.55 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.55 => P )
% 0.32/0.55 = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 0.32/0.55 => P5 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % imp_le_cong
% 0.32/0.55 thf(fact_2996_conj__le__cong,axiom,
% 0.32/0.55 ! [X: int,X8: int,P: $o,P5: $o] :
% 0.32/0.55 ( ( X = X8 )
% 0.32/0.55 => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 0.32/0.55 => ( P = P5 ) )
% 0.32/0.55 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.55 & P )
% 0.32/0.55 = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 0.32/0.55 & P5 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % conj_le_cong
% 0.32/0.55 thf(fact_2997_int__power__div__base,axiom,
% 0.32/0.55 ! [M: nat,K: int] :
% 0.32/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.55 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.55 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.32/0.55 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % int_power_div_base
% 0.32/0.55 thf(fact_2998_set__bit__Suc,axiom,
% 0.32/0.55 ! [N: nat,A: code_integer] :
% 0.32/0.55 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 0.32/0.55 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_bit_Suc
% 0.32/0.55 thf(fact_2999_set__bit__Suc,axiom,
% 0.32/0.55 ! [N: nat,A: int] :
% 0.32/0.55 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 0.32/0.55 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_bit_Suc
% 0.32/0.55 thf(fact_3000_set__bit__Suc,axiom,
% 0.32/0.55 ! [N: nat,A: nat] :
% 0.32/0.55 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 0.32/0.55 = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_bit_Suc
% 0.32/0.55 thf(fact_3001_aset_I2_J,axiom,
% 0.32/0.55 ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.32/0.55 ( ! [X3: int] :
% 0.32/0.55 ( ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb2: int] :
% 0.32/0.55 ( ( member_int @ Xb2 @ A2 )
% 0.32/0.55 => ( X3
% 0.32/0.55 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.55 => ( ( P @ X3 )
% 0.32/0.55 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: int] :
% 0.32/0.55 ( ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb2: int] :
% 0.32/0.55 ( ( member_int @ Xb2 @ A2 )
% 0.32/0.55 => ( X3
% 0.32/0.55 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.55 => ( ( Q @ X3 )
% 0.32/0.55 => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( ( P @ X5 )
% 0.32/0.55 | ( Q @ X5 ) )
% 0.32/0.55 => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
% 0.32/0.55 | ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % aset(2)
% 0.32/0.55 thf(fact_3002_aset_I1_J,axiom,
% 0.32/0.55 ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
% 0.32/0.55 ( ! [X3: int] :
% 0.32/0.55 ( ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb2: int] :
% 0.32/0.55 ( ( member_int @ Xb2 @ A2 )
% 0.32/0.55 => ( X3
% 0.32/0.55 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.55 => ( ( P @ X3 )
% 0.32/0.55 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: int] :
% 0.32/0.55 ( ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb2: int] :
% 0.32/0.55 ( ( member_int @ Xb2 @ A2 )
% 0.32/0.55 => ( X3
% 0.32/0.55 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.55 => ( ( Q @ X3 )
% 0.32/0.55 => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( ( P @ X5 )
% 0.32/0.55 & ( Q @ X5 ) )
% 0.32/0.55 => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
% 0.32/0.55 & ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % aset(1)
% 0.32/0.55 thf(fact_3003_bset_I2_J,axiom,
% 0.32/0.55 ! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
% 0.32/0.55 ( ! [X3: int] :
% 0.32/0.55 ( ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb2: int] :
% 0.32/0.55 ( ( member_int @ Xb2 @ B5 )
% 0.32/0.55 => ( X3
% 0.32/0.55 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.55 => ( ( P @ X3 )
% 0.32/0.55 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: int] :
% 0.32/0.55 ( ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb2: int] :
% 0.32/0.55 ( ( member_int @ Xb2 @ B5 )
% 0.32/0.55 => ( X3
% 0.32/0.55 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.55 => ( ( Q @ X3 )
% 0.32/0.55 => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( ( P @ X5 )
% 0.32/0.55 | ( Q @ X5 ) )
% 0.32/0.55 => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
% 0.32/0.55 | ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % bset(2)
% 0.32/0.55 thf(fact_3004_bset_I1_J,axiom,
% 0.32/0.55 ! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
% 0.32/0.55 ( ! [X3: int] :
% 0.32/0.55 ( ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb2: int] :
% 0.32/0.55 ( ( member_int @ Xb2 @ B5 )
% 0.32/0.55 => ( X3
% 0.32/0.55 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.55 => ( ( P @ X3 )
% 0.32/0.55 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.32/0.55 => ( ! [X3: int] :
% 0.32/0.55 ( ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb2: int] :
% 0.32/0.55 ( ( member_int @ Xb2 @ B5 )
% 0.32/0.55 => ( X3
% 0.32/0.55 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 0.32/0.55 => ( ( Q @ X3 )
% 0.32/0.55 => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( ( P @ X5 )
% 0.32/0.55 & ( Q @ X5 ) )
% 0.32/0.55 => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
% 0.32/0.55 & ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % bset(1)
% 0.32/0.55 thf(fact_3005_invar__vebt_Ointros_I3_J,axiom,
% 0.32/0.55 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.32/0.55 ( ! [X3: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.32/0.55 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.32/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.55 => ( ( M
% 0.32/0.55 = ( suc @ N ) )
% 0.32/0.55 => ( ( Deg
% 0.32/0.55 = ( plus_plus_nat @ N @ M ) )
% 0.32/0.55 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.32/0.55 => ( ! [X3: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.55 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % invar_vebt.intros(3)
% 0.32/0.55 thf(fact_3006_minusinfinity,axiom,
% 0.32/0.55 ! [D: int,P1: int > $o,P: int > $o] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.55 => ( ! [X3: int,K3: int] :
% 0.32/0.55 ( ( P1 @ X3 )
% 0.32/0.55 = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.32/0.55 => ( ? [Z3: int] :
% 0.32/0.55 ! [X3: int] :
% 0.32/0.55 ( ( ord_less_int @ X3 @ Z3 )
% 0.32/0.55 => ( ( P @ X3 )
% 0.32/0.55 = ( P1 @ X3 ) ) )
% 0.32/0.55 => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.32/0.55 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minusinfinity
% 0.32/0.55 thf(fact_3007_plusinfinity,axiom,
% 0.32/0.55 ! [D: int,P5: int > $o,P: int > $o] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.55 => ( ! [X3: int,K3: int] :
% 0.32/0.55 ( ( P5 @ X3 )
% 0.32/0.55 = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.32/0.55 => ( ? [Z3: int] :
% 0.32/0.55 ! [X3: int] :
% 0.32/0.55 ( ( ord_less_int @ Z3 @ X3 )
% 0.32/0.55 => ( ( P @ X3 )
% 0.32/0.55 = ( P5 @ X3 ) ) )
% 0.32/0.55 => ( ? [X_12: int] : ( P5 @ X_12 )
% 0.32/0.55 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % plusinfinity
% 0.32/0.55 thf(fact_3008_incr__mult__lemma,axiom,
% 0.32/0.55 ! [D: int,P: int > $o,K: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.55 => ( ! [X3: int] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 => ( P @ ( plus_plus_int @ X3 @ D ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ( P @ X5 )
% 0.32/0.55 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % incr_mult_lemma
% 0.32/0.55 thf(fact_3009_invar__vebt_Ointros_I5_J,axiom,
% 0.32/0.55 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.32/0.55 ( ! [X3: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X3 @ N ) )
% 0.32/0.55 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.32/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.55 => ( ( M
% 0.32/0.55 = ( suc @ N ) )
% 0.32/0.55 => ( ( Deg
% 0.32/0.55 = ( plus_plus_nat @ N @ M ) )
% 0.32/0.55 => ( ! [I2: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.55 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
% 0.32/0.55 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.32/0.55 => ( ( ( Mi = Ma )
% 0.32/0.55 => ! [X3: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.55 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.32/0.55 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.32/0.55 => ( ( ( Mi != Ma )
% 0.32/0.55 => ! [I2: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.32/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.32/0.55 = I2 )
% 0.32/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.32/0.55 & ! [X3: nat] :
% 0.32/0.55 ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 0.32/0.55 = I2 )
% 0.32/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 0.32/0.55 => ( ( ord_less_nat @ Mi @ X3 )
% 0.32/0.55 & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % invar_vebt.intros(5)
% 0.32/0.55 thf(fact_3010_decr__mult__lemma,axiom,
% 0.32/0.55 ! [D: int,P: int > $o,K: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.55 => ( ! [X3: int] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 => ( P @ ( minus_minus_int @ X3 @ D ) ) )
% 0.32/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ( P @ X5 )
% 0.32/0.55 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % decr_mult_lemma
% 0.32/0.55 thf(fact_3011_periodic__finite__ex,axiom,
% 0.32/0.55 ! [D: int,P: int > $o] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.55 => ( ! [X3: int,K3: int] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 0.32/0.55 => ( ( ? [X6: int] : ( P @ X6 ) )
% 0.32/0.55 = ( ? [X2: int] :
% 0.32/0.55 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.32/0.55 & ( P @ X2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % periodic_finite_ex
% 0.32/0.55 thf(fact_3012_bset_I3_J,axiom,
% 0.32/0.55 ! [D4: int,T: int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.55 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( X5 = T )
% 0.32/0.55 => ( ( minus_minus_int @ X5 @ D4 )
% 0.32/0.55 = T ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % bset(3)
% 0.32/0.55 thf(fact_3013_bset_I4_J,axiom,
% 0.32/0.55 ! [D4: int,T: int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.55 => ( ( member_int @ T @ B5 )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( X5 != T )
% 0.32/0.55 => ( ( minus_minus_int @ X5 @ D4 )
% 0.32/0.55 != T ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % bset(4)
% 0.32/0.55 thf(fact_3014_bset_I5_J,axiom,
% 0.32/0.55 ! [D4: int,B5: set_int,T: int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( ord_less_int @ X5 @ T )
% 0.32/0.55 => ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % bset(5)
% 0.32/0.55 thf(fact_3015_bset_I7_J,axiom,
% 0.32/0.55 ! [D4: int,T: int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.55 => ( ( member_int @ T @ B5 )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( ord_less_int @ T @ X5 )
% 0.32/0.55 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % bset(7)
% 0.32/0.55 thf(fact_3016_aset_I3_J,axiom,
% 0.32/0.55 ! [D4: int,T: int,A2: set_int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.55 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( X5 = T )
% 0.32/0.55 => ( ( plus_plus_int @ X5 @ D4 )
% 0.32/0.55 = T ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % aset(3)
% 0.32/0.55 thf(fact_3017_aset_I4_J,axiom,
% 0.32/0.55 ! [D4: int,T: int,A2: set_int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.55 => ( ( member_int @ T @ A2 )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( X5 != T )
% 0.32/0.55 => ( ( plus_plus_int @ X5 @ D4 )
% 0.32/0.55 != T ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % aset(4)
% 0.32/0.55 thf(fact_3018_aset_I5_J,axiom,
% 0.32/0.55 ! [D4: int,T: int,A2: set_int] :
% 0.32/0.55 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.32/0.55 => ( ( member_int @ T @ A2 )
% 0.32/0.55 => ! [X5: int] :
% 0.32/0.55 ( ! [Xa3: int] :
% 0.32/0.55 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.55 => ! [Xb3: int] :
% 0.32/0.55 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.55 => ( X5
% 0.32/0.55 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.55 => ( ( ord_less_int @ X5 @ T )
% 0.32/0.55 => ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % aset(5)
% 0.32/0.55 thf(fact_3019_vebt__insert_Osimps_I4_J,axiom,
% 0.32/0.55 ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.32/0.55 ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_insert.simps(4)
% 0.32/0.55 thf(fact_3020_invar__vebt_Ocases,axiom,
% 0.32/0.55 ! [A1: vEBT_VEBT,A22: nat] :
% 0.32/0.55 ( ( vEBT_invar_vebt @ A1 @ A22 )
% 0.32/0.55 => ( ( ? [A5: $o,B3: $o] :
% 0.32/0.55 ( A1
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ( A22
% 0.32/0.55 != ( suc @ zero_zero_nat ) ) )
% 0.32/0.55 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.32/0.55 ( ( A1
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.55 => ( ( A22 = Deg2 )
% 0.32/0.55 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.32/0.55 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.32/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.32/0.55 => ( ( M4 = N4 )
% 0.32/0.55 => ( ( Deg2
% 0.32/0.55 = ( plus_plus_nat @ N4 @ M4 ) )
% 0.32/0.55 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.32/0.55 => ~ ! [X5: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.55 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.32/0.55 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.32/0.55 ( ( A1
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.55 => ( ( A22 = Deg2 )
% 0.32/0.55 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.32/0.55 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.32/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.32/0.55 => ( ( M4
% 0.32/0.55 = ( suc @ N4 ) )
% 0.32/0.55 => ( ( Deg2
% 0.32/0.55 = ( plus_plus_nat @ N4 @ M4 ) )
% 0.32/0.55 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.32/0.55 => ~ ! [X5: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.55 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.32/0.55 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.32/0.55 ( ( A1
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.55 => ( ( A22 = Deg2 )
% 0.32/0.55 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.32/0.55 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.32/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.32/0.55 => ( ( M4 = N4 )
% 0.32/0.55 => ( ( Deg2
% 0.32/0.55 = ( plus_plus_nat @ N4 @ M4 ) )
% 0.32/0.55 => ( ! [I3: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.32/0.55 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.32/0.55 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.32/0.55 => ( ( ( Mi2 = Ma2 )
% 0.32/0.55 => ! [X5: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.55 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.32/0.55 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.55 => ~ ( ( Mi2 != Ma2 )
% 0.32/0.55 => ! [I3: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.32/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 0.32/0.55 = I3 )
% 0.32/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 0.32/0.55 & ! [X5: nat] :
% 0.32/0.55 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 0.32/0.55 = I3 )
% 0.32/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 0.32/0.55 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.32/0.55 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.55 => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.32/0.55 ( ( A1
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.55 => ( ( A22 = Deg2 )
% 0.32/0.55 => ( ! [X5: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 0.32/0.55 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.32/0.55 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.32/0.55 => ( ( M4
% 0.32/0.55 = ( suc @ N4 ) )
% 0.32/0.55 => ( ( Deg2
% 0.32/0.55 = ( plus_plus_nat @ N4 @ M4 ) )
% 0.32/0.55 => ( ! [I3: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.32/0.55 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 0.32/0.55 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.32/0.55 => ( ( ( Mi2 = Ma2 )
% 0.32/0.55 => ! [X5: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.55 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 0.32/0.55 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.32/0.55 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.55 => ~ ( ( Mi2 != Ma2 )
% 0.32/0.55 => ! [I3: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.32/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 0.32/0.55 = I3 )
% 0.32/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 0.32/0.55 & ! [X5: nat] :
% 0.32/0.55 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 0.32/0.55 = I3 )
% 0.32/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 0.32/0.55 => ( ( ord_less_nat @ Mi2 @ X5 )
% 0.32/0.55 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % invar_vebt.cases
% 0.32/0.55 thf(fact_3021_invar__vebt_Osimps,axiom,
% 0.32/0.55 ( vEBT_invar_vebt
% 0.32/0.55 = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 0.32/0.55 ( ( ? [A3: $o,B4: $o] :
% 0.32/0.55 ( A12
% 0.32/0.55 = ( vEBT_Leaf @ A3 @ B4 ) )
% 0.32/0.55 & ( A23
% 0.32/0.55 = ( suc @ zero_zero_nat ) ) )
% 0.32/0.55 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 0.32/0.55 ( ( A12
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary3 ) )
% 0.32/0.55 & ! [X2: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 0.32/0.55 & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 0.32/0.55 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.55 & ( A23
% 0.32/0.55 = ( plus_plus_nat @ N2 @ N2 ) )
% 0.32/0.55 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.32/0.55 & ! [X2: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.55 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.55 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 0.32/0.55 ( ( A12
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary3 ) )
% 0.32/0.55 & ! [X2: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 0.32/0.55 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 0.32/0.55 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.32/0.55 & ( A23
% 0.32/0.55 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.32/0.55 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 0.32/0.55 & ! [X2: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.55 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.55 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.32/0.55 ( ( A12
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
% 0.32/0.55 & ! [X2: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 0.32/0.55 & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 0.32/0.55 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.55 & ( A23
% 0.32/0.55 = ( plus_plus_nat @ N2 @ N2 ) )
% 0.32/0.55 & ! [I4: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.55 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
% 0.32/0.55 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.32/0.55 & ( ( Mi3 = Ma3 )
% 0.32/0.55 => ! [X2: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.55 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.55 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.55 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.32/0.55 & ( ( Mi3 != Ma3 )
% 0.32/0.55 => ! [I4: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.32/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 0.32/0.55 = I4 )
% 0.32/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 0.32/0.55 & ! [X2: nat] :
% 0.32/0.55 ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
% 0.32/0.55 = I4 )
% 0.32/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.55 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
% 0.32/0.55 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.32/0.55 ( ( A12
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
% 0.32/0.55 & ! [X2: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.55 => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 0.32/0.55 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 0.32/0.55 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.32/0.55 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.32/0.55 & ( A23
% 0.32/0.55 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.32/0.55 & ! [I4: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
% 0.32/0.55 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 0.32/0.55 & ( ( Mi3 = Ma3 )
% 0.32/0.55 => ! [X2: vEBT_VEBT] :
% 0.32/0.55 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.32/0.55 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.55 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.55 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.32/0.55 & ( ( Mi3 != Ma3 )
% 0.32/0.55 => ! [I4: nat] :
% 0.32/0.55 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.32/0.55 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 0.32/0.55 = I4 )
% 0.32/0.55 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 0.32/0.55 & ! [X2: nat] :
% 0.32/0.55 ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
% 0.32/0.55 = I4 )
% 0.32/0.55 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
% 0.32/0.55 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.55 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % invar_vebt.simps
% 0.32/0.55 thf(fact_3022_vebt__member_Osimps_I4_J,axiom,
% 0.32/0.55 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 0.32/0.55 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_member.simps(4)
% 0.32/0.55 thf(fact_3023_psubsetI,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( A2 != B5 )
% 0.32/0.55 => ( ord_less_set_int @ A2 @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubsetI
% 0.32/0.55 thf(fact_3024_Diff__eq__empty__iff,axiom,
% 0.32/0.55 ! [A2: set_real,B5: set_real] :
% 0.32/0.55 ( ( ( minus_minus_set_real @ A2 @ B5 )
% 0.32/0.55 = bot_bot_set_real )
% 0.32/0.55 = ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_eq_empty_iff
% 0.32/0.55 thf(fact_3025_Diff__eq__empty__iff,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( ( minus_minus_set_nat @ A2 @ B5 )
% 0.32/0.55 = bot_bot_set_nat )
% 0.32/0.55 = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_eq_empty_iff
% 0.32/0.55 thf(fact_3026_Diff__eq__empty__iff,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( ( minus_minus_set_int @ A2 @ B5 )
% 0.32/0.55 = bot_bot_set_int )
% 0.32/0.55 = ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_eq_empty_iff
% 0.32/0.55 thf(fact_3027_empty__subsetI,axiom,
% 0.32/0.55 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % empty_subsetI
% 0.32/0.55 thf(fact_3028_empty__subsetI,axiom,
% 0.32/0.55 ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % empty_subsetI
% 0.32/0.55 thf(fact_3029_empty__subsetI,axiom,
% 0.32/0.55 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % empty_subsetI
% 0.32/0.55 thf(fact_3030_Leaf__0__not,axiom,
% 0.32/0.55 ! [A: $o,B: $o] :
% 0.32/0.55 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 0.32/0.55
% 0.32/0.55 % Leaf_0_not
% 0.32/0.55 thf(fact_3031_deg1Leaf,axiom,
% 0.32/0.55 ! [T: vEBT_VEBT] :
% 0.32/0.55 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.32/0.55 = ( ? [A3: $o,B4: $o] :
% 0.32/0.55 ( T
% 0.32/0.55 = ( vEBT_Leaf @ A3 @ B4 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % deg1Leaf
% 0.32/0.55 thf(fact_3032_deg__1__Leaf,axiom,
% 0.32/0.55 ! [T: vEBT_VEBT] :
% 0.32/0.55 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.32/0.55 => ? [A5: $o,B3: $o] :
% 0.32/0.55 ( T
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % deg_1_Leaf
% 0.32/0.55 thf(fact_3033_deg__1__Leafy,axiom,
% 0.32/0.55 ! [T: vEBT_VEBT,N: nat] :
% 0.32/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.32/0.55 => ( ( N = one_one_nat )
% 0.32/0.55 => ? [A5: $o,B3: $o] :
% 0.32/0.55 ( T
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % deg_1_Leafy
% 0.32/0.55 thf(fact_3034_subsetI,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ! [X3: nat] :
% 0.32/0.55 ( ( member_nat @ X3 @ A2 )
% 0.32/0.55 => ( member_nat @ X3 @ B5 ) )
% 0.32/0.55 => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetI
% 0.32/0.55 thf(fact_3035_subsetI,axiom,
% 0.32/0.55 ! [A2: set_real,B5: set_real] :
% 0.32/0.55 ( ! [X3: real] :
% 0.32/0.55 ( ( member_real @ X3 @ A2 )
% 0.32/0.55 => ( member_real @ X3 @ B5 ) )
% 0.32/0.55 => ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetI
% 0.32/0.55 thf(fact_3036_subsetI,axiom,
% 0.32/0.55 ! [A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ! [X3: complex] :
% 0.32/0.55 ( ( member_complex @ X3 @ A2 )
% 0.32/0.55 => ( member_complex @ X3 @ B5 ) )
% 0.32/0.55 => ( ord_le211207098394363844omplex @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetI
% 0.32/0.55 thf(fact_3037_subsetI,axiom,
% 0.32/0.55 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ! [X3: produc859450856879609959at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ X3 @ A2 )
% 0.32/0.55 => ( member8206827879206165904at_nat @ X3 @ B5 ) )
% 0.32/0.55 => ( ord_le3000389064537975527at_nat @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetI
% 0.32/0.55 thf(fact_3038_subsetI,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ! [X3: int] :
% 0.32/0.55 ( ( member_int @ X3 @ A2 )
% 0.32/0.55 => ( member_int @ X3 @ B5 ) )
% 0.32/0.55 => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetI
% 0.32/0.55 thf(fact_3039_subset__antisym,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.55 => ( A2 = B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_antisym
% 0.32/0.55 thf(fact_3040_Diff__idemp,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ B5 )
% 0.32/0.55 = ( minus_minus_set_nat @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_idemp
% 0.32/0.55 thf(fact_3041_Diff__iff,axiom,
% 0.32/0.55 ! [C: real,A2: set_real,B5: set_real] :
% 0.32/0.55 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
% 0.32/0.55 = ( ( member_real @ C @ A2 )
% 0.32/0.55 & ~ ( member_real @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_iff
% 0.32/0.55 thf(fact_3042_Diff__iff,axiom,
% 0.32/0.55 ! [C: int,A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.55 = ( ( member_int @ C @ A2 )
% 0.32/0.55 & ~ ( member_int @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_iff
% 0.32/0.55 thf(fact_3043_Diff__iff,axiom,
% 0.32/0.55 ! [C: complex,A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.55 = ( ( member_complex @ C @ A2 )
% 0.32/0.55 & ~ ( member_complex @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_iff
% 0.32/0.55 thf(fact_3044_Diff__iff,axiom,
% 0.32/0.55 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) )
% 0.32/0.55 = ( ( member8206827879206165904at_nat @ C @ A2 )
% 0.32/0.55 & ~ ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_iff
% 0.32/0.55 thf(fact_3045_Diff__iff,axiom,
% 0.32/0.55 ! [C: nat,A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.55 = ( ( member_nat @ C @ A2 )
% 0.32/0.55 & ~ ( member_nat @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_iff
% 0.32/0.55 thf(fact_3046_DiffI,axiom,
% 0.32/0.55 ! [C: real,A2: set_real,B5: set_real] :
% 0.32/0.55 ( ( member_real @ C @ A2 )
% 0.32/0.55 => ( ~ ( member_real @ C @ B5 )
% 0.32/0.55 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffI
% 0.32/0.55 thf(fact_3047_DiffI,axiom,
% 0.32/0.55 ! [C: int,A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( member_int @ C @ A2 )
% 0.32/0.55 => ( ~ ( member_int @ C @ B5 )
% 0.32/0.55 => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffI
% 0.32/0.55 thf(fact_3048_DiffI,axiom,
% 0.32/0.55 ! [C: complex,A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ( member_complex @ C @ A2 )
% 0.32/0.55 => ( ~ ( member_complex @ C @ B5 )
% 0.32/0.55 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffI
% 0.32/0.55 thf(fact_3049_DiffI,axiom,
% 0.32/0.55 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ C @ A2 )
% 0.32/0.55 => ( ~ ( member8206827879206165904at_nat @ C @ B5 )
% 0.32/0.55 => ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffI
% 0.32/0.55 thf(fact_3050_DiffI,axiom,
% 0.32/0.55 ! [C: nat,A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( member_nat @ C @ A2 )
% 0.32/0.55 => ( ~ ( member_nat @ C @ B5 )
% 0.32/0.55 => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffI
% 0.32/0.55 thf(fact_3051_subset__empty,axiom,
% 0.32/0.55 ! [A2: set_nat] :
% 0.32/0.55 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.32/0.55 = ( A2 = bot_bot_set_nat ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_empty
% 0.32/0.55 thf(fact_3052_subset__empty,axiom,
% 0.32/0.55 ! [A2: set_real] :
% 0.32/0.55 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 0.32/0.55 = ( A2 = bot_bot_set_real ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_empty
% 0.32/0.55 thf(fact_3053_subset__empty,axiom,
% 0.32/0.55 ! [A2: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.32/0.55 = ( A2 = bot_bot_set_int ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_empty
% 0.32/0.55 thf(fact_3054_Diff__cancel,axiom,
% 0.32/0.55 ! [A2: set_int] :
% 0.32/0.55 ( ( minus_minus_set_int @ A2 @ A2 )
% 0.32/0.55 = bot_bot_set_int ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_cancel
% 0.32/0.55 thf(fact_3055_Diff__cancel,axiom,
% 0.32/0.55 ! [A2: set_real] :
% 0.32/0.55 ( ( minus_minus_set_real @ A2 @ A2 )
% 0.32/0.55 = bot_bot_set_real ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_cancel
% 0.32/0.55 thf(fact_3056_Diff__cancel,axiom,
% 0.32/0.55 ! [A2: set_nat] :
% 0.32/0.55 ( ( minus_minus_set_nat @ A2 @ A2 )
% 0.32/0.55 = bot_bot_set_nat ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_cancel
% 0.32/0.55 thf(fact_3057_empty__Diff,axiom,
% 0.32/0.55 ! [A2: set_int] :
% 0.32/0.55 ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 0.32/0.55 = bot_bot_set_int ) ).
% 0.32/0.55
% 0.32/0.55 % empty_Diff
% 0.32/0.55 thf(fact_3058_empty__Diff,axiom,
% 0.32/0.55 ! [A2: set_real] :
% 0.32/0.55 ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 0.32/0.55 = bot_bot_set_real ) ).
% 0.32/0.55
% 0.32/0.55 % empty_Diff
% 0.32/0.55 thf(fact_3059_empty__Diff,axiom,
% 0.32/0.55 ! [A2: set_nat] :
% 0.32/0.55 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 0.32/0.55 = bot_bot_set_nat ) ).
% 0.32/0.55
% 0.32/0.55 % empty_Diff
% 0.32/0.55 thf(fact_3060_Diff__empty,axiom,
% 0.32/0.55 ! [A2: set_int] :
% 0.32/0.55 ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 0.32/0.55 = A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_empty
% 0.32/0.55 thf(fact_3061_Diff__empty,axiom,
% 0.32/0.55 ! [A2: set_real] :
% 0.32/0.55 ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 0.32/0.55 = A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_empty
% 0.32/0.55 thf(fact_3062_Diff__empty,axiom,
% 0.32/0.55 ! [A2: set_nat] :
% 0.32/0.55 ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 0.32/0.55 = A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_empty
% 0.32/0.55 thf(fact_3063_DiffD2,axiom,
% 0.32/0.55 ! [C: real,A2: set_real,B5: set_real] :
% 0.32/0.55 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( member_real @ C @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD2
% 0.32/0.55 thf(fact_3064_DiffD2,axiom,
% 0.32/0.55 ! [C: int,A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( member_int @ C @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD2
% 0.32/0.55 thf(fact_3065_DiffD2,axiom,
% 0.32/0.55 ! [C: complex,A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( member_complex @ C @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD2
% 0.32/0.55 thf(fact_3066_DiffD2,axiom,
% 0.32/0.55 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( member8206827879206165904at_nat @ C @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD2
% 0.32/0.55 thf(fact_3067_DiffD2,axiom,
% 0.32/0.55 ! [C: nat,A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( member_nat @ C @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD2
% 0.32/0.55 thf(fact_3068_DiffD1,axiom,
% 0.32/0.55 ! [C: real,A2: set_real,B5: set_real] :
% 0.32/0.55 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
% 0.32/0.55 => ( member_real @ C @ A2 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD1
% 0.32/0.55 thf(fact_3069_DiffD1,axiom,
% 0.32/0.55 ! [C: int,A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.55 => ( member_int @ C @ A2 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD1
% 0.32/0.55 thf(fact_3070_DiffD1,axiom,
% 0.32/0.55 ! [C: complex,A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.55 => ( member_complex @ C @ A2 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD1
% 0.32/0.55 thf(fact_3071_DiffD1,axiom,
% 0.32/0.55 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) )
% 0.32/0.55 => ( member8206827879206165904at_nat @ C @ A2 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD1
% 0.32/0.55 thf(fact_3072_DiffD1,axiom,
% 0.32/0.55 ! [C: nat,A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.55 => ( member_nat @ C @ A2 ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffD1
% 0.32/0.55 thf(fact_3073_DiffE,axiom,
% 0.32/0.55 ! [C: real,A2: set_real,B5: set_real] :
% 0.32/0.55 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( ( member_real @ C @ A2 )
% 0.32/0.55 => ( member_real @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffE
% 0.32/0.55 thf(fact_3074_DiffE,axiom,
% 0.32/0.55 ! [C: int,A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( ( member_int @ C @ A2 )
% 0.32/0.55 => ( member_int @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffE
% 0.32/0.55 thf(fact_3075_DiffE,axiom,
% 0.32/0.55 ! [C: complex,A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( ( member_complex @ C @ A2 )
% 0.32/0.55 => ( member_complex @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffE
% 0.32/0.55 thf(fact_3076_DiffE,axiom,
% 0.32/0.55 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( ( member8206827879206165904at_nat @ C @ A2 )
% 0.32/0.55 => ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffE
% 0.32/0.55 thf(fact_3077_DiffE,axiom,
% 0.32/0.55 ! [C: nat,A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.55 => ~ ( ( member_nat @ C @ A2 )
% 0.32/0.55 => ( member_nat @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % DiffE
% 0.32/0.55 thf(fact_3078_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 0.32/0.55 ! [Uu: $o] :
% 0.32/0.55 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.minNull.simps(3)
% 0.32/0.55 thf(fact_3079_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 0.32/0.55 ! [Uv: $o] :
% 0.32/0.55 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.minNull.simps(2)
% 0.32/0.55 thf(fact_3080_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 0.32/0.55 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.minNull.simps(1)
% 0.32/0.55 thf(fact_3081_vebt__member_Osimps_I1_J,axiom,
% 0.32/0.55 ! [A: $o,B: $o,X: nat] :
% 0.32/0.55 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.32/0.55 = ( ( ( X = zero_zero_nat )
% 0.32/0.55 => A )
% 0.32/0.55 & ( ( X != zero_zero_nat )
% 0.32/0.55 => ( ( ( X = one_one_nat )
% 0.32/0.55 => B )
% 0.32/0.55 & ( X = one_one_nat ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_member.simps(1)
% 0.32/0.55 thf(fact_3082_vebt__insert_Osimps_I1_J,axiom,
% 0.32/0.55 ! [X: nat,A: $o,B: $o] :
% 0.32/0.55 ( ( ( X = zero_zero_nat )
% 0.32/0.55 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.32/0.55 = ( vEBT_Leaf @ $true @ B ) ) )
% 0.32/0.55 & ( ( X != zero_zero_nat )
% 0.32/0.55 => ( ( ( X = one_one_nat )
% 0.32/0.55 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.32/0.55 = ( vEBT_Leaf @ A @ $true ) ) )
% 0.32/0.55 & ( ( X != one_one_nat )
% 0.32/0.55 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.32/0.55 = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_insert.simps(1)
% 0.32/0.55 thf(fact_3083_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT] :
% 0.32/0.55 ( ~ ( vEBT_VEBT_minNull @ X )
% 0.32/0.55 => ( ! [Uv2: $o] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.32/0.55 => ( ! [Uu2: $o] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.32/0.55 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.minNull.elims(3)
% 0.32/0.55 thf(fact_3084_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT] :
% 0.32/0.55 ( ( vEBT_VEBT_minNull @ X )
% 0.32/0.55 => ( ( X
% 0.32/0.55 != ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.55 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.minNull.elims(2)
% 0.32/0.55 thf(fact_3085_invar__vebt_Ointros_I1_J,axiom,
% 0.32/0.55 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 0.32/0.55
% 0.32/0.55 % invar_vebt.intros(1)
% 0.32/0.55 thf(fact_3086_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Y3: $o] :
% 0.32/0.55 ( ( ( vEBT_VEBT_minNull @ X )
% 0.32/0.55 = Y3 )
% 0.32/0.55 => ( ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.55 => ~ Y3 )
% 0.32/0.55 => ( ( ? [Uv2: $o] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.32/0.55 => Y3 )
% 0.32/0.55 => ( ( ? [Uu2: $o] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.32/0.55 => Y3 )
% 0.32/0.55 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.32/0.55 => ~ Y3 )
% 0.32/0.55 => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 0.32/0.55 => Y3 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.minNull.elims(1)
% 0.32/0.55 thf(fact_3087_vebt__succ_Osimps_I2_J,axiom,
% 0.32/0.55 ! [Uv: $o,Uw: $o,N: nat] :
% 0.32/0.55 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
% 0.32/0.55 = none_nat ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_succ.simps(2)
% 0.32/0.55 thf(fact_3088_Diff__mono,axiom,
% 0.32/0.55 ! [A2: set_nat,C4: set_nat,D4: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.32/0.55 => ( ( ord_less_eq_set_nat @ D4 @ B5 )
% 0.32/0.55 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_mono
% 0.32/0.55 thf(fact_3089_Diff__mono,axiom,
% 0.32/0.55 ! [A2: set_int,C4: set_int,D4: set_int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ C4 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ D4 @ B5 )
% 0.32/0.55 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_mono
% 0.32/0.55 thf(fact_3090_Diff__subset,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_subset
% 0.32/0.55 thf(fact_3091_Diff__subset,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_subset
% 0.32/0.55 thf(fact_3092_double__diff,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat,C4: set_nat] :
% 0.32/0.55 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.55 => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.32/0.55 => ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.32/0.55 = A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % double_diff
% 0.32/0.55 thf(fact_3093_double__diff,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int,C4: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 0.32/0.55 => ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A2 ) )
% 0.32/0.55 = A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % double_diff
% 0.32/0.55 thf(fact_3094_in__mono,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat,X: nat] :
% 0.32/0.55 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.55 => ( ( member_nat @ X @ A2 )
% 0.32/0.55 => ( member_nat @ X @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % in_mono
% 0.32/0.55 thf(fact_3095_in__mono,axiom,
% 0.32/0.55 ! [A2: set_real,B5: set_real,X: real] :
% 0.32/0.55 ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.55 => ( ( member_real @ X @ A2 )
% 0.32/0.55 => ( member_real @ X @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % in_mono
% 0.32/0.55 thf(fact_3096_in__mono,axiom,
% 0.32/0.55 ! [A2: set_complex,B5: set_complex,X: complex] :
% 0.32/0.55 ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.55 => ( ( member_complex @ X @ A2 )
% 0.32/0.55 => ( member_complex @ X @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % in_mono
% 0.32/0.55 thf(fact_3097_in__mono,axiom,
% 0.32/0.55 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat] :
% 0.32/0.55 ( ( ord_le3000389064537975527at_nat @ A2 @ B5 )
% 0.32/0.55 => ( ( member8206827879206165904at_nat @ X @ A2 )
% 0.32/0.55 => ( member8206827879206165904at_nat @ X @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % in_mono
% 0.32/0.55 thf(fact_3098_in__mono,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int,X: int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( member_int @ X @ A2 )
% 0.32/0.55 => ( member_int @ X @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % in_mono
% 0.32/0.55 thf(fact_3099_subsetD,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat,C: nat] :
% 0.32/0.55 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.55 => ( ( member_nat @ C @ A2 )
% 0.32/0.55 => ( member_nat @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetD
% 0.32/0.55 thf(fact_3100_subsetD,axiom,
% 0.32/0.55 ! [A2: set_real,B5: set_real,C: real] :
% 0.32/0.55 ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.55 => ( ( member_real @ C @ A2 )
% 0.32/0.55 => ( member_real @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetD
% 0.32/0.55 thf(fact_3101_subsetD,axiom,
% 0.32/0.55 ! [A2: set_complex,B5: set_complex,C: complex] :
% 0.32/0.55 ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.55 => ( ( member_complex @ C @ A2 )
% 0.32/0.55 => ( member_complex @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetD
% 0.32/0.55 thf(fact_3102_subsetD,axiom,
% 0.32/0.55 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
% 0.32/0.55 ( ( ord_le3000389064537975527at_nat @ A2 @ B5 )
% 0.32/0.55 => ( ( member8206827879206165904at_nat @ C @ A2 )
% 0.32/0.55 => ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetD
% 0.32/0.55 thf(fact_3103_subsetD,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int,C: int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( member_int @ C @ A2 )
% 0.32/0.55 => ( member_int @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subsetD
% 0.32/0.55 thf(fact_3104_equalityE,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( A2 = B5 )
% 0.32/0.55 => ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ~ ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % equalityE
% 0.32/0.55 thf(fact_3105_subset__eq,axiom,
% 0.32/0.55 ( ord_less_eq_set_nat
% 0.32/0.55 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.32/0.55 ! [X2: nat] :
% 0.32/0.55 ( ( member_nat @ X2 @ A6 )
% 0.32/0.55 => ( member_nat @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_eq
% 0.32/0.55 thf(fact_3106_subset__eq,axiom,
% 0.32/0.55 ( ord_less_eq_set_real
% 0.32/0.55 = ( ^ [A6: set_real,B6: set_real] :
% 0.32/0.55 ! [X2: real] :
% 0.32/0.55 ( ( member_real @ X2 @ A6 )
% 0.32/0.55 => ( member_real @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_eq
% 0.32/0.55 thf(fact_3107_subset__eq,axiom,
% 0.32/0.55 ( ord_le211207098394363844omplex
% 0.32/0.55 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.32/0.55 ! [X2: complex] :
% 0.32/0.55 ( ( member_complex @ X2 @ A6 )
% 0.32/0.55 => ( member_complex @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_eq
% 0.32/0.55 thf(fact_3108_subset__eq,axiom,
% 0.32/0.55 ( ord_le3000389064537975527at_nat
% 0.32/0.55 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ! [X2: produc859450856879609959at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ X2 @ A6 )
% 0.32/0.55 => ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_eq
% 0.32/0.55 thf(fact_3109_subset__eq,axiom,
% 0.32/0.55 ( ord_less_eq_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ! [X2: int] :
% 0.32/0.55 ( ( member_int @ X2 @ A6 )
% 0.32/0.55 => ( member_int @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_eq
% 0.32/0.55 thf(fact_3110_equalityD1,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( A2 = B5 )
% 0.32/0.55 => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % equalityD1
% 0.32/0.55 thf(fact_3111_equalityD2,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( A2 = B5 )
% 0.32/0.55 => ( ord_less_eq_set_int @ B5 @ A2 ) ) ).
% 0.32/0.55
% 0.32/0.55 % equalityD2
% 0.32/0.55 thf(fact_3112_subset__iff,axiom,
% 0.32/0.55 ( ord_less_eq_set_nat
% 0.32/0.55 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.32/0.55 ! [T2: nat] :
% 0.32/0.55 ( ( member_nat @ T2 @ A6 )
% 0.32/0.55 => ( member_nat @ T2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_iff
% 0.32/0.55 thf(fact_3113_subset__iff,axiom,
% 0.32/0.55 ( ord_less_eq_set_real
% 0.32/0.55 = ( ^ [A6: set_real,B6: set_real] :
% 0.32/0.55 ! [T2: real] :
% 0.32/0.55 ( ( member_real @ T2 @ A6 )
% 0.32/0.55 => ( member_real @ T2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_iff
% 0.32/0.55 thf(fact_3114_subset__iff,axiom,
% 0.32/0.55 ( ord_le211207098394363844omplex
% 0.32/0.55 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.32/0.55 ! [T2: complex] :
% 0.32/0.55 ( ( member_complex @ T2 @ A6 )
% 0.32/0.55 => ( member_complex @ T2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_iff
% 0.32/0.55 thf(fact_3115_subset__iff,axiom,
% 0.32/0.55 ( ord_le3000389064537975527at_nat
% 0.32/0.55 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ! [T2: produc859450856879609959at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ T2 @ A6 )
% 0.32/0.55 => ( member8206827879206165904at_nat @ T2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_iff
% 0.32/0.55 thf(fact_3116_subset__iff,axiom,
% 0.32/0.55 ( ord_less_eq_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ! [T2: int] :
% 0.32/0.55 ( ( member_int @ T2 @ A6 )
% 0.32/0.55 => ( member_int @ T2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_iff
% 0.32/0.55 thf(fact_3117_subset__refl,axiom,
% 0.32/0.55 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % subset_refl
% 0.32/0.55 thf(fact_3118_Collect__mono,axiom,
% 0.32/0.55 ! [P: complex > $o,Q: complex > $o] :
% 0.32/0.55 ( ! [X3: complex] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 => ( Q @ X3 ) )
% 0.32/0.55 => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono
% 0.32/0.55 thf(fact_3119_Collect__mono,axiom,
% 0.32/0.55 ! [P: real > $o,Q: real > $o] :
% 0.32/0.55 ( ! [X3: real] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 => ( Q @ X3 ) )
% 0.32/0.55 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono
% 0.32/0.55 thf(fact_3120_Collect__mono,axiom,
% 0.32/0.55 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.32/0.55 ( ! [X3: list_nat] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 => ( Q @ X3 ) )
% 0.32/0.55 => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono
% 0.32/0.55 thf(fact_3121_Collect__mono,axiom,
% 0.32/0.55 ! [P: nat > $o,Q: nat > $o] :
% 0.32/0.55 ( ! [X3: nat] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 => ( Q @ X3 ) )
% 0.32/0.55 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono
% 0.32/0.55 thf(fact_3122_Collect__mono,axiom,
% 0.32/0.55 ! [P: int > $o,Q: int > $o] :
% 0.32/0.55 ( ! [X3: int] :
% 0.32/0.55 ( ( P @ X3 )
% 0.32/0.55 => ( Q @ X3 ) )
% 0.32/0.55 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono
% 0.32/0.55 thf(fact_3123_subset__trans,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int,C4: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 0.32/0.55 => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_trans
% 0.32/0.55 thf(fact_3124_set__eq__subset,axiom,
% 0.32/0.55 ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.32/0.55 & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_eq_subset
% 0.32/0.55 thf(fact_3125_Collect__mono__iff,axiom,
% 0.32/0.55 ! [P: complex > $o,Q: complex > $o] :
% 0.32/0.55 ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 0.32/0.55 = ( ! [X2: complex] :
% 0.32/0.55 ( ( P @ X2 )
% 0.32/0.55 => ( Q @ X2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono_iff
% 0.32/0.55 thf(fact_3126_Collect__mono__iff,axiom,
% 0.32/0.55 ! [P: real > $o,Q: real > $o] :
% 0.32/0.55 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.32/0.55 = ( ! [X2: real] :
% 0.32/0.55 ( ( P @ X2 )
% 0.32/0.55 => ( Q @ X2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono_iff
% 0.32/0.55 thf(fact_3127_Collect__mono__iff,axiom,
% 0.32/0.55 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.32/0.55 ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 0.32/0.55 = ( ! [X2: list_nat] :
% 0.32/0.55 ( ( P @ X2 )
% 0.32/0.55 => ( Q @ X2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono_iff
% 0.32/0.55 thf(fact_3128_Collect__mono__iff,axiom,
% 0.32/0.55 ! [P: nat > $o,Q: nat > $o] :
% 0.32/0.55 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.32/0.55 = ( ! [X2: nat] :
% 0.32/0.55 ( ( P @ X2 )
% 0.32/0.55 => ( Q @ X2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono_iff
% 0.32/0.55 thf(fact_3129_Collect__mono__iff,axiom,
% 0.32/0.55 ! [P: int > $o,Q: int > $o] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.32/0.55 = ( ! [X2: int] :
% 0.32/0.55 ( ( P @ X2 )
% 0.32/0.55 => ( Q @ X2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_mono_iff
% 0.32/0.55 thf(fact_3130_psubsetD,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat,C: nat] :
% 0.32/0.55 ( ( ord_less_set_nat @ A2 @ B5 )
% 0.32/0.55 => ( ( member_nat @ C @ A2 )
% 0.32/0.55 => ( member_nat @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubsetD
% 0.32/0.55 thf(fact_3131_psubsetD,axiom,
% 0.32/0.55 ! [A2: set_real,B5: set_real,C: real] :
% 0.32/0.55 ( ( ord_less_set_real @ A2 @ B5 )
% 0.32/0.55 => ( ( member_real @ C @ A2 )
% 0.32/0.55 => ( member_real @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubsetD
% 0.32/0.55 thf(fact_3132_psubsetD,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int,C: int] :
% 0.32/0.55 ( ( ord_less_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( member_int @ C @ A2 )
% 0.32/0.55 => ( member_int @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubsetD
% 0.32/0.55 thf(fact_3133_psubsetD,axiom,
% 0.32/0.55 ! [A2: set_complex,B5: set_complex,C: complex] :
% 0.32/0.55 ( ( ord_less_set_complex @ A2 @ B5 )
% 0.32/0.55 => ( ( member_complex @ C @ A2 )
% 0.32/0.55 => ( member_complex @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubsetD
% 0.32/0.55 thf(fact_3134_psubsetD,axiom,
% 0.32/0.55 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
% 0.32/0.55 ( ( ord_le6428140832669894131at_nat @ A2 @ B5 )
% 0.32/0.55 => ( ( member8206827879206165904at_nat @ C @ A2 )
% 0.32/0.55 => ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubsetD
% 0.32/0.55 thf(fact_3135_psubset__imp__ex__mem,axiom,
% 0.32/0.55 ! [A2: set_real,B5: set_real] :
% 0.32/0.55 ( ( ord_less_set_real @ A2 @ B5 )
% 0.32/0.55 => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubset_imp_ex_mem
% 0.32/0.55 thf(fact_3136_psubset__imp__ex__mem,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_set_int @ A2 @ B5 )
% 0.32/0.55 => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubset_imp_ex_mem
% 0.32/0.55 thf(fact_3137_psubset__imp__ex__mem,axiom,
% 0.32/0.55 ! [A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ( ord_less_set_complex @ A2 @ B5 )
% 0.32/0.55 => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubset_imp_ex_mem
% 0.32/0.55 thf(fact_3138_psubset__imp__ex__mem,axiom,
% 0.32/0.55 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ( ord_le6428140832669894131at_nat @ A2 @ B5 )
% 0.32/0.55 => ? [B3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ B5 @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubset_imp_ex_mem
% 0.32/0.55 thf(fact_3139_psubset__imp__ex__mem,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( ord_less_set_nat @ A2 @ B5 )
% 0.32/0.55 => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubset_imp_ex_mem
% 0.32/0.55 thf(fact_3140_vebt__succ_Osimps_I1_J,axiom,
% 0.32/0.55 ! [B: $o,Uu: $o] :
% 0.32/0.55 ( ( B
% 0.32/0.55 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 0.32/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.55 & ( ~ B
% 0.32/0.55 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 0.32/0.55 = none_nat ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_succ.simps(1)
% 0.32/0.55 thf(fact_3141_vebt__mint_Osimps_I1_J,axiom,
% 0.32/0.55 ! [A: $o,B: $o] :
% 0.32/0.55 ( ( A
% 0.32/0.55 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.32/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.55 & ( ~ A
% 0.32/0.55 => ( ( B
% 0.32/0.55 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.32/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.55 & ( ~ B
% 0.32/0.55 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.32/0.55 = none_nat ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_mint.simps(1)
% 0.32/0.55 thf(fact_3142_vebt__maxt_Osimps_I1_J,axiom,
% 0.32/0.55 ! [B: $o,A: $o] :
% 0.32/0.55 ( ( B
% 0.32/0.55 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.32/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.55 & ( ~ B
% 0.32/0.55 => ( ( A
% 0.32/0.55 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.32/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.55 & ( ~ A
% 0.32/0.55 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.32/0.55 = none_nat ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_maxt.simps(1)
% 0.32/0.55 thf(fact_3143_vebt__succ_Ocases,axiom,
% 0.32/0.55 ! [X: produc9072475918466114483BT_nat] :
% 0.32/0.55 ( ! [Uu2: $o,B3: $o] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) )
% 0.32/0.55 => ( ! [Uv2: $o,Uw2: $o,N4: nat] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) )
% 0.32/0.55 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
% 0.32/0.55 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
% 0.32/0.55 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
% 0.32/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_succ.cases
% 0.32/0.55 thf(fact_3144_not__psubset__empty,axiom,
% 0.32/0.55 ! [A2: set_nat] :
% 0.32/0.55 ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 0.32/0.55
% 0.32/0.55 % not_psubset_empty
% 0.32/0.55 thf(fact_3145_not__psubset__empty,axiom,
% 0.32/0.55 ! [A2: set_int] :
% 0.32/0.55 ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 0.32/0.55
% 0.32/0.55 % not_psubset_empty
% 0.32/0.55 thf(fact_3146_not__psubset__empty,axiom,
% 0.32/0.55 ! [A2: set_real] :
% 0.32/0.55 ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 0.32/0.55
% 0.32/0.55 % not_psubset_empty
% 0.32/0.55 thf(fact_3147_subset__iff__psubset__eq,axiom,
% 0.32/0.55 ( ord_less_eq_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ( ( ord_less_set_int @ A6 @ B6 )
% 0.32/0.55 | ( A6 = B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_iff_psubset_eq
% 0.32/0.55 thf(fact_3148_subset__psubset__trans,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int,C4: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( ord_less_set_int @ B5 @ C4 )
% 0.32/0.55 => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_psubset_trans
% 0.32/0.55 thf(fact_3149_subset__not__subset__eq,axiom,
% 0.32/0.55 ( ord_less_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.32/0.55 & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % subset_not_subset_eq
% 0.32/0.55 thf(fact_3150_psubset__subset__trans,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int,C4: set_int] :
% 0.32/0.55 ( ( ord_less_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 0.32/0.55 => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubset_subset_trans
% 0.32/0.55 thf(fact_3151_psubset__imp__subset,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubset_imp_subset
% 0.32/0.55 thf(fact_3152_psubset__eq,axiom,
% 0.32/0.55 ( ord_less_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.32/0.55 & ( A6 != B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubset_eq
% 0.32/0.55 thf(fact_3153_psubsetE,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_set_int @ A2 @ B5 )
% 0.32/0.55 => ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % psubsetE
% 0.32/0.55 thf(fact_3154_vebt__mint_Oelims,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Y3: option_nat] :
% 0.32/0.55 ( ( ( vEBT_vebt_mint @ X )
% 0.32/0.55 = Y3 )
% 0.32/0.55 => ( ! [A5: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ~ ( ( A5
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.55 & ( ~ A5
% 0.32/0.55 => ( ( B3
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.55 & ( ~ B3
% 0.32/0.55 => ( Y3 = none_nat ) ) ) ) ) )
% 0.32/0.55 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.55 => ( Y3 != none_nat ) )
% 0.32/0.55 => ~ ! [Mi2: nat] :
% 0.32/0.55 ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.55 => ( Y3
% 0.32/0.55 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_mint.elims
% 0.32/0.55 thf(fact_3155_vebt__maxt_Oelims,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Y3: option_nat] :
% 0.32/0.55 ( ( ( vEBT_vebt_maxt @ X )
% 0.32/0.55 = Y3 )
% 0.32/0.55 => ( ! [A5: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ~ ( ( B3
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.55 & ( ~ B3
% 0.32/0.55 => ( ( A5
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.55 & ( ~ A5
% 0.32/0.55 => ( Y3 = none_nat ) ) ) ) ) )
% 0.32/0.55 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.55 => ( Y3 != none_nat ) )
% 0.32/0.55 => ~ ! [Mi2: nat,Ma2: nat] :
% 0.32/0.55 ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.55 => ( Y3
% 0.32/0.55 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_maxt.elims
% 0.32/0.55 thf(fact_3156_vebt__insert_Osimps_I2_J,axiom,
% 0.32/0.55 ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 0.32/0.55 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
% 0.32/0.55 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_insert.simps(2)
% 0.32/0.55 thf(fact_3157_vebt__member_Osimps_I2_J,axiom,
% 0.32/0.55 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
% 0.32/0.55 ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_member.simps(2)
% 0.32/0.55 thf(fact_3158_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 0.32/0.55 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.32/0.55 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.minNull.simps(5)
% 0.32/0.55 thf(fact_3159_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 0.32/0.55 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.minNull.simps(4)
% 0.32/0.55 thf(fact_3160_vebt__insert_Osimps_I3_J,axiom,
% 0.32/0.55 ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 0.32/0.55 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
% 0.32/0.55 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_insert.simps(3)
% 0.32/0.55 thf(fact_3161_vebt__member_Osimps_I3_J,axiom,
% 0.32/0.55 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
% 0.32/0.55 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_member.simps(3)
% 0.32/0.55 thf(fact_3162_finite__Diff,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( finite_finite_int @ A2 )
% 0.32/0.55 => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Diff
% 0.32/0.55 thf(fact_3163_finite__Diff,axiom,
% 0.32/0.55 ! [A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.55 => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Diff
% 0.32/0.55 thf(fact_3164_finite__Diff,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( finite_finite_nat @ A2 )
% 0.32/0.55 => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Diff
% 0.32/0.55 thf(fact_3165_finite__Diff2,axiom,
% 0.32/0.55 ! [B5: set_int,A2: set_int] :
% 0.32/0.55 ( ( finite_finite_int @ B5 )
% 0.32/0.55 => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.55 = ( finite_finite_int @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Diff2
% 0.32/0.55 thf(fact_3166_finite__Diff2,axiom,
% 0.32/0.55 ! [B5: set_complex,A2: set_complex] :
% 0.32/0.55 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.55 => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.55 = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Diff2
% 0.32/0.55 thf(fact_3167_finite__Diff2,axiom,
% 0.32/0.55 ! [B5: set_nat,A2: set_nat] :
% 0.32/0.55 ( ( finite_finite_nat @ B5 )
% 0.32/0.55 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.55 = ( finite_finite_nat @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Diff2
% 0.32/0.55 thf(fact_3168_succ__greatereq__min,axiom,
% 0.32/0.55 ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.32/0.55 => ( ( ord_less_eq_nat @ Mi @ X )
% 0.32/0.55 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.55 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.55 @ ( if_option_nat
% 0.32/0.55 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 != none_nat )
% 0.32/0.55 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.32/0.55 @ ( if_option_nat
% 0.32/0.55 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.55 = none_nat )
% 0.32/0.55 @ none_nat
% 0.32/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.55 @ none_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % succ_greatereq_min
% 0.32/0.55 thf(fact_3169_succ__less__length__list,axiom,
% 0.32/0.55 ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.32/0.55 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.32/0.55 => ( ( ord_less_eq_nat @ Mi @ X )
% 0.32/0.55 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.55 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.55 = ( if_option_nat
% 0.32/0.55 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 != none_nat )
% 0.32/0.55 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.32/0.55 @ ( if_option_nat
% 0.32/0.55 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.55 = none_nat )
% 0.32/0.55 @ none_nat
% 0.32/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % succ_less_length_list
% 0.32/0.55 thf(fact_3170_unset__bit__0,axiom,
% 0.32/0.55 ! [A: int] :
% 0.32/0.55 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.32/0.55 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % unset_bit_0
% 0.32/0.55 thf(fact_3171_unset__bit__0,axiom,
% 0.32/0.55 ! [A: nat] :
% 0.32/0.55 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.32/0.55 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % unset_bit_0
% 0.32/0.55 thf(fact_3172_buildup__nothing__in__leaf,axiom,
% 0.32/0.55 ! [N: nat,X: nat] :
% 0.32/0.55 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.32/0.55
% 0.32/0.55 % buildup_nothing_in_leaf
% 0.32/0.55 thf(fact_3173_unset__bit__Suc,axiom,
% 0.32/0.55 ! [N: nat,A: code_integer] :
% 0.32/0.55 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 0.32/0.55 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % unset_bit_Suc
% 0.32/0.55 thf(fact_3174_unset__bit__Suc,axiom,
% 0.32/0.55 ! [N: nat,A: int] :
% 0.32/0.55 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 0.32/0.55 = ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % unset_bit_Suc
% 0.32/0.55 thf(fact_3175_unset__bit__Suc,axiom,
% 0.32/0.55 ! [N: nat,A: nat] :
% 0.32/0.55 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 0.32/0.55 = ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % unset_bit_Suc
% 0.32/0.55 thf(fact_3176_set__vebt_H__def,axiom,
% 0.32/0.55 ( vEBT_VEBT_set_vebt
% 0.32/0.55 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_vebt'_def
% 0.32/0.55 thf(fact_3177_finite__interval__int1,axiom,
% 0.32/0.55 ! [A: int,B: int] :
% 0.32/0.55 ( finite_finite_int
% 0.32/0.55 @ ( collect_int
% 0.32/0.55 @ ^ [I4: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ A @ I4 )
% 0.32/0.55 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_interval_int1
% 0.32/0.55 thf(fact_3178_finite__interval__int4,axiom,
% 0.32/0.55 ! [A: int,B: int] :
% 0.32/0.55 ( finite_finite_int
% 0.32/0.55 @ ( collect_int
% 0.32/0.55 @ ^ [I4: int] :
% 0.32/0.55 ( ( ord_less_int @ A @ I4 )
% 0.32/0.55 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_interval_int4
% 0.32/0.55 thf(fact_3179_unset__bit__nonnegative__int__iff,axiom,
% 0.32/0.55 ! [N: nat,K: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 0.32/0.55 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.55
% 0.32/0.55 % unset_bit_nonnegative_int_iff
% 0.32/0.55 thf(fact_3180_unset__bit__negative__int__iff,axiom,
% 0.32/0.55 ! [N: nat,K: int] :
% 0.32/0.55 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.55 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.55
% 0.32/0.55 % unset_bit_negative_int_iff
% 0.32/0.55 thf(fact_3181_finite__Collect__subsets,axiom,
% 0.32/0.55 ! [A2: set_nat] :
% 0.32/0.55 ( ( finite_finite_nat @ A2 )
% 0.32/0.55 => ( finite1152437895449049373et_nat
% 0.32/0.55 @ ( collect_set_nat
% 0.32/0.55 @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Collect_subsets
% 0.32/0.55 thf(fact_3182_finite__Collect__subsets,axiom,
% 0.32/0.55 ! [A2: set_complex] :
% 0.32/0.55 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.55 => ( finite6551019134538273531omplex
% 0.32/0.55 @ ( collect_set_complex
% 0.32/0.55 @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Collect_subsets
% 0.32/0.55 thf(fact_3183_finite__Collect__subsets,axiom,
% 0.32/0.55 ! [A2: set_int] :
% 0.32/0.55 ( ( finite_finite_int @ A2 )
% 0.32/0.55 => ( finite6197958912794628473et_int
% 0.32/0.55 @ ( collect_set_int
% 0.32/0.55 @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Collect_subsets
% 0.32/0.55 thf(fact_3184_finite__interval__int3,axiom,
% 0.32/0.55 ! [A: int,B: int] :
% 0.32/0.55 ( finite_finite_int
% 0.32/0.55 @ ( collect_int
% 0.32/0.55 @ ^ [I4: int] :
% 0.32/0.55 ( ( ord_less_int @ A @ I4 )
% 0.32/0.55 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_interval_int3
% 0.32/0.55 thf(fact_3185_finite__interval__int2,axiom,
% 0.32/0.55 ! [A: int,B: int] :
% 0.32/0.55 ( finite_finite_int
% 0.32/0.55 @ ( collect_int
% 0.32/0.55 @ ^ [I4: int] :
% 0.32/0.55 ( ( ord_less_eq_int @ A @ I4 )
% 0.32/0.55 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_interval_int2
% 0.32/0.55 thf(fact_3186_finite__Collect__less__nat,axiom,
% 0.32/0.55 ! [K: nat] :
% 0.32/0.55 ( finite_finite_nat
% 0.32/0.55 @ ( collect_nat
% 0.32/0.55 @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Collect_less_nat
% 0.32/0.55 thf(fact_3187_finite__Collect__le__nat,axiom,
% 0.32/0.55 ! [K: nat] :
% 0.32/0.55 ( finite_finite_nat
% 0.32/0.55 @ ( collect_nat
% 0.32/0.55 @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_Collect_le_nat
% 0.32/0.55 thf(fact_3188_minus__set__def,axiom,
% 0.32/0.55 ( minus_8321449233255521966at_nat
% 0.32/0.55 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( collec7088162979684241874at_nat
% 0.32/0.55 @ ( minus_4666906530369457071_nat_o
% 0.32/0.55 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minus_set_def
% 0.32/0.55 thf(fact_3189_minus__set__def,axiom,
% 0.32/0.55 ( minus_811609699411566653omplex
% 0.32/0.55 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.32/0.55 ( collect_complex
% 0.32/0.55 @ ( minus_8727706125548526216plex_o
% 0.32/0.55 @ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minus_set_def
% 0.32/0.55 thf(fact_3190_minus__set__def,axiom,
% 0.32/0.55 ( minus_minus_set_real
% 0.32/0.55 = ( ^ [A6: set_real,B6: set_real] :
% 0.32/0.55 ( collect_real
% 0.32/0.55 @ ( minus_minus_real_o
% 0.32/0.55 @ ^ [X2: real] : ( member_real @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minus_set_def
% 0.32/0.55 thf(fact_3191_minus__set__def,axiom,
% 0.32/0.55 ( minus_7954133019191499631st_nat
% 0.32/0.55 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.32/0.55 ( collect_list_nat
% 0.32/0.55 @ ( minus_1139252259498527702_nat_o
% 0.32/0.55 @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minus_set_def
% 0.32/0.55 thf(fact_3192_minus__set__def,axiom,
% 0.32/0.55 ( minus_minus_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ( collect_int
% 0.32/0.55 @ ( minus_minus_int_o
% 0.32/0.55 @ ^ [X2: int] : ( member_int @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minus_set_def
% 0.32/0.55 thf(fact_3193_minus__set__def,axiom,
% 0.32/0.55 ( minus_minus_set_nat
% 0.32/0.55 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.32/0.55 ( collect_nat
% 0.32/0.55 @ ( minus_minus_nat_o
% 0.32/0.55 @ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % minus_set_def
% 0.32/0.55 thf(fact_3194_set__diff__eq,axiom,
% 0.32/0.55 ( minus_8321449233255521966at_nat
% 0.32/0.55 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( collec7088162979684241874at_nat
% 0.32/0.55 @ ^ [X2: produc859450856879609959at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ X2 @ A6 )
% 0.32/0.55 & ~ ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_diff_eq
% 0.32/0.55 thf(fact_3195_set__diff__eq,axiom,
% 0.32/0.55 ( minus_811609699411566653omplex
% 0.32/0.55 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.32/0.55 ( collect_complex
% 0.32/0.55 @ ^ [X2: complex] :
% 0.32/0.55 ( ( member_complex @ X2 @ A6 )
% 0.32/0.55 & ~ ( member_complex @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_diff_eq
% 0.32/0.55 thf(fact_3196_set__diff__eq,axiom,
% 0.32/0.55 ( minus_minus_set_real
% 0.32/0.55 = ( ^ [A6: set_real,B6: set_real] :
% 0.32/0.55 ( collect_real
% 0.32/0.55 @ ^ [X2: real] :
% 0.32/0.55 ( ( member_real @ X2 @ A6 )
% 0.32/0.55 & ~ ( member_real @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_diff_eq
% 0.32/0.55 thf(fact_3197_set__diff__eq,axiom,
% 0.32/0.55 ( minus_7954133019191499631st_nat
% 0.32/0.55 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.32/0.55 ( collect_list_nat
% 0.32/0.55 @ ^ [X2: list_nat] :
% 0.32/0.55 ( ( member_list_nat @ X2 @ A6 )
% 0.32/0.55 & ~ ( member_list_nat @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_diff_eq
% 0.32/0.55 thf(fact_3198_set__diff__eq,axiom,
% 0.32/0.55 ( minus_minus_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ( collect_int
% 0.32/0.55 @ ^ [X2: int] :
% 0.32/0.55 ( ( member_int @ X2 @ A6 )
% 0.32/0.55 & ~ ( member_int @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_diff_eq
% 0.32/0.55 thf(fact_3199_set__diff__eq,axiom,
% 0.32/0.55 ( minus_minus_set_nat
% 0.32/0.55 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.32/0.55 ( collect_nat
% 0.32/0.55 @ ^ [X2: nat] :
% 0.32/0.55 ( ( member_nat @ X2 @ A6 )
% 0.32/0.55 & ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_diff_eq
% 0.32/0.55 thf(fact_3200_less__set__def,axiom,
% 0.32/0.55 ( ord_less_set_nat
% 0.32/0.55 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.32/0.55 ( ord_less_nat_o
% 0.32/0.55 @ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_set_def
% 0.32/0.55 thf(fact_3201_less__set__def,axiom,
% 0.32/0.55 ( ord_less_set_real
% 0.32/0.55 = ( ^ [A6: set_real,B6: set_real] :
% 0.32/0.55 ( ord_less_real_o
% 0.32/0.55 @ ^ [X2: real] : ( member_real @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_set_def
% 0.32/0.55 thf(fact_3202_less__set__def,axiom,
% 0.32/0.55 ( ord_less_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ( ord_less_int_o
% 0.32/0.55 @ ^ [X2: int] : ( member_int @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_set_def
% 0.32/0.55 thf(fact_3203_less__set__def,axiom,
% 0.32/0.55 ( ord_less_set_complex
% 0.32/0.55 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.32/0.55 ( ord_less_complex_o
% 0.32/0.55 @ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_set_def
% 0.32/0.55 thf(fact_3204_less__set__def,axiom,
% 0.32/0.55 ( ord_le6428140832669894131at_nat
% 0.32/0.55 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ord_le7432393201274343594_nat_o
% 0.32/0.55 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_set_def
% 0.32/0.55 thf(fact_3205_Collect__subset,axiom,
% 0.32/0.55 ! [A2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
% 0.32/0.55 ( ord_le3000389064537975527at_nat
% 0.32/0.55 @ ( collec7088162979684241874at_nat
% 0.32/0.55 @ ^ [X2: produc859450856879609959at_nat] :
% 0.32/0.55 ( ( member8206827879206165904at_nat @ X2 @ A2 )
% 0.32/0.55 & ( P @ X2 ) ) )
% 0.32/0.55 @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_subset
% 0.32/0.55 thf(fact_3206_Collect__subset,axiom,
% 0.32/0.55 ! [A2: set_complex,P: complex > $o] :
% 0.32/0.55 ( ord_le211207098394363844omplex
% 0.32/0.55 @ ( collect_complex
% 0.32/0.55 @ ^ [X2: complex] :
% 0.32/0.55 ( ( member_complex @ X2 @ A2 )
% 0.32/0.55 & ( P @ X2 ) ) )
% 0.32/0.55 @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_subset
% 0.32/0.55 thf(fact_3207_Collect__subset,axiom,
% 0.32/0.55 ! [A2: set_real,P: real > $o] :
% 0.32/0.55 ( ord_less_eq_set_real
% 0.32/0.55 @ ( collect_real
% 0.32/0.55 @ ^ [X2: real] :
% 0.32/0.55 ( ( member_real @ X2 @ A2 )
% 0.32/0.55 & ( P @ X2 ) ) )
% 0.32/0.55 @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_subset
% 0.32/0.55 thf(fact_3208_Collect__subset,axiom,
% 0.32/0.55 ! [A2: set_list_nat,P: list_nat > $o] :
% 0.32/0.55 ( ord_le6045566169113846134st_nat
% 0.32/0.55 @ ( collect_list_nat
% 0.32/0.55 @ ^ [X2: list_nat] :
% 0.32/0.55 ( ( member_list_nat @ X2 @ A2 )
% 0.32/0.55 & ( P @ X2 ) ) )
% 0.32/0.55 @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_subset
% 0.32/0.55 thf(fact_3209_Collect__subset,axiom,
% 0.32/0.55 ! [A2: set_nat,P: nat > $o] :
% 0.32/0.55 ( ord_less_eq_set_nat
% 0.32/0.55 @ ( collect_nat
% 0.32/0.55 @ ^ [X2: nat] :
% 0.32/0.55 ( ( member_nat @ X2 @ A2 )
% 0.32/0.55 & ( P @ X2 ) ) )
% 0.32/0.55 @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_subset
% 0.32/0.55 thf(fact_3210_Collect__subset,axiom,
% 0.32/0.55 ! [A2: set_int,P: int > $o] :
% 0.32/0.55 ( ord_less_eq_set_int
% 0.32/0.55 @ ( collect_int
% 0.32/0.55 @ ^ [X2: int] :
% 0.32/0.55 ( ( member_int @ X2 @ A2 )
% 0.32/0.55 & ( P @ X2 ) ) )
% 0.32/0.55 @ A2 ) ).
% 0.32/0.55
% 0.32/0.55 % Collect_subset
% 0.32/0.55 thf(fact_3211_less__eq__set__def,axiom,
% 0.32/0.55 ( ord_less_eq_set_nat
% 0.32/0.55 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.32/0.55 ( ord_less_eq_nat_o
% 0.32/0.55 @ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_eq_set_def
% 0.32/0.55 thf(fact_3212_less__eq__set__def,axiom,
% 0.32/0.55 ( ord_less_eq_set_real
% 0.32/0.55 = ( ^ [A6: set_real,B6: set_real] :
% 0.32/0.55 ( ord_less_eq_real_o
% 0.32/0.55 @ ^ [X2: real] : ( member_real @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_eq_set_def
% 0.32/0.55 thf(fact_3213_less__eq__set__def,axiom,
% 0.32/0.55 ( ord_le211207098394363844omplex
% 0.32/0.55 = ( ^ [A6: set_complex,B6: set_complex] :
% 0.32/0.55 ( ord_le4573692005234683329plex_o
% 0.32/0.55 @ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_eq_set_def
% 0.32/0.55 thf(fact_3214_less__eq__set__def,axiom,
% 0.32/0.55 ( ord_le3000389064537975527at_nat
% 0.32/0.55 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.32/0.55 ( ord_le7858099551454983350_nat_o
% 0.32/0.55 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_eq_set_def
% 0.32/0.55 thf(fact_3215_less__eq__set__def,axiom,
% 0.32/0.55 ( ord_less_eq_set_int
% 0.32/0.55 = ( ^ [A6: set_int,B6: set_int] :
% 0.32/0.55 ( ord_less_eq_int_o
% 0.32/0.55 @ ^ [X2: int] : ( member_int @ X2 @ A6 )
% 0.32/0.55 @ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % less_eq_set_def
% 0.32/0.55 thf(fact_3216_mult__commute__abs,axiom,
% 0.32/0.55 ! [C: real] :
% 0.32/0.55 ( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
% 0.32/0.55 = ( times_times_real @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_commute_abs
% 0.32/0.55 thf(fact_3217_mult__commute__abs,axiom,
% 0.32/0.55 ! [C: rat] :
% 0.32/0.55 ( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
% 0.32/0.55 = ( times_times_rat @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_commute_abs
% 0.32/0.55 thf(fact_3218_mult__commute__abs,axiom,
% 0.32/0.55 ! [C: nat] :
% 0.32/0.55 ( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
% 0.32/0.55 = ( times_times_nat @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_commute_abs
% 0.32/0.55 thf(fact_3219_mult__commute__abs,axiom,
% 0.32/0.55 ! [C: int] :
% 0.32/0.55 ( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
% 0.32/0.55 = ( times_times_int @ C ) ) ).
% 0.32/0.55
% 0.32/0.55 % mult_commute_abs
% 0.32/0.55 thf(fact_3220_lambda__zero,axiom,
% 0.32/0.55 ( ( ^ [H: complex] : zero_zero_complex )
% 0.32/0.55 = ( times_times_complex @ zero_zero_complex ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_zero
% 0.32/0.55 thf(fact_3221_lambda__zero,axiom,
% 0.32/0.55 ( ( ^ [H: real] : zero_zero_real )
% 0.32/0.55 = ( times_times_real @ zero_zero_real ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_zero
% 0.32/0.55 thf(fact_3222_lambda__zero,axiom,
% 0.32/0.55 ( ( ^ [H: rat] : zero_zero_rat )
% 0.32/0.55 = ( times_times_rat @ zero_zero_rat ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_zero
% 0.32/0.55 thf(fact_3223_lambda__zero,axiom,
% 0.32/0.55 ( ( ^ [H: nat] : zero_zero_nat )
% 0.32/0.55 = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_zero
% 0.32/0.55 thf(fact_3224_lambda__zero,axiom,
% 0.32/0.55 ( ( ^ [H: int] : zero_zero_int )
% 0.32/0.55 = ( times_times_int @ zero_zero_int ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_zero
% 0.32/0.55 thf(fact_3225_lambda__one,axiom,
% 0.32/0.55 ( ( ^ [X2: complex] : X2 )
% 0.32/0.55 = ( times_times_complex @ one_one_complex ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_one
% 0.32/0.55 thf(fact_3226_lambda__one,axiom,
% 0.32/0.55 ( ( ^ [X2: real] : X2 )
% 0.32/0.55 = ( times_times_real @ one_one_real ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_one
% 0.32/0.55 thf(fact_3227_lambda__one,axiom,
% 0.32/0.55 ( ( ^ [X2: rat] : X2 )
% 0.32/0.55 = ( times_times_rat @ one_one_rat ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_one
% 0.32/0.55 thf(fact_3228_lambda__one,axiom,
% 0.32/0.55 ( ( ^ [X2: nat] : X2 )
% 0.32/0.55 = ( times_times_nat @ one_one_nat ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_one
% 0.32/0.55 thf(fact_3229_lambda__one,axiom,
% 0.32/0.55 ( ( ^ [X2: int] : X2 )
% 0.32/0.55 = ( times_times_int @ one_one_int ) ) ).
% 0.32/0.55
% 0.32/0.55 % lambda_one
% 0.32/0.55 thf(fact_3230_finite__M__bounded__by__nat,axiom,
% 0.32/0.55 ! [P: nat > $o,I: nat] :
% 0.32/0.55 ( finite_finite_nat
% 0.32/0.55 @ ( collect_nat
% 0.32/0.55 @ ^ [K2: nat] :
% 0.32/0.55 ( ( P @ K2 )
% 0.32/0.55 & ( ord_less_nat @ K2 @ I ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_M_bounded_by_nat
% 0.32/0.55 thf(fact_3231_finite__less__ub,axiom,
% 0.32/0.55 ! [F: nat > nat,U: nat] :
% 0.32/0.55 ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
% 0.32/0.55 => ( finite_finite_nat
% 0.32/0.55 @ ( collect_nat
% 0.32/0.55 @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_less_ub
% 0.32/0.55 thf(fact_3232_numeral__code_I2_J,axiom,
% 0.32/0.55 ! [N: num] :
% 0.32/0.55 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 0.32/0.55 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % numeral_code(2)
% 0.32/0.55 thf(fact_3233_numeral__code_I2_J,axiom,
% 0.32/0.55 ! [N: num] :
% 0.32/0.55 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 0.32/0.55 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % numeral_code(2)
% 0.32/0.55 thf(fact_3234_numeral__code_I2_J,axiom,
% 0.32/0.55 ! [N: num] :
% 0.32/0.55 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 0.32/0.55 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % numeral_code(2)
% 0.32/0.55 thf(fact_3235_numeral__code_I2_J,axiom,
% 0.32/0.55 ! [N: num] :
% 0.32/0.55 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.32/0.55 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % numeral_code(2)
% 0.32/0.55 thf(fact_3236_numeral__code_I2_J,axiom,
% 0.32/0.55 ! [N: num] :
% 0.32/0.55 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 0.32/0.55 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % numeral_code(2)
% 0.32/0.55 thf(fact_3237_set__vebt__def,axiom,
% 0.32/0.55 ( vEBT_set_vebt
% 0.32/0.55 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % set_vebt_def
% 0.32/0.55 thf(fact_3238_unset__bit__less__eq,axiom,
% 0.32/0.55 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 0.32/0.55
% 0.32/0.55 % unset_bit_less_eq
% 0.32/0.55 thf(fact_3239_power__numeral__even,axiom,
% 0.32/0.55 ! [Z: complex,W: num] :
% 0.32/0.55 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.32/0.55 = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_numeral_even
% 0.32/0.55 thf(fact_3240_power__numeral__even,axiom,
% 0.32/0.55 ! [Z: real,W: num] :
% 0.32/0.55 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.32/0.55 = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_numeral_even
% 0.32/0.55 thf(fact_3241_power__numeral__even,axiom,
% 0.32/0.55 ! [Z: rat,W: num] :
% 0.32/0.55 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.32/0.55 = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_numeral_even
% 0.32/0.55 thf(fact_3242_power__numeral__even,axiom,
% 0.32/0.55 ! [Z: nat,W: num] :
% 0.32/0.55 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.32/0.55 = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_numeral_even
% 0.32/0.55 thf(fact_3243_power__numeral__even,axiom,
% 0.32/0.55 ! [Z: int,W: num] :
% 0.32/0.55 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 0.32/0.55 = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % power_numeral_even
% 0.32/0.55 thf(fact_3244_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.32/0.55 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 0.32/0.55 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
% 0.32/0.55 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.32/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.naive_member.simps(3)
% 0.32/0.55 thf(fact_3245_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.32/0.55 ! [A: $o,B: $o,X: nat] :
% 0.32/0.55 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 0.32/0.55 = ( ( ( X = zero_zero_nat )
% 0.32/0.55 => A )
% 0.32/0.55 & ( ( X != zero_zero_nat )
% 0.32/0.55 => ( ( ( X = one_one_nat )
% 0.32/0.55 => B )
% 0.32/0.55 & ( X = one_one_nat ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.naive_member.simps(1)
% 0.32/0.55 thf(fact_3246_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.55 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.32/0.55 => ( ! [A5: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ( ( ( Xa = zero_zero_nat )
% 0.32/0.55 => A5 )
% 0.32/0.55 & ( ( Xa != zero_zero_nat )
% 0.32/0.55 => ( ( ( Xa = one_one_nat )
% 0.32/0.55 => B3 )
% 0.32/0.55 & ( Xa = one_one_nat ) ) ) ) )
% 0.32/0.55 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.32/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.55 ( ? [S3: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.32/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.naive_member.elims(3)
% 0.32/0.55 thf(fact_3247_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.55 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.32/0.55 => ( ! [A5: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ~ ( ( ( Xa = zero_zero_nat )
% 0.32/0.55 => A5 )
% 0.32/0.55 & ( ( Xa != zero_zero_nat )
% 0.32/0.55 => ( ( ( Xa = one_one_nat )
% 0.32/0.55 => B3 )
% 0.32/0.55 & ( Xa = one_one_nat ) ) ) ) )
% 0.32/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.55 ( ? [S3: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.32/0.55 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.naive_member.elims(2)
% 0.32/0.55 thf(fact_3248_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.32/0.55 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.32/0.55 = Y3 )
% 0.32/0.55 => ( ! [A5: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( ~ ( ( ( Xa = zero_zero_nat )
% 0.32/0.55 => A5 )
% 0.32/0.55 & ( ( Xa != zero_zero_nat )
% 0.32/0.55 => ( ( ( Xa = one_one_nat )
% 0.32/0.55 => B3 )
% 0.32/0.55 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 0.32/0.55 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.32/0.55 => Y3 )
% 0.32/0.55 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.55 ( ? [S3: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % VEBT_internal.naive_member.elims(1)
% 0.32/0.55 thf(fact_3249_vebt__member_Osimps_I5_J,axiom,
% 0.32/0.55 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.32/0.55 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.32/0.55 = ( ( X != Mi )
% 0.32/0.55 => ( ( X != Ma )
% 0.32/0.55 => ( ~ ( ord_less_nat @ X @ Mi )
% 0.32/0.55 & ( ~ ( ord_less_nat @ X @ Mi )
% 0.32/0.55 => ( ~ ( ord_less_nat @ Ma @ X )
% 0.32/0.55 & ( ~ ( ord_less_nat @ Ma @ X )
% 0.32/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_member.simps(5)
% 0.32/0.55 thf(fact_3250_finite__has__minimal2,axiom,
% 0.32/0.55 ! [A2: set_real,A: real] :
% 0.32/0.55 ( ( finite_finite_real @ A2 )
% 0.32/0.55 => ( ( member_real @ A @ A2 )
% 0.32/0.55 => ? [X3: real] :
% 0.32/0.55 ( ( member_real @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_real @ X3 @ A )
% 0.32/0.55 & ! [Xa2: real] :
% 0.32/0.55 ( ( member_real @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_real @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal2
% 0.32/0.55 thf(fact_3251_finite__has__minimal2,axiom,
% 0.32/0.55 ! [A2: set_set_int,A: set_int] :
% 0.32/0.55 ( ( finite6197958912794628473et_int @ A2 )
% 0.32/0.55 => ( ( member_set_int @ A @ A2 )
% 0.32/0.55 => ? [X3: set_int] :
% 0.32/0.55 ( ( member_set_int @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_set_int @ X3 @ A )
% 0.32/0.55 & ! [Xa2: set_int] :
% 0.32/0.55 ( ( member_set_int @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal2
% 0.32/0.55 thf(fact_3252_finite__has__minimal2,axiom,
% 0.32/0.55 ! [A2: set_rat,A: rat] :
% 0.32/0.55 ( ( finite_finite_rat @ A2 )
% 0.32/0.55 => ( ( member_rat @ A @ A2 )
% 0.32/0.55 => ? [X3: rat] :
% 0.32/0.55 ( ( member_rat @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_rat @ X3 @ A )
% 0.32/0.55 & ! [Xa2: rat] :
% 0.32/0.55 ( ( member_rat @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_rat @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal2
% 0.32/0.55 thf(fact_3253_finite__has__minimal2,axiom,
% 0.32/0.55 ! [A2: set_num,A: num] :
% 0.32/0.55 ( ( finite_finite_num @ A2 )
% 0.32/0.55 => ( ( member_num @ A @ A2 )
% 0.32/0.55 => ? [X3: num] :
% 0.32/0.55 ( ( member_num @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_num @ X3 @ A )
% 0.32/0.55 & ! [Xa2: num] :
% 0.32/0.55 ( ( member_num @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_num @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal2
% 0.32/0.55 thf(fact_3254_finite__has__minimal2,axiom,
% 0.32/0.55 ! [A2: set_nat,A: nat] :
% 0.32/0.55 ( ( finite_finite_nat @ A2 )
% 0.32/0.55 => ( ( member_nat @ A @ A2 )
% 0.32/0.55 => ? [X3: nat] :
% 0.32/0.55 ( ( member_nat @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_nat @ X3 @ A )
% 0.32/0.55 & ! [Xa2: nat] :
% 0.32/0.55 ( ( member_nat @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal2
% 0.32/0.55 thf(fact_3255_finite__has__minimal2,axiom,
% 0.32/0.55 ! [A2: set_int,A: int] :
% 0.32/0.55 ( ( finite_finite_int @ A2 )
% 0.32/0.55 => ( ( member_int @ A @ A2 )
% 0.32/0.55 => ? [X3: int] :
% 0.32/0.55 ( ( member_int @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_int @ X3 @ A )
% 0.32/0.55 & ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_int @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal2
% 0.32/0.55 thf(fact_3256_finite__has__maximal2,axiom,
% 0.32/0.55 ! [A2: set_real,A: real] :
% 0.32/0.55 ( ( finite_finite_real @ A2 )
% 0.32/0.55 => ( ( member_real @ A @ A2 )
% 0.32/0.55 => ? [X3: real] :
% 0.32/0.55 ( ( member_real @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_real @ A @ X3 )
% 0.32/0.55 & ! [Xa2: real] :
% 0.32/0.55 ( ( member_real @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_real @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal2
% 0.32/0.55 thf(fact_3257_finite__has__maximal2,axiom,
% 0.32/0.55 ! [A2: set_set_int,A: set_int] :
% 0.32/0.55 ( ( finite6197958912794628473et_int @ A2 )
% 0.32/0.55 => ( ( member_set_int @ A @ A2 )
% 0.32/0.55 => ? [X3: set_int] :
% 0.32/0.55 ( ( member_set_int @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_set_int @ A @ X3 )
% 0.32/0.55 & ! [Xa2: set_int] :
% 0.32/0.55 ( ( member_set_int @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal2
% 0.32/0.55 thf(fact_3258_finite__has__maximal2,axiom,
% 0.32/0.55 ! [A2: set_rat,A: rat] :
% 0.32/0.55 ( ( finite_finite_rat @ A2 )
% 0.32/0.55 => ( ( member_rat @ A @ A2 )
% 0.32/0.55 => ? [X3: rat] :
% 0.32/0.55 ( ( member_rat @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_rat @ A @ X3 )
% 0.32/0.55 & ! [Xa2: rat] :
% 0.32/0.55 ( ( member_rat @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_rat @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal2
% 0.32/0.55 thf(fact_3259_finite__has__maximal2,axiom,
% 0.32/0.55 ! [A2: set_num,A: num] :
% 0.32/0.55 ( ( finite_finite_num @ A2 )
% 0.32/0.55 => ( ( member_num @ A @ A2 )
% 0.32/0.55 => ? [X3: num] :
% 0.32/0.55 ( ( member_num @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_num @ A @ X3 )
% 0.32/0.55 & ! [Xa2: num] :
% 0.32/0.55 ( ( member_num @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_num @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal2
% 0.32/0.55 thf(fact_3260_finite__has__maximal2,axiom,
% 0.32/0.55 ! [A2: set_nat,A: nat] :
% 0.32/0.55 ( ( finite_finite_nat @ A2 )
% 0.32/0.55 => ( ( member_nat @ A @ A2 )
% 0.32/0.55 => ? [X3: nat] :
% 0.32/0.55 ( ( member_nat @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_nat @ A @ X3 )
% 0.32/0.55 & ! [Xa2: nat] :
% 0.32/0.55 ( ( member_nat @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_nat @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal2
% 0.32/0.55 thf(fact_3261_finite__has__maximal2,axiom,
% 0.32/0.55 ! [A2: set_int,A: int] :
% 0.32/0.55 ( ( finite_finite_int @ A2 )
% 0.32/0.55 => ( ( member_int @ A @ A2 )
% 0.32/0.55 => ? [X3: int] :
% 0.32/0.55 ( ( member_int @ X3 @ A2 )
% 0.32/0.55 & ( ord_less_eq_int @ A @ X3 )
% 0.32/0.55 & ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_int @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal2
% 0.32/0.55 thf(fact_3262_rev__finite__subset,axiom,
% 0.32/0.55 ! [B5: set_nat,A2: set_nat] :
% 0.32/0.55 ( ( finite_finite_nat @ B5 )
% 0.32/0.55 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.55 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % rev_finite_subset
% 0.32/0.55 thf(fact_3263_rev__finite__subset,axiom,
% 0.32/0.55 ! [B5: set_complex,A2: set_complex] :
% 0.32/0.55 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.55 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.55 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % rev_finite_subset
% 0.32/0.55 thf(fact_3264_rev__finite__subset,axiom,
% 0.32/0.55 ! [B5: set_int,A2: set_int] :
% 0.32/0.55 ( ( finite_finite_int @ B5 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( finite_finite_int @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % rev_finite_subset
% 0.32/0.55 thf(fact_3265_infinite__super,axiom,
% 0.32/0.55 ! [S2: set_nat,T3: set_nat] :
% 0.32/0.55 ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.55 => ( ~ ( finite_finite_nat @ S2 )
% 0.32/0.55 => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % infinite_super
% 0.32/0.55 thf(fact_3266_infinite__super,axiom,
% 0.32/0.55 ! [S2: set_complex,T3: set_complex] :
% 0.32/0.55 ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.55 => ( ~ ( finite3207457112153483333omplex @ S2 )
% 0.32/0.55 => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % infinite_super
% 0.32/0.55 thf(fact_3267_infinite__super,axiom,
% 0.32/0.55 ! [S2: set_int,T3: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.32/0.55 => ( ~ ( finite_finite_int @ S2 )
% 0.32/0.55 => ~ ( finite_finite_int @ T3 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % infinite_super
% 0.32/0.55 thf(fact_3268_finite__subset,axiom,
% 0.32/0.55 ! [A2: set_nat,B5: set_nat] :
% 0.32/0.55 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.55 => ( ( finite_finite_nat @ B5 )
% 0.32/0.55 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_subset
% 0.32/0.55 thf(fact_3269_finite__subset,axiom,
% 0.32/0.55 ! [A2: set_complex,B5: set_complex] :
% 0.32/0.55 ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.55 => ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.55 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_subset
% 0.32/0.55 thf(fact_3270_finite__subset,axiom,
% 0.32/0.55 ! [A2: set_int,B5: set_int] :
% 0.32/0.55 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.55 => ( ( finite_finite_int @ B5 )
% 0.32/0.55 => ( finite_finite_int @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_subset
% 0.32/0.55 thf(fact_3271_Diff__infinite__finite,axiom,
% 0.32/0.55 ! [T3: set_int,S2: set_int] :
% 0.32/0.55 ( ( finite_finite_int @ T3 )
% 0.32/0.55 => ( ~ ( finite_finite_int @ S2 )
% 0.32/0.55 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T3 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_infinite_finite
% 0.32/0.55 thf(fact_3272_Diff__infinite__finite,axiom,
% 0.32/0.55 ! [T3: set_complex,S2: set_complex] :
% 0.32/0.55 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.55 => ( ~ ( finite3207457112153483333omplex @ S2 )
% 0.32/0.55 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T3 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_infinite_finite
% 0.32/0.55 thf(fact_3273_Diff__infinite__finite,axiom,
% 0.32/0.55 ! [T3: set_nat,S2: set_nat] :
% 0.32/0.55 ( ( finite_finite_nat @ T3 )
% 0.32/0.55 => ( ~ ( finite_finite_nat @ S2 )
% 0.32/0.55 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % Diff_infinite_finite
% 0.32/0.55 thf(fact_3274_finite__psubset__induct,axiom,
% 0.32/0.55 ! [A2: set_nat,P: set_nat > $o] :
% 0.32/0.55 ( ( finite_finite_nat @ A2 )
% 0.32/0.55 => ( ! [A7: set_nat] :
% 0.32/0.55 ( ( finite_finite_nat @ A7 )
% 0.32/0.55 => ( ! [B7: set_nat] :
% 0.32/0.55 ( ( ord_less_set_nat @ B7 @ A7 )
% 0.32/0.55 => ( P @ B7 ) )
% 0.32/0.55 => ( P @ A7 ) ) )
% 0.32/0.55 => ( P @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_psubset_induct
% 0.32/0.55 thf(fact_3275_finite__psubset__induct,axiom,
% 0.32/0.55 ! [A2: set_int,P: set_int > $o] :
% 0.32/0.55 ( ( finite_finite_int @ A2 )
% 0.32/0.55 => ( ! [A7: set_int] :
% 0.32/0.55 ( ( finite_finite_int @ A7 )
% 0.32/0.55 => ( ! [B7: set_int] :
% 0.32/0.55 ( ( ord_less_set_int @ B7 @ A7 )
% 0.32/0.55 => ( P @ B7 ) )
% 0.32/0.55 => ( P @ A7 ) ) )
% 0.32/0.55 => ( P @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_psubset_induct
% 0.32/0.55 thf(fact_3276_finite__psubset__induct,axiom,
% 0.32/0.55 ! [A2: set_complex,P: set_complex > $o] :
% 0.32/0.55 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.55 => ( ! [A7: set_complex] :
% 0.32/0.55 ( ( finite3207457112153483333omplex @ A7 )
% 0.32/0.55 => ( ! [B7: set_complex] :
% 0.32/0.55 ( ( ord_less_set_complex @ B7 @ A7 )
% 0.32/0.55 => ( P @ B7 ) )
% 0.32/0.55 => ( P @ A7 ) ) )
% 0.32/0.55 => ( P @ A2 ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_psubset_induct
% 0.32/0.55 thf(fact_3277_vebt__member_Oelims_I2_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.55 ( ( vEBT_vebt_member @ X @ Xa )
% 0.32/0.55 => ( ! [A5: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ~ ( ( ( Xa = zero_zero_nat )
% 0.32/0.55 => A5 )
% 0.32/0.55 & ( ( Xa != zero_zero_nat )
% 0.32/0.55 => ( ( ( Xa = one_one_nat )
% 0.32/0.55 => B3 )
% 0.32/0.55 & ( Xa = one_one_nat ) ) ) ) )
% 0.32/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.55 ( ? [Summary2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.55 => ~ ( ( Xa != Mi2 )
% 0.32/0.55 => ( ( Xa != Ma2 )
% 0.32/0.55 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.55 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_member.elims(2)
% 0.32/0.55 thf(fact_3278_vebt__member_Oelims_I3_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.55 ( ~ ( vEBT_vebt_member @ X @ Xa )
% 0.32/0.55 => ( ! [A5: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ( ( ( Xa = zero_zero_nat )
% 0.32/0.55 => A5 )
% 0.32/0.55 & ( ( Xa != zero_zero_nat )
% 0.32/0.55 => ( ( ( Xa = one_one_nat )
% 0.32/0.55 => B3 )
% 0.32/0.55 & ( Xa = one_one_nat ) ) ) ) )
% 0.32/0.55 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.55 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.32/0.55 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.32/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.55 ( ? [Summary2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.55 => ( ( Xa != Mi2 )
% 0.32/0.55 => ( ( Xa != Ma2 )
% 0.32/0.55 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.55 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_member.elims(3)
% 0.32/0.55 thf(fact_3279_vebt__member_Oelims_I1_J,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.32/0.55 ( ( ( vEBT_vebt_member @ X @ Xa )
% 0.32/0.55 = Y3 )
% 0.32/0.55 => ( ! [A5: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( ~ ( ( ( Xa = zero_zero_nat )
% 0.32/0.55 => A5 )
% 0.32/0.55 & ( ( Xa != zero_zero_nat )
% 0.32/0.55 => ( ( ( Xa = one_one_nat )
% 0.32/0.55 => B3 )
% 0.32/0.55 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 0.32/0.55 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.55 => Y3 )
% 0.32/0.55 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.32/0.55 => Y3 )
% 0.32/0.55 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.32/0.55 => Y3 )
% 0.32/0.55 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.55 ( ? [Summary2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( ~ ( ( Xa != Mi2 )
% 0.32/0.55 => ( ( Xa != Ma2 )
% 0.32/0.55 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.55 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.55 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.55 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.55 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.55 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_member.elims(1)
% 0.32/0.55 thf(fact_3280_finite__has__minimal,axiom,
% 0.32/0.55 ! [A2: set_real] :
% 0.32/0.55 ( ( finite_finite_real @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_real )
% 0.32/0.55 => ? [X3: real] :
% 0.32/0.55 ( ( member_real @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: real] :
% 0.32/0.55 ( ( member_real @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_real @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal
% 0.32/0.55 thf(fact_3281_finite__has__minimal,axiom,
% 0.32/0.55 ! [A2: set_set_int] :
% 0.32/0.55 ( ( finite6197958912794628473et_int @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_set_int )
% 0.32/0.55 => ? [X3: set_int] :
% 0.32/0.55 ( ( member_set_int @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: set_int] :
% 0.32/0.55 ( ( member_set_int @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal
% 0.32/0.55 thf(fact_3282_finite__has__minimal,axiom,
% 0.32/0.55 ! [A2: set_rat] :
% 0.32/0.55 ( ( finite_finite_rat @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_rat )
% 0.32/0.55 => ? [X3: rat] :
% 0.32/0.55 ( ( member_rat @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: rat] :
% 0.32/0.55 ( ( member_rat @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_rat @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal
% 0.32/0.55 thf(fact_3283_finite__has__minimal,axiom,
% 0.32/0.55 ! [A2: set_num] :
% 0.32/0.55 ( ( finite_finite_num @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_num )
% 0.32/0.55 => ? [X3: num] :
% 0.32/0.55 ( ( member_num @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: num] :
% 0.32/0.55 ( ( member_num @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_num @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal
% 0.32/0.55 thf(fact_3284_finite__has__minimal,axiom,
% 0.32/0.55 ! [A2: set_nat] :
% 0.32/0.55 ( ( finite_finite_nat @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_nat )
% 0.32/0.55 => ? [X3: nat] :
% 0.32/0.55 ( ( member_nat @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: nat] :
% 0.32/0.55 ( ( member_nat @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal
% 0.32/0.55 thf(fact_3285_finite__has__minimal,axiom,
% 0.32/0.55 ! [A2: set_int] :
% 0.32/0.55 ( ( finite_finite_int @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_int )
% 0.32/0.55 => ? [X3: int] :
% 0.32/0.55 ( ( member_int @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_int @ Xa2 @ X3 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_minimal
% 0.32/0.55 thf(fact_3286_finite__has__maximal,axiom,
% 0.32/0.55 ! [A2: set_real] :
% 0.32/0.55 ( ( finite_finite_real @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_real )
% 0.32/0.55 => ? [X3: real] :
% 0.32/0.55 ( ( member_real @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: real] :
% 0.32/0.55 ( ( member_real @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_real @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal
% 0.32/0.55 thf(fact_3287_finite__has__maximal,axiom,
% 0.32/0.55 ! [A2: set_set_int] :
% 0.32/0.55 ( ( finite6197958912794628473et_int @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_set_int )
% 0.32/0.55 => ? [X3: set_int] :
% 0.32/0.55 ( ( member_set_int @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: set_int] :
% 0.32/0.55 ( ( member_set_int @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal
% 0.32/0.55 thf(fact_3288_finite__has__maximal,axiom,
% 0.32/0.55 ! [A2: set_rat] :
% 0.32/0.55 ( ( finite_finite_rat @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_rat )
% 0.32/0.55 => ? [X3: rat] :
% 0.32/0.55 ( ( member_rat @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: rat] :
% 0.32/0.55 ( ( member_rat @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_rat @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal
% 0.32/0.55 thf(fact_3289_finite__has__maximal,axiom,
% 0.32/0.55 ! [A2: set_num] :
% 0.32/0.55 ( ( finite_finite_num @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_num )
% 0.32/0.55 => ? [X3: num] :
% 0.32/0.55 ( ( member_num @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: num] :
% 0.32/0.55 ( ( member_num @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_num @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal
% 0.32/0.55 thf(fact_3290_finite__has__maximal,axiom,
% 0.32/0.55 ! [A2: set_nat] :
% 0.32/0.55 ( ( finite_finite_nat @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_nat )
% 0.32/0.55 => ? [X3: nat] :
% 0.32/0.55 ( ( member_nat @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: nat] :
% 0.32/0.55 ( ( member_nat @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_nat @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal
% 0.32/0.55 thf(fact_3291_finite__has__maximal,axiom,
% 0.32/0.55 ! [A2: set_int] :
% 0.32/0.55 ( ( finite_finite_int @ A2 )
% 0.32/0.55 => ( ( A2 != bot_bot_set_int )
% 0.32/0.55 => ? [X3: int] :
% 0.32/0.55 ( ( member_int @ X3 @ A2 )
% 0.32/0.55 & ! [Xa2: int] :
% 0.32/0.55 ( ( member_int @ Xa2 @ A2 )
% 0.32/0.55 => ( ( ord_less_eq_int @ X3 @ Xa2 )
% 0.32/0.55 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % finite_has_maximal
% 0.32/0.55 thf(fact_3292_vebt__succ_Osimps_I6_J,axiom,
% 0.32/0.55 ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.32/0.55 ( ( ( ord_less_nat @ X @ Mi )
% 0.32/0.55 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.32/0.55 = ( some_nat @ Mi ) ) )
% 0.32/0.55 & ( ~ ( ord_less_nat @ X @ Mi )
% 0.32/0.55 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.32/0.55 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.55 @ ( if_option_nat
% 0.32/0.55 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 != none_nat )
% 0.32/0.55 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.55 @ ( if_option_nat
% 0.32/0.55 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.55 = none_nat )
% 0.32/0.55 @ none_nat
% 0.32/0.55 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.55 @ none_nat ) ) ) ) ).
% 0.32/0.55
% 0.32/0.55 % vebt_succ.simps(6)
% 0.32/0.55 thf(fact_3293_vebt__succ_Oelims,axiom,
% 0.32/0.55 ! [X: vEBT_VEBT,Xa: nat,Y3: option_nat] :
% 0.32/0.55 ( ( ( vEBT_vebt_succ @ X @ Xa )
% 0.32/0.55 = Y3 )
% 0.32/0.55 => ( ! [Uu2: $o,B3: $o] :
% 0.32/0.55 ( ( X
% 0.32/0.55 = ( vEBT_Leaf @ Uu2 @ B3 ) )
% 0.32/0.55 => ( ( Xa = zero_zero_nat )
% 0.32/0.55 => ~ ( ( B3
% 0.32/0.55 => ( Y3
% 0.32/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.55 & ( ~ B3
% 0.32/0.55 => ( Y3 = none_nat ) ) ) ) )
% 0.32/0.55 => ( ( ? [Uv2: $o,Uw2: $o] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.32/0.55 => ( ? [N4: nat] :
% 0.32/0.55 ( Xa
% 0.32/0.55 = ( suc @ N4 ) )
% 0.32/0.55 => ( Y3 != none_nat ) ) )
% 0.32/0.55 => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.55 => ( Y3 != none_nat ) )
% 0.32/0.55 => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.32/0.55 => ( Y3 != none_nat ) )
% 0.32/0.55 => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 0.32/0.55 ( X
% 0.32/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 0.32/0.55 => ( Y3 != none_nat ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 => ( Y3
% 0.32/0.56 = ( some_nat @ Mi2 ) ) )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 => ( Y3
% 0.32/0.56 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 @ ( if_option_nat
% 0.32/0.56 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 != none_nat )
% 0.32/0.56 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 @ ( if_option_nat
% 0.32/0.56 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = none_nat )
% 0.32/0.56 @ none_nat
% 0.32/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_succ.elims
% 0.32/0.56 thf(fact_3294_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.32/0.56 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.32/0.56 = Y3 )
% 0.32/0.56 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.56 => Y3 )
% 0.32/0.56 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.32/0.56 => Y3 )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.32/0.56 => ( Y3
% 0.32/0.56 = ( ~ ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 ) ) ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ( Y3
% 0.32/0.56 = ( ~ ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ( Y3
% 0.32/0.56 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.elims(1)
% 0.32/0.56 thf(fact_3295_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.56 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 0.32/0.56 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.32/0.56 => ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.elims(3)
% 0.32/0.56 thf(fact_3296_finite__nth__roots,axiom,
% 0.32/0.56 ! [N: nat,C: complex] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [Z2: complex] :
% 0.32/0.56 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.56 = C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_nth_roots
% 0.32/0.56 thf(fact_3297_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.56 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat] :
% 0.32/0.56 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.32/0.56 => ~ ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vc2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ~ ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.32/0.56 ( ? [Vd2: vEBT_VEBT] :
% 0.32/0.56 ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.elims(2)
% 0.32/0.56 thf(fact_3298_member__valid__both__member__options,axiom,
% 0.32/0.56 ! [Tree: vEBT_VEBT,N: nat,X: nat] :
% 0.32/0.56 ( ( vEBT_invar_vebt @ Tree @ N )
% 0.32/0.56 => ( ( vEBT_vebt_member @ Tree @ X )
% 0.32/0.56 => ( ( vEBT_V5719532721284313246member @ Tree @ X )
% 0.32/0.56 | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % member_valid_both_member_options
% 0.32/0.56 thf(fact_3299_set__conv__nth,axiom,
% 0.32/0.56 ( set_complex2
% 0.32/0.56 = ( ^ [Xs: list_complex] :
% 0.32/0.56 ( collect_complex
% 0.32/0.56 @ ^ [Uu3: complex] :
% 0.32/0.56 ? [I4: nat] :
% 0.32/0.56 ( ( Uu3
% 0.32/0.56 = ( nth_complex @ Xs @ I4 ) )
% 0.32/0.56 & ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_conv_nth
% 0.32/0.56 thf(fact_3300_set__conv__nth,axiom,
% 0.32/0.56 ( set_real2
% 0.32/0.56 = ( ^ [Xs: list_real] :
% 0.32/0.56 ( collect_real
% 0.32/0.56 @ ^ [Uu3: real] :
% 0.32/0.56 ? [I4: nat] :
% 0.32/0.56 ( ( Uu3
% 0.32/0.56 = ( nth_real @ Xs @ I4 ) )
% 0.32/0.56 & ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_conv_nth
% 0.32/0.56 thf(fact_3301_set__conv__nth,axiom,
% 0.32/0.56 ( set_list_nat2
% 0.32/0.56 = ( ^ [Xs: list_list_nat] :
% 0.32/0.56 ( collect_list_nat
% 0.32/0.56 @ ^ [Uu3: list_nat] :
% 0.32/0.56 ? [I4: nat] :
% 0.32/0.56 ( ( Uu3
% 0.32/0.56 = ( nth_list_nat @ Xs @ I4 ) )
% 0.32/0.56 & ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_conv_nth
% 0.32/0.56 thf(fact_3302_set__conv__nth,axiom,
% 0.32/0.56 ( set_VEBT_VEBT2
% 0.32/0.56 = ( ^ [Xs: list_VEBT_VEBT] :
% 0.32/0.56 ( collect_VEBT_VEBT
% 0.32/0.56 @ ^ [Uu3: vEBT_VEBT] :
% 0.32/0.56 ? [I4: nat] :
% 0.32/0.56 ( ( Uu3
% 0.32/0.56 = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.32/0.56 & ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_conv_nth
% 0.32/0.56 thf(fact_3303_set__conv__nth,axiom,
% 0.32/0.56 ( set_o2
% 0.32/0.56 = ( ^ [Xs: list_o] :
% 0.32/0.56 ( collect_o
% 0.32/0.56 @ ^ [Uu3: $o] :
% 0.32/0.56 ? [I4: nat] :
% 0.32/0.56 ( ( Uu3
% 0.32/0.56 = ( nth_o @ Xs @ I4 ) )
% 0.32/0.56 & ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_conv_nth
% 0.32/0.56 thf(fact_3304_set__conv__nth,axiom,
% 0.32/0.56 ( set_nat2
% 0.32/0.56 = ( ^ [Xs: list_nat] :
% 0.32/0.56 ( collect_nat
% 0.32/0.56 @ ^ [Uu3: nat] :
% 0.32/0.56 ? [I4: nat] :
% 0.32/0.56 ( ( Uu3
% 0.32/0.56 = ( nth_nat @ Xs @ I4 ) )
% 0.32/0.56 & ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_conv_nth
% 0.32/0.56 thf(fact_3305_set__conv__nth,axiom,
% 0.32/0.56 ( set_int2
% 0.32/0.56 = ( ^ [Xs: list_int] :
% 0.32/0.56 ( collect_int
% 0.32/0.56 @ ^ [Uu3: int] :
% 0.32/0.56 ? [I4: nat] :
% 0.32/0.56 ( ( Uu3
% 0.32/0.56 = ( nth_int @ Xs @ I4 ) )
% 0.32/0.56 & ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % set_conv_nth
% 0.32/0.56 thf(fact_3306_finite__lists__length__le,axiom,
% 0.32/0.56 ! [A2: set_complex,N: nat] :
% 0.32/0.56 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.56 => ( finite8712137658972009173omplex
% 0.32/0.56 @ ( collect_list_complex
% 0.32/0.56 @ ^ [Xs: list_complex] :
% 0.32/0.56 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_le
% 0.32/0.56 thf(fact_3307_finite__lists__length__le,axiom,
% 0.32/0.56 ! [A2: set_VEBT_VEBT,N: nat] :
% 0.32/0.56 ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.32/0.56 => ( finite3004134309566078307T_VEBT
% 0.32/0.56 @ ( collec5608196760682091941T_VEBT
% 0.32/0.56 @ ^ [Xs: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_le
% 0.32/0.56 thf(fact_3308_finite__lists__length__le,axiom,
% 0.32/0.56 ! [A2: set_o,N: nat] :
% 0.32/0.56 ( ( finite_finite_o @ A2 )
% 0.32/0.56 => ( finite_finite_list_o
% 0.32/0.56 @ ( collect_list_o
% 0.32/0.56 @ ^ [Xs: list_o] :
% 0.32/0.56 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_le
% 0.32/0.56 thf(fact_3309_finite__lists__length__le,axiom,
% 0.32/0.56 ! [A2: set_nat,N: nat] :
% 0.32/0.56 ( ( finite_finite_nat @ A2 )
% 0.32/0.56 => ( finite8100373058378681591st_nat
% 0.32/0.56 @ ( collect_list_nat
% 0.32/0.56 @ ^ [Xs: list_nat] :
% 0.32/0.56 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_le
% 0.32/0.56 thf(fact_3310_finite__lists__length__le,axiom,
% 0.32/0.56 ! [A2: set_int,N: nat] :
% 0.32/0.56 ( ( finite_finite_int @ A2 )
% 0.32/0.56 => ( finite3922522038869484883st_int
% 0.32/0.56 @ ( collect_list_int
% 0.32/0.56 @ ^ [Xs: list_int] :
% 0.32/0.56 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_le
% 0.32/0.56 thf(fact_3311_vebt__succ_Opelims,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat,Y3: option_nat] :
% 0.32/0.56 ( ( ( vEBT_vebt_succ @ X @ Xa )
% 0.32/0.56 = Y3 )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [Uu2: $o,B3: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uu2 @ B3 ) )
% 0.32/0.56 => ( ( Xa = zero_zero_nat )
% 0.32/0.56 => ( ( ( B3
% 0.32/0.56 => ( Y3
% 0.32/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.56 & ( ~ B3
% 0.32/0.56 => ( Y3 = none_nat ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
% 0.32/0.56 => ( ! [Uv2: $o,Uw2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ! [N4: nat] :
% 0.32/0.56 ( ( Xa
% 0.32/0.56 = ( suc @ N4 ) )
% 0.32/0.56 => ( ( Y3 = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
% 0.32/0.56 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.56 => ( ( Y3 = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.32/0.56 => ( ( Y3 = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 0.32/0.56 => ( ( Y3 = none_nat )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 => ( Y3
% 0.32/0.56 = ( some_nat @ Mi2 ) ) )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 => ( Y3
% 0.32/0.56 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 @ ( if_option_nat
% 0.32/0.56 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 != none_nat )
% 0.32/0.56 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.32/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 @ ( if_option_nat
% 0.32/0.56 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.56 = none_nat )
% 0.32/0.56 @ none_nat
% 0.32/0.56 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 @ none_nat ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_succ.pelims
% 0.32/0.56 thf(fact_3312_buildup__nothing__in__min__max,axiom,
% 0.32/0.56 ! [N: nat,X: nat] :
% 0.32/0.56 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 0.32/0.56
% 0.32/0.56 % buildup_nothing_in_min_max
% 0.32/0.56 thf(fact_3313_both__member__options__def,axiom,
% 0.32/0.56 ( vEBT_V8194947554948674370ptions
% 0.32/0.56 = ( ^ [T2: vEBT_VEBT,X2: nat] :
% 0.32/0.56 ( ( vEBT_V5719532721284313246member @ T2 @ X2 )
% 0.32/0.56 | ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % both_member_options_def
% 0.32/0.56 thf(fact_3314_subset__code_I1_J,axiom,
% 0.32/0.56 ! [Xs2: list_nat,B5: set_nat] :
% 0.32/0.56 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
% 0.32/0.56 = ( ! [X2: nat] :
% 0.32/0.56 ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 0.32/0.56 => ( member_nat @ X2 @ B5 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % subset_code(1)
% 0.32/0.56 thf(fact_3315_subset__code_I1_J,axiom,
% 0.32/0.56 ! [Xs2: list_real,B5: set_real] :
% 0.32/0.56 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
% 0.32/0.56 = ( ! [X2: real] :
% 0.32/0.56 ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
% 0.32/0.56 => ( member_real @ X2 @ B5 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % subset_code(1)
% 0.32/0.56 thf(fact_3316_subset__code_I1_J,axiom,
% 0.32/0.56 ! [Xs2: list_complex,B5: set_complex] :
% 0.32/0.56 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B5 )
% 0.32/0.56 = ( ! [X2: complex] :
% 0.32/0.56 ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
% 0.32/0.56 => ( member_complex @ X2 @ B5 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % subset_code(1)
% 0.32/0.56 thf(fact_3317_subset__code_I1_J,axiom,
% 0.32/0.56 ! [Xs2: list_P8469869581646625389at_nat,B5: set_Pr8693737435421807431at_nat] :
% 0.32/0.56 ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs2 ) @ B5 )
% 0.32/0.56 = ( ! [X2: produc859450856879609959at_nat] :
% 0.32/0.56 ( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 0.32/0.56 => ( member8206827879206165904at_nat @ X2 @ B5 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % subset_code(1)
% 0.32/0.56 thf(fact_3318_subset__code_I1_J,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
% 0.32/0.56 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B5 )
% 0.32/0.56 = ( ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.32/0.56 => ( member_VEBT_VEBT @ X2 @ B5 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % subset_code(1)
% 0.32/0.56 thf(fact_3319_subset__code_I1_J,axiom,
% 0.32/0.56 ! [Xs2: list_int,B5: set_int] :
% 0.32/0.56 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B5 )
% 0.32/0.56 = ( ! [X2: int] :
% 0.32/0.56 ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 0.32/0.56 => ( member_int @ X2 @ B5 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % subset_code(1)
% 0.32/0.56 thf(fact_3320_length__induct,axiom,
% 0.32/0.56 ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
% 0.32/0.56 ( ! [Xs3: list_VEBT_VEBT] :
% 0.32/0.56 ( ! [Ys: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.32/0.56 => ( P @ Ys ) )
% 0.32/0.56 => ( P @ Xs3 ) )
% 0.32/0.56 => ( P @ Xs2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_induct
% 0.32/0.56 thf(fact_3321_length__induct,axiom,
% 0.32/0.56 ! [P: list_o > $o,Xs2: list_o] :
% 0.32/0.56 ( ! [Xs3: list_o] :
% 0.32/0.56 ( ! [Ys: list_o] :
% 0.32/0.56 ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
% 0.32/0.56 => ( P @ Ys ) )
% 0.32/0.56 => ( P @ Xs3 ) )
% 0.32/0.56 => ( P @ Xs2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_induct
% 0.32/0.56 thf(fact_3322_length__induct,axiom,
% 0.32/0.56 ! [P: list_nat > $o,Xs2: list_nat] :
% 0.32/0.56 ( ! [Xs3: list_nat] :
% 0.32/0.56 ( ! [Ys: list_nat] :
% 0.32/0.56 ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
% 0.32/0.56 => ( P @ Ys ) )
% 0.32/0.56 => ( P @ Xs3 ) )
% 0.32/0.56 => ( P @ Xs2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_induct
% 0.32/0.56 thf(fact_3323_length__induct,axiom,
% 0.32/0.56 ! [P: list_int > $o,Xs2: list_int] :
% 0.32/0.56 ( ! [Xs3: list_int] :
% 0.32/0.56 ( ! [Ys: list_int] :
% 0.32/0.56 ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs3 ) )
% 0.32/0.56 => ( P @ Ys ) )
% 0.32/0.56 => ( P @ Xs3 ) )
% 0.32/0.56 => ( P @ Xs2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_induct
% 0.32/0.56 thf(fact_3324_finite__maxlen,axiom,
% 0.32/0.56 ! [M5: set_list_VEBT_VEBT] :
% 0.32/0.56 ( ( finite3004134309566078307T_VEBT @ M5 )
% 0.32/0.56 => ? [N4: nat] :
% 0.32/0.56 ! [X5: list_VEBT_VEBT] :
% 0.32/0.56 ( ( member2936631157270082147T_VEBT @ X5 @ M5 )
% 0.32/0.56 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N4 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_maxlen
% 0.32/0.56 thf(fact_3325_finite__maxlen,axiom,
% 0.32/0.56 ! [M5: set_list_o] :
% 0.32/0.56 ( ( finite_finite_list_o @ M5 )
% 0.32/0.56 => ? [N4: nat] :
% 0.32/0.56 ! [X5: list_o] :
% 0.32/0.56 ( ( member_list_o @ X5 @ M5 )
% 0.32/0.56 => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N4 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_maxlen
% 0.32/0.56 thf(fact_3326_finite__maxlen,axiom,
% 0.32/0.56 ! [M5: set_list_nat] :
% 0.32/0.56 ( ( finite8100373058378681591st_nat @ M5 )
% 0.32/0.56 => ? [N4: nat] :
% 0.32/0.56 ! [X5: list_nat] :
% 0.32/0.56 ( ( member_list_nat @ X5 @ M5 )
% 0.32/0.56 => ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N4 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_maxlen
% 0.32/0.56 thf(fact_3327_finite__maxlen,axiom,
% 0.32/0.56 ! [M5: set_list_int] :
% 0.32/0.56 ( ( finite3922522038869484883st_int @ M5 )
% 0.32/0.56 => ? [N4: nat] :
% 0.32/0.56 ! [X5: list_int] :
% 0.32/0.56 ( ( member_list_int @ X5 @ M5 )
% 0.32/0.56 => ( ord_less_nat @ ( size_size_list_int @ X5 ) @ N4 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_maxlen
% 0.32/0.56 thf(fact_3328_list__eq__iff__nth__eq,axiom,
% 0.32/0.56 ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : ( Y6 = Z4 ) )
% 0.32/0.56 = ( ^ [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.32/0.56 = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.32/0.56 & ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.32/0.56 => ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 0.32/0.56 = ( nth_VEBT_VEBT @ Ys2 @ I4 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_eq_iff_nth_eq
% 0.32/0.56 thf(fact_3329_list__eq__iff__nth__eq,axiom,
% 0.32/0.56 ( ( ^ [Y6: list_o,Z4: list_o] : ( Y6 = Z4 ) )
% 0.32/0.56 = ( ^ [Xs: list_o,Ys2: list_o] :
% 0.32/0.56 ( ( ( size_size_list_o @ Xs )
% 0.32/0.56 = ( size_size_list_o @ Ys2 ) )
% 0.32/0.56 & ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.32/0.56 => ( ( nth_o @ Xs @ I4 )
% 0.32/0.56 = ( nth_o @ Ys2 @ I4 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_eq_iff_nth_eq
% 0.32/0.56 thf(fact_3330_list__eq__iff__nth__eq,axiom,
% 0.32/0.56 ( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
% 0.32/0.56 = ( ^ [Xs: list_nat,Ys2: list_nat] :
% 0.32/0.56 ( ( ( size_size_list_nat @ Xs )
% 0.32/0.56 = ( size_size_list_nat @ Ys2 ) )
% 0.32/0.56 & ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.32/0.56 => ( ( nth_nat @ Xs @ I4 )
% 0.32/0.56 = ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_eq_iff_nth_eq
% 0.32/0.56 thf(fact_3331_list__eq__iff__nth__eq,axiom,
% 0.32/0.56 ( ( ^ [Y6: list_int,Z4: list_int] : ( Y6 = Z4 ) )
% 0.32/0.56 = ( ^ [Xs: list_int,Ys2: list_int] :
% 0.32/0.56 ( ( ( size_size_list_int @ Xs )
% 0.32/0.56 = ( size_size_list_int @ Ys2 ) )
% 0.32/0.56 & ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.32/0.56 => ( ( nth_int @ Xs @ I4 )
% 0.32/0.56 = ( nth_int @ Ys2 @ I4 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_eq_iff_nth_eq
% 0.32/0.56 thf(fact_3332_Skolem__list__nth,axiom,
% 0.32/0.56 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.32/0.56 ( ( ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ K )
% 0.32/0.56 => ? [X6: vEBT_VEBT] : ( P @ I4 @ X6 ) ) )
% 0.32/0.56 = ( ? [Xs: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.32/0.56 = K )
% 0.32/0.56 & ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ K )
% 0.32/0.56 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Skolem_list_nth
% 0.32/0.56 thf(fact_3333_Skolem__list__nth,axiom,
% 0.32/0.56 ! [K: nat,P: nat > $o > $o] :
% 0.32/0.56 ( ( ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ K )
% 0.32/0.56 => ? [X6: $o] : ( P @ I4 @ X6 ) ) )
% 0.32/0.56 = ( ? [Xs: list_o] :
% 0.32/0.56 ( ( ( size_size_list_o @ Xs )
% 0.32/0.56 = K )
% 0.32/0.56 & ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ K )
% 0.32/0.56 => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Skolem_list_nth
% 0.32/0.56 thf(fact_3334_Skolem__list__nth,axiom,
% 0.32/0.56 ! [K: nat,P: nat > nat > $o] :
% 0.32/0.56 ( ( ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ K )
% 0.32/0.56 => ? [X6: nat] : ( P @ I4 @ X6 ) ) )
% 0.32/0.56 = ( ? [Xs: list_nat] :
% 0.32/0.56 ( ( ( size_size_list_nat @ Xs )
% 0.32/0.56 = K )
% 0.32/0.56 & ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ K )
% 0.32/0.56 => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Skolem_list_nth
% 0.32/0.56 thf(fact_3335_Skolem__list__nth,axiom,
% 0.32/0.56 ! [K: nat,P: nat > int > $o] :
% 0.32/0.56 ( ( ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ K )
% 0.32/0.56 => ? [X6: int] : ( P @ I4 @ X6 ) ) )
% 0.32/0.56 = ( ? [Xs: list_int] :
% 0.32/0.56 ( ( ( size_size_list_int @ Xs )
% 0.32/0.56 = K )
% 0.32/0.56 & ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ K )
% 0.32/0.56 => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Skolem_list_nth
% 0.32/0.56 thf(fact_3336_nth__equalityI,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.32/0.56 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.32/0.56 => ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.56 => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
% 0.32/0.56 = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) )
% 0.32/0.56 => ( Xs2 = Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_equalityI
% 0.32/0.56 thf(fact_3337_nth__equalityI,axiom,
% 0.32/0.56 ! [Xs2: list_o,Ys3: list_o] :
% 0.32/0.56 ( ( ( size_size_list_o @ Xs2 )
% 0.32/0.56 = ( size_size_list_o @ Ys3 ) )
% 0.32/0.56 => ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.56 => ( ( nth_o @ Xs2 @ I2 )
% 0.32/0.56 = ( nth_o @ Ys3 @ I2 ) ) )
% 0.32/0.56 => ( Xs2 = Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_equalityI
% 0.32/0.56 thf(fact_3338_nth__equalityI,axiom,
% 0.32/0.56 ! [Xs2: list_nat,Ys3: list_nat] :
% 0.32/0.56 ( ( ( size_size_list_nat @ Xs2 )
% 0.32/0.56 = ( size_size_list_nat @ Ys3 ) )
% 0.32/0.56 => ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.56 => ( ( nth_nat @ Xs2 @ I2 )
% 0.32/0.56 = ( nth_nat @ Ys3 @ I2 ) ) )
% 0.32/0.56 => ( Xs2 = Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_equalityI
% 0.32/0.56 thf(fact_3339_nth__equalityI,axiom,
% 0.32/0.56 ! [Xs2: list_int,Ys3: list_int] :
% 0.32/0.56 ( ( ( size_size_list_int @ Xs2 )
% 0.32/0.56 = ( size_size_list_int @ Ys3 ) )
% 0.32/0.56 => ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.56 => ( ( nth_int @ Xs2 @ I2 )
% 0.32/0.56 = ( nth_int @ Ys3 @ I2 ) ) )
% 0.32/0.56 => ( Xs2 = Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_equalityI
% 0.32/0.56 thf(fact_3340_ex__nat__less,axiom,
% 0.32/0.56 ! [N: nat,P: nat > $o] :
% 0.32/0.56 ( ( ? [M3: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ M3 @ N )
% 0.32/0.56 & ( P @ M3 ) ) )
% 0.32/0.56 = ( ? [X2: nat] :
% 0.32/0.56 ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.56 & ( P @ X2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % ex_nat_less
% 0.32/0.56 thf(fact_3341_all__nat__less,axiom,
% 0.32/0.56 ! [N: nat,P: nat > $o] :
% 0.32/0.56 ( ( ! [M3: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ M3 @ N )
% 0.32/0.56 => ( P @ M3 ) ) )
% 0.32/0.56 = ( ! [X2: nat] :
% 0.32/0.56 ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.56 => ( P @ X2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_nat_less
% 0.32/0.56 thf(fact_3342_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: real,Xs2: list_real] :
% 0.32/0.56 ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3343_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: complex,Xs2: list_complex] :
% 0.32/0.56 ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3344_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
% 0.32/0.56 ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_s3679842834875189465at_nat @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3345_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3346_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: $o,Xs2: list_o] :
% 0.32/0.56 ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3347_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: nat,Xs2: list_nat] :
% 0.32/0.56 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3348_length__pos__if__in__set,axiom,
% 0.32/0.56 ! [X: int,Xs2: list_int] :
% 0.32/0.56 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 0.32/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_pos_if_in_set
% 0.32/0.56 thf(fact_3349_all__set__conv__all__nth,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.32/0.56 ( ( ! [X2: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X2 ) ) )
% 0.32/0.56 = ( ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_set_conv_all_nth
% 0.32/0.56 thf(fact_3350_all__set__conv__all__nth,axiom,
% 0.32/0.56 ! [Xs2: list_o,P: $o > $o] :
% 0.32/0.56 ( ( ! [X2: $o] :
% 0.32/0.56 ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X2 ) ) )
% 0.32/0.56 = ( ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_set_conv_all_nth
% 0.32/0.56 thf(fact_3351_all__set__conv__all__nth,axiom,
% 0.32/0.56 ! [Xs2: list_nat,P: nat > $o] :
% 0.32/0.56 ( ( ! [X2: nat] :
% 0.32/0.56 ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X2 ) ) )
% 0.32/0.56 = ( ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_set_conv_all_nth
% 0.32/0.56 thf(fact_3352_all__set__conv__all__nth,axiom,
% 0.32/0.56 ! [Xs2: list_int,P: int > $o] :
% 0.32/0.56 ( ( ! [X2: int] :
% 0.32/0.56 ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X2 ) ) )
% 0.32/0.56 = ( ! [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_set_conv_all_nth
% 0.32/0.56 thf(fact_3353_all__nth__imp__all__set,axiom,
% 0.32/0.56 ! [Xs2: list_real,P: real > $o,X: real] :
% 0.32/0.56 ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
% 0.32/0.56 => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_nth_imp_all_set
% 0.32/0.56 thf(fact_3354_all__nth__imp__all__set,axiom,
% 0.32/0.56 ! [Xs2: list_complex,P: complex > $o,X: complex] :
% 0.32/0.56 ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_complex @ Xs2 @ I2 ) ) )
% 0.32/0.56 => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_nth_imp_all_set
% 0.32/0.56 thf(fact_3355_all__nth__imp__all__set,axiom,
% 0.32/0.56 ! [Xs2: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
% 0.32/0.56 ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ I2 ) ) )
% 0.32/0.56 => ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 0.32/0.56 => ( P @ X ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_nth_imp_all_set
% 0.32/0.56 thf(fact_3356_all__nth__imp__all__set,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 0.32/0.56 ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
% 0.32/0.56 => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_nth_imp_all_set
% 0.32/0.56 thf(fact_3357_all__nth__imp__all__set,axiom,
% 0.32/0.56 ! [Xs2: list_o,P: $o > $o,X: $o] :
% 0.32/0.56 ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
% 0.32/0.56 => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_nth_imp_all_set
% 0.32/0.56 thf(fact_3358_all__nth__imp__all__set,axiom,
% 0.32/0.56 ! [Xs2: list_nat,P: nat > $o,X: nat] :
% 0.32/0.56 ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
% 0.32/0.56 => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_nth_imp_all_set
% 0.32/0.56 thf(fact_3359_all__nth__imp__all__set,axiom,
% 0.32/0.56 ! [Xs2: list_int,P: int > $o,X: int] :
% 0.32/0.56 ( ! [I2: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.56 => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
% 0.32/0.56 => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % all_nth_imp_all_set
% 0.32/0.56 thf(fact_3360_in__set__conv__nth,axiom,
% 0.32/0.56 ! [X: real,Xs2: list_real] :
% 0.32/0.56 ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 0.32/0.56 = ( ? [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
% 0.32/0.56 & ( ( nth_real @ Xs2 @ I4 )
% 0.32/0.56 = X ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % in_set_conv_nth
% 0.32/0.56 thf(fact_3361_in__set__conv__nth,axiom,
% 0.32/0.56 ! [X: complex,Xs2: list_complex] :
% 0.32/0.56 ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 0.32/0.56 = ( ? [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.32/0.56 & ( ( nth_complex @ Xs2 @ I4 )
% 0.32/0.56 = X ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % in_set_conv_nth
% 0.32/0.56 thf(fact_3362_in__set__conv__nth,axiom,
% 0.32/0.56 ! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
% 0.32/0.56 ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 0.32/0.56 = ( ? [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 0.32/0.56 & ( ( nth_Pr6744343527793145070at_nat @ Xs2 @ I4 )
% 0.32/0.56 = X ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % in_set_conv_nth
% 0.32/0.56 thf(fact_3363_in__set__conv__nth,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.32/0.56 = ( ? [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.56 & ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 0.32/0.56 = X ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % in_set_conv_nth
% 0.32/0.56 thf(fact_3364_in__set__conv__nth,axiom,
% 0.32/0.56 ! [X: $o,Xs2: list_o] :
% 0.32/0.56 ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 0.32/0.56 = ( ? [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.56 & ( ( nth_o @ Xs2 @ I4 )
% 0.32/0.56 = X ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % in_set_conv_nth
% 0.32/0.56 thf(fact_3365_in__set__conv__nth,axiom,
% 0.32/0.56 ! [X: nat,Xs2: list_nat] :
% 0.32/0.56 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 0.32/0.56 = ( ? [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.56 & ( ( nth_nat @ Xs2 @ I4 )
% 0.32/0.56 = X ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % in_set_conv_nth
% 0.32/0.56 thf(fact_3366_in__set__conv__nth,axiom,
% 0.32/0.56 ! [X: int,Xs2: list_int] :
% 0.32/0.56 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 0.32/0.56 = ( ? [I4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.56 & ( ( nth_int @ Xs2 @ I4 )
% 0.32/0.56 = X ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % in_set_conv_nth
% 0.32/0.56 thf(fact_3367_list__ball__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.56 => ( ! [X3: vEBT_VEBT] :
% 0.32/0.56 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X3 ) )
% 0.32/0.56 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_ball_nth
% 0.32/0.56 thf(fact_3368_list__ball__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_o,P: $o > $o] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.56 => ( ! [X3: $o] :
% 0.32/0.56 ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X3 ) )
% 0.32/0.56 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_ball_nth
% 0.32/0.56 thf(fact_3369_list__ball__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_nat,P: nat > $o] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.56 => ( ! [X3: nat] :
% 0.32/0.56 ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X3 ) )
% 0.32/0.56 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_ball_nth
% 0.32/0.56 thf(fact_3370_list__ball__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_int,P: int > $o] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.56 => ( ! [X3: int] :
% 0.32/0.56 ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 0.32/0.56 => ( P @ X3 ) )
% 0.32/0.56 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_ball_nth
% 0.32/0.56 thf(fact_3371_nth__mem,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_real] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.32/0.56 => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_mem
% 0.32/0.56 thf(fact_3372_nth__mem,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_complex] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.32/0.56 => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_mem
% 0.32/0.56 thf(fact_3373_nth__mem,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_P8469869581646625389at_nat] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 0.32/0.56 => ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ N ) @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_mem
% 0.32/0.56 thf(fact_3374_nth__mem,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.56 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_mem
% 0.32/0.56 thf(fact_3375_nth__mem,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_o] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.56 => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_mem
% 0.32/0.56 thf(fact_3376_nth__mem,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_nat] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.56 => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_mem
% 0.32/0.56 thf(fact_3377_nth__mem,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_int] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.56 => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_mem
% 0.32/0.56 thf(fact_3378_finite__lists__length__eq,axiom,
% 0.32/0.56 ! [A2: set_complex,N: nat] :
% 0.32/0.56 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.56 => ( finite8712137658972009173omplex
% 0.32/0.56 @ ( collect_list_complex
% 0.32/0.56 @ ^ [Xs: list_complex] :
% 0.32/0.56 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ( size_s3451745648224563538omplex @ Xs )
% 0.32/0.56 = N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_eq
% 0.32/0.56 thf(fact_3379_finite__lists__length__eq,axiom,
% 0.32/0.56 ! [A2: set_VEBT_VEBT,N: nat] :
% 0.32/0.56 ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.32/0.56 => ( finite3004134309566078307T_VEBT
% 0.32/0.56 @ ( collec5608196760682091941T_VEBT
% 0.32/0.56 @ ^ [Xs: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.32/0.56 = N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_eq
% 0.32/0.56 thf(fact_3380_finite__lists__length__eq,axiom,
% 0.32/0.56 ! [A2: set_o,N: nat] :
% 0.32/0.56 ( ( finite_finite_o @ A2 )
% 0.32/0.56 => ( finite_finite_list_o
% 0.32/0.56 @ ( collect_list_o
% 0.32/0.56 @ ^ [Xs: list_o] :
% 0.32/0.56 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ( size_size_list_o @ Xs )
% 0.32/0.56 = N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_eq
% 0.32/0.56 thf(fact_3381_finite__lists__length__eq,axiom,
% 0.32/0.56 ! [A2: set_nat,N: nat] :
% 0.32/0.56 ( ( finite_finite_nat @ A2 )
% 0.32/0.56 => ( finite8100373058378681591st_nat
% 0.32/0.56 @ ( collect_list_nat
% 0.32/0.56 @ ^ [Xs: list_nat] :
% 0.32/0.56 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ( size_size_list_nat @ Xs )
% 0.32/0.56 = N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_eq
% 0.32/0.56 thf(fact_3382_finite__lists__length__eq,axiom,
% 0.32/0.56 ! [A2: set_int,N: nat] :
% 0.32/0.56 ( ( finite_finite_int @ A2 )
% 0.32/0.56 => ( finite3922522038869484883st_int
% 0.32/0.56 @ ( collect_list_int
% 0.32/0.56 @ ^ [Xs: list_int] :
% 0.32/0.56 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 0.32/0.56 & ( ( size_size_list_int @ Xs )
% 0.32/0.56 = N ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_lists_length_eq
% 0.32/0.56 thf(fact_3383_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.32/0.56 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
% 0.32/0.56 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
% 0.32/0.56 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.simps(5)
% 0.32/0.56 thf(fact_3384_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.32/0.56 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 0.32/0.56 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
% 0.32/0.56 = ( ( X = Mi )
% 0.32/0.56 | ( X = Ma )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.simps(4)
% 0.32/0.56 thf(fact_3385_vebt__member_Opelims_I3_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.56 ( ~ ( vEBT_vebt_member @ X @ Xa )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [A5: $o,B3: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) )
% 0.32/0.56 => ( ( ( Xa = zero_zero_nat )
% 0.32/0.56 => A5 )
% 0.32/0.56 & ( ( Xa != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa = one_one_nat )
% 0.32/0.56 => B3 )
% 0.32/0.56 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.32/0.56 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.32/0.56 => ( ( Xa != Mi2 )
% 0.32/0.56 => ( ( Xa != Ma2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_member.pelims(3)
% 0.32/0.56 thf(fact_3386_vebt__member_Opelims_I1_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.32/0.56 ( ( ( vEBT_vebt_member @ X @ Xa )
% 0.32/0.56 = Y3 )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [A5: $o,B3: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.56 => ( ( Y3
% 0.32/0.56 = ( ( ( Xa = zero_zero_nat )
% 0.32/0.56 => A5 )
% 0.32/0.56 & ( ( Xa != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa = one_one_nat )
% 0.32/0.56 => B3 )
% 0.32/0.56 & ( Xa = one_one_nat ) ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ( ~ Y3
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.32/0.56 => ( ~ Y3
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.32/0.56 => ( ~ Y3
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( Y3
% 0.32/0.56 = ( ( Xa != Mi2 )
% 0.32/0.56 => ( ( Xa != Ma2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_member.pelims(1)
% 0.32/0.56 thf(fact_3387_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.56 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [A5: $o,B3: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) )
% 0.32/0.56 => ( ( ( Xa = zero_zero_nat )
% 0.32/0.56 => A5 )
% 0.32/0.56 & ( ( Xa != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa = one_one_nat )
% 0.32/0.56 => B3 )
% 0.32/0.56 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.32/0.56 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
% 0.32/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.naive_member.pelims(3)
% 0.32/0.56 thf(fact_3388_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.56 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [A5: $o,B3: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) )
% 0.32/0.56 => ~ ( ( ( Xa = zero_zero_nat )
% 0.32/0.56 => A5 )
% 0.32/0.56 & ( ( Xa != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa = one_one_nat )
% 0.32/0.56 => B3 )
% 0.32/0.56 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.32/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
% 0.32/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.naive_member.pelims(2)
% 0.32/0.56 thf(fact_3389_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.32/0.56 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 0.32/0.56 = Y3 )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [A5: $o,B3: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.56 => ( ( Y3
% 0.32/0.56 = ( ( ( Xa = zero_zero_nat )
% 0.32/0.56 => A5 )
% 0.32/0.56 & ( ( Xa != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa = one_one_nat )
% 0.32/0.56 => B3 )
% 0.32/0.56 & ( Xa = one_one_nat ) ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.32/0.56 => ( ~ Y3
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 0.32/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 0.32/0.56 => ( ( Y3
% 0.32/0.56 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.naive_member.pelims(1)
% 0.32/0.56 thf(fact_3390_vebt__member_Opelims_I2_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.56 ( ( vEBT_vebt_member @ X @ Xa )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [A5: $o,B3: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) )
% 0.32/0.56 => ~ ( ( ( Xa = zero_zero_nat )
% 0.32/0.56 => A5 )
% 0.32/0.56 & ( ( Xa != zero_zero_nat )
% 0.32/0.56 => ( ( ( Xa = one_one_nat )
% 0.32/0.56 => B3 )
% 0.32/0.56 & ( Xa = one_one_nat ) ) ) ) ) )
% 0.32/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.32/0.56 => ~ ( ( Xa != Mi2 )
% 0.32/0.56 => ( ( Xa != Ma2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 0.32/0.56 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.56 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % vebt_member.pelims(2)
% 0.32/0.56 thf(fact_3391_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.32/0.56 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.32/0.56 = Y3 )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.56 => ( ~ Y3
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.32/0.56 => ( ~ Y3
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.32/0.56 => ( ( Y3
% 0.32/0.56 = ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ( ( Y3
% 0.32/0.56 = ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ( ( Y3
% 0.32/0.56 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.pelims(1)
% 0.32/0.56 thf(fact_3392_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.56 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
% 0.32/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.32/0.56 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) )
% 0.32/0.56 => ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 ) ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 0.32/0.56 => ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 0.32/0.56 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.pelims(3)
% 0.32/0.56 thf(fact_3393_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.32/0.56 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.56 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) )
% 0.32/0.56 => ~ ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 ) ) ) )
% 0.32/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 0.32/0.56 => ~ ( ( Xa = Mi2 )
% 0.32/0.56 | ( Xa = Ma2 )
% 0.32/0.56 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.32/0.56 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.32/0.56 ( ( X
% 0.32/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 0.32/0.56 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 0.32/0.56 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.56 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.56 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % VEBT_internal.membermima.pelims(2)
% 0.32/0.56 thf(fact_3394_finite__roots__unity,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.32/0.56 => ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [Z2: real] :
% 0.32/0.56 ( ( power_power_real @ Z2 @ N )
% 0.32/0.56 = one_one_real ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_roots_unity
% 0.32/0.56 thf(fact_3395_finite__roots__unity,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.32/0.56 => ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [Z2: complex] :
% 0.32/0.56 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.56 = one_one_complex ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % finite_roots_unity
% 0.32/0.56 thf(fact_3396_flip__bit__Suc,axiom,
% 0.32/0.56 ! [N: nat,A: code_integer] :
% 0.32/0.56 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % flip_bit_Suc
% 0.32/0.56 thf(fact_3397_flip__bit__Suc,axiom,
% 0.32/0.56 ! [N: nat,A: int] :
% 0.32/0.56 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 0.32/0.56 = ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % flip_bit_Suc
% 0.32/0.56 thf(fact_3398_flip__bit__Suc,axiom,
% 0.32/0.56 ! [N: nat,A: nat] :
% 0.32/0.56 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 0.32/0.56 = ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % flip_bit_Suc
% 0.32/0.56 thf(fact_3399_Bolzano,axiom,
% 0.32/0.56 ! [A: real,B: real,P: real > real > $o] :
% 0.32/0.56 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.56 => ( ! [A5: real,B3: real,C2: real] :
% 0.32/0.56 ( ( P @ A5 @ B3 )
% 0.32/0.56 => ( ( P @ B3 @ C2 )
% 0.32/0.56 => ( ( ord_less_eq_real @ A5 @ B3 )
% 0.32/0.56 => ( ( ord_less_eq_real @ B3 @ C2 )
% 0.32/0.56 => ( P @ A5 @ C2 ) ) ) ) )
% 0.32/0.56 => ( ! [X3: real] :
% 0.32/0.56 ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.56 => ( ( ord_less_eq_real @ X3 @ B )
% 0.32/0.56 => ? [D5: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ D5 )
% 0.32/0.56 & ! [A5: real,B3: real] :
% 0.32/0.56 ( ( ( ord_less_eq_real @ A5 @ X3 )
% 0.32/0.56 & ( ord_less_eq_real @ X3 @ B3 )
% 0.32/0.56 & ( ord_less_real @ ( minus_minus_real @ B3 @ A5 ) @ D5 ) )
% 0.32/0.56 => ( P @ A5 @ B3 ) ) ) ) )
% 0.32/0.56 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % Bolzano
% 0.32/0.56 thf(fact_3400_mult__le__cancel__iff1,axiom,
% 0.32/0.56 ! [Z: real,X: real,Y3: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 0.32/0.56 = ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff1
% 0.32/0.56 thf(fact_3401_mult__le__cancel__iff1,axiom,
% 0.32/0.56 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.32/0.56 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 0.32/0.56 = ( ord_less_eq_rat @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff1
% 0.32/0.56 thf(fact_3402_mult__le__cancel__iff1,axiom,
% 0.32/0.56 ! [Z: int,X: int,Y3: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.56 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 0.32/0.56 = ( ord_less_eq_int @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff1
% 0.32/0.56 thf(fact_3403_mult__le__cancel__iff2,axiom,
% 0.32/0.56 ! [Z: real,X: real,Y3: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.32/0.56 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y3 ) )
% 0.32/0.56 = ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff2
% 0.32/0.56 thf(fact_3404_mult__le__cancel__iff2,axiom,
% 0.32/0.56 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.32/0.56 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y3 ) )
% 0.32/0.56 = ( ord_less_eq_rat @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff2
% 0.32/0.56 thf(fact_3405_mult__le__cancel__iff2,axiom,
% 0.32/0.56 ! [Z: int,X: int,Y3: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.56 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y3 ) )
% 0.32/0.56 = ( ord_less_eq_int @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_le_cancel_iff2
% 0.32/0.56 thf(fact_3406_divides__aux__eq,axiom,
% 0.32/0.56 ! [Q2: nat,R: nat] :
% 0.32/0.56 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 0.32/0.56 = ( R = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % divides_aux_eq
% 0.32/0.56 thf(fact_3407_divides__aux__eq,axiom,
% 0.32/0.56 ! [Q2: int,R: int] :
% 0.32/0.56 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.56 = ( R = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % divides_aux_eq
% 0.32/0.56 thf(fact_3408_flip__bit__nonnegative__int__iff,axiom,
% 0.32/0.56 ! [N: nat,K: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 0.32/0.56 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.56
% 0.32/0.56 % flip_bit_nonnegative_int_iff
% 0.32/0.56 thf(fact_3409_flip__bit__negative__int__iff,axiom,
% 0.32/0.56 ! [N: nat,K: int] :
% 0.32/0.56 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.56 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % flip_bit_negative_int_iff
% 0.32/0.56 thf(fact_3410_mult__less__iff1,axiom,
% 0.32/0.56 ! [Z: real,X: real,Y3: real] :
% 0.32/0.56 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.32/0.56 => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 0.32/0.56 = ( ord_less_real @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_iff1
% 0.32/0.56 thf(fact_3411_mult__less__iff1,axiom,
% 0.32/0.56 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.56 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.32/0.56 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 0.32/0.56 = ( ord_less_rat @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_iff1
% 0.32/0.56 thf(fact_3412_mult__less__iff1,axiom,
% 0.32/0.56 ! [Z: int,X: int,Y3: int] :
% 0.32/0.56 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.56 => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 0.32/0.56 = ( ord_less_int @ X @ Y3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % mult_less_iff1
% 0.32/0.56 thf(fact_3413_arcosh__1,axiom,
% 0.32/0.56 ( ( arcosh_real @ one_one_real )
% 0.32/0.56 = zero_zero_real ) ).
% 0.32/0.56
% 0.32/0.56 % arcosh_1
% 0.32/0.56 thf(fact_3414_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_num,Ys3: list_num] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys3 ) ) ) @ ( nth_num @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3415_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3416_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,Ys3: list_o] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3417_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,Ys3: list_nat] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3418_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_VEBT_VEBT,Ys3: list_int] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3419_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_o,Ys3: list_VEBT_VEBT] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3420_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_o,Ys3: list_o] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3421_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_o,Ys3: list_nat] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3422_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_o,Ys3: list_int] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3423_product__nth,axiom,
% 0.32/0.56 ! [N: nat,Xs2: list_nat,Ys3: list_num] :
% 0.32/0.56 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_num @ Ys3 ) ) )
% 0.32/0.56 => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs2 @ Ys3 ) @ N )
% 0.32/0.56 = ( product_Pair_nat_num @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys3 ) ) ) @ ( nth_num @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys3 ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % product_nth
% 0.32/0.56 thf(fact_3424_neg__eucl__rel__int__mult__2,axiom,
% 0.32/0.56 ! [B: int,A: int,Q2: int,R: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.32/0.56 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.56 => ( 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 ) ) @ R ) @ one_one_int ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % neg_eucl_rel_int_mult_2
% 0.32/0.56 thf(fact_3425_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_real,X: real > complex,Y3: real > complex] :
% 0.32/0.56 ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_complex ) ) ) )
% 0.32/0.56 => ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_complex ) ) ) )
% 0.32/0.56 => ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_complex ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3426_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_nat,X: nat > complex,Y3: nat > complex] :
% 0.32/0.56 ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_complex ) ) ) )
% 0.32/0.56 => ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_complex ) ) ) )
% 0.32/0.56 => ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_complex ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3427_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_int,X: int > complex,Y3: int > complex] :
% 0.32/0.56 ( ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_complex ) ) ) )
% 0.32/0.56 => ( ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_complex ) ) ) )
% 0.32/0.56 => ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_complex ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3428_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_complex,X: complex > complex,Y3: complex > complex] :
% 0.32/0.56 ( ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_complex ) ) ) )
% 0.32/0.56 => ( ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_complex ) ) ) )
% 0.32/0.56 => ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_complex ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3429_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_real,X: real > real,Y3: real > real] :
% 0.32/0.56 ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_real ) ) ) )
% 0.32/0.56 => ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_real ) ) ) )
% 0.32/0.56 => ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3430_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_nat,X: nat > real,Y3: nat > real] :
% 0.32/0.56 ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_real ) ) ) )
% 0.32/0.56 => ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_real ) ) ) )
% 0.32/0.56 => ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3431_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_int,X: int > real,Y3: int > real] :
% 0.32/0.56 ( ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_real ) ) ) )
% 0.32/0.56 => ( ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_real ) ) ) )
% 0.32/0.56 => ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3432_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_complex,X: complex > real,Y3: complex > real] :
% 0.32/0.56 ( ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_real ) ) ) )
% 0.32/0.56 => ( ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_real ) ) ) )
% 0.32/0.56 => ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3433_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_real,X: real > rat,Y3: real > rat] :
% 0.32/0.56 ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_rat ) ) ) )
% 0.32/0.56 => ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_rat ) ) ) )
% 0.32/0.56 => ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_rat @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_rat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3434_prod_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_nat,X: nat > rat,Y3: nat > rat] :
% 0.32/0.56 ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != one_one_rat ) ) ) )
% 0.32/0.56 => ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != one_one_rat ) ) ) )
% 0.32/0.56 => ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( times_times_rat @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != one_one_rat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % prod.finite_Collect_op
% 0.32/0.56 thf(fact_3435_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_real,X: real > complex,Y3: real > complex] :
% 0.32/0.56 ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_complex ) ) ) )
% 0.32/0.56 => ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_complex ) ) ) )
% 0.32/0.56 => ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_complex ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3436_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_nat,X: nat > complex,Y3: nat > complex] :
% 0.32/0.56 ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_complex ) ) ) )
% 0.32/0.56 => ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_complex ) ) ) )
% 0.32/0.56 => ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_complex ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3437_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_int,X: int > complex,Y3: int > complex] :
% 0.32/0.56 ( ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_complex ) ) ) )
% 0.32/0.56 => ( ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_complex ) ) ) )
% 0.32/0.56 => ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_complex ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3438_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_complex,X: complex > complex,Y3: complex > complex] :
% 0.32/0.56 ( ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_complex ) ) ) )
% 0.32/0.56 => ( ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_complex ) ) ) )
% 0.32/0.56 => ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_complex ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3439_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_real,X: real > real,Y3: real > real] :
% 0.32/0.56 ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_real ) ) ) )
% 0.32/0.56 => ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_real ) ) ) )
% 0.32/0.56 => ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3440_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_nat,X: nat > real,Y3: nat > real] :
% 0.32/0.56 ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_real ) ) ) )
% 0.32/0.56 => ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_real ) ) ) )
% 0.32/0.56 => ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3441_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_int,X: int > real,Y3: int > real] :
% 0.32/0.56 ( ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_real ) ) ) )
% 0.32/0.56 => ( ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_real ) ) ) )
% 0.32/0.56 => ( finite_finite_int
% 0.32/0.56 @ ( collect_int
% 0.32/0.56 @ ^ [I4: int] :
% 0.32/0.56 ( ( member_int @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3442_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_complex,X: complex > real,Y3: complex > real] :
% 0.32/0.56 ( ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_real ) ) ) )
% 0.32/0.56 => ( ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_real ) ) ) )
% 0.32/0.56 => ( finite3207457112153483333omplex
% 0.32/0.56 @ ( collect_complex
% 0.32/0.56 @ ^ [I4: complex] :
% 0.32/0.56 ( ( member_complex @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3443_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_real,X: real > rat,Y3: real > rat] :
% 0.32/0.56 ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_rat ) ) ) )
% 0.32/0.56 => ( ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_rat ) ) ) )
% 0.32/0.56 => ( finite_finite_real
% 0.32/0.56 @ ( collect_real
% 0.32/0.56 @ ^ [I4: real] :
% 0.32/0.56 ( ( member_real @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3444_sum_Ofinite__Collect__op,axiom,
% 0.32/0.56 ! [I5: set_nat,X: nat > rat,Y3: nat > rat] :
% 0.32/0.56 ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( X @ I4 )
% 0.32/0.56 != zero_zero_rat ) ) ) )
% 0.32/0.56 => ( ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( Y3 @ I4 )
% 0.32/0.56 != zero_zero_rat ) ) ) )
% 0.32/0.56 => ( finite_finite_nat
% 0.32/0.56 @ ( collect_nat
% 0.32/0.56 @ ^ [I4: nat] :
% 0.32/0.56 ( ( member_nat @ I4 @ I5 )
% 0.32/0.56 & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 0.32/0.56 != zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % sum.finite_Collect_op
% 0.32/0.56 thf(fact_3445_pos__eucl__rel__int__mult__2,axiom,
% 0.32/0.56 ! [B: int,A: int,Q2: int,R: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.56 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.56 => ( 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 ) ) @ R ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pos_eucl_rel_int_mult_2
% 0.32/0.56 thf(fact_3446_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.32/0.56 ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3447_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,Ys3: list_o] :
% 0.32/0.56 ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3448_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,Ys3: list_nat] :
% 0.32/0.56 ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3449_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,Ys3: list_int] :
% 0.32/0.56 ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3450_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_o,Ys3: list_VEBT_VEBT] :
% 0.32/0.56 ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3451_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_o,Ys3: list_o] :
% 0.32/0.56 ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3452_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_o,Ys3: list_nat] :
% 0.32/0.56 ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3453_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_o,Ys3: list_int] :
% 0.32/0.56 ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3454_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_nat,Ys3: list_VEBT_VEBT] :
% 0.32/0.56 ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3455_length__product,axiom,
% 0.32/0.56 ! [Xs2: list_nat,Ys3: list_o] :
% 0.32/0.56 ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys3 ) )
% 0.32/0.56 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % length_product
% 0.32/0.56 thf(fact_3456_unique__quotient,axiom,
% 0.32/0.56 ! [A: int,B: int,Q2: int,R: int,Q4: int,R3: int] :
% 0.32/0.56 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.56 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
% 0.32/0.56 => ( Q2 = Q4 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_quotient
% 0.32/0.56 thf(fact_3457_unique__remainder,axiom,
% 0.32/0.56 ! [A: int,B: int,Q2: int,R: int,Q4: int,R3: int] :
% 0.32/0.56 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.56 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
% 0.32/0.56 => ( R = R3 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unique_remainder
% 0.32/0.56 thf(fact_3458_eucl__rel__int__by0,axiom,
% 0.32/0.56 ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 0.32/0.56
% 0.32/0.56 % eucl_rel_int_by0
% 0.32/0.56 thf(fact_3459_div__int__unique,axiom,
% 0.32/0.56 ! [K: int,L2: int,Q2: int,R: int] :
% 0.32/0.56 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.56 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.56 = Q2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_int_unique
% 0.32/0.56 thf(fact_3460_mod__int__unique,axiom,
% 0.32/0.56 ! [K: int,L2: int,Q2: int,R: int] :
% 0.32/0.56 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.56 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.56 = R ) ) ).
% 0.32/0.56
% 0.32/0.56 % mod_int_unique
% 0.32/0.56 thf(fact_3461_eucl__rel__int__dividesI,axiom,
% 0.32/0.56 ! [L2: int,K: int,Q2: int] :
% 0.32/0.56 ( ( L2 != zero_zero_int )
% 0.32/0.56 => ( ( K
% 0.32/0.56 = ( times_times_int @ Q2 @ L2 ) )
% 0.32/0.56 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % eucl_rel_int_dividesI
% 0.32/0.56 thf(fact_3462_eucl__rel__int,axiom,
% 0.32/0.56 ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % eucl_rel_int
% 0.32/0.56 thf(fact_3463_eucl__rel__int__iff,axiom,
% 0.32/0.56 ! [K: int,L2: int,Q2: int,R: int] :
% 0.32/0.56 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.56 = ( ( K
% 0.32/0.56 = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R ) )
% 0.32/0.56 & ( ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.56 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 0.32/0.56 & ( ord_less_int @ R @ L2 ) ) )
% 0.32/0.56 & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
% 0.32/0.56 => ( ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.56 => ( ( ord_less_int @ L2 @ R )
% 0.32/0.56 & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
% 0.32/0.56 & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.56 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % eucl_rel_int_iff
% 0.32/0.56 thf(fact_3464_insert__simp__norm,axiom,
% 0.32/0.56 ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.32/0.56 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.56 => ( ( ord_less_nat @ Mi @ X )
% 0.32/0.56 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.32/0.56 => ( ( X != Ma )
% 0.32/0.56 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % insert_simp_norm
% 0.32/0.56 thf(fact_3465_insert__simp__excp,axiom,
% 0.32/0.56 ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.32/0.56 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.56 => ( ( ord_less_nat @ X @ Mi )
% 0.32/0.56 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.32/0.56 => ( ( X != Ma )
% 0.32/0.56 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 0.32/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % insert_simp_excp
% 0.32/0.56 thf(fact_3466_gcd__nat__induct,axiom,
% 0.32/0.56 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.32/0.56 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 0.32/0.56 => ( ! [M4: nat,N4: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.56 => ( ( P @ N4 @ ( modulo_modulo_nat @ M4 @ N4 ) )
% 0.32/0.56 => ( P @ M4 @ N4 ) ) )
% 0.32/0.56 => ( P @ M @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % gcd_nat_induct
% 0.32/0.56 thf(fact_3467_concat__bit__Suc,axiom,
% 0.32/0.56 ! [N: nat,K: int,L2: int] :
% 0.32/0.56 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L2 )
% 0.32/0.56 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % concat_bit_Suc
% 0.32/0.56 thf(fact_3468_even__succ__mod__exp,axiom,
% 0.32/0.56 ! [A: nat,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_succ_mod_exp
% 0.32/0.56 thf(fact_3469_even__succ__mod__exp,axiom,
% 0.32/0.56 ! [A: int,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_succ_mod_exp
% 0.32/0.56 thf(fact_3470_even__succ__mod__exp,axiom,
% 0.32/0.56 ! [A: code_integer,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_succ_mod_exp
% 0.32/0.56 thf(fact_3471_option_Osize__gen_I2_J,axiom,
% 0.32/0.56 ! [X: nat > nat,X22: nat] :
% 0.32/0.56 ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
% 0.32/0.56 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % option.size_gen(2)
% 0.32/0.56 thf(fact_3472_option_Osize__gen_I2_J,axiom,
% 0.32/0.56 ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 0.32/0.56 ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.32/0.56 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % option.size_gen(2)
% 0.32/0.56 thf(fact_3473_option_Osize__gen_I2_J,axiom,
% 0.32/0.56 ! [X: num > nat,X22: num] :
% 0.32/0.56 ( ( size_option_num @ X @ ( some_num @ X22 ) )
% 0.32/0.56 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % option.size_gen(2)
% 0.32/0.56 thf(fact_3474_order__refl,axiom,
% 0.32/0.56 ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% 0.32/0.56
% 0.32/0.56 % order_refl
% 0.32/0.56 thf(fact_3475_order__refl,axiom,
% 0.32/0.56 ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 0.32/0.56
% 0.32/0.56 % order_refl
% 0.32/0.56 thf(fact_3476_order__refl,axiom,
% 0.32/0.56 ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 0.32/0.56
% 0.32/0.56 % order_refl
% 0.32/0.56 thf(fact_3477_order__refl,axiom,
% 0.32/0.56 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.32/0.56
% 0.32/0.56 % order_refl
% 0.32/0.56 thf(fact_3478_order__refl,axiom,
% 0.32/0.56 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 0.32/0.56
% 0.32/0.56 % order_refl
% 0.32/0.56 thf(fact_3479_dual__order_Orefl,axiom,
% 0.32/0.56 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.32/0.56
% 0.32/0.56 % dual_order.refl
% 0.32/0.56 thf(fact_3480_dual__order_Orefl,axiom,
% 0.32/0.56 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.32/0.56
% 0.32/0.56 % dual_order.refl
% 0.32/0.56 thf(fact_3481_dual__order_Orefl,axiom,
% 0.32/0.56 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.32/0.56
% 0.32/0.56 % dual_order.refl
% 0.32/0.56 thf(fact_3482_dual__order_Orefl,axiom,
% 0.32/0.56 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.32/0.56
% 0.32/0.56 % dual_order.refl
% 0.32/0.56 thf(fact_3483_dual__order_Orefl,axiom,
% 0.32/0.56 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.32/0.56
% 0.32/0.56 % dual_order.refl
% 0.32/0.56 thf(fact_3484_nat__dvd__1__iff__1,axiom,
% 0.32/0.56 ! [M: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.32/0.56 = ( M = one_one_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_dvd_1_iff_1
% 0.32/0.56 thf(fact_3485_dvd__0__left__iff,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.32/0.56 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_left_iff
% 0.32/0.56 thf(fact_3486_dvd__0__left__iff,axiom,
% 0.32/0.56 ! [A: complex] :
% 0.32/0.56 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.32/0.56 = ( A = zero_zero_complex ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_left_iff
% 0.32/0.56 thf(fact_3487_dvd__0__left__iff,axiom,
% 0.32/0.56 ! [A: real] :
% 0.32/0.56 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.32/0.56 = ( A = zero_zero_real ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_left_iff
% 0.32/0.56 thf(fact_3488_dvd__0__left__iff,axiom,
% 0.32/0.56 ! [A: rat] :
% 0.32/0.56 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.32/0.56 = ( A = zero_zero_rat ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_left_iff
% 0.32/0.56 thf(fact_3489_dvd__0__left__iff,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.32/0.56 = ( A = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_left_iff
% 0.32/0.56 thf(fact_3490_dvd__0__left__iff,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.32/0.56 = ( A = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_left_iff
% 0.32/0.56 thf(fact_3491_dvd__0__right,axiom,
% 0.32/0.56 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_right
% 0.32/0.56 thf(fact_3492_dvd__0__right,axiom,
% 0.32/0.56 ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_right
% 0.32/0.56 thf(fact_3493_dvd__0__right,axiom,
% 0.32/0.56 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_right
% 0.32/0.56 thf(fact_3494_dvd__0__right,axiom,
% 0.32/0.56 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_right
% 0.32/0.56 thf(fact_3495_dvd__0__right,axiom,
% 0.32/0.56 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_right
% 0.32/0.56 thf(fact_3496_dvd__0__right,axiom,
% 0.32/0.56 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_0_right
% 0.32/0.56 thf(fact_3497_dvd__add__triv__left__iff,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_left_iff
% 0.32/0.56 thf(fact_3498_dvd__add__triv__left__iff,axiom,
% 0.32/0.56 ! [A: real,B: real] :
% 0.32/0.56 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_left_iff
% 0.32/0.56 thf(fact_3499_dvd__add__triv__left__iff,axiom,
% 0.32/0.56 ! [A: rat,B: rat] :
% 0.32/0.56 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_left_iff
% 0.32/0.56 thf(fact_3500_dvd__add__triv__left__iff,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_left_iff
% 0.32/0.56 thf(fact_3501_dvd__add__triv__left__iff,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.32/0.56 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_left_iff
% 0.32/0.56 thf(fact_3502_dvd__add__triv__right__iff,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_right_iff
% 0.32/0.56 thf(fact_3503_dvd__add__triv__right__iff,axiom,
% 0.32/0.56 ! [A: real,B: real] :
% 0.32/0.56 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.32/0.56 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_right_iff
% 0.32/0.56 thf(fact_3504_dvd__add__triv__right__iff,axiom,
% 0.32/0.56 ! [A: rat,B: rat] :
% 0.32/0.56 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.32/0.56 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_right_iff
% 0.32/0.56 thf(fact_3505_dvd__add__triv__right__iff,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.32/0.56 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_right_iff
% 0.32/0.56 thf(fact_3506_dvd__add__triv__right__iff,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.32/0.56 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_triv_right_iff
% 0.32/0.56 thf(fact_3507_dvd__1__iff__1,axiom,
% 0.32/0.56 ! [M: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.32/0.56 = ( M
% 0.32/0.56 = ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_1_iff_1
% 0.32/0.56 thf(fact_3508_dvd__1__left,axiom,
% 0.32/0.56 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_1_left
% 0.32/0.56 thf(fact_3509_div__dvd__div,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_dvd_div
% 0.32/0.56 thf(fact_3510_div__dvd__div,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.56 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.56 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.32/0.56 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_dvd_div
% 0.32/0.56 thf(fact_3511_div__dvd__div,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.56 => ( ( dvd_dvd_int @ A @ C )
% 0.32/0.56 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.32/0.56 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_dvd_div
% 0.32/0.56 thf(fact_3512_max_Oabsorb1,axiom,
% 0.32/0.56 ! [B: extended_enat,A: extended_enat] :
% 0.32/0.56 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.32/0.56 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb1
% 0.32/0.56 thf(fact_3513_max_Oabsorb1,axiom,
% 0.32/0.56 ! [B: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.56 => ( ( ord_max_rat @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb1
% 0.32/0.56 thf(fact_3514_max_Oabsorb1,axiom,
% 0.32/0.56 ! [B: num,A: num] :
% 0.32/0.56 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.56 => ( ( ord_max_num @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb1
% 0.32/0.56 thf(fact_3515_max_Oabsorb1,axiom,
% 0.32/0.56 ! [B: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.56 => ( ( ord_max_nat @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb1
% 0.32/0.56 thf(fact_3516_max_Oabsorb1,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.56 => ( ( ord_max_int @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb1
% 0.32/0.56 thf(fact_3517_max_Oabsorb2,axiom,
% 0.32/0.56 ! [A: extended_enat,B: extended_enat] :
% 0.32/0.56 ( ( ord_le2932123472753598470d_enat @ A @ B )
% 0.32/0.56 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb2
% 0.32/0.56 thf(fact_3518_max_Oabsorb2,axiom,
% 0.32/0.56 ! [A: rat,B: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.56 => ( ( ord_max_rat @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb2
% 0.32/0.56 thf(fact_3519_max_Oabsorb2,axiom,
% 0.32/0.56 ! [A: num,B: num] :
% 0.32/0.56 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.56 => ( ( ord_max_num @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb2
% 0.32/0.56 thf(fact_3520_max_Oabsorb2,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.56 => ( ( ord_max_nat @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb2
% 0.32/0.56 thf(fact_3521_max_Oabsorb2,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.56 => ( ( ord_max_int @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb2
% 0.32/0.56 thf(fact_3522_max_Obounded__iff,axiom,
% 0.32/0.56 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 0.32/0.56 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 0.32/0.56 = ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.32/0.56 & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.bounded_iff
% 0.32/0.56 thf(fact_3523_max_Obounded__iff,axiom,
% 0.32/0.56 ! [B: rat,C: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.32/0.56 = ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.56 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.bounded_iff
% 0.32/0.56 thf(fact_3524_max_Obounded__iff,axiom,
% 0.32/0.56 ! [B: num,C: num,A: num] :
% 0.32/0.56 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.32/0.56 = ( ( ord_less_eq_num @ B @ A )
% 0.32/0.56 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.bounded_iff
% 0.32/0.56 thf(fact_3525_max_Obounded__iff,axiom,
% 0.32/0.56 ! [B: nat,C: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.32/0.56 = ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.56 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.bounded_iff
% 0.32/0.56 thf(fact_3526_max_Obounded__iff,axiom,
% 0.32/0.56 ! [B: int,C: int,A: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.32/0.56 = ( ( ord_less_eq_int @ B @ A )
% 0.32/0.56 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.bounded_iff
% 0.32/0.56 thf(fact_3527_nat__mult__dvd__cancel__disj,axiom,
% 0.32/0.56 ! [K: nat,M: nat,N: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.56 = ( ( K = zero_zero_nat )
% 0.32/0.56 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % nat_mult_dvd_cancel_disj
% 0.32/0.56 thf(fact_3528_max_Oabsorb3,axiom,
% 0.32/0.56 ! [B: extended_enat,A: extended_enat] :
% 0.32/0.56 ( ( ord_le72135733267957522d_enat @ B @ A )
% 0.32/0.56 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb3
% 0.32/0.56 thf(fact_3529_max_Oabsorb3,axiom,
% 0.32/0.56 ! [B: real,A: real] :
% 0.32/0.56 ( ( ord_less_real @ B @ A )
% 0.32/0.56 => ( ( ord_max_real @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb3
% 0.32/0.56 thf(fact_3530_max_Oabsorb3,axiom,
% 0.32/0.56 ! [B: rat,A: rat] :
% 0.32/0.56 ( ( ord_less_rat @ B @ A )
% 0.32/0.56 => ( ( ord_max_rat @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb3
% 0.32/0.56 thf(fact_3531_max_Oabsorb3,axiom,
% 0.32/0.56 ! [B: num,A: num] :
% 0.32/0.56 ( ( ord_less_num @ B @ A )
% 0.32/0.56 => ( ( ord_max_num @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb3
% 0.32/0.56 thf(fact_3532_max_Oabsorb3,axiom,
% 0.32/0.56 ! [B: nat,A: nat] :
% 0.32/0.56 ( ( ord_less_nat @ B @ A )
% 0.32/0.56 => ( ( ord_max_nat @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb3
% 0.32/0.56 thf(fact_3533_max_Oabsorb3,axiom,
% 0.32/0.56 ! [B: int,A: int] :
% 0.32/0.56 ( ( ord_less_int @ B @ A )
% 0.32/0.56 => ( ( ord_max_int @ A @ B )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb3
% 0.32/0.56 thf(fact_3534_max_Oabsorb4,axiom,
% 0.32/0.56 ! [A: extended_enat,B: extended_enat] :
% 0.32/0.56 ( ( ord_le72135733267957522d_enat @ A @ B )
% 0.32/0.56 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb4
% 0.32/0.56 thf(fact_3535_max_Oabsorb4,axiom,
% 0.32/0.56 ! [A: real,B: real] :
% 0.32/0.56 ( ( ord_less_real @ A @ B )
% 0.32/0.56 => ( ( ord_max_real @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb4
% 0.32/0.56 thf(fact_3536_max_Oabsorb4,axiom,
% 0.32/0.56 ! [A: rat,B: rat] :
% 0.32/0.56 ( ( ord_less_rat @ A @ B )
% 0.32/0.56 => ( ( ord_max_rat @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb4
% 0.32/0.56 thf(fact_3537_max_Oabsorb4,axiom,
% 0.32/0.56 ! [A: num,B: num] :
% 0.32/0.56 ( ( ord_less_num @ A @ B )
% 0.32/0.56 => ( ( ord_max_num @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb4
% 0.32/0.56 thf(fact_3538_max_Oabsorb4,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ord_less_nat @ A @ B )
% 0.32/0.56 => ( ( ord_max_nat @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb4
% 0.32/0.56 thf(fact_3539_max_Oabsorb4,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( ord_less_int @ A @ B )
% 0.32/0.56 => ( ( ord_max_int @ A @ B )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % max.absorb4
% 0.32/0.56 thf(fact_3540_max__less__iff__conj,axiom,
% 0.32/0.56 ! [X: extended_enat,Y3: extended_enat,Z: extended_enat] :
% 0.32/0.56 ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y3 ) @ Z )
% 0.32/0.56 = ( ( ord_le72135733267957522d_enat @ X @ Z )
% 0.32/0.56 & ( ord_le72135733267957522d_enat @ Y3 @ Z ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_less_iff_conj
% 0.32/0.56 thf(fact_3541_max__less__iff__conj,axiom,
% 0.32/0.56 ! [X: real,Y3: real,Z: real] :
% 0.32/0.56 ( ( ord_less_real @ ( ord_max_real @ X @ Y3 ) @ Z )
% 0.32/0.56 = ( ( ord_less_real @ X @ Z )
% 0.32/0.56 & ( ord_less_real @ Y3 @ Z ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_less_iff_conj
% 0.32/0.56 thf(fact_3542_max__less__iff__conj,axiom,
% 0.32/0.56 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.56 ( ( ord_less_rat @ ( ord_max_rat @ X @ Y3 ) @ Z )
% 0.32/0.56 = ( ( ord_less_rat @ X @ Z )
% 0.32/0.56 & ( ord_less_rat @ Y3 @ Z ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_less_iff_conj
% 0.32/0.56 thf(fact_3543_max__less__iff__conj,axiom,
% 0.32/0.56 ! [X: num,Y3: num,Z: num] :
% 0.32/0.56 ( ( ord_less_num @ ( ord_max_num @ X @ Y3 ) @ Z )
% 0.32/0.56 = ( ( ord_less_num @ X @ Z )
% 0.32/0.56 & ( ord_less_num @ Y3 @ Z ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_less_iff_conj
% 0.32/0.56 thf(fact_3544_max__less__iff__conj,axiom,
% 0.32/0.56 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.56 ( ( ord_less_nat @ ( ord_max_nat @ X @ Y3 ) @ Z )
% 0.32/0.56 = ( ( ord_less_nat @ X @ Z )
% 0.32/0.56 & ( ord_less_nat @ Y3 @ Z ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_less_iff_conj
% 0.32/0.56 thf(fact_3545_max__less__iff__conj,axiom,
% 0.32/0.56 ! [X: int,Y3: int,Z: int] :
% 0.32/0.56 ( ( ord_less_int @ ( ord_max_int @ X @ Y3 ) @ Z )
% 0.32/0.56 = ( ( ord_less_int @ X @ Z )
% 0.32/0.56 & ( ord_less_int @ Y3 @ Z ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_less_iff_conj
% 0.32/0.56 thf(fact_3546_max__Suc__Suc,axiom,
% 0.32/0.56 ! [M: nat,N: nat] :
% 0.32/0.56 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.56 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_Suc_Suc
% 0.32/0.56 thf(fact_3547_max__nat_Oeq__neutr__iff,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( ( ord_max_nat @ A @ B )
% 0.32/0.56 = zero_zero_nat )
% 0.32/0.56 = ( ( A = zero_zero_nat )
% 0.32/0.56 & ( B = zero_zero_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_nat.eq_neutr_iff
% 0.32/0.56 thf(fact_3548_max__nat_Oleft__neutral,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % max_nat.left_neutral
% 0.32/0.56 thf(fact_3549_max__nat_Oneutr__eq__iff,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( zero_zero_nat
% 0.32/0.56 = ( ord_max_nat @ A @ B ) )
% 0.32/0.56 = ( ( A = zero_zero_nat )
% 0.32/0.56 & ( B = zero_zero_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_nat.neutr_eq_iff
% 0.32/0.56 thf(fact_3550_max__nat_Oright__neutral,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.32/0.56 = A ) ).
% 0.32/0.56
% 0.32/0.56 % max_nat.right_neutral
% 0.32/0.56 thf(fact_3551_max__0L,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.32/0.56 = N ) ).
% 0.32/0.56
% 0.32/0.56 % max_0L
% 0.32/0.56 thf(fact_3552_max__0R,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.32/0.56 = N ) ).
% 0.32/0.56
% 0.32/0.56 % max_0R
% 0.32/0.56 thf(fact_3553_concat__bit__0,axiom,
% 0.32/0.56 ! [K: int,L2: int] :
% 0.32/0.56 ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
% 0.32/0.56 = L2 ) ).
% 0.32/0.56
% 0.32/0.56 % concat_bit_0
% 0.32/0.56 thf(fact_3554_dvd__mult__cancel__left,axiom,
% 0.32/0.56 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.32/0.56 = ( ( C = zero_z3403309356797280102nteger )
% 0.32/0.56 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_left
% 0.32/0.56 thf(fact_3555_dvd__mult__cancel__left,axiom,
% 0.32/0.56 ! [C: complex,A: complex,B: complex] :
% 0.32/0.56 ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.32/0.56 = ( ( C = zero_zero_complex )
% 0.32/0.56 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_left
% 0.32/0.56 thf(fact_3556_dvd__mult__cancel__left,axiom,
% 0.32/0.56 ! [C: real,A: real,B: real] :
% 0.32/0.56 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.32/0.56 = ( ( C = zero_zero_real )
% 0.32/0.56 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_left
% 0.32/0.56 thf(fact_3557_dvd__mult__cancel__left,axiom,
% 0.32/0.56 ! [C: rat,A: rat,B: rat] :
% 0.32/0.56 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.32/0.56 = ( ( C = zero_zero_rat )
% 0.32/0.56 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_left
% 0.32/0.56 thf(fact_3558_dvd__mult__cancel__left,axiom,
% 0.32/0.56 ! [C: int,A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.56 = ( ( C = zero_zero_int )
% 0.32/0.56 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_left
% 0.32/0.56 thf(fact_3559_dvd__mult__cancel__right,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.56 = ( ( C = zero_z3403309356797280102nteger )
% 0.32/0.56 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_right
% 0.32/0.56 thf(fact_3560_dvd__mult__cancel__right,axiom,
% 0.32/0.56 ! [A: complex,C: complex,B: complex] :
% 0.32/0.56 ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.32/0.56 = ( ( C = zero_zero_complex )
% 0.32/0.56 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_right
% 0.32/0.56 thf(fact_3561_dvd__mult__cancel__right,axiom,
% 0.32/0.56 ! [A: real,C: real,B: real] :
% 0.32/0.56 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.32/0.56 = ( ( C = zero_zero_real )
% 0.32/0.56 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_right
% 0.32/0.56 thf(fact_3562_dvd__mult__cancel__right,axiom,
% 0.32/0.56 ! [A: rat,C: rat,B: rat] :
% 0.32/0.56 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.32/0.56 = ( ( C = zero_zero_rat )
% 0.32/0.56 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_right
% 0.32/0.56 thf(fact_3563_dvd__mult__cancel__right,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.32/0.56 = ( ( C = zero_zero_int )
% 0.32/0.56 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_cancel_right
% 0.32/0.56 thf(fact_3564_dvd__times__left__cancel__iff,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_times_left_cancel_iff
% 0.32/0.56 thf(fact_3565_dvd__times__left__cancel__iff,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( A != zero_zero_nat )
% 0.32/0.56 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 0.32/0.56 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_times_left_cancel_iff
% 0.32/0.56 thf(fact_3566_dvd__times__left__cancel__iff,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( A != zero_zero_int )
% 0.32/0.56 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 0.32/0.56 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_times_left_cancel_iff
% 0.32/0.56 thf(fact_3567_dvd__times__right__cancel__iff,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_times_right_cancel_iff
% 0.32/0.56 thf(fact_3568_dvd__times__right__cancel__iff,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( A != zero_zero_nat )
% 0.32/0.56 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 0.32/0.56 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_times_right_cancel_iff
% 0.32/0.56 thf(fact_3569_dvd__times__right__cancel__iff,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( A != zero_zero_int )
% 0.32/0.56 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 0.32/0.56 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_times_right_cancel_iff
% 0.32/0.56 thf(fact_3570_unit__prod,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.56 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_prod
% 0.32/0.56 thf(fact_3571_unit__prod,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.56 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_prod
% 0.32/0.56 thf(fact_3572_unit__prod,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.56 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.56 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_prod
% 0.32/0.56 thf(fact_3573_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_right_iff
% 0.32/0.56 thf(fact_3574_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.56 ! [A: real,B: real,C: real] :
% 0.32/0.56 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 0.32/0.56 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_right_iff
% 0.32/0.56 thf(fact_3575_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.56 ! [A: rat,B: rat,C: rat] :
% 0.32/0.56 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 0.32/0.56 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_right_iff
% 0.32/0.56 thf(fact_3576_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.56 ! [A: nat,B: nat,C: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 0.32/0.56 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_right_iff
% 0.32/0.56 thf(fact_3577_dvd__add__times__triv__right__iff,axiom,
% 0.32/0.56 ! [A: int,B: int,C: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 0.32/0.56 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_right_iff
% 0.32/0.56 thf(fact_3578_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.56 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_left_iff
% 0.32/0.56 thf(fact_3579_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.56 ! [A: real,C: real,B: real] :
% 0.32/0.56 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 0.32/0.56 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_left_iff
% 0.32/0.56 thf(fact_3580_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.56 ! [A: rat,C: rat,B: rat] :
% 0.32/0.56 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 0.32/0.56 = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_left_iff
% 0.32/0.56 thf(fact_3581_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.56 ! [A: nat,C: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 0.32/0.56 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_left_iff
% 0.32/0.56 thf(fact_3582_dvd__add__times__triv__left__iff,axiom,
% 0.32/0.56 ! [A: int,C: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 0.32/0.56 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_add_times_triv_left_iff
% 0.32/0.56 thf(fact_3583_dvd__div__mult__self,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.56 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_div_mult_self
% 0.32/0.56 thf(fact_3584_dvd__div__mult__self,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.56 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_div_mult_self
% 0.32/0.56 thf(fact_3585_dvd__div__mult__self,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.56 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_div_mult_self
% 0.32/0.56 thf(fact_3586_dvd__mult__div__cancel,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.56 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_div_cancel
% 0.32/0.56 thf(fact_3587_dvd__mult__div__cancel,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.56 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_div_cancel
% 0.32/0.56 thf(fact_3588_dvd__mult__div__cancel,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.56 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_mult_div_cancel
% 0.32/0.56 thf(fact_3589_unit__div__1__div__1,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.56 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_1_div_1
% 0.32/0.56 thf(fact_3590_unit__div__1__div__1,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_1_div_1
% 0.32/0.56 thf(fact_3591_unit__div__1__div__1,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.56 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.32/0.56 = A ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_1_div_1
% 0.32/0.56 thf(fact_3592_unit__div__1__unit,axiom,
% 0.32/0.56 ! [A: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.56 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_1_unit
% 0.32/0.56 thf(fact_3593_unit__div__1__unit,axiom,
% 0.32/0.56 ! [A: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.56 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_1_unit
% 0.32/0.56 thf(fact_3594_unit__div__1__unit,axiom,
% 0.32/0.56 ! [A: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.56 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_1_unit
% 0.32/0.56 thf(fact_3595_unit__div,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.56 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div
% 0.32/0.56 thf(fact_3596_unit__div,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.56 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div
% 0.32/0.56 thf(fact_3597_unit__div,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.56 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.56 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div
% 0.32/0.56 thf(fact_3598_div__add,axiom,
% 0.32/0.56 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.56 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.56 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_add
% 0.32/0.56 thf(fact_3599_div__add,axiom,
% 0.32/0.56 ! [C: nat,A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.56 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.56 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.56 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_add
% 0.32/0.56 thf(fact_3600_div__add,axiom,
% 0.32/0.56 ! [C: int,A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.56 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.56 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.56 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_add
% 0.32/0.56 thf(fact_3601_div__diff,axiom,
% 0.32/0.56 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.56 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.56 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.32/0.56 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_diff
% 0.32/0.56 thf(fact_3602_div__diff,axiom,
% 0.32/0.56 ! [C: int,A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.56 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.56 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.32/0.56 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % div_diff
% 0.32/0.56 thf(fact_3603_dvd__imp__mod__0,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.56 => ( ( modulo_modulo_nat @ B @ A )
% 0.32/0.56 = zero_zero_nat ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_imp_mod_0
% 0.32/0.56 thf(fact_3604_dvd__imp__mod__0,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.56 => ( ( modulo_modulo_int @ B @ A )
% 0.32/0.56 = zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_imp_mod_0
% 0.32/0.56 thf(fact_3605_dvd__imp__mod__0,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.56 => ( ( modulo364778990260209775nteger @ B @ A )
% 0.32/0.56 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.56
% 0.32/0.56 % dvd_imp_mod_0
% 0.32/0.56 thf(fact_3606_max__number__of_I1_J,axiom,
% 0.32/0.56 ! [U: num,V: num] :
% 0.32/0.56 ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.32/0.56 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.32/0.56 = ( numera1916890842035813515d_enat @ V ) ) )
% 0.32/0.56 & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.32/0.56 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.32/0.56 = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_number_of(1)
% 0.32/0.56 thf(fact_3607_max__number__of_I1_J,axiom,
% 0.32/0.56 ! [U: num,V: num] :
% 0.32/0.56 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.56 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.56 = ( numeral_numeral_real @ V ) ) )
% 0.32/0.56 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.56 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.56 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_number_of(1)
% 0.32/0.56 thf(fact_3608_max__number__of_I1_J,axiom,
% 0.32/0.56 ! [U: num,V: num] :
% 0.32/0.56 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.56 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.56 = ( numeral_numeral_rat @ V ) ) )
% 0.32/0.56 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.56 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.56 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_number_of(1)
% 0.32/0.56 thf(fact_3609_max__number__of_I1_J,axiom,
% 0.32/0.56 ! [U: num,V: num] :
% 0.32/0.56 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.32/0.56 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.32/0.56 = ( numeral_numeral_nat @ V ) ) )
% 0.32/0.56 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.32/0.56 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.32/0.56 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_number_of(1)
% 0.32/0.56 thf(fact_3610_max__number__of_I1_J,axiom,
% 0.32/0.56 ! [U: num,V: num] :
% 0.32/0.56 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.56 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.56 = ( numeral_numeral_int @ V ) ) )
% 0.32/0.56 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.56 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.56 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_number_of(1)
% 0.32/0.56 thf(fact_3611_max__0__1_I4_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
% 0.32/0.56 = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(4)
% 0.32/0.56 thf(fact_3612_max__0__1_I4_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 0.32/0.56 = ( numeral_numeral_real @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(4)
% 0.32/0.56 thf(fact_3613_max__0__1_I4_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 0.32/0.56 = ( numeral_numeral_rat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(4)
% 0.32/0.56 thf(fact_3614_max__0__1_I4_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 0.32/0.56 = ( numeral_numeral_nat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(4)
% 0.32/0.56 thf(fact_3615_max__0__1_I4_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 0.32/0.56 = ( numeral_numeral_int @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(4)
% 0.32/0.56 thf(fact_3616_max__0__1_I3_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 0.32/0.56 = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(3)
% 0.32/0.56 thf(fact_3617_max__0__1_I3_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 0.32/0.56 = ( numeral_numeral_real @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(3)
% 0.32/0.56 thf(fact_3618_max__0__1_I3_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 0.32/0.56 = ( numeral_numeral_rat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(3)
% 0.32/0.56 thf(fact_3619_max__0__1_I3_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 0.32/0.56 = ( numeral_numeral_nat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(3)
% 0.32/0.56 thf(fact_3620_max__0__1_I3_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 0.32/0.56 = ( numeral_numeral_int @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(3)
% 0.32/0.56 thf(fact_3621_max__0__1_I1_J,axiom,
% 0.32/0.56 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 0.32/0.56 = one_one_real ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(1)
% 0.32/0.56 thf(fact_3622_max__0__1_I1_J,axiom,
% 0.32/0.56 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 0.32/0.56 = one_one_rat ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(1)
% 0.32/0.56 thf(fact_3623_max__0__1_I1_J,axiom,
% 0.32/0.56 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.32/0.56 = one_one_nat ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(1)
% 0.32/0.56 thf(fact_3624_max__0__1_I1_J,axiom,
% 0.32/0.56 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
% 0.32/0.56 = one_on7984719198319812577d_enat ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(1)
% 0.32/0.56 thf(fact_3625_max__0__1_I1_J,axiom,
% 0.32/0.56 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.32/0.56 = one_one_int ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(1)
% 0.32/0.56 thf(fact_3626_max__0__1_I2_J,axiom,
% 0.32/0.56 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 0.32/0.56 = one_one_real ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(2)
% 0.32/0.56 thf(fact_3627_max__0__1_I2_J,axiom,
% 0.32/0.56 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 0.32/0.56 = one_one_rat ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(2)
% 0.32/0.56 thf(fact_3628_max__0__1_I2_J,axiom,
% 0.32/0.56 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.32/0.56 = one_one_nat ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(2)
% 0.32/0.56 thf(fact_3629_max__0__1_I2_J,axiom,
% 0.32/0.56 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
% 0.32/0.56 = one_on7984719198319812577d_enat ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(2)
% 0.32/0.56 thf(fact_3630_max__0__1_I2_J,axiom,
% 0.32/0.56 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.32/0.56 = one_one_int ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(2)
% 0.32/0.56 thf(fact_3631_max__0__1_I6_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
% 0.32/0.56 = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(6)
% 0.32/0.56 thf(fact_3632_max__0__1_I6_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
% 0.32/0.56 = ( numeral_numeral_real @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(6)
% 0.32/0.56 thf(fact_3633_max__0__1_I6_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
% 0.32/0.56 = ( numeral_numeral_rat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(6)
% 0.32/0.56 thf(fact_3634_max__0__1_I6_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
% 0.32/0.56 = ( numeral_numeral_nat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(6)
% 0.32/0.56 thf(fact_3635_max__0__1_I6_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
% 0.32/0.56 = ( numeral_numeral_int @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(6)
% 0.32/0.56 thf(fact_3636_max__0__1_I5_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 0.32/0.56 = ( numera1916890842035813515d_enat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(5)
% 0.32/0.56 thf(fact_3637_max__0__1_I5_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 0.32/0.56 = ( numeral_numeral_real @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(5)
% 0.32/0.56 thf(fact_3638_max__0__1_I5_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 0.32/0.56 = ( numeral_numeral_rat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(5)
% 0.32/0.56 thf(fact_3639_max__0__1_I5_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 0.32/0.56 = ( numeral_numeral_nat @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(5)
% 0.32/0.56 thf(fact_3640_max__0__1_I5_J,axiom,
% 0.32/0.56 ! [X: num] :
% 0.32/0.56 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 0.32/0.56 = ( numeral_numeral_int @ X ) ) ).
% 0.32/0.56
% 0.32/0.56 % max_0_1(5)
% 0.32/0.56 thf(fact_3641_list__update__beyond,axiom,
% 0.32/0.56 ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
% 0.32/0.56 => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
% 0.32/0.56 = Xs2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_update_beyond
% 0.32/0.56 thf(fact_3642_list__update__beyond,axiom,
% 0.32/0.56 ! [Xs2: list_o,I: nat,X: $o] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
% 0.32/0.56 => ( ( list_update_o @ Xs2 @ I @ X )
% 0.32/0.56 = Xs2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_update_beyond
% 0.32/0.56 thf(fact_3643_list__update__beyond,axiom,
% 0.32/0.56 ! [Xs2: list_nat,I: nat,X: nat] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
% 0.32/0.56 => ( ( list_update_nat @ Xs2 @ I @ X )
% 0.32/0.56 = Xs2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_update_beyond
% 0.32/0.56 thf(fact_3644_list__update__beyond,axiom,
% 0.32/0.56 ! [Xs2: list_int,I: nat,X: int] :
% 0.32/0.56 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
% 0.32/0.56 => ( ( list_update_int @ Xs2 @ I @ X )
% 0.32/0.56 = Xs2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % list_update_beyond
% 0.32/0.56 thf(fact_3645_concat__bit__nonnegative__iff,axiom,
% 0.32/0.56 ! [N: nat,K: int,L2: int] :
% 0.32/0.56 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L2 ) )
% 0.32/0.56 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).
% 0.32/0.56
% 0.32/0.56 % concat_bit_nonnegative_iff
% 0.32/0.56 thf(fact_3646_concat__bit__negative__iff,axiom,
% 0.32/0.56 ! [N: nat,K: int,L2: int] :
% 0.32/0.56 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L2 ) @ zero_zero_int )
% 0.32/0.56 = ( ord_less_int @ L2 @ zero_zero_int ) ) ).
% 0.32/0.56
% 0.32/0.56 % concat_bit_negative_iff
% 0.32/0.56 thf(fact_3647_unit__div__mult__self,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.56 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_mult_self
% 0.32/0.56 thf(fact_3648_unit__div__mult__self,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_mult_self
% 0.32/0.56 thf(fact_3649_unit__div__mult__self,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.56 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.32/0.56 = B ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_div_mult_self
% 0.32/0.56 thf(fact_3650_unit__mult__div__div,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.56 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.32/0.56 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_mult_div_div
% 0.32/0.56 thf(fact_3651_unit__mult__div__div,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.56 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.32/0.56 = ( divide_divide_nat @ B @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_mult_div_div
% 0.32/0.56 thf(fact_3652_unit__mult__div__div,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.56 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 0.32/0.56 = ( divide_divide_int @ B @ A ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % unit_mult_div_div
% 0.32/0.56 thf(fact_3653_even__Suc__Suc__iff,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 0.32/0.56 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_Suc_Suc_iff
% 0.32/0.56 thf(fact_3654_even__Suc,axiom,
% 0.32/0.56 ! [N: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 0.32/0.56 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_Suc
% 0.32/0.56 thf(fact_3655_pow__divides__pow__iff,axiom,
% 0.32/0.56 ! [N: nat,A: nat,B: nat] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 0.32/0.56 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pow_divides_pow_iff
% 0.32/0.56 thf(fact_3656_pow__divides__pow__iff,axiom,
% 0.32/0.56 ! [N: nat,A: int,B: int] :
% 0.32/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.56 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 0.32/0.56 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % pow_divides_pow_iff
% 0.32/0.56 thf(fact_3657_nth__list__update__eq,axiom,
% 0.32/0.56 ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.32/0.56 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.56 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I )
% 0.32/0.56 = X ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_list_update_eq
% 0.32/0.56 thf(fact_3658_nth__list__update__eq,axiom,
% 0.32/0.56 ! [I: nat,Xs2: list_o,X: $o] :
% 0.32/0.56 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.56 => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ I )
% 0.32/0.56 = X ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_list_update_eq
% 0.32/0.56 thf(fact_3659_nth__list__update__eq,axiom,
% 0.32/0.56 ! [I: nat,Xs2: list_nat,X: nat] :
% 0.32/0.56 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.56 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
% 0.32/0.56 = X ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_list_update_eq
% 0.32/0.56 thf(fact_3660_nth__list__update__eq,axiom,
% 0.32/0.56 ! [I: nat,Xs2: list_int,X: int] :
% 0.32/0.56 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.56 => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ I )
% 0.32/0.56 = X ) ) ).
% 0.32/0.56
% 0.32/0.56 % nth_list_update_eq
% 0.32/0.56 thf(fact_3661_even__mult__iff,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.32/0.56 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mult_iff
% 0.32/0.56 thf(fact_3662_even__mult__iff,axiom,
% 0.32/0.56 ! [A: nat,B: nat] :
% 0.32/0.56 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 0.32/0.56 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mult_iff
% 0.32/0.56 thf(fact_3663_even__mult__iff,axiom,
% 0.32/0.56 ! [A: int,B: int] :
% 0.32/0.56 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 0.32/0.56 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_mult_iff
% 0.32/0.56 thf(fact_3664_even__add,axiom,
% 0.32/0.56 ! [A: code_integer,B: code_integer] :
% 0.32/0.56 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.56 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.56 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.56
% 0.32/0.56 % even_add
% 0.32/0.57 thf(fact_3665_even__add,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.57 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_add
% 0.32/0.57 thf(fact_3666_even__add,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 0.32/0.57 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_add
% 0.32/0.57 thf(fact_3667_odd__add,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 0.32/0.57 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.57 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_add
% 0.32/0.57 thf(fact_3668_odd__add,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 0.32/0.57 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.57 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_add
% 0.32/0.57 thf(fact_3669_odd__add,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 0.32/0.57 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.57 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_add
% 0.32/0.57 thf(fact_3670_even__mod__2__iff,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.57 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_mod_2_iff
% 0.32/0.57 thf(fact_3671_even__mod__2__iff,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.57 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_mod_2_iff
% 0.32/0.57 thf(fact_3672_even__mod__2__iff,axiom,
% 0.32/0.57 ! [A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_mod_2_iff
% 0.32/0.57 thf(fact_3673_even__Suc__div__two,axiom,
% 0.32/0.57 ! [N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.57 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_Suc_div_two
% 0.32/0.57 thf(fact_3674_odd__Suc__div__two,axiom,
% 0.32/0.57 ! [N: nat] :
% 0.32/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.57 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_Suc_div_two
% 0.32/0.57 thf(fact_3675_set__swap,axiom,
% 0.32/0.57 ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
% 0.32/0.57 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.57 => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.57 => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
% 0.32/0.57 = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_swap
% 0.32/0.57 thf(fact_3676_set__swap,axiom,
% 0.32/0.57 ! [I: nat,Xs2: list_o,J: nat] :
% 0.32/0.57 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.57 => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.57 => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
% 0.32/0.57 = ( set_o2 @ Xs2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_swap
% 0.32/0.57 thf(fact_3677_set__swap,axiom,
% 0.32/0.57 ! [I: nat,Xs2: list_nat,J: nat] :
% 0.32/0.57 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.57 => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.57 => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
% 0.32/0.57 = ( set_nat2 @ Xs2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_swap
% 0.32/0.57 thf(fact_3678_set__swap,axiom,
% 0.32/0.57 ! [I: nat,Xs2: list_int,J: nat] :
% 0.32/0.57 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.57 => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.57 => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
% 0.32/0.57 = ( set_int2 @ Xs2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_swap
% 0.32/0.57 thf(fact_3679_zero__le__power__eq__numeral,axiom,
% 0.32/0.57 ! [A: real,W: num] :
% 0.32/0.57 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.57 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zero_le_power_eq_numeral
% 0.32/0.57 thf(fact_3680_zero__le__power__eq__numeral,axiom,
% 0.32/0.57 ! [A: rat,W: num] :
% 0.32/0.57 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.57 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zero_le_power_eq_numeral
% 0.32/0.57 thf(fact_3681_zero__le__power__eq__numeral,axiom,
% 0.32/0.57 ! [A: int,W: num] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.57 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zero_le_power_eq_numeral
% 0.32/0.57 thf(fact_3682_power__less__zero__eq__numeral,axiom,
% 0.32/0.57 ! [A: real,W: num] :
% 0.32/0.57 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_less_zero_eq_numeral
% 0.32/0.57 thf(fact_3683_power__less__zero__eq__numeral,axiom,
% 0.32/0.57 ! [A: rat,W: num] :
% 0.32/0.57 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_less_zero_eq_numeral
% 0.32/0.57 thf(fact_3684_power__less__zero__eq__numeral,axiom,
% 0.32/0.57 ! [A: int,W: num] :
% 0.32/0.57 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_less_zero_eq_numeral
% 0.32/0.57 thf(fact_3685_power__less__zero__eq,axiom,
% 0.32/0.57 ! [A: real,N: nat] :
% 0.32/0.57 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.57 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_less_zero_eq
% 0.32/0.57 thf(fact_3686_power__less__zero__eq,axiom,
% 0.32/0.57 ! [A: rat,N: nat] :
% 0.32/0.57 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.57 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_less_zero_eq
% 0.32/0.57 thf(fact_3687_power__less__zero__eq,axiom,
% 0.32/0.57 ! [A: int,N: nat] :
% 0.32/0.57 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.57 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_less_zero_eq
% 0.32/0.57 thf(fact_3688_even__plus__one__iff,axiom,
% 0.32/0.57 ! [A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_plus_one_iff
% 0.32/0.57 thf(fact_3689_even__plus__one__iff,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_plus_one_iff
% 0.32/0.57 thf(fact_3690_even__plus__one__iff,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_plus_one_iff
% 0.32/0.57 thf(fact_3691_even__diff,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_diff
% 0.32/0.57 thf(fact_3692_even__diff,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 0.32/0.57 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_diff
% 0.32/0.57 thf(fact_3693_odd__Suc__minus__one,axiom,
% 0.32/0.57 ! [N: nat] :
% 0.32/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.57 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.32/0.57 = N ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_Suc_minus_one
% 0.32/0.57 thf(fact_3694_even__diff__nat,axiom,
% 0.32/0.57 ! [M: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.57 = ( ( ord_less_nat @ M @ N )
% 0.32/0.57 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_diff_nat
% 0.32/0.57 thf(fact_3695_zero__less__power__eq__numeral,axiom,
% 0.32/0.57 ! [A: real,W: num] :
% 0.32/0.57 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.57 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( A != zero_zero_real ) )
% 0.32/0.57 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zero_less_power_eq_numeral
% 0.32/0.57 thf(fact_3696_zero__less__power__eq__numeral,axiom,
% 0.32/0.57 ! [A: rat,W: num] :
% 0.32/0.57 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.57 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( A != zero_zero_rat ) )
% 0.32/0.57 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zero_less_power_eq_numeral
% 0.32/0.57 thf(fact_3697_zero__less__power__eq__numeral,axiom,
% 0.32/0.57 ! [A: int,W: num] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 0.32/0.57 = ( ( ( numeral_numeral_nat @ W )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( A != zero_zero_int ) )
% 0.32/0.57 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zero_less_power_eq_numeral
% 0.32/0.57 thf(fact_3698_even__succ__div__2,axiom,
% 0.32/0.57 ! [A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_2
% 0.32/0.57 thf(fact_3699_even__succ__div__2,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_2
% 0.32/0.57 thf(fact_3700_even__succ__div__2,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_2
% 0.32/0.57 thf(fact_3701_even__succ__div__two,axiom,
% 0.32/0.57 ! [A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_two
% 0.32/0.57 thf(fact_3702_even__succ__div__two,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_two
% 0.32/0.57 thf(fact_3703_even__succ__div__two,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_two
% 0.32/0.57 thf(fact_3704_odd__succ__div__two,axiom,
% 0.32/0.57 ! [A: code_integer] :
% 0.32/0.57 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_succ_div_two
% 0.32/0.57 thf(fact_3705_odd__succ__div__two,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_succ_div_two
% 0.32/0.57 thf(fact_3706_odd__succ__div__two,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_succ_div_two
% 0.32/0.57 thf(fact_3707_even__power,axiom,
% 0.32/0.57 ! [A: code_integer,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.32/0.57 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_power
% 0.32/0.57 thf(fact_3708_even__power,axiom,
% 0.32/0.57 ! [A: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 0.32/0.57 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_power
% 0.32/0.57 thf(fact_3709_even__power,axiom,
% 0.32/0.57 ! [A: int,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 0.32/0.57 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_power
% 0.32/0.57 thf(fact_3710_odd__two__times__div__two__nat,axiom,
% 0.32/0.57 ! [N: nat] :
% 0.32/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.57 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.57 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_two_times_div_two_nat
% 0.32/0.57 thf(fact_3711_odd__two__times__div__two__succ,axiom,
% 0.32/0.57 ! [A: code_integer] :
% 0.32/0.57 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.32/0.57 = A ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_two_times_div_two_succ
% 0.32/0.57 thf(fact_3712_odd__two__times__div__two__succ,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.32/0.57 = A ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_two_times_div_two_succ
% 0.32/0.57 thf(fact_3713_odd__two__times__div__two__succ,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.32/0.57 = A ) ) ).
% 0.32/0.57
% 0.32/0.57 % odd_two_times_div_two_succ
% 0.32/0.57 thf(fact_3714_power__le__zero__eq__numeral,axiom,
% 0.32/0.57 ! [A: real,W: num] :
% 0.32/0.57 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 0.32/0.57 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.32/0.57 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_le_zero_eq_numeral
% 0.32/0.57 thf(fact_3715_power__le__zero__eq__numeral,axiom,
% 0.32/0.57 ! [A: rat,W: num] :
% 0.32/0.57 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 0.32/0.57 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.32/0.57 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_le_zero_eq_numeral
% 0.32/0.57 thf(fact_3716_power__le__zero__eq__numeral,axiom,
% 0.32/0.57 ! [A: int,W: num] :
% 0.32/0.57 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 0.32/0.57 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.32/0.57 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.57 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_le_zero_eq_numeral
% 0.32/0.57 thf(fact_3717_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.57 ! [N: nat] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 0.32/0.57 = ( N = zero_zero_nat ) ) ).
% 0.32/0.57
% 0.32/0.57 % semiring_parity_class.even_mask_iff
% 0.32/0.57 thf(fact_3718_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.57 ! [N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 0.32/0.57 = ( N = zero_zero_nat ) ) ).
% 0.32/0.57
% 0.32/0.57 % semiring_parity_class.even_mask_iff
% 0.32/0.57 thf(fact_3719_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.32/0.57 ! [N: nat] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.32/0.57 = ( N = zero_zero_nat ) ) ).
% 0.32/0.57
% 0.32/0.57 % semiring_parity_class.even_mask_iff
% 0.32/0.57 thf(fact_3720_even__succ__div__exp,axiom,
% 0.32/0.57 ! [A: code_integer,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.57 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_exp
% 0.32/0.57 thf(fact_3721_even__succ__div__exp,axiom,
% 0.32/0.57 ! [A: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.57 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.57 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_exp
% 0.32/0.57 thf(fact_3722_even__succ__div__exp,axiom,
% 0.32/0.57 ! [A: int,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.57 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.57 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_succ_div_exp
% 0.32/0.57 thf(fact_3723_division__decomp,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.57 => ? [B8: nat,C5: nat] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( times_times_nat @ B8 @ C5 ) )
% 0.32/0.57 & ( dvd_dvd_nat @ B8 @ B )
% 0.32/0.57 & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % division_decomp
% 0.32/0.57 thf(fact_3724_division__decomp,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.57 => ? [B8: int,C5: int] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( times_times_int @ B8 @ C5 ) )
% 0.32/0.57 & ( dvd_dvd_int @ B8 @ B )
% 0.32/0.57 & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % division_decomp
% 0.32/0.57 thf(fact_3725_dvd__productE,axiom,
% 0.32/0.57 ! [P6: nat,A: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A @ B ) )
% 0.32/0.57 => ~ ! [X3: nat,Y4: nat] :
% 0.32/0.57 ( ( P6
% 0.32/0.57 = ( times_times_nat @ X3 @ Y4 ) )
% 0.32/0.57 => ( ( dvd_dvd_nat @ X3 @ A )
% 0.32/0.57 => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_productE
% 0.32/0.57 thf(fact_3726_dvd__productE,axiom,
% 0.32/0.57 ! [P6: int,A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ P6 @ ( times_times_int @ A @ B ) )
% 0.32/0.57 => ~ ! [X3: int,Y4: int] :
% 0.32/0.57 ( ( P6
% 0.32/0.57 = ( times_times_int @ X3 @ Y4 ) )
% 0.32/0.57 => ( ( dvd_dvd_int @ X3 @ A )
% 0.32/0.57 => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_productE
% 0.32/0.57 thf(fact_3727_dvd__trans,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ B @ C )
% 0.32/0.57 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_trans
% 0.32/0.57 thf(fact_3728_dvd__trans,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ B @ C )
% 0.32/0.57 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_trans
% 0.32/0.57 thf(fact_3729_dvd__trans,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ B @ C )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_trans
% 0.32/0.57 thf(fact_3730_dvd__refl,axiom,
% 0.32/0.57 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_refl
% 0.32/0.57 thf(fact_3731_dvd__refl,axiom,
% 0.32/0.57 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_refl
% 0.32/0.57 thf(fact_3732_dvd__refl,axiom,
% 0.32/0.57 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_refl
% 0.32/0.57 thf(fact_3733_max__absorb2,axiom,
% 0.32/0.57 ! [X: extended_enat,Y3: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ X @ Y3 )
% 0.32/0.57 => ( ( ord_ma741700101516333627d_enat @ X @ Y3 )
% 0.32/0.57 = Y3 ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb2
% 0.32/0.57 thf(fact_3734_max__absorb2,axiom,
% 0.32/0.57 ! [X: set_int,Y3: set_int] :
% 0.32/0.57 ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.57 => ( ( ord_max_set_int @ X @ Y3 )
% 0.32/0.57 = Y3 ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb2
% 0.32/0.57 thf(fact_3735_max__absorb2,axiom,
% 0.32/0.57 ! [X: rat,Y3: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.57 => ( ( ord_max_rat @ X @ Y3 )
% 0.32/0.57 = Y3 ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb2
% 0.32/0.57 thf(fact_3736_max__absorb2,axiom,
% 0.32/0.57 ! [X: num,Y3: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.57 => ( ( ord_max_num @ X @ Y3 )
% 0.32/0.57 = Y3 ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb2
% 0.32/0.57 thf(fact_3737_max__absorb2,axiom,
% 0.32/0.57 ! [X: nat,Y3: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.57 => ( ( ord_max_nat @ X @ Y3 )
% 0.32/0.57 = Y3 ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb2
% 0.32/0.57 thf(fact_3738_max__absorb2,axiom,
% 0.32/0.57 ! [X: int,Y3: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.57 => ( ( ord_max_int @ X @ Y3 )
% 0.32/0.57 = Y3 ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb2
% 0.32/0.57 thf(fact_3739_max__absorb1,axiom,
% 0.32/0.57 ! [Y3: extended_enat,X: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ Y3 @ X )
% 0.32/0.57 => ( ( ord_ma741700101516333627d_enat @ X @ Y3 )
% 0.32/0.57 = X ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb1
% 0.32/0.57 thf(fact_3740_max__absorb1,axiom,
% 0.32/0.57 ! [Y3: set_int,X: set_int] :
% 0.32/0.57 ( ( ord_less_eq_set_int @ Y3 @ X )
% 0.32/0.57 => ( ( ord_max_set_int @ X @ Y3 )
% 0.32/0.57 = X ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb1
% 0.32/0.57 thf(fact_3741_max__absorb1,axiom,
% 0.32/0.57 ! [Y3: rat,X: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ Y3 @ X )
% 0.32/0.57 => ( ( ord_max_rat @ X @ Y3 )
% 0.32/0.57 = X ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb1
% 0.32/0.57 thf(fact_3742_max__absorb1,axiom,
% 0.32/0.57 ! [Y3: num,X: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ Y3 @ X )
% 0.32/0.57 => ( ( ord_max_num @ X @ Y3 )
% 0.32/0.57 = X ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb1
% 0.32/0.57 thf(fact_3743_max__absorb1,axiom,
% 0.32/0.57 ! [Y3: nat,X: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ Y3 @ X )
% 0.32/0.57 => ( ( ord_max_nat @ X @ Y3 )
% 0.32/0.57 = X ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb1
% 0.32/0.57 thf(fact_3744_max__absorb1,axiom,
% 0.32/0.57 ! [Y3: int,X: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ Y3 @ X )
% 0.32/0.57 => ( ( ord_max_int @ X @ Y3 )
% 0.32/0.57 = X ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_absorb1
% 0.32/0.57 thf(fact_3745_max__def,axiom,
% 0.32/0.57 ( ord_ma741700101516333627d_enat
% 0.32/0.57 = ( ^ [A3: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def
% 0.32/0.57 thf(fact_3746_max__def,axiom,
% 0.32/0.57 ( ord_max_set_int
% 0.32/0.57 = ( ^ [A3: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def
% 0.32/0.57 thf(fact_3747_max__def,axiom,
% 0.32/0.57 ( ord_max_rat
% 0.32/0.57 = ( ^ [A3: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def
% 0.32/0.57 thf(fact_3748_max__def,axiom,
% 0.32/0.57 ( ord_max_num
% 0.32/0.57 = ( ^ [A3: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def
% 0.32/0.57 thf(fact_3749_max__def,axiom,
% 0.32/0.57 ( ord_max_nat
% 0.32/0.57 = ( ^ [A3: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def
% 0.32/0.57 thf(fact_3750_max__def,axiom,
% 0.32/0.57 ( ord_max_int
% 0.32/0.57 = ( ^ [A3: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def
% 0.32/0.57 thf(fact_3751_max_OcoboundedI2,axiom,
% 0.32/0.57 ! [C: extended_enat,B: extended_enat,A: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ C @ B )
% 0.32/0.57 => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI2
% 0.32/0.57 thf(fact_3752_max_OcoboundedI2,axiom,
% 0.32/0.57 ! [C: rat,B: rat,A: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ C @ B )
% 0.32/0.57 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI2
% 0.32/0.57 thf(fact_3753_max_OcoboundedI2,axiom,
% 0.32/0.57 ! [C: num,B: num,A: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ C @ B )
% 0.32/0.57 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI2
% 0.32/0.57 thf(fact_3754_max_OcoboundedI2,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ C @ B )
% 0.32/0.57 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI2
% 0.32/0.57 thf(fact_3755_max_OcoboundedI2,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ C @ B )
% 0.32/0.57 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI2
% 0.32/0.57 thf(fact_3756_max_OcoboundedI1,axiom,
% 0.32/0.57 ! [C: extended_enat,A: extended_enat,B: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.32/0.57 => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI1
% 0.32/0.57 thf(fact_3757_max_OcoboundedI1,axiom,
% 0.32/0.57 ! [C: rat,A: rat,B: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ C @ A )
% 0.32/0.57 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI1
% 0.32/0.57 thf(fact_3758_max_OcoboundedI1,axiom,
% 0.32/0.57 ! [C: num,A: num,B: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ C @ A )
% 0.32/0.57 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI1
% 0.32/0.57 thf(fact_3759_max_OcoboundedI1,axiom,
% 0.32/0.57 ! [C: nat,A: nat,B: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.57 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI1
% 0.32/0.57 thf(fact_3760_max_OcoboundedI1,axiom,
% 0.32/0.57 ! [C: int,A: int,B: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ C @ A )
% 0.32/0.57 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.coboundedI1
% 0.32/0.57 thf(fact_3761_max_Oabsorb__iff2,axiom,
% 0.32/0.57 ( ord_le2932123472753598470d_enat
% 0.32/0.57 = ( ^ [A3: extended_enat,B4: extended_enat] :
% 0.32/0.57 ( ( ord_ma741700101516333627d_enat @ A3 @ B4 )
% 0.32/0.57 = B4 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff2
% 0.32/0.57 thf(fact_3762_max_Oabsorb__iff2,axiom,
% 0.32/0.57 ( ord_less_eq_rat
% 0.32/0.57 = ( ^ [A3: rat,B4: rat] :
% 0.32/0.57 ( ( ord_max_rat @ A3 @ B4 )
% 0.32/0.57 = B4 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff2
% 0.32/0.57 thf(fact_3763_max_Oabsorb__iff2,axiom,
% 0.32/0.57 ( ord_less_eq_num
% 0.32/0.57 = ( ^ [A3: num,B4: num] :
% 0.32/0.57 ( ( ord_max_num @ A3 @ B4 )
% 0.32/0.57 = B4 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff2
% 0.32/0.57 thf(fact_3764_max_Oabsorb__iff2,axiom,
% 0.32/0.57 ( ord_less_eq_nat
% 0.32/0.57 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.57 ( ( ord_max_nat @ A3 @ B4 )
% 0.32/0.57 = B4 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff2
% 0.32/0.57 thf(fact_3765_max_Oabsorb__iff2,axiom,
% 0.32/0.57 ( ord_less_eq_int
% 0.32/0.57 = ( ^ [A3: int,B4: int] :
% 0.32/0.57 ( ( ord_max_int @ A3 @ B4 )
% 0.32/0.57 = B4 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff2
% 0.32/0.57 thf(fact_3766_max_Oabsorb__iff1,axiom,
% 0.32/0.57 ( ord_le2932123472753598470d_enat
% 0.32/0.57 = ( ^ [B4: extended_enat,A3: extended_enat] :
% 0.32/0.57 ( ( ord_ma741700101516333627d_enat @ A3 @ B4 )
% 0.32/0.57 = A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff1
% 0.32/0.57 thf(fact_3767_max_Oabsorb__iff1,axiom,
% 0.32/0.57 ( ord_less_eq_rat
% 0.32/0.57 = ( ^ [B4: rat,A3: rat] :
% 0.32/0.57 ( ( ord_max_rat @ A3 @ B4 )
% 0.32/0.57 = A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff1
% 0.32/0.57 thf(fact_3768_max_Oabsorb__iff1,axiom,
% 0.32/0.57 ( ord_less_eq_num
% 0.32/0.57 = ( ^ [B4: num,A3: num] :
% 0.32/0.57 ( ( ord_max_num @ A3 @ B4 )
% 0.32/0.57 = A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff1
% 0.32/0.57 thf(fact_3769_max_Oabsorb__iff1,axiom,
% 0.32/0.57 ( ord_less_eq_nat
% 0.32/0.57 = ( ^ [B4: nat,A3: nat] :
% 0.32/0.57 ( ( ord_max_nat @ A3 @ B4 )
% 0.32/0.57 = A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff1
% 0.32/0.57 thf(fact_3770_max_Oabsorb__iff1,axiom,
% 0.32/0.57 ( ord_less_eq_int
% 0.32/0.57 = ( ^ [B4: int,A3: int] :
% 0.32/0.57 ( ( ord_max_int @ A3 @ B4 )
% 0.32/0.57 = A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.absorb_iff1
% 0.32/0.57 thf(fact_3771_le__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: extended_enat,X: extended_enat,Y3: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_le2932123472753598470d_enat @ Z @ X )
% 0.32/0.57 | ( ord_le2932123472753598470d_enat @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_max_iff_disj
% 0.32/0.57 thf(fact_3772_le__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_eq_rat @ Z @ X )
% 0.32/0.57 | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_max_iff_disj
% 0.32/0.57 thf(fact_3773_le__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: num,X: num,Y3: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_eq_num @ Z @ X )
% 0.32/0.57 | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_max_iff_disj
% 0.32/0.57 thf(fact_3774_le__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: nat,X: nat,Y3: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_eq_nat @ Z @ X )
% 0.32/0.57 | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_max_iff_disj
% 0.32/0.57 thf(fact_3775_le__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: int,X: int,Y3: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_eq_int @ Z @ X )
% 0.32/0.57 | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_max_iff_disj
% 0.32/0.57 thf(fact_3776_max_Ocobounded2,axiom,
% 0.32/0.57 ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded2
% 0.32/0.57 thf(fact_3777_max_Ocobounded2,axiom,
% 0.32/0.57 ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded2
% 0.32/0.57 thf(fact_3778_max_Ocobounded2,axiom,
% 0.32/0.57 ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded2
% 0.32/0.57 thf(fact_3779_max_Ocobounded2,axiom,
% 0.32/0.57 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded2
% 0.32/0.57 thf(fact_3780_max_Ocobounded2,axiom,
% 0.32/0.57 ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded2
% 0.32/0.57 thf(fact_3781_max_Ocobounded1,axiom,
% 0.32/0.57 ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded1
% 0.32/0.57 thf(fact_3782_max_Ocobounded1,axiom,
% 0.32/0.57 ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded1
% 0.32/0.57 thf(fact_3783_max_Ocobounded1,axiom,
% 0.32/0.57 ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded1
% 0.32/0.57 thf(fact_3784_max_Ocobounded1,axiom,
% 0.32/0.57 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded1
% 0.32/0.57 thf(fact_3785_max_Ocobounded1,axiom,
% 0.32/0.57 ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.cobounded1
% 0.32/0.57 thf(fact_3786_max_Oorder__iff,axiom,
% 0.32/0.57 ( ord_le2932123472753598470d_enat
% 0.32/0.57 = ( ^ [B4: extended_enat,A3: extended_enat] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( ord_ma741700101516333627d_enat @ A3 @ B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.order_iff
% 0.32/0.57 thf(fact_3787_max_Oorder__iff,axiom,
% 0.32/0.57 ( ord_less_eq_rat
% 0.32/0.57 = ( ^ [B4: rat,A3: rat] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( ord_max_rat @ A3 @ B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.order_iff
% 0.32/0.57 thf(fact_3788_max_Oorder__iff,axiom,
% 0.32/0.57 ( ord_less_eq_num
% 0.32/0.57 = ( ^ [B4: num,A3: num] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( ord_max_num @ A3 @ B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.order_iff
% 0.32/0.57 thf(fact_3789_max_Oorder__iff,axiom,
% 0.32/0.57 ( ord_less_eq_nat
% 0.32/0.57 = ( ^ [B4: nat,A3: nat] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( ord_max_nat @ A3 @ B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.order_iff
% 0.32/0.57 thf(fact_3790_max_Oorder__iff,axiom,
% 0.32/0.57 ( ord_less_eq_int
% 0.32/0.57 = ( ^ [B4: int,A3: int] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( ord_max_int @ A3 @ B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.order_iff
% 0.32/0.57 thf(fact_3791_max_OboundedI,axiom,
% 0.32/0.57 ! [B: extended_enat,A: extended_enat,C: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.32/0.57 => ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.32/0.57 => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedI
% 0.32/0.57 thf(fact_3792_max_OboundedI,axiom,
% 0.32/0.57 ! [B: rat,A: rat,C: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.57 => ( ( ord_less_eq_rat @ C @ A )
% 0.32/0.57 => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedI
% 0.32/0.57 thf(fact_3793_max_OboundedI,axiom,
% 0.32/0.57 ! [B: num,A: num,C: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.57 => ( ( ord_less_eq_num @ C @ A )
% 0.32/0.57 => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedI
% 0.32/0.57 thf(fact_3794_max_OboundedI,axiom,
% 0.32/0.57 ! [B: nat,A: nat,C: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.57 => ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.57 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedI
% 0.32/0.57 thf(fact_3795_max_OboundedI,axiom,
% 0.32/0.57 ! [B: int,A: int,C: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.57 => ( ( ord_less_eq_int @ C @ A )
% 0.32/0.57 => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedI
% 0.32/0.57 thf(fact_3796_max_OboundedE,axiom,
% 0.32/0.57 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.32/0.57 => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedE
% 0.32/0.57 thf(fact_3797_max_OboundedE,axiom,
% 0.32/0.57 ! [B: rat,C: rat,A: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.57 => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedE
% 0.32/0.57 thf(fact_3798_max_OboundedE,axiom,
% 0.32/0.57 ! [B: num,C: num,A: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_eq_num @ B @ A )
% 0.32/0.57 => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedE
% 0.32/0.57 thf(fact_3799_max_OboundedE,axiom,
% 0.32/0.57 ! [B: nat,C: nat,A: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.57 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedE
% 0.32/0.57 thf(fact_3800_max_OboundedE,axiom,
% 0.32/0.57 ! [B: int,C: int,A: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_eq_int @ B @ A )
% 0.32/0.57 => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.boundedE
% 0.32/0.57 thf(fact_3801_max_OorderI,axiom,
% 0.32/0.57 ! [A: extended_enat,B: extended_enat] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( ord_ma741700101516333627d_enat @ A @ B ) )
% 0.32/0.57 => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderI
% 0.32/0.57 thf(fact_3802_max_OorderI,axiom,
% 0.32/0.57 ! [A: rat,B: rat] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( ord_max_rat @ A @ B ) )
% 0.32/0.57 => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderI
% 0.32/0.57 thf(fact_3803_max_OorderI,axiom,
% 0.32/0.57 ! [A: num,B: num] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( ord_max_num @ A @ B ) )
% 0.32/0.57 => ( ord_less_eq_num @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderI
% 0.32/0.57 thf(fact_3804_max_OorderI,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( ord_max_nat @ A @ B ) )
% 0.32/0.57 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderI
% 0.32/0.57 thf(fact_3805_max_OorderI,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( ord_max_int @ A @ B ) )
% 0.32/0.57 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderI
% 0.32/0.57 thf(fact_3806_max_OorderE,axiom,
% 0.32/0.57 ! [B: extended_enat,A: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 0.32/0.57 => ( A
% 0.32/0.57 = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderE
% 0.32/0.57 thf(fact_3807_max_OorderE,axiom,
% 0.32/0.57 ! [B: rat,A: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.57 => ( A
% 0.32/0.57 = ( ord_max_rat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderE
% 0.32/0.57 thf(fact_3808_max_OorderE,axiom,
% 0.32/0.57 ! [B: num,A: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.57 => ( A
% 0.32/0.57 = ( ord_max_num @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderE
% 0.32/0.57 thf(fact_3809_max_OorderE,axiom,
% 0.32/0.57 ! [B: nat,A: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.57 => ( A
% 0.32/0.57 = ( ord_max_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderE
% 0.32/0.57 thf(fact_3810_max_OorderE,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.57 => ( A
% 0.32/0.57 = ( ord_max_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.orderE
% 0.32/0.57 thf(fact_3811_max_Omono,axiom,
% 0.32/0.57 ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
% 0.32/0.57 ( ( ord_le2932123472753598470d_enat @ C @ A )
% 0.32/0.57 => ( ( ord_le2932123472753598470d_enat @ D @ B )
% 0.32/0.57 => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.mono
% 0.32/0.57 thf(fact_3812_max_Omono,axiom,
% 0.32/0.57 ! [C: rat,A: rat,D: rat,B: rat] :
% 0.32/0.57 ( ( ord_less_eq_rat @ C @ A )
% 0.32/0.57 => ( ( ord_less_eq_rat @ D @ B )
% 0.32/0.57 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.mono
% 0.32/0.57 thf(fact_3813_max_Omono,axiom,
% 0.32/0.57 ! [C: num,A: num,D: num,B: num] :
% 0.32/0.57 ( ( ord_less_eq_num @ C @ A )
% 0.32/0.57 => ( ( ord_less_eq_num @ D @ B )
% 0.32/0.57 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.mono
% 0.32/0.57 thf(fact_3814_max_Omono,axiom,
% 0.32/0.57 ! [C: nat,A: nat,D: nat,B: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ C @ A )
% 0.32/0.57 => ( ( ord_less_eq_nat @ D @ B )
% 0.32/0.57 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.mono
% 0.32/0.57 thf(fact_3815_max_Omono,axiom,
% 0.32/0.57 ! [C: int,A: int,D: int,B: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ C @ A )
% 0.32/0.57 => ( ( ord_less_eq_int @ D @ B )
% 0.32/0.57 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.mono
% 0.32/0.57 thf(fact_3816_max_Ostrict__coboundedI2,axiom,
% 0.32/0.57 ! [C: extended_enat,B: extended_enat,A: extended_enat] :
% 0.32/0.57 ( ( ord_le72135733267957522d_enat @ C @ B )
% 0.32/0.57 => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI2
% 0.32/0.57 thf(fact_3817_max_Ostrict__coboundedI2,axiom,
% 0.32/0.57 ! [C: real,B: real,A: real] :
% 0.32/0.57 ( ( ord_less_real @ C @ B )
% 0.32/0.57 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI2
% 0.32/0.57 thf(fact_3818_max_Ostrict__coboundedI2,axiom,
% 0.32/0.57 ! [C: rat,B: rat,A: rat] :
% 0.32/0.57 ( ( ord_less_rat @ C @ B )
% 0.32/0.57 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI2
% 0.32/0.57 thf(fact_3819_max_Ostrict__coboundedI2,axiom,
% 0.32/0.57 ! [C: num,B: num,A: num] :
% 0.32/0.57 ( ( ord_less_num @ C @ B )
% 0.32/0.57 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI2
% 0.32/0.57 thf(fact_3820_max_Ostrict__coboundedI2,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( ord_less_nat @ C @ B )
% 0.32/0.57 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI2
% 0.32/0.57 thf(fact_3821_max_Ostrict__coboundedI2,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( ord_less_int @ C @ B )
% 0.32/0.57 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI2
% 0.32/0.57 thf(fact_3822_max_Ostrict__coboundedI1,axiom,
% 0.32/0.57 ! [C: extended_enat,A: extended_enat,B: extended_enat] :
% 0.32/0.57 ( ( ord_le72135733267957522d_enat @ C @ A )
% 0.32/0.57 => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI1
% 0.32/0.57 thf(fact_3823_max_Ostrict__coboundedI1,axiom,
% 0.32/0.57 ! [C: real,A: real,B: real] :
% 0.32/0.57 ( ( ord_less_real @ C @ A )
% 0.32/0.57 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI1
% 0.32/0.57 thf(fact_3824_max_Ostrict__coboundedI1,axiom,
% 0.32/0.57 ! [C: rat,A: rat,B: rat] :
% 0.32/0.57 ( ( ord_less_rat @ C @ A )
% 0.32/0.57 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI1
% 0.32/0.57 thf(fact_3825_max_Ostrict__coboundedI1,axiom,
% 0.32/0.57 ! [C: num,A: num,B: num] :
% 0.32/0.57 ( ( ord_less_num @ C @ A )
% 0.32/0.57 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI1
% 0.32/0.57 thf(fact_3826_max_Ostrict__coboundedI1,axiom,
% 0.32/0.57 ! [C: nat,A: nat,B: nat] :
% 0.32/0.57 ( ( ord_less_nat @ C @ A )
% 0.32/0.57 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI1
% 0.32/0.57 thf(fact_3827_max_Ostrict__coboundedI1,axiom,
% 0.32/0.57 ! [C: int,A: int,B: int] :
% 0.32/0.57 ( ( ord_less_int @ C @ A )
% 0.32/0.57 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_coboundedI1
% 0.32/0.57 thf(fact_3828_max_Ostrict__order__iff,axiom,
% 0.32/0.57 ( ord_le72135733267957522d_enat
% 0.32/0.57 = ( ^ [B4: extended_enat,A3: extended_enat] :
% 0.32/0.57 ( ( A3
% 0.32/0.57 = ( ord_ma741700101516333627d_enat @ A3 @ B4 ) )
% 0.32/0.57 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_order_iff
% 0.32/0.57 thf(fact_3829_max_Ostrict__order__iff,axiom,
% 0.32/0.57 ( ord_less_real
% 0.32/0.57 = ( ^ [B4: real,A3: real] :
% 0.32/0.57 ( ( A3
% 0.32/0.57 = ( ord_max_real @ A3 @ B4 ) )
% 0.32/0.57 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_order_iff
% 0.32/0.57 thf(fact_3830_max_Ostrict__order__iff,axiom,
% 0.32/0.57 ( ord_less_rat
% 0.32/0.57 = ( ^ [B4: rat,A3: rat] :
% 0.32/0.57 ( ( A3
% 0.32/0.57 = ( ord_max_rat @ A3 @ B4 ) )
% 0.32/0.57 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_order_iff
% 0.32/0.57 thf(fact_3831_max_Ostrict__order__iff,axiom,
% 0.32/0.57 ( ord_less_num
% 0.32/0.57 = ( ^ [B4: num,A3: num] :
% 0.32/0.57 ( ( A3
% 0.32/0.57 = ( ord_max_num @ A3 @ B4 ) )
% 0.32/0.57 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_order_iff
% 0.32/0.57 thf(fact_3832_max_Ostrict__order__iff,axiom,
% 0.32/0.57 ( ord_less_nat
% 0.32/0.57 = ( ^ [B4: nat,A3: nat] :
% 0.32/0.57 ( ( A3
% 0.32/0.57 = ( ord_max_nat @ A3 @ B4 ) )
% 0.32/0.57 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_order_iff
% 0.32/0.57 thf(fact_3833_max_Ostrict__order__iff,axiom,
% 0.32/0.57 ( ord_less_int
% 0.32/0.57 = ( ^ [B4: int,A3: int] :
% 0.32/0.57 ( ( A3
% 0.32/0.57 = ( ord_max_int @ A3 @ B4 ) )
% 0.32/0.57 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_order_iff
% 0.32/0.57 thf(fact_3834_max_Ostrict__boundedE,axiom,
% 0.32/0.57 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 0.32/0.57 ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
% 0.32/0.57 => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_boundedE
% 0.32/0.57 thf(fact_3835_max_Ostrict__boundedE,axiom,
% 0.32/0.57 ! [B: real,C: real,A: real] :
% 0.32/0.57 ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_real @ B @ A )
% 0.32/0.57 => ~ ( ord_less_real @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_boundedE
% 0.32/0.57 thf(fact_3836_max_Ostrict__boundedE,axiom,
% 0.32/0.57 ! [B: rat,C: rat,A: rat] :
% 0.32/0.57 ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_rat @ B @ A )
% 0.32/0.57 => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_boundedE
% 0.32/0.57 thf(fact_3837_max_Ostrict__boundedE,axiom,
% 0.32/0.57 ! [B: num,C: num,A: num] :
% 0.32/0.57 ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_num @ B @ A )
% 0.32/0.57 => ~ ( ord_less_num @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_boundedE
% 0.32/0.57 thf(fact_3838_max_Ostrict__boundedE,axiom,
% 0.32/0.57 ! [B: nat,C: nat,A: nat] :
% 0.32/0.57 ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_nat @ B @ A )
% 0.32/0.57 => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_boundedE
% 0.32/0.57 thf(fact_3839_max_Ostrict__boundedE,axiom,
% 0.32/0.57 ! [B: int,C: int,A: int] :
% 0.32/0.57 ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 0.32/0.57 => ~ ( ( ord_less_int @ B @ A )
% 0.32/0.57 => ~ ( ord_less_int @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max.strict_boundedE
% 0.32/0.57 thf(fact_3840_less__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: extended_enat,X: extended_enat,Y3: extended_enat] :
% 0.32/0.57 ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_le72135733267957522d_enat @ Z @ X )
% 0.32/0.57 | ( ord_le72135733267957522d_enat @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % less_max_iff_disj
% 0.32/0.57 thf(fact_3841_less__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: real,X: real,Y3: real] :
% 0.32/0.57 ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_real @ Z @ X )
% 0.32/0.57 | ( ord_less_real @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % less_max_iff_disj
% 0.32/0.57 thf(fact_3842_less__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_rat @ Z @ X )
% 0.32/0.57 | ( ord_less_rat @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % less_max_iff_disj
% 0.32/0.57 thf(fact_3843_less__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: num,X: num,Y3: num] :
% 0.32/0.57 ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_num @ Z @ X )
% 0.32/0.57 | ( ord_less_num @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % less_max_iff_disj
% 0.32/0.57 thf(fact_3844_less__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: nat,X: nat,Y3: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_nat @ Z @ X )
% 0.32/0.57 | ( ord_less_nat @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % less_max_iff_disj
% 0.32/0.57 thf(fact_3845_less__max__iff__disj,axiom,
% 0.32/0.57 ! [Z: int,X: int,Y3: int] :
% 0.32/0.57 ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y3 ) )
% 0.32/0.57 = ( ( ord_less_int @ Z @ X )
% 0.32/0.57 | ( ord_less_int @ Z @ Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % less_max_iff_disj
% 0.32/0.57 thf(fact_3846_max__add__distrib__left,axiom,
% 0.32/0.57 ! [X: real,Y3: real,Z: real] :
% 0.32/0.57 ( ( plus_plus_real @ ( ord_max_real @ X @ Y3 ) @ Z )
% 0.32/0.57 = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_add_distrib_left
% 0.32/0.57 thf(fact_3847_max__add__distrib__left,axiom,
% 0.32/0.57 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.57 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y3 ) @ Z )
% 0.32/0.57 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_add_distrib_left
% 0.32/0.57 thf(fact_3848_max__add__distrib__left,axiom,
% 0.32/0.57 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.57 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y3 ) @ Z )
% 0.32/0.57 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_add_distrib_left
% 0.32/0.57 thf(fact_3849_max__add__distrib__left,axiom,
% 0.32/0.57 ! [X: int,Y3: int,Z: int] :
% 0.32/0.57 ( ( plus_plus_int @ ( ord_max_int @ X @ Y3 ) @ Z )
% 0.32/0.57 = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_add_distrib_left
% 0.32/0.57 thf(fact_3850_max__add__distrib__right,axiom,
% 0.32/0.57 ! [X: real,Y3: real,Z: real] :
% 0.32/0.57 ( ( plus_plus_real @ X @ ( ord_max_real @ Y3 @ Z ) )
% 0.32/0.57 = ( ord_max_real @ ( plus_plus_real @ X @ Y3 ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_add_distrib_right
% 0.32/0.57 thf(fact_3851_max__add__distrib__right,axiom,
% 0.32/0.57 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.57 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y3 @ Z ) )
% 0.32/0.57 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_add_distrib_right
% 0.32/0.57 thf(fact_3852_max__add__distrib__right,axiom,
% 0.32/0.57 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.57 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y3 @ Z ) )
% 0.32/0.57 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_add_distrib_right
% 0.32/0.57 thf(fact_3853_max__add__distrib__right,axiom,
% 0.32/0.57 ! [X: int,Y3: int,Z: int] :
% 0.32/0.57 ( ( plus_plus_int @ X @ ( ord_max_int @ Y3 @ Z ) )
% 0.32/0.57 = ( ord_max_int @ ( plus_plus_int @ X @ Y3 ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_add_distrib_right
% 0.32/0.57 thf(fact_3854_max__diff__distrib__left,axiom,
% 0.32/0.57 ! [X: real,Y3: real,Z: real] :
% 0.32/0.57 ( ( minus_minus_real @ ( ord_max_real @ X @ Y3 ) @ Z )
% 0.32/0.57 = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_diff_distrib_left
% 0.32/0.57 thf(fact_3855_max__diff__distrib__left,axiom,
% 0.32/0.57 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.57 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y3 ) @ Z )
% 0.32/0.57 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_diff_distrib_left
% 0.32/0.57 thf(fact_3856_max__diff__distrib__left,axiom,
% 0.32/0.57 ! [X: int,Y3: int,Z: int] :
% 0.32/0.57 ( ( minus_minus_int @ ( ord_max_int @ X @ Y3 ) @ Z )
% 0.32/0.57 = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_diff_distrib_left
% 0.32/0.57 thf(fact_3857_nat__add__max__left,axiom,
% 0.32/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.57 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.32/0.57 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % nat_add_max_left
% 0.32/0.57 thf(fact_3858_nat__add__max__right,axiom,
% 0.32/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.57 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.32/0.57 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % nat_add_max_right
% 0.32/0.57 thf(fact_3859_dvd__0__left,axiom,
% 0.32/0.57 ! [A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.32/0.57 => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_0_left
% 0.32/0.57 thf(fact_3860_dvd__0__left,axiom,
% 0.32/0.57 ! [A: complex] :
% 0.32/0.57 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.32/0.57 => ( A = zero_zero_complex ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_0_left
% 0.32/0.57 thf(fact_3861_dvd__0__left,axiom,
% 0.32/0.57 ! [A: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.32/0.57 => ( A = zero_zero_real ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_0_left
% 0.32/0.57 thf(fact_3862_dvd__0__left,axiom,
% 0.32/0.57 ! [A: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.32/0.57 => ( A = zero_zero_rat ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_0_left
% 0.32/0.57 thf(fact_3863_dvd__0__left,axiom,
% 0.32/0.57 ! [A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.32/0.57 => ( A = zero_zero_nat ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_0_left
% 0.32/0.57 thf(fact_3864_dvd__0__left,axiom,
% 0.32/0.57 ! [A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.32/0.57 => ( A = zero_zero_int ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_0_left
% 0.32/0.57 thf(fact_3865_nat__mult__max__left,axiom,
% 0.32/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.57 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 0.32/0.57 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % nat_mult_max_left
% 0.32/0.57 thf(fact_3866_nat__mult__max__right,axiom,
% 0.32/0.57 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.57 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 0.32/0.57 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % nat_mult_max_right
% 0.32/0.57 thf(fact_3867_dvdE,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.57 => ~ ! [K3: code_integer] :
% 0.32/0.57 ( A
% 0.32/0.57 != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdE
% 0.32/0.57 thf(fact_3868_dvdE,axiom,
% 0.32/0.57 ! [B: real,A: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ B @ A )
% 0.32/0.57 => ~ ! [K3: real] :
% 0.32/0.57 ( A
% 0.32/0.57 != ( times_times_real @ B @ K3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdE
% 0.32/0.57 thf(fact_3869_dvdE,axiom,
% 0.32/0.57 ! [B: rat,A: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ B @ A )
% 0.32/0.57 => ~ ! [K3: rat] :
% 0.32/0.57 ( A
% 0.32/0.57 != ( times_times_rat @ B @ K3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdE
% 0.32/0.57 thf(fact_3870_dvdE,axiom,
% 0.32/0.57 ! [B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.57 => ~ ! [K3: nat] :
% 0.32/0.57 ( A
% 0.32/0.57 != ( times_times_nat @ B @ K3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdE
% 0.32/0.57 thf(fact_3871_dvdE,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.57 => ~ ! [K3: int] :
% 0.32/0.57 ( A
% 0.32/0.57 != ( times_times_int @ B @ K3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdE
% 0.32/0.57 thf(fact_3872_dvdI,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,K: code_integer] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( times_3573771949741848930nteger @ B @ K ) )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdI
% 0.32/0.57 thf(fact_3873_dvdI,axiom,
% 0.32/0.57 ! [A: real,B: real,K: real] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( times_times_real @ B @ K ) )
% 0.32/0.57 => ( dvd_dvd_real @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdI
% 0.32/0.57 thf(fact_3874_dvdI,axiom,
% 0.32/0.57 ! [A: rat,B: rat,K: rat] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( times_times_rat @ B @ K ) )
% 0.32/0.57 => ( dvd_dvd_rat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdI
% 0.32/0.57 thf(fact_3875_dvdI,axiom,
% 0.32/0.57 ! [A: nat,B: nat,K: nat] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( times_times_nat @ B @ K ) )
% 0.32/0.57 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdI
% 0.32/0.57 thf(fact_3876_dvdI,axiom,
% 0.32/0.57 ! [A: int,B: int,K: int] :
% 0.32/0.57 ( ( A
% 0.32/0.57 = ( times_times_int @ B @ K ) )
% 0.32/0.57 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvdI
% 0.32/0.57 thf(fact_3877_dvd__def,axiom,
% 0.32/0.57 ( dvd_dvd_Code_integer
% 0.32/0.57 = ( ^ [B4: code_integer,A3: code_integer] :
% 0.32/0.57 ? [K2: code_integer] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( times_3573771949741848930nteger @ B4 @ K2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_def
% 0.32/0.57 thf(fact_3878_dvd__def,axiom,
% 0.32/0.57 ( dvd_dvd_real
% 0.32/0.57 = ( ^ [B4: real,A3: real] :
% 0.32/0.57 ? [K2: real] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( times_times_real @ B4 @ K2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_def
% 0.32/0.57 thf(fact_3879_dvd__def,axiom,
% 0.32/0.57 ( dvd_dvd_rat
% 0.32/0.57 = ( ^ [B4: rat,A3: rat] :
% 0.32/0.57 ? [K2: rat] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( times_times_rat @ B4 @ K2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_def
% 0.32/0.57 thf(fact_3880_dvd__def,axiom,
% 0.32/0.57 ( dvd_dvd_nat
% 0.32/0.57 = ( ^ [B4: nat,A3: nat] :
% 0.32/0.57 ? [K2: nat] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( times_times_nat @ B4 @ K2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_def
% 0.32/0.57 thf(fact_3881_dvd__def,axiom,
% 0.32/0.57 ( dvd_dvd_int
% 0.32/0.57 = ( ^ [B4: int,A3: int] :
% 0.32/0.57 ? [K2: int] :
% 0.32/0.57 ( A3
% 0.32/0.57 = ( times_times_int @ B4 @ K2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_def
% 0.32/0.57 thf(fact_3882_dvd__mult,axiom,
% 0.32/0.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult
% 0.32/0.57 thf(fact_3883_dvd__mult,axiom,
% 0.32/0.57 ! [A: real,C: real,B: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ A @ C )
% 0.32/0.57 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult
% 0.32/0.57 thf(fact_3884_dvd__mult,axiom,
% 0.32/0.57 ! [A: rat,C: rat,B: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.57 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult
% 0.32/0.57 thf(fact_3885_dvd__mult,axiom,
% 0.32/0.57 ! [A: nat,C: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.57 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult
% 0.32/0.57 thf(fact_3886_dvd__mult,axiom,
% 0.32/0.57 ! [A: int,C: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ C )
% 0.32/0.57 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult
% 0.32/0.57 thf(fact_3887_dvd__mult2,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult2
% 0.32/0.57 thf(fact_3888_dvd__mult2,axiom,
% 0.32/0.57 ! [A: real,B: real,C: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.57 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult2
% 0.32/0.57 thf(fact_3889_dvd__mult2,axiom,
% 0.32/0.57 ! [A: rat,B: rat,C: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.57 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult2
% 0.32/0.57 thf(fact_3890_dvd__mult2,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult2
% 0.32/0.57 thf(fact_3891_dvd__mult2,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult2
% 0.32/0.57 thf(fact_3892_dvd__mult__left,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_left
% 0.32/0.57 thf(fact_3893_dvd__mult__left,axiom,
% 0.32/0.57 ! [A: real,B: real,C: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_real @ A @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_left
% 0.32/0.57 thf(fact_3894_dvd__mult__left,axiom,
% 0.32/0.57 ! [A: rat,B: rat,C: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_rat @ A @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_left
% 0.32/0.57 thf(fact_3895_dvd__mult__left,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_nat @ A @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_left
% 0.32/0.57 thf(fact_3896_dvd__mult__left,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_int @ A @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_left
% 0.32/0.57 thf(fact_3897_dvd__triv__left,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_left
% 0.32/0.57 thf(fact_3898_dvd__triv__left,axiom,
% 0.32/0.57 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_left
% 0.32/0.57 thf(fact_3899_dvd__triv__left,axiom,
% 0.32/0.57 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_left
% 0.32/0.57 thf(fact_3900_dvd__triv__left,axiom,
% 0.32/0.57 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_left
% 0.32/0.57 thf(fact_3901_dvd__triv__left,axiom,
% 0.32/0.57 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_left
% 0.32/0.57 thf(fact_3902_mult__dvd__mono,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_dvd_mono
% 0.32/0.57 thf(fact_3903_mult__dvd__mono,axiom,
% 0.32/0.57 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_real @ C @ D )
% 0.32/0.57 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_dvd_mono
% 0.32/0.57 thf(fact_3904_mult__dvd__mono,axiom,
% 0.32/0.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_rat @ C @ D )
% 0.32/0.57 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_dvd_mono
% 0.32/0.57 thf(fact_3905_mult__dvd__mono,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ C @ D )
% 0.32/0.57 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_dvd_mono
% 0.32/0.57 thf(fact_3906_mult__dvd__mono,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ C @ D )
% 0.32/0.57 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_dvd_mono
% 0.32/0.57 thf(fact_3907_dvd__mult__right,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_right
% 0.32/0.57 thf(fact_3908_dvd__mult__right,axiom,
% 0.32/0.57 ! [A: real,B: real,C: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_real @ B @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_right
% 0.32/0.57 thf(fact_3909_dvd__mult__right,axiom,
% 0.32/0.57 ! [A: rat,B: rat,C: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_rat @ B @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_right
% 0.32/0.57 thf(fact_3910_dvd__mult__right,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_nat @ B @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_right
% 0.32/0.57 thf(fact_3911_dvd__mult__right,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.57 => ( dvd_dvd_int @ B @ C ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_right
% 0.32/0.57 thf(fact_3912_dvd__triv__right,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_right
% 0.32/0.57 thf(fact_3913_dvd__triv__right,axiom,
% 0.32/0.57 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_right
% 0.32/0.57 thf(fact_3914_dvd__triv__right,axiom,
% 0.32/0.57 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_right
% 0.32/0.57 thf(fact_3915_dvd__triv__right,axiom,
% 0.32/0.57 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_right
% 0.32/0.57 thf(fact_3916_dvd__triv__right,axiom,
% 0.32/0.57 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_triv_right
% 0.32/0.57 thf(fact_3917_one__dvd,axiom,
% 0.32/0.57 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.32/0.57
% 0.32/0.57 % one_dvd
% 0.32/0.57 thf(fact_3918_one__dvd,axiom,
% 0.32/0.57 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.32/0.57
% 0.32/0.57 % one_dvd
% 0.32/0.57 thf(fact_3919_one__dvd,axiom,
% 0.32/0.57 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.32/0.57
% 0.32/0.57 % one_dvd
% 0.32/0.57 thf(fact_3920_one__dvd,axiom,
% 0.32/0.57 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.32/0.57
% 0.32/0.57 % one_dvd
% 0.32/0.57 thf(fact_3921_one__dvd,axiom,
% 0.32/0.57 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.32/0.57
% 0.32/0.57 % one_dvd
% 0.32/0.57 thf(fact_3922_one__dvd,axiom,
% 0.32/0.57 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.32/0.57
% 0.32/0.57 % one_dvd
% 0.32/0.57 thf(fact_3923_unit__imp__dvd,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_imp_dvd
% 0.32/0.57 thf(fact_3924_unit__imp__dvd,axiom,
% 0.32/0.57 ! [B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_imp_dvd
% 0.32/0.57 thf(fact_3925_unit__imp__dvd,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_imp_dvd
% 0.32/0.57 thf(fact_3926_dvd__unit__imp__unit,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_unit_imp_unit
% 0.32/0.57 thf(fact_3927_dvd__unit__imp__unit,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_unit_imp_unit
% 0.32/0.57 thf(fact_3928_dvd__unit__imp__unit,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_unit_imp_unit
% 0.32/0.57 thf(fact_3929_dvd__add,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add
% 0.32/0.57 thf(fact_3930_dvd__add,axiom,
% 0.32/0.57 ! [A: real,B: real,C: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_real @ A @ C )
% 0.32/0.57 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add
% 0.32/0.57 thf(fact_3931_dvd__add,axiom,
% 0.32/0.57 ! [A: rat,B: rat,C: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.57 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add
% 0.32/0.57 thf(fact_3932_dvd__add,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.57 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add
% 0.32/0.57 thf(fact_3933_dvd__add,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ C )
% 0.32/0.57 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add
% 0.32/0.57 thf(fact_3934_dvd__add__left__iff,axiom,
% 0.32/0.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ C )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_left_iff
% 0.32/0.57 thf(fact_3935_dvd__add__left__iff,axiom,
% 0.32/0.57 ! [A: real,C: real,B: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ A @ C )
% 0.32/0.57 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_left_iff
% 0.32/0.57 thf(fact_3936_dvd__add__left__iff,axiom,
% 0.32/0.57 ! [A: rat,C: rat,B: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ A @ C )
% 0.32/0.57 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_left_iff
% 0.32/0.57 thf(fact_3937_dvd__add__left__iff,axiom,
% 0.32/0.57 ! [A: nat,C: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ C )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_left_iff
% 0.32/0.57 thf(fact_3938_dvd__add__left__iff,axiom,
% 0.32/0.57 ! [A: int,C: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ C )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_left_iff
% 0.32/0.57 thf(fact_3939_dvd__add__right__iff,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_right_iff
% 0.32/0.57 thf(fact_3940_dvd__add__right__iff,axiom,
% 0.32/0.57 ! [A: real,B: real,C: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_right_iff
% 0.32/0.57 thf(fact_3941_dvd__add__right__iff,axiom,
% 0.32/0.57 ! [A: rat,B: rat,C: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_right_iff
% 0.32/0.57 thf(fact_3942_dvd__add__right__iff,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_right_iff
% 0.32/0.57 thf(fact_3943_dvd__add__right__iff,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_add_right_iff
% 0.32/0.57 thf(fact_3944_dvd__diff,axiom,
% 0.32/0.57 ! [X: code_integer,Y3: code_integer,Z: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ X @ Y3 )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ X @ Z )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diff
% 0.32/0.57 thf(fact_3945_dvd__diff,axiom,
% 0.32/0.57 ! [X: real,Y3: real,Z: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ X @ Y3 )
% 0.32/0.57 => ( ( dvd_dvd_real @ X @ Z )
% 0.32/0.57 => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diff
% 0.32/0.57 thf(fact_3946_dvd__diff,axiom,
% 0.32/0.57 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ X @ Y3 )
% 0.32/0.57 => ( ( dvd_dvd_rat @ X @ Z )
% 0.32/0.57 => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diff
% 0.32/0.57 thf(fact_3947_dvd__diff,axiom,
% 0.32/0.57 ! [X: int,Y3: int,Z: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ X @ Y3 )
% 0.32/0.57 => ( ( dvd_dvd_int @ X @ Z )
% 0.32/0.57 => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diff
% 0.32/0.57 thf(fact_3948_dvd__diff__commute,axiom,
% 0.32/0.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diff_commute
% 0.32/0.57 thf(fact_3949_dvd__diff__commute,axiom,
% 0.32/0.57 ! [A: int,C: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.32/0.57 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diff_commute
% 0.32/0.57 thf(fact_3950_div__div__div__same,axiom,
% 0.32/0.57 ! [D: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ D @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 0.32/0.57 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_div_div_same
% 0.32/0.57 thf(fact_3951_div__div__div__same,axiom,
% 0.32/0.57 ! [D: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ D @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.57 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 0.32/0.57 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_div_div_same
% 0.32/0.57 thf(fact_3952_div__div__div__same,axiom,
% 0.32/0.57 ! [D: int,B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ D @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.57 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 0.32/0.57 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_div_div_same
% 0.32/0.57 thf(fact_3953_dvd__div__eq__cancel,axiom,
% 0.32/0.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.57 ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.32/0.57 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_cancel
% 0.32/0.57 thf(fact_3954_dvd__div__eq__cancel,axiom,
% 0.32/0.57 ! [A: complex,C: complex,B: complex] :
% 0.32/0.57 ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.32/0.57 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.57 => ( ( dvd_dvd_complex @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_complex @ C @ B )
% 0.32/0.57 => ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_cancel
% 0.32/0.57 thf(fact_3955_dvd__div__eq__cancel,axiom,
% 0.32/0.57 ! [A: real,C: real,B: real] :
% 0.32/0.57 ( ( ( divide_divide_real @ A @ C )
% 0.32/0.57 = ( divide_divide_real @ B @ C ) )
% 0.32/0.57 => ( ( dvd_dvd_real @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_real @ C @ B )
% 0.32/0.57 => ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_cancel
% 0.32/0.57 thf(fact_3956_dvd__div__eq__cancel,axiom,
% 0.32/0.57 ! [A: rat,C: rat,B: rat] :
% 0.32/0.57 ( ( ( divide_divide_rat @ A @ C )
% 0.32/0.57 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.57 => ( ( dvd_dvd_rat @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_rat @ C @ B )
% 0.32/0.57 => ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_cancel
% 0.32/0.57 thf(fact_3957_dvd__div__eq__cancel,axiom,
% 0.32/0.57 ! [A: nat,C: nat,B: nat] :
% 0.32/0.57 ( ( ( divide_divide_nat @ A @ C )
% 0.32/0.57 = ( divide_divide_nat @ B @ C ) )
% 0.32/0.57 => ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_cancel
% 0.32/0.57 thf(fact_3958_dvd__div__eq__cancel,axiom,
% 0.32/0.57 ! [A: int,C: int,B: int] :
% 0.32/0.57 ( ( ( divide_divide_int @ A @ C )
% 0.32/0.57 = ( divide_divide_int @ B @ C ) )
% 0.32/0.57 => ( ( dvd_dvd_int @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_cancel
% 0.32/0.57 thf(fact_3959_dvd__div__eq__iff,axiom,
% 0.32/0.57 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.32/0.57 = ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.57 = ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_iff
% 0.32/0.57 thf(fact_3960_dvd__div__eq__iff,axiom,
% 0.32/0.57 ! [C: complex,A: complex,B: complex] :
% 0.32/0.57 ( ( dvd_dvd_complex @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_complex @ C @ B )
% 0.32/0.57 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.32/0.57 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.57 = ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_iff
% 0.32/0.57 thf(fact_3961_dvd__div__eq__iff,axiom,
% 0.32/0.57 ! [C: real,A: real,B: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_real @ C @ B )
% 0.32/0.57 => ( ( ( divide_divide_real @ A @ C )
% 0.32/0.57 = ( divide_divide_real @ B @ C ) )
% 0.32/0.57 = ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_iff
% 0.32/0.57 thf(fact_3962_dvd__div__eq__iff,axiom,
% 0.32/0.57 ! [C: rat,A: rat,B: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_rat @ C @ B )
% 0.32/0.57 => ( ( ( divide_divide_rat @ A @ C )
% 0.32/0.57 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.57 = ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_iff
% 0.32/0.57 thf(fact_3963_dvd__div__eq__iff,axiom,
% 0.32/0.57 ! [C: nat,A: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( ( ( divide_divide_nat @ A @ C )
% 0.32/0.57 = ( divide_divide_nat @ B @ C ) )
% 0.32/0.57 = ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_iff
% 0.32/0.57 thf(fact_3964_dvd__div__eq__iff,axiom,
% 0.32/0.57 ! [C: int,A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.57 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( ( ( divide_divide_int @ A @ C )
% 0.32/0.57 = ( divide_divide_int @ B @ C ) )
% 0.32/0.57 = ( A = B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_iff
% 0.32/0.57 thf(fact_3965_dvd__power__same,axiom,
% 0.32/0.57 ! [X: code_integer,Y3: code_integer,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ X @ Y3 )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_same
% 0.32/0.57 thf(fact_3966_dvd__power__same,axiom,
% 0.32/0.57 ! [X: nat,Y3: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ X @ Y3 )
% 0.32/0.57 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_same
% 0.32/0.57 thf(fact_3967_dvd__power__same,axiom,
% 0.32/0.57 ! [X: real,Y3: real,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_real @ X @ Y3 )
% 0.32/0.57 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_same
% 0.32/0.57 thf(fact_3968_dvd__power__same,axiom,
% 0.32/0.57 ! [X: int,Y3: int,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_int @ X @ Y3 )
% 0.32/0.57 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_same
% 0.32/0.57 thf(fact_3969_dvd__power__same,axiom,
% 0.32/0.57 ! [X: complex,Y3: complex,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_complex @ X @ Y3 )
% 0.32/0.57 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_same
% 0.32/0.57 thf(fact_3970_mod__mod__cancel,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 0.32/0.57 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_mod_cancel
% 0.32/0.57 thf(fact_3971_mod__mod__cancel,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 0.32/0.57 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_mod_cancel
% 0.32/0.57 thf(fact_3972_mod__mod__cancel,axiom,
% 0.32/0.57 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 0.32/0.57 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_mod_cancel
% 0.32/0.57 thf(fact_3973_dvd__mod,axiom,
% 0.32/0.57 ! [K: nat,M: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.57 => ( ( dvd_dvd_nat @ K @ N )
% 0.32/0.57 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod
% 0.32/0.57 thf(fact_3974_dvd__mod,axiom,
% 0.32/0.57 ! [K: int,M: int,N: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ K @ M )
% 0.32/0.57 => ( ( dvd_dvd_int @ K @ N )
% 0.32/0.57 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod
% 0.32/0.57 thf(fact_3975_dvd__mod,axiom,
% 0.32/0.57 ! [K: code_integer,M: code_integer,N: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ K @ M )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ K @ N )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod
% 0.32/0.57 thf(fact_3976_dvd__mod__imp__dvd,axiom,
% 0.32/0.57 ! [C: nat,A: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.57 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod_imp_dvd
% 0.32/0.57 thf(fact_3977_dvd__mod__imp__dvd,axiom,
% 0.32/0.57 ! [C: int,A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.57 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod_imp_dvd
% 0.32/0.57 thf(fact_3978_dvd__mod__imp__dvd,axiom,
% 0.32/0.57 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod_imp_dvd
% 0.32/0.57 thf(fact_3979_dvd__mod__iff,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.57 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod_iff
% 0.32/0.57 thf(fact_3980_dvd__mod__iff,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.32/0.57 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod_iff
% 0.32/0.57 thf(fact_3981_dvd__mod__iff,axiom,
% 0.32/0.57 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mod_iff
% 0.32/0.57 thf(fact_3982_dvd__diff__nat,axiom,
% 0.32/0.57 ! [K: nat,M: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.57 => ( ( dvd_dvd_nat @ K @ N )
% 0.32/0.57 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diff_nat
% 0.32/0.57 thf(fact_3983_zdvd__zdiffD,axiom,
% 0.32/0.57 ! [K: int,M: int,N: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 0.32/0.57 => ( ( dvd_dvd_int @ K @ N )
% 0.32/0.57 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdvd_zdiffD
% 0.32/0.57 thf(fact_3984_dvd__pos__nat,axiom,
% 0.32/0.57 ! [N: nat,M: nat] :
% 0.32/0.57 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.57 => ( ( dvd_dvd_nat @ M @ N )
% 0.32/0.57 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_pos_nat
% 0.32/0.57 thf(fact_3985_max__def__raw,axiom,
% 0.32/0.57 ( ord_ma741700101516333627d_enat
% 0.32/0.57 = ( ^ [A3: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3986_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_set_int
% 0.32/0.57 = ( ^ [A3: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3987_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_rat
% 0.32/0.57 = ( ^ [A3: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3988_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_num
% 0.32/0.57 = ( ^ [A3: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3989_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_nat
% 0.32/0.57 = ( ^ [A3: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3990_max__def__raw,axiom,
% 0.32/0.57 ( ord_max_int
% 0.32/0.57 = ( ^ [A3: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % max_def_raw
% 0.32/0.57 thf(fact_3991_bezout__lemma__nat,axiom,
% 0.32/0.57 ! [D: nat,A: nat,B: nat,X: nat,Y3: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ D @ A )
% 0.32/0.57 => ( ( dvd_dvd_nat @ D @ B )
% 0.32/0.57 => ( ( ( ( times_times_nat @ A @ X )
% 0.32/0.57 = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
% 0.32/0.57 | ( ( times_times_nat @ B @ X )
% 0.32/0.57 = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 0.32/0.57 => ? [X3: nat,Y4: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ D @ A )
% 0.32/0.57 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.57 & ( ( ( times_times_nat @ A @ X3 )
% 0.32/0.57 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 0.32/0.57 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
% 0.32/0.57 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bezout_lemma_nat
% 0.32/0.57 thf(fact_3992_bezout__add__nat,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ? [D3: nat,X3: nat,Y4: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ D3 @ A )
% 0.32/0.57 & ( dvd_dvd_nat @ D3 @ B )
% 0.32/0.57 & ( ( ( times_times_nat @ A @ X3 )
% 0.32/0.57 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) )
% 0.32/0.57 | ( ( times_times_nat @ B @ X3 )
% 0.32/0.57 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bezout_add_nat
% 0.32/0.57 thf(fact_3993_bezout1__nat,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ? [D3: nat,X3: nat,Y4: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ D3 @ A )
% 0.32/0.57 & ( dvd_dvd_nat @ D3 @ B )
% 0.32/0.57 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.32/0.57 = D3 )
% 0.32/0.57 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.32/0.57 = D3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bezout1_nat
% 0.32/0.57 thf(fact_3994_subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: complex,B: complex] :
% 0.32/0.57 ( ( ord_le211207098394363844omplex
% 0.32/0.57 @ ( collect_complex
% 0.32/0.57 @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
% 0.32/0.57 @ ( collect_complex
% 0.32/0.57 @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
% 0.32/0.57 = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % subset_divisors_dvd
% 0.32/0.57 thf(fact_3995_subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: real,B: real] :
% 0.32/0.57 ( ( ord_less_eq_set_real
% 0.32/0.57 @ ( collect_real
% 0.32/0.57 @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 0.32/0.57 @ ( collect_real
% 0.32/0.57 @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
% 0.32/0.57 = ( dvd_dvd_real @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % subset_divisors_dvd
% 0.32/0.57 thf(fact_3996_subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( ord_less_eq_set_nat
% 0.32/0.57 @ ( collect_nat
% 0.32/0.57 @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 0.32/0.57 @ ( collect_nat
% 0.32/0.57 @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % subset_divisors_dvd
% 0.32/0.57 thf(fact_3997_subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( ord_le7084787975880047091nteger
% 0.32/0.57 @ ( collect_Code_integer
% 0.32/0.57 @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 0.32/0.57 @ ( collect_Code_integer
% 0.32/0.57 @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % subset_divisors_dvd
% 0.32/0.57 thf(fact_3998_subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ord_less_eq_set_int
% 0.32/0.57 @ ( collect_int
% 0.32/0.57 @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 0.32/0.57 @ ( collect_int
% 0.32/0.57 @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
% 0.32/0.57 = ( dvd_dvd_int @ A @ B ) ) ).
% 0.32/0.57
% 0.32/0.57 % subset_divisors_dvd
% 0.32/0.57 thf(fact_3999_concat__bit__assoc,axiom,
% 0.32/0.57 ! [N: nat,K: int,M: nat,L2: int,R: int] :
% 0.32/0.57 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R ) )
% 0.32/0.57 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R ) ) ).
% 0.32/0.57
% 0.32/0.57 % concat_bit_assoc
% 0.32/0.57 thf(fact_4000_strict__subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: complex,B: complex] :
% 0.32/0.57 ( ( ord_less_set_complex
% 0.32/0.57 @ ( collect_complex
% 0.32/0.57 @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
% 0.32/0.57 @ ( collect_complex
% 0.32/0.57 @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
% 0.32/0.57 = ( ( dvd_dvd_complex @ A @ B )
% 0.32/0.57 & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % strict_subset_divisors_dvd
% 0.32/0.57 thf(fact_4001_strict__subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: real,B: real] :
% 0.32/0.57 ( ( ord_less_set_real
% 0.32/0.57 @ ( collect_real
% 0.32/0.57 @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 0.32/0.57 @ ( collect_real
% 0.32/0.57 @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
% 0.32/0.57 = ( ( dvd_dvd_real @ A @ B )
% 0.32/0.57 & ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % strict_subset_divisors_dvd
% 0.32/0.57 thf(fact_4002_strict__subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( ord_less_set_nat
% 0.32/0.57 @ ( collect_nat
% 0.32/0.57 @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 0.32/0.57 @ ( collect_nat
% 0.32/0.57 @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
% 0.32/0.57 = ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % strict_subset_divisors_dvd
% 0.32/0.57 thf(fact_4003_strict__subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ord_less_set_int
% 0.32/0.57 @ ( collect_int
% 0.32/0.57 @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 0.32/0.57 @ ( collect_int
% 0.32/0.57 @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
% 0.32/0.57 = ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % strict_subset_divisors_dvd
% 0.32/0.57 thf(fact_4004_strict__subset__divisors__dvd,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( ord_le1307284697595431911nteger
% 0.32/0.57 @ ( collect_Code_integer
% 0.32/0.57 @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 0.32/0.57 @ ( collect_Code_integer
% 0.32/0.57 @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
% 0.32/0.57 = ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % strict_subset_divisors_dvd
% 0.32/0.57 thf(fact_4005_nat__minus__add__max,axiom,
% 0.32/0.57 ! [N: nat,M: nat] :
% 0.32/0.57 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.32/0.57 = ( ord_max_nat @ N @ M ) ) ).
% 0.32/0.57
% 0.32/0.57 % nat_minus_add_max
% 0.32/0.57 thf(fact_4006_not__is__unit__0,axiom,
% 0.32/0.57 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.32/0.57
% 0.32/0.57 % not_is_unit_0
% 0.32/0.57 thf(fact_4007_not__is__unit__0,axiom,
% 0.32/0.57 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.32/0.57
% 0.32/0.57 % not_is_unit_0
% 0.32/0.57 thf(fact_4008_not__is__unit__0,axiom,
% 0.32/0.57 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.32/0.57
% 0.32/0.57 % not_is_unit_0
% 0.32/0.57 thf(fact_4009_minf_I10_J,axiom,
% 0.32/0.57 ! [D: code_integer,S: code_integer] :
% 0.32/0.57 ? [Z5: code_integer] :
% 0.32/0.57 ! [X5: code_integer] :
% 0.32/0.57 ( ( ord_le6747313008572928689nteger @ X5 @ Z5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(10)
% 0.32/0.57 thf(fact_4010_minf_I10_J,axiom,
% 0.32/0.57 ! [D: real,S: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(10)
% 0.32/0.57 thf(fact_4011_minf_I10_J,axiom,
% 0.32/0.57 ! [D: rat,S: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(10)
% 0.32/0.57 thf(fact_4012_minf_I10_J,axiom,
% 0.32/0.57 ! [D: nat,S: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(10)
% 0.32/0.57 thf(fact_4013_minf_I10_J,axiom,
% 0.32/0.57 ! [D: int,S: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(10)
% 0.32/0.57 thf(fact_4014_minf_I9_J,axiom,
% 0.32/0.57 ! [D: code_integer,S: code_integer] :
% 0.32/0.57 ? [Z5: code_integer] :
% 0.32/0.57 ! [X5: code_integer] :
% 0.32/0.57 ( ( ord_le6747313008572928689nteger @ X5 @ Z5 )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(9)
% 0.32/0.57 thf(fact_4015_minf_I9_J,axiom,
% 0.32/0.57 ! [D: real,S: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ X5 @ Z5 )
% 0.32/0.57 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(9)
% 0.32/0.57 thf(fact_4016_minf_I9_J,axiom,
% 0.32/0.57 ! [D: rat,S: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ X5 @ Z5 )
% 0.32/0.57 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(9)
% 0.32/0.57 thf(fact_4017_minf_I9_J,axiom,
% 0.32/0.57 ! [D: nat,S: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ X5 @ Z5 )
% 0.32/0.57 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(9)
% 0.32/0.57 thf(fact_4018_minf_I9_J,axiom,
% 0.32/0.57 ! [D: int,S: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ X5 @ Z5 )
% 0.32/0.57 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % minf(9)
% 0.32/0.57 thf(fact_4019_pinf_I10_J,axiom,
% 0.32/0.57 ! [D: code_integer,S: code_integer] :
% 0.32/0.57 ? [Z5: code_integer] :
% 0.32/0.57 ! [X5: code_integer] :
% 0.32/0.57 ( ( ord_le6747313008572928689nteger @ Z5 @ X5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(10)
% 0.32/0.57 thf(fact_4020_pinf_I10_J,axiom,
% 0.32/0.57 ! [D: real,S: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(10)
% 0.32/0.57 thf(fact_4021_pinf_I10_J,axiom,
% 0.32/0.57 ! [D: rat,S: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(10)
% 0.32/0.57 thf(fact_4022_pinf_I10_J,axiom,
% 0.32/0.57 ! [D: nat,S: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(10)
% 0.32/0.57 thf(fact_4023_pinf_I10_J,axiom,
% 0.32/0.57 ! [D: int,S: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 0.32/0.57 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(10)
% 0.32/0.57 thf(fact_4024_pinf_I9_J,axiom,
% 0.32/0.57 ! [D: code_integer,S: code_integer] :
% 0.32/0.57 ? [Z5: code_integer] :
% 0.32/0.57 ! [X5: code_integer] :
% 0.32/0.57 ( ( ord_le6747313008572928689nteger @ Z5 @ X5 )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(9)
% 0.32/0.57 thf(fact_4025_pinf_I9_J,axiom,
% 0.32/0.57 ! [D: real,S: real] :
% 0.32/0.57 ? [Z5: real] :
% 0.32/0.57 ! [X5: real] :
% 0.32/0.57 ( ( ord_less_real @ Z5 @ X5 )
% 0.32/0.57 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(9)
% 0.32/0.57 thf(fact_4026_pinf_I9_J,axiom,
% 0.32/0.57 ! [D: rat,S: rat] :
% 0.32/0.57 ? [Z5: rat] :
% 0.32/0.57 ! [X5: rat] :
% 0.32/0.57 ( ( ord_less_rat @ Z5 @ X5 )
% 0.32/0.57 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(9)
% 0.32/0.57 thf(fact_4027_pinf_I9_J,axiom,
% 0.32/0.57 ! [D: nat,S: nat] :
% 0.32/0.57 ? [Z5: nat] :
% 0.32/0.57 ! [X5: nat] :
% 0.32/0.57 ( ( ord_less_nat @ Z5 @ X5 )
% 0.32/0.57 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(9)
% 0.32/0.57 thf(fact_4028_pinf_I9_J,axiom,
% 0.32/0.57 ! [D: int,S: int] :
% 0.32/0.57 ? [Z5: int] :
% 0.32/0.57 ! [X5: int] :
% 0.32/0.57 ( ( ord_less_int @ Z5 @ X5 )
% 0.32/0.57 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 0.32/0.57 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % pinf(9)
% 0.32/0.57 thf(fact_4029_dvd__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.57 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.57 = zero_z3403309356797280102nteger )
% 0.32/0.57 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_0_iff
% 0.32/0.57 thf(fact_4030_dvd__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: complex,A: complex] :
% 0.32/0.57 ( ( dvd_dvd_complex @ B @ A )
% 0.32/0.57 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.57 = zero_zero_complex )
% 0.32/0.57 = ( A = zero_zero_complex ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_0_iff
% 0.32/0.57 thf(fact_4031_dvd__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: real,A: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ B @ A )
% 0.32/0.57 => ( ( ( divide_divide_real @ A @ B )
% 0.32/0.57 = zero_zero_real )
% 0.32/0.57 = ( A = zero_zero_real ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_0_iff
% 0.32/0.57 thf(fact_4032_dvd__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: rat,A: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ B @ A )
% 0.32/0.57 => ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.57 = zero_zero_rat )
% 0.32/0.57 = ( A = zero_zero_rat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_0_iff
% 0.32/0.57 thf(fact_4033_dvd__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.57 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 = ( A = zero_zero_nat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_0_iff
% 0.32/0.57 thf(fact_4034_dvd__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.57 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.57 = zero_zero_int )
% 0.32/0.57 = ( A = zero_zero_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_0_iff
% 0.32/0.57 thf(fact_4035_unit__mult__right__cancel,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.57 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 0.32/0.57 = ( times_3573771949741848930nteger @ C @ A ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_mult_right_cancel
% 0.32/0.57 thf(fact_4036_unit__mult__right__cancel,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.57 => ( ( ( times_times_nat @ B @ A )
% 0.32/0.57 = ( times_times_nat @ C @ A ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_mult_right_cancel
% 0.32/0.57 thf(fact_4037_unit__mult__right__cancel,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.57 => ( ( ( times_times_int @ B @ A )
% 0.32/0.57 = ( times_times_int @ C @ A ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_mult_right_cancel
% 0.32/0.57 thf(fact_4038_unit__mult__left__cancel,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.57 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 0.32/0.57 = ( times_3573771949741848930nteger @ A @ C ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_mult_left_cancel
% 0.32/0.57 thf(fact_4039_unit__mult__left__cancel,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.57 => ( ( ( times_times_nat @ A @ B )
% 0.32/0.57 = ( times_times_nat @ A @ C ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_mult_left_cancel
% 0.32/0.57 thf(fact_4040_unit__mult__left__cancel,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.57 => ( ( ( times_times_int @ A @ B )
% 0.32/0.57 = ( times_times_int @ A @ C ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_mult_left_cancel
% 0.32/0.57 thf(fact_4041_mult__unit__dvd__iff_H,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_unit_dvd_iff'
% 0.32/0.57 thf(fact_4042_mult__unit__dvd__iff_H,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_unit_dvd_iff'
% 0.32/0.57 thf(fact_4043_mult__unit__dvd__iff_H,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_unit_dvd_iff'
% 0.32/0.57 thf(fact_4044_dvd__mult__unit__iff_H,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_unit_iff'
% 0.32/0.57 thf(fact_4045_dvd__mult__unit__iff_H,axiom,
% 0.32/0.57 ! [B: nat,A: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_unit_iff'
% 0.32/0.57 thf(fact_4046_dvd__mult__unit__iff_H,axiom,
% 0.32/0.57 ! [B: int,A: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_unit_iff'
% 0.32/0.57 thf(fact_4047_mult__unit__dvd__iff,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_unit_dvd_iff
% 0.32/0.57 thf(fact_4048_mult__unit__dvd__iff,axiom,
% 0.32/0.57 ! [B: nat,A: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_unit_dvd_iff
% 0.32/0.57 thf(fact_4049_mult__unit__dvd__iff,axiom,
% 0.32/0.57 ! [B: int,A: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mult_unit_dvd_iff
% 0.32/0.57 thf(fact_4050_dvd__mult__unit__iff,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_unit_iff
% 0.32/0.57 thf(fact_4051_dvd__mult__unit__iff,axiom,
% 0.32/0.57 ! [B: nat,A: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_unit_iff
% 0.32/0.57 thf(fact_4052_dvd__mult__unit__iff,axiom,
% 0.32/0.57 ! [B: int,A: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 0.32/0.57 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_unit_iff
% 0.32/0.57 thf(fact_4053_is__unit__mult__iff,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 0.32/0.57 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.57 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % is_unit_mult_iff
% 0.32/0.57 thf(fact_4054_is__unit__mult__iff,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 0.32/0.57 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.57 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % is_unit_mult_iff
% 0.32/0.57 thf(fact_4055_is__unit__mult__iff,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 0.32/0.57 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.57 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % is_unit_mult_iff
% 0.32/0.57 thf(fact_4056_dvd__div__mult,axiom,
% 0.32/0.57 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 0.32/0.57 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_mult
% 0.32/0.57 thf(fact_4057_dvd__div__mult,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 0.32/0.57 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_mult
% 0.32/0.57 thf(fact_4058_dvd__div__mult,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 0.32/0.57 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_mult
% 0.32/0.57 thf(fact_4059_div__mult__swap,axiom,
% 0.32/0.57 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.57 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_mult_swap
% 0.32/0.57 thf(fact_4060_div__mult__swap,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.57 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_mult_swap
% 0.32/0.57 thf(fact_4061_div__mult__swap,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.57 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_mult_swap
% 0.32/0.57 thf(fact_4062_div__div__eq__right,axiom,
% 0.32/0.57 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.57 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_div_eq_right
% 0.32/0.57 thf(fact_4063_div__div__eq__right,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.57 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.57 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_div_eq_right
% 0.32/0.57 thf(fact_4064_div__div__eq__right,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.57 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.57 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_div_eq_right
% 0.32/0.57 thf(fact_4065_dvd__div__mult2__eq,axiom,
% 0.32/0.57 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.57 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_mult2_eq
% 0.32/0.57 thf(fact_4066_dvd__div__mult2__eq,axiom,
% 0.32/0.57 ! [B: nat,C: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 0.32/0.57 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.57 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_mult2_eq
% 0.32/0.57 thf(fact_4067_dvd__div__mult2__eq,axiom,
% 0.32/0.57 ! [B: int,C: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 0.32/0.57 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.57 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_mult2_eq
% 0.32/0.57 thf(fact_4068_dvd__mult__imp__div,axiom,
% 0.32/0.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_imp_div
% 0.32/0.57 thf(fact_4069_dvd__mult__imp__div,axiom,
% 0.32/0.57 ! [A: nat,C: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 0.32/0.57 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_imp_div
% 0.32/0.57 thf(fact_4070_dvd__mult__imp__div,axiom,
% 0.32/0.57 ! [A: int,C: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 0.32/0.57 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_mult_imp_div
% 0.32/0.57 thf(fact_4071_div__mult__div__if__dvd,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.32/0.57 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.32/0.57 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_mult_div_if_dvd
% 0.32/0.57 thf(fact_4072_div__mult__div__if__dvd,axiom,
% 0.32/0.57 ! [B: nat,A: nat,D: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.57 => ( ( dvd_dvd_nat @ D @ C )
% 0.32/0.57 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 0.32/0.57 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_mult_div_if_dvd
% 0.32/0.57 thf(fact_4073_div__mult__div__if__dvd,axiom,
% 0.32/0.57 ! [B: int,A: int,D: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.57 => ( ( dvd_dvd_int @ D @ C )
% 0.32/0.57 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 0.32/0.57 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_mult_div_if_dvd
% 0.32/0.57 thf(fact_4074_dvd__div__unit__iff,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_unit_iff
% 0.32/0.57 thf(fact_4075_dvd__div__unit__iff,axiom,
% 0.32/0.57 ! [B: nat,A: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_unit_iff
% 0.32/0.57 thf(fact_4076_dvd__div__unit__iff,axiom,
% 0.32/0.57 ! [B: int,A: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 0.32/0.57 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_unit_iff
% 0.32/0.57 thf(fact_4077_div__unit__dvd__iff,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_unit_dvd_iff
% 0.32/0.57 thf(fact_4078_div__unit__dvd__iff,axiom,
% 0.32/0.57 ! [B: nat,A: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_unit_dvd_iff
% 0.32/0.57 thf(fact_4079_div__unit__dvd__iff,axiom,
% 0.32/0.57 ! [B: int,A: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_unit_dvd_iff
% 0.32/0.57 thf(fact_4080_unit__div__cancel,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.57 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.57 = ( divide6298287555418463151nteger @ C @ A ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_div_cancel
% 0.32/0.57 thf(fact_4081_unit__div__cancel,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.57 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.57 = ( divide_divide_nat @ C @ A ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_div_cancel
% 0.32/0.57 thf(fact_4082_unit__div__cancel,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.57 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.57 = ( divide_divide_int @ C @ A ) )
% 0.32/0.57 = ( B = C ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_div_cancel
% 0.32/0.57 thf(fact_4083_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.57 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ A )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.57 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_plus_div_distrib_dvd_left
% 0.32/0.57 thf(fact_4084_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.57 ! [C: nat,A: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ A )
% 0.32/0.57 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.57 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_plus_div_distrib_dvd_left
% 0.32/0.57 thf(fact_4085_div__plus__div__distrib__dvd__left,axiom,
% 0.32/0.57 ! [C: int,A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ A )
% 0.32/0.57 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.57 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_plus_div_distrib_dvd_left
% 0.32/0.57 thf(fact_4086_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.57 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.32/0.57 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_plus_div_distrib_dvd_right
% 0.32/0.57 thf(fact_4087_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.32/0.57 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_plus_div_distrib_dvd_right
% 0.32/0.57 thf(fact_4088_div__plus__div__distrib__dvd__right,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.32/0.57 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_plus_div_distrib_dvd_right
% 0.32/0.57 thf(fact_4089_div__power,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.57 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 0.32/0.57 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_power
% 0.32/0.57 thf(fact_4090_div__power,axiom,
% 0.32/0.57 ! [B: nat,A: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.57 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 0.32/0.57 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_power
% 0.32/0.57 thf(fact_4091_div__power,axiom,
% 0.32/0.57 ! [B: int,A: int,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.57 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 0.32/0.57 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_power
% 0.32/0.57 thf(fact_4092_mod__eq__0__iff__dvd,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 = ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_eq_0_iff_dvd
% 0.32/0.57 thf(fact_4093_mod__eq__0__iff__dvd,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.57 = zero_zero_int )
% 0.32/0.57 = ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_eq_0_iff_dvd
% 0.32/0.57 thf(fact_4094_mod__eq__0__iff__dvd,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.57 = zero_z3403309356797280102nteger )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_eq_0_iff_dvd
% 0.32/0.57 thf(fact_4095_dvd__eq__mod__eq__0,axiom,
% 0.32/0.57 ( dvd_dvd_nat
% 0.32/0.57 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.57 ( ( modulo_modulo_nat @ B4 @ A3 )
% 0.32/0.57 = zero_zero_nat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_eq_mod_eq_0
% 0.32/0.57 thf(fact_4096_dvd__eq__mod__eq__0,axiom,
% 0.32/0.57 ( dvd_dvd_int
% 0.32/0.57 = ( ^ [A3: int,B4: int] :
% 0.32/0.57 ( ( modulo_modulo_int @ B4 @ A3 )
% 0.32/0.57 = zero_zero_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_eq_mod_eq_0
% 0.32/0.57 thf(fact_4097_dvd__eq__mod__eq__0,axiom,
% 0.32/0.57 ( dvd_dvd_Code_integer
% 0.32/0.57 = ( ^ [A3: code_integer,B4: code_integer] :
% 0.32/0.57 ( ( modulo364778990260209775nteger @ B4 @ A3 )
% 0.32/0.57 = zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_eq_mod_eq_0
% 0.32/0.57 thf(fact_4098_mod__0__imp__dvd,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_0_imp_dvd
% 0.32/0.57 thf(fact_4099_mod__0__imp__dvd,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.57 = zero_zero_int )
% 0.32/0.57 => ( dvd_dvd_int @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_0_imp_dvd
% 0.32/0.57 thf(fact_4100_mod__0__imp__dvd,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.57 = zero_z3403309356797280102nteger )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_0_imp_dvd
% 0.32/0.57 thf(fact_4101_dvd__power__le,axiom,
% 0.32/0.57 ! [X: code_integer,Y3: code_integer,N: nat,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ X @ Y3 )
% 0.32/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_le
% 0.32/0.57 thf(fact_4102_dvd__power__le,axiom,
% 0.32/0.57 ! [X: nat,Y3: nat,N: nat,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ X @ Y3 )
% 0.32/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.57 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_le
% 0.32/0.57 thf(fact_4103_dvd__power__le,axiom,
% 0.32/0.57 ! [X: real,Y3: real,N: nat,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_real @ X @ Y3 )
% 0.32/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.57 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_le
% 0.32/0.57 thf(fact_4104_dvd__power__le,axiom,
% 0.32/0.57 ! [X: int,Y3: int,N: nat,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_int @ X @ Y3 )
% 0.32/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.57 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_le
% 0.32/0.57 thf(fact_4105_dvd__power__le,axiom,
% 0.32/0.57 ! [X: complex,Y3: complex,N: nat,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_complex @ X @ Y3 )
% 0.32/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.57 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_power_le
% 0.32/0.57 thf(fact_4106_power__le__dvd,axiom,
% 0.32/0.57 ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 0.32/0.57 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_le_dvd
% 0.32/0.57 thf(fact_4107_power__le__dvd,axiom,
% 0.32/0.57 ! [A: nat,N: nat,B: nat,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 0.32/0.57 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_le_dvd
% 0.32/0.57 thf(fact_4108_power__le__dvd,axiom,
% 0.32/0.57 ! [A: real,N: nat,B: real,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 0.32/0.57 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_le_dvd
% 0.32/0.57 thf(fact_4109_power__le__dvd,axiom,
% 0.32/0.57 ! [A: int,N: nat,B: int,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 0.32/0.57 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_le_dvd
% 0.32/0.57 thf(fact_4110_power__le__dvd,axiom,
% 0.32/0.57 ! [A: complex,N: nat,B: complex,M: nat] :
% 0.32/0.57 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 0.32/0.57 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % power_le_dvd
% 0.32/0.57 thf(fact_4111_le__imp__power__dvd,axiom,
% 0.32/0.57 ! [M: nat,N: nat,A: code_integer] :
% 0.32/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_imp_power_dvd
% 0.32/0.57 thf(fact_4112_le__imp__power__dvd,axiom,
% 0.32/0.57 ! [M: nat,N: nat,A: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_imp_power_dvd
% 0.32/0.57 thf(fact_4113_le__imp__power__dvd,axiom,
% 0.32/0.57 ! [M: nat,N: nat,A: real] :
% 0.32/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_imp_power_dvd
% 0.32/0.57 thf(fact_4114_le__imp__power__dvd,axiom,
% 0.32/0.57 ! [M: nat,N: nat,A: int] :
% 0.32/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_imp_power_dvd
% 0.32/0.57 thf(fact_4115_le__imp__power__dvd,axiom,
% 0.32/0.57 ! [M: nat,N: nat,A: complex] :
% 0.32/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % le_imp_power_dvd
% 0.32/0.57 thf(fact_4116_dvd__minus__mod,axiom,
% 0.32/0.57 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_minus_mod
% 0.32/0.57 thf(fact_4117_dvd__minus__mod,axiom,
% 0.32/0.57 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_minus_mod
% 0.32/0.57 thf(fact_4118_dvd__minus__mod,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_minus_mod
% 0.32/0.57 thf(fact_4119_mod__eq__dvd__iff,axiom,
% 0.32/0.57 ! [A: int,C: int,B: int] :
% 0.32/0.57 ( ( ( modulo_modulo_int @ A @ C )
% 0.32/0.57 = ( modulo_modulo_int @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_eq_dvd_iff
% 0.32/0.57 thf(fact_4120_mod__eq__dvd__iff,axiom,
% 0.32/0.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.32/0.57 ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.32/0.57 = ( modulo364778990260209775nteger @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % mod_eq_dvd_iff
% 0.32/0.57 thf(fact_4121_nat__dvd__not__less,axiom,
% 0.32/0.57 ! [M: nat,N: nat] :
% 0.32/0.57 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.57 => ( ( ord_less_nat @ M @ N )
% 0.32/0.57 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % nat_dvd_not_less
% 0.32/0.57 thf(fact_4122_bezout__add__strong__nat,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( A != zero_zero_nat )
% 0.32/0.57 => ? [D3: nat,X3: nat,Y4: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ D3 @ A )
% 0.32/0.57 & ( dvd_dvd_nat @ D3 @ B )
% 0.32/0.57 & ( ( times_times_nat @ A @ X3 )
% 0.32/0.57 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % bezout_add_strong_nat
% 0.32/0.57 thf(fact_4123_set__update__subsetI,axiom,
% 0.32/0.57 ! [Xs2: list_nat,A2: set_nat,X: nat,I: nat] :
% 0.32/0.57 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 0.32/0.57 => ( ( member_nat @ X @ A2 )
% 0.32/0.57 => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_update_subsetI
% 0.32/0.57 thf(fact_4124_set__update__subsetI,axiom,
% 0.32/0.57 ! [Xs2: list_real,A2: set_real,X: real,I: nat] :
% 0.32/0.57 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
% 0.32/0.57 => ( ( member_real @ X @ A2 )
% 0.32/0.57 => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_update_subsetI
% 0.32/0.57 thf(fact_4125_set__update__subsetI,axiom,
% 0.32/0.57 ! [Xs2: list_complex,A2: set_complex,X: complex,I: nat] :
% 0.32/0.57 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 0.32/0.57 => ( ( member_complex @ X @ A2 )
% 0.32/0.57 => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_update_subsetI
% 0.32/0.57 thf(fact_4126_set__update__subsetI,axiom,
% 0.32/0.57 ! [Xs2: list_P8469869581646625389at_nat,A2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,I: nat] :
% 0.32/0.57 ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs2 ) @ A2 )
% 0.32/0.57 => ( ( member8206827879206165904at_nat @ X @ A2 )
% 0.32/0.57 => ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_update_subsetI
% 0.32/0.57 thf(fact_4127_set__update__subsetI,axiom,
% 0.32/0.57 ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
% 0.32/0.57 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 0.32/0.57 => ( ( member_VEBT_VEBT @ X @ A2 )
% 0.32/0.57 => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_update_subsetI
% 0.32/0.57 thf(fact_4128_set__update__subsetI,axiom,
% 0.32/0.57 ! [Xs2: list_int,A2: set_int,X: int,I: nat] :
% 0.32/0.57 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 0.32/0.57 => ( ( member_int @ X @ A2 )
% 0.32/0.57 => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % set_update_subsetI
% 0.32/0.57 thf(fact_4129_dvd__minus__self,axiom,
% 0.32/0.57 ! [M: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.57 = ( ( ord_less_nat @ N @ M )
% 0.32/0.57 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_minus_self
% 0.32/0.57 thf(fact_4130_zdvd__antisym__nonneg,axiom,
% 0.32/0.57 ! [M: int,N: int] :
% 0.32/0.57 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.32/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.32/0.57 => ( ( dvd_dvd_int @ M @ N )
% 0.32/0.57 => ( ( dvd_dvd_int @ N @ M )
% 0.32/0.57 => ( M = N ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdvd_antisym_nonneg
% 0.32/0.57 thf(fact_4131_less__eq__dvd__minus,axiom,
% 0.32/0.57 ! [M: nat,N: nat] :
% 0.32/0.57 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.57 => ( ( dvd_dvd_nat @ M @ N )
% 0.32/0.57 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % less_eq_dvd_minus
% 0.32/0.57 thf(fact_4132_dvd__diffD1,axiom,
% 0.32/0.57 ! [K: nat,M: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.57 => ( ( dvd_dvd_nat @ K @ M )
% 0.32/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.57 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diffD1
% 0.32/0.57 thf(fact_4133_dvd__diffD,axiom,
% 0.32/0.57 ! [K: nat,M: nat,N: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.57 => ( ( dvd_dvd_nat @ K @ N )
% 0.32/0.57 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.57 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_diffD
% 0.32/0.57 thf(fact_4134_zdvd__not__zless,axiom,
% 0.32/0.57 ! [M: int,N: int] :
% 0.32/0.57 ( ( ord_less_int @ zero_zero_int @ M )
% 0.32/0.57 => ( ( ord_less_int @ M @ N )
% 0.32/0.57 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdvd_not_zless
% 0.32/0.57 thf(fact_4135_zdvd__mult__cancel,axiom,
% 0.32/0.57 ! [K: int,M: int,N: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 0.32/0.57 => ( ( K != zero_zero_int )
% 0.32/0.57 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdvd_mult_cancel
% 0.32/0.57 thf(fact_4136_zdvd__mono,axiom,
% 0.32/0.57 ! [K: int,M: int,T: int] :
% 0.32/0.57 ( ( K != zero_zero_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ M @ T )
% 0.32/0.57 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdvd_mono
% 0.32/0.57 thf(fact_4137_zdvd__reduce,axiom,
% 0.32/0.57 ! [K: int,N: int,M: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 0.32/0.57 = ( dvd_dvd_int @ K @ N ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdvd_reduce
% 0.32/0.57 thf(fact_4138_zdvd__period,axiom,
% 0.32/0.57 ! [A: int,D: int,X: int,T: int,C: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ D )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
% 0.32/0.57 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % zdvd_period
% 0.32/0.57 thf(fact_4139_finite__divisors__nat,axiom,
% 0.32/0.57 ! [M: nat] :
% 0.32/0.57 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.57 => ( finite_finite_nat
% 0.32/0.57 @ ( collect_nat
% 0.32/0.57 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % finite_divisors_nat
% 0.32/0.57 thf(fact_4140_div2__even__ext__nat,axiom,
% 0.32/0.57 ! [X: nat,Y3: nat] :
% 0.32/0.57 ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.57 = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.57 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 0.32/0.57 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
% 0.32/0.57 => ( X = Y3 ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div2_even_ext_nat
% 0.32/0.57 thf(fact_4141_unit__dvdE,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.57 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.57 => ! [C2: code_integer] :
% 0.32/0.57 ( B
% 0.32/0.57 != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_dvdE
% 0.32/0.57 thf(fact_4142_unit__dvdE,axiom,
% 0.32/0.57 ! [A: nat,B: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.57 => ~ ( ( A != zero_zero_nat )
% 0.32/0.57 => ! [C2: nat] :
% 0.32/0.57 ( B
% 0.32/0.57 != ( times_times_nat @ A @ C2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_dvdE
% 0.32/0.57 thf(fact_4143_unit__dvdE,axiom,
% 0.32/0.57 ! [A: int,B: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.57 => ~ ( ( A != zero_zero_int )
% 0.32/0.57 => ! [C2: int] :
% 0.32/0.57 ( B
% 0.32/0.57 != ( times_times_int @ A @ C2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_dvdE
% 0.32/0.57 thf(fact_4144_unity__coeff__ex,axiom,
% 0.32/0.57 ! [P: code_integer > $o,L2: code_integer] :
% 0.32/0.57 ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X2 ) ) )
% 0.32/0.57 = ( ? [X2: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
% 0.32/0.57 & ( P @ X2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unity_coeff_ex
% 0.32/0.57 thf(fact_4145_unity__coeff__ex,axiom,
% 0.32/0.57 ! [P: complex > $o,L2: complex] :
% 0.32/0.57 ( ( ? [X2: complex] : ( P @ ( times_times_complex @ L2 @ X2 ) ) )
% 0.32/0.57 = ( ? [X2: complex] :
% 0.32/0.57 ( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
% 0.32/0.57 & ( P @ X2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unity_coeff_ex
% 0.32/0.57 thf(fact_4146_unity__coeff__ex,axiom,
% 0.32/0.57 ! [P: real > $o,L2: real] :
% 0.32/0.57 ( ( ? [X2: real] : ( P @ ( times_times_real @ L2 @ X2 ) ) )
% 0.32/0.57 = ( ? [X2: real] :
% 0.32/0.57 ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X2 @ zero_zero_real ) )
% 0.32/0.57 & ( P @ X2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unity_coeff_ex
% 0.32/0.57 thf(fact_4147_unity__coeff__ex,axiom,
% 0.32/0.57 ! [P: rat > $o,L2: rat] :
% 0.32/0.57 ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L2 @ X2 ) ) )
% 0.32/0.57 = ( ? [X2: rat] :
% 0.32/0.57 ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
% 0.32/0.57 & ( P @ X2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unity_coeff_ex
% 0.32/0.57 thf(fact_4148_unity__coeff__ex,axiom,
% 0.32/0.57 ! [P: nat > $o,L2: nat] :
% 0.32/0.57 ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L2 @ X2 ) ) )
% 0.32/0.57 = ( ? [X2: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
% 0.32/0.57 & ( P @ X2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unity_coeff_ex
% 0.32/0.57 thf(fact_4149_unity__coeff__ex,axiom,
% 0.32/0.57 ! [P: int > $o,L2: int] :
% 0.32/0.57 ( ( ? [X2: int] : ( P @ ( times_times_int @ L2 @ X2 ) ) )
% 0.32/0.57 = ( ? [X2: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X2 @ zero_zero_int ) )
% 0.32/0.57 & ( P @ X2 ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unity_coeff_ex
% 0.32/0.57 thf(fact_4150_dvd__div__eq__mult,axiom,
% 0.32/0.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.32/0.57 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.57 = C )
% 0.32/0.57 = ( B
% 0.32/0.57 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_mult
% 0.32/0.57 thf(fact_4151_dvd__div__eq__mult,axiom,
% 0.32/0.57 ! [A: nat,B: nat,C: nat] :
% 0.32/0.57 ( ( A != zero_zero_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.57 = C )
% 0.32/0.57 = ( B
% 0.32/0.57 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_mult
% 0.32/0.57 thf(fact_4152_dvd__div__eq__mult,axiom,
% 0.32/0.57 ! [A: int,B: int,C: int] :
% 0.32/0.57 ( ( A != zero_zero_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.57 = C )
% 0.32/0.57 = ( B
% 0.32/0.57 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_eq_mult
% 0.32/0.57 thf(fact_4153_div__dvd__iff__mult,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.57 ( ( B != zero_z3403309356797280102nteger )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_dvd_iff_mult
% 0.32/0.57 thf(fact_4154_div__dvd__iff__mult,axiom,
% 0.32/0.57 ! [B: nat,A: nat,C: nat] :
% 0.32/0.57 ( ( B != zero_zero_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.57 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_dvd_iff_mult
% 0.32/0.57 thf(fact_4155_div__dvd__iff__mult,axiom,
% 0.32/0.57 ! [B: int,A: int,C: int] :
% 0.32/0.57 ( ( B != zero_zero_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.57 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.57 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % div_dvd_iff_mult
% 0.32/0.57 thf(fact_4156_dvd__div__iff__mult,axiom,
% 0.32/0.57 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( C != zero_z3403309356797280102nteger )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_iff_mult
% 0.32/0.57 thf(fact_4157_dvd__div__iff__mult,axiom,
% 0.32/0.57 ! [C: nat,B: nat,A: nat] :
% 0.32/0.57 ( ( C != zero_zero_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_iff_mult
% 0.32/0.57 thf(fact_4158_dvd__div__iff__mult,axiom,
% 0.32/0.57 ! [C: int,B: int,A: int] :
% 0.32/0.57 ( ( C != zero_zero_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ C @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.57 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_iff_mult
% 0.32/0.57 thf(fact_4159_dvd__div__div__eq__mult,axiom,
% 0.32/0.57 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.32/0.57 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.57 => ( ( C != zero_z3403309356797280102nteger )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.32/0.57 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.32/0.57 = ( divide6298287555418463151nteger @ D @ C ) )
% 0.32/0.57 = ( ( times_3573771949741848930nteger @ B @ C )
% 0.32/0.57 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_div_eq_mult
% 0.32/0.57 thf(fact_4160_dvd__div__div__eq__mult,axiom,
% 0.32/0.57 ! [A: nat,C: nat,B: nat,D: nat] :
% 0.32/0.57 ( ( A != zero_zero_nat )
% 0.32/0.57 => ( ( C != zero_zero_nat )
% 0.32/0.57 => ( ( dvd_dvd_nat @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_nat @ C @ D )
% 0.32/0.57 => ( ( ( divide_divide_nat @ B @ A )
% 0.32/0.57 = ( divide_divide_nat @ D @ C ) )
% 0.32/0.57 = ( ( times_times_nat @ B @ C )
% 0.32/0.57 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_div_eq_mult
% 0.32/0.57 thf(fact_4161_dvd__div__div__eq__mult,axiom,
% 0.32/0.57 ! [A: int,C: int,B: int,D: int] :
% 0.32/0.57 ( ( A != zero_zero_int )
% 0.32/0.57 => ( ( C != zero_zero_int )
% 0.32/0.57 => ( ( dvd_dvd_int @ A @ B )
% 0.32/0.57 => ( ( dvd_dvd_int @ C @ D )
% 0.32/0.57 => ( ( ( divide_divide_int @ B @ A )
% 0.32/0.57 = ( divide_divide_int @ D @ C ) )
% 0.32/0.57 = ( ( times_times_int @ B @ C )
% 0.32/0.57 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % dvd_div_div_eq_mult
% 0.32/0.57 thf(fact_4162_unit__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: code_integer,A: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.57 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.57 = zero_z3403309356797280102nteger )
% 0.32/0.57 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_div_eq_0_iff
% 0.32/0.57 thf(fact_4163_unit__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: nat,A: nat] :
% 0.32/0.57 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.57 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.57 = zero_zero_nat )
% 0.32/0.57 = ( A = zero_zero_nat ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_div_eq_0_iff
% 0.32/0.57 thf(fact_4164_unit__div__eq__0__iff,axiom,
% 0.32/0.57 ! [B: int,A: int] :
% 0.32/0.57 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.57 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.57 = zero_zero_int )
% 0.32/0.57 = ( A = zero_zero_int ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % unit_div_eq_0_iff
% 0.32/0.57 thf(fact_4165_even__numeral,axiom,
% 0.32/0.57 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_numeral
% 0.32/0.57 thf(fact_4166_even__numeral,axiom,
% 0.32/0.57 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_numeral
% 0.32/0.57 thf(fact_4167_even__numeral,axiom,
% 0.32/0.57 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 0.32/0.57
% 0.32/0.57 % even_numeral
% 0.32/0.57 thf(fact_4168_inf__period_I4_J,axiom,
% 0.32/0.57 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 0.32/0.57 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 0.32/0.58 => ! [X5: code_integer,K4: code_integer] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % inf_period(4)
% 0.32/0.58 thf(fact_4169_inf__period_I4_J,axiom,
% 0.32/0.58 ! [D: real,D4: real,T: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ D @ D4 )
% 0.32/0.58 => ! [X5: real,K4: real] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % inf_period(4)
% 0.32/0.58 thf(fact_4170_inf__period_I4_J,axiom,
% 0.32/0.58 ! [D: rat,D4: rat,T: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ D @ D4 )
% 0.32/0.58 => ! [X5: rat,K4: rat] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % inf_period(4)
% 0.32/0.58 thf(fact_4171_inf__period_I4_J,axiom,
% 0.32/0.58 ! [D: int,D4: int,T: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ D @ D4 )
% 0.32/0.58 => ! [X5: int,K4: int] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % inf_period(4)
% 0.32/0.58 thf(fact_4172_inf__period_I3_J,axiom,
% 0.32/0.58 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 0.32/0.58 => ! [X5: code_integer,K4: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % inf_period(3)
% 0.32/0.58 thf(fact_4173_inf__period_I3_J,axiom,
% 0.32/0.58 ! [D: real,D4: real,T: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ D @ D4 )
% 0.32/0.58 => ! [X5: real,K4: real] :
% 0.32/0.58 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 0.32/0.58 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % inf_period(3)
% 0.32/0.58 thf(fact_4174_inf__period_I3_J,axiom,
% 0.32/0.58 ! [D: rat,D4: rat,T: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ D @ D4 )
% 0.32/0.58 => ! [X5: rat,K4: rat] :
% 0.32/0.58 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 0.32/0.58 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % inf_period(3)
% 0.32/0.58 thf(fact_4175_inf__period_I3_J,axiom,
% 0.32/0.58 ! [D: int,D4: int,T: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ D @ D4 )
% 0.32/0.58 => ! [X5: int,K4: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.58 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % inf_period(3)
% 0.32/0.58 thf(fact_4176_is__unit__div__mult2__eq,axiom,
% 0.32/0.58 ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult2_eq
% 0.32/0.58 thf(fact_4177_is__unit__div__mult2__eq,axiom,
% 0.32/0.58 ! [B: nat,C: nat,A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.58 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.58 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult2_eq
% 0.32/0.58 thf(fact_4178_is__unit__div__mult2__eq,axiom,
% 0.32/0.58 ! [B: int,C: int,A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.58 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.58 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult2_eq
% 0.32/0.58 thf(fact_4179_unit__div__mult__swap,axiom,
% 0.32/0.58 ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.58 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_mult_swap
% 0.32/0.58 thf(fact_4180_unit__div__mult__swap,axiom,
% 0.32/0.58 ! [C: nat,A: nat,B: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.58 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.32/0.58 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_mult_swap
% 0.32/0.58 thf(fact_4181_unit__div__mult__swap,axiom,
% 0.32/0.58 ! [C: int,A: int,B: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.58 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.32/0.58 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_mult_swap
% 0.32/0.58 thf(fact_4182_unit__div__commute,axiom,
% 0.32/0.58 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.32/0.58 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_commute
% 0.32/0.58 thf(fact_4183_unit__div__commute,axiom,
% 0.32/0.58 ! [B: nat,A: nat,C: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.32/0.58 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_commute
% 0.32/0.58 thf(fact_4184_unit__div__commute,axiom,
% 0.32/0.58 ! [B: int,A: int,C: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.32/0.58 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_div_commute
% 0.32/0.58 thf(fact_4185_div__mult__unit2,axiom,
% 0.32/0.58 ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_mult_unit2
% 0.32/0.58 thf(fact_4186_div__mult__unit2,axiom,
% 0.32/0.58 ! [C: nat,B: nat,A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ B @ A )
% 0.32/0.58 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.32/0.58 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_mult_unit2
% 0.32/0.58 thf(fact_4187_div__mult__unit2,axiom,
% 0.32/0.58 ! [C: int,B: int,A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ C @ one_one_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ B @ A )
% 0.32/0.58 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.32/0.58 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % div_mult_unit2
% 0.32/0.58 thf(fact_4188_unit__eq__div2,axiom,
% 0.32/0.58 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( ( A
% 0.32/0.58 = ( divide6298287555418463151nteger @ C @ B ) )
% 0.32/0.58 = ( ( times_3573771949741848930nteger @ A @ B )
% 0.32/0.58 = C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_eq_div2
% 0.32/0.58 thf(fact_4189_unit__eq__div2,axiom,
% 0.32/0.58 ! [B: nat,A: nat,C: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( ( A
% 0.32/0.58 = ( divide_divide_nat @ C @ B ) )
% 0.32/0.58 = ( ( times_times_nat @ A @ B )
% 0.32/0.58 = C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_eq_div2
% 0.32/0.58 thf(fact_4190_unit__eq__div2,axiom,
% 0.32/0.58 ! [B: int,A: int,C: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( ( A
% 0.32/0.58 = ( divide_divide_int @ C @ B ) )
% 0.32/0.58 = ( ( times_times_int @ A @ B )
% 0.32/0.58 = C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_eq_div2
% 0.32/0.58 thf(fact_4191_unit__eq__div1,axiom,
% 0.32/0.58 ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.32/0.58 = C )
% 0.32/0.58 = ( A
% 0.32/0.58 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_eq_div1
% 0.32/0.58 thf(fact_4192_unit__eq__div1,axiom,
% 0.32/0.58 ! [B: nat,A: nat,C: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( ( ( divide_divide_nat @ A @ B )
% 0.32/0.58 = C )
% 0.32/0.58 = ( A
% 0.32/0.58 = ( times_times_nat @ C @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_eq_div1
% 0.32/0.58 thf(fact_4193_unit__eq__div1,axiom,
% 0.32/0.58 ! [B: int,A: int,C: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( ( ( divide_divide_int @ A @ B )
% 0.32/0.58 = C )
% 0.32/0.58 = ( A
% 0.32/0.58 = ( times_times_int @ C @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_eq_div1
% 0.32/0.58 thf(fact_4194_unit__imp__mod__eq__0,axiom,
% 0.32/0.58 ! [B: nat,A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( ( modulo_modulo_nat @ A @ B )
% 0.32/0.58 = zero_zero_nat ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_imp_mod_eq_0
% 0.32/0.58 thf(fact_4195_unit__imp__mod__eq__0,axiom,
% 0.32/0.58 ! [B: int,A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( ( modulo_modulo_int @ A @ B )
% 0.32/0.58 = zero_zero_int ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_imp_mod_eq_0
% 0.32/0.58 thf(fact_4196_unit__imp__mod__eq__0,axiom,
% 0.32/0.58 ! [B: code_integer,A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.58 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.58
% 0.32/0.58 % unit_imp_mod_eq_0
% 0.32/0.58 thf(fact_4197_is__unit__power__iff,axiom,
% 0.32/0.58 ! [A: code_integer,N: nat] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 0.32/0.58 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.58 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_power_iff
% 0.32/0.58 thf(fact_4198_is__unit__power__iff,axiom,
% 0.32/0.58 ! [A: nat,N: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 0.32/0.58 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.58 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_power_iff
% 0.32/0.58 thf(fact_4199_is__unit__power__iff,axiom,
% 0.32/0.58 ! [A: int,N: nat] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 0.32/0.58 = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.58 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_power_iff
% 0.32/0.58 thf(fact_4200_dvd__imp__le,axiom,
% 0.32/0.58 ! [K: nat,N: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ K @ N )
% 0.32/0.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.58 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_imp_le
% 0.32/0.58 thf(fact_4201_nat__mult__dvd__cancel1,axiom,
% 0.32/0.58 ! [K: nat,M: nat,N: nat] :
% 0.32/0.58 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.58 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nat_mult_dvd_cancel1
% 0.32/0.58 thf(fact_4202_dvd__mult__cancel,axiom,
% 0.32/0.58 ! [K: nat,M: nat,N: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.32/0.58 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.58 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel
% 0.32/0.58 thf(fact_4203_set__update__memI,axiom,
% 0.32/0.58 ! [N: nat,Xs2: list_real,X: real] :
% 0.32/0.58 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 0.32/0.58 => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % set_update_memI
% 0.32/0.58 thf(fact_4204_set__update__memI,axiom,
% 0.32/0.58 ! [N: nat,Xs2: list_complex,X: complex] :
% 0.32/0.58 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 0.32/0.58 => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % set_update_memI
% 0.32/0.58 thf(fact_4205_set__update__memI,axiom,
% 0.32/0.58 ! [N: nat,Xs2: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
% 0.32/0.58 ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 0.32/0.58 => ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ Xs2 @ N @ X ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % set_update_memI
% 0.32/0.58 thf(fact_4206_set__update__memI,axiom,
% 0.32/0.58 ! [N: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.32/0.58 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.58 => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % set_update_memI
% 0.32/0.58 thf(fact_4207_set__update__memI,axiom,
% 0.32/0.58 ! [N: nat,Xs2: list_o,X: $o] :
% 0.32/0.58 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.58 => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % set_update_memI
% 0.32/0.58 thf(fact_4208_set__update__memI,axiom,
% 0.32/0.58 ! [N: nat,Xs2: list_nat,X: nat] :
% 0.32/0.58 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.58 => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % set_update_memI
% 0.32/0.58 thf(fact_4209_set__update__memI,axiom,
% 0.32/0.58 ! [N: nat,Xs2: list_int,X: int] :
% 0.32/0.58 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.58 => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % set_update_memI
% 0.32/0.58 thf(fact_4210_zdvd__imp__le,axiom,
% 0.32/0.58 ! [Z: int,N: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ Z @ N )
% 0.32/0.58 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.32/0.58 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zdvd_imp_le
% 0.32/0.58 thf(fact_4211_nle__le,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 0.32/0.58 = ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.58 & ( B != A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nle_le
% 0.32/0.58 thf(fact_4212_nle__le,axiom,
% 0.32/0.58 ! [A: num,B: num] :
% 0.32/0.58 ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 0.32/0.58 = ( ( ord_less_eq_num @ B @ A )
% 0.32/0.58 & ( B != A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nle_le
% 0.32/0.58 thf(fact_4213_nle__le,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 0.32/0.58 = ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.58 & ( B != A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nle_le
% 0.32/0.58 thf(fact_4214_nle__le,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 0.32/0.58 = ( ( ord_less_eq_int @ B @ A )
% 0.32/0.58 & ( B != A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nle_le
% 0.32/0.58 thf(fact_4215_le__cases3,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.58 ( ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 0.32/0.58 => ( ( ( ord_less_eq_rat @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_eq_rat @ X @ Z ) )
% 0.32/0.58 => ( ( ( ord_less_eq_rat @ X @ Z )
% 0.32/0.58 => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 0.32/0.58 => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 0.32/0.58 => ~ ( ord_less_eq_rat @ Y3 @ X ) )
% 0.32/0.58 => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.32/0.58 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 0.32/0.58 => ~ ( ( ord_less_eq_rat @ Z @ X )
% 0.32/0.58 => ~ ( ord_less_eq_rat @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % le_cases3
% 0.32/0.58 thf(fact_4216_le__cases3,axiom,
% 0.32/0.58 ! [X: num,Y3: num,Z: num] :
% 0.32/0.58 ( ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 0.32/0.58 => ( ( ( ord_less_eq_num @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_eq_num @ X @ Z ) )
% 0.32/0.58 => ( ( ( ord_less_eq_num @ X @ Z )
% 0.32/0.58 => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 0.32/0.58 => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 0.32/0.58 => ~ ( ord_less_eq_num @ Y3 @ X ) )
% 0.32/0.58 => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 0.32/0.58 => ~ ( ord_less_eq_num @ Z @ X ) )
% 0.32/0.58 => ~ ( ( ord_less_eq_num @ Z @ X )
% 0.32/0.58 => ~ ( ord_less_eq_num @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % le_cases3
% 0.32/0.58 thf(fact_4217_le__cases3,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.58 ( ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 0.32/0.58 => ( ( ( ord_less_eq_nat @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 0.32/0.58 => ( ( ( ord_less_eq_nat @ X @ Z )
% 0.32/0.58 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 0.32/0.58 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 0.32/0.58 => ~ ( ord_less_eq_nat @ Y3 @ X ) )
% 0.32/0.58 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.32/0.58 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 0.32/0.58 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 0.32/0.58 => ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % le_cases3
% 0.32/0.58 thf(fact_4218_le__cases3,axiom,
% 0.32/0.58 ! [X: int,Y3: int,Z: int] :
% 0.32/0.58 ( ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 0.32/0.58 => ( ( ( ord_less_eq_int @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_eq_int @ X @ Z ) )
% 0.32/0.58 => ( ( ( ord_less_eq_int @ X @ Z )
% 0.32/0.58 => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 0.32/0.58 => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 0.32/0.58 => ~ ( ord_less_eq_int @ Y3 @ X ) )
% 0.32/0.58 => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 0.32/0.58 => ~ ( ord_less_eq_int @ Z @ X ) )
% 0.32/0.58 => ~ ( ( ord_less_eq_int @ Z @ X )
% 0.32/0.58 => ~ ( ord_less_eq_int @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % le_cases3
% 0.32/0.58 thf(fact_4219_order__class_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [X2: set_int,Y: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ X2 @ Y )
% 0.32/0.58 & ( ord_less_eq_set_int @ Y @ X2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_class.order_eq_iff
% 0.32/0.58 thf(fact_4220_order__class_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [X2: rat,Y: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X2 @ Y )
% 0.32/0.58 & ( ord_less_eq_rat @ Y @ X2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_class.order_eq_iff
% 0.32/0.58 thf(fact_4221_order__class_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [X2: num,Y: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X2 @ Y )
% 0.32/0.58 & ( ord_less_eq_num @ Y @ X2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_class.order_eq_iff
% 0.32/0.58 thf(fact_4222_order__class_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [X2: nat,Y: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/0.58 & ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_class.order_eq_iff
% 0.32/0.58 thf(fact_4223_order__class_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [X2: int,Y: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ X2 @ Y )
% 0.32/0.58 & ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_class.order_eq_iff
% 0.32/0.58 thf(fact_4224_ord__eq__le__trans,axiom,
% 0.32/0.58 ! [A: set_int,B: set_int,C: set_int] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ B @ C )
% 0.32/0.58 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_trans
% 0.32/0.58 thf(fact_4225_ord__eq__le__trans,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_trans
% 0.32/0.58 thf(fact_4226_ord__eq__le__trans,axiom,
% 0.32/0.58 ! [A: num,B: num,C: num] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_trans
% 0.32/0.58 thf(fact_4227_ord__eq__le__trans,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_trans
% 0.32/0.58 thf(fact_4228_ord__eq__le__trans,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.58 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_trans
% 0.32/0.58 thf(fact_4229_ord__le__eq__trans,axiom,
% 0.32/0.58 ! [A: set_int,B: set_int,C: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_trans
% 0.32/0.58 thf(fact_4230_ord__le__eq__trans,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_trans
% 0.32/0.58 thf(fact_4231_ord__le__eq__trans,axiom,
% 0.32/0.58 ! [A: num,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_trans
% 0.32/0.58 thf(fact_4232_ord__le__eq__trans,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_trans
% 0.32/0.58 thf(fact_4233_ord__le__eq__trans,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_trans
% 0.32/0.58 thf(fact_4234_order__antisym,axiom,
% 0.32/0.58 ! [X: set_int,Y3: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ Y3 @ X )
% 0.32/0.58 => ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym
% 0.32/0.58 thf(fact_4235_order__antisym,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_rat @ Y3 @ X )
% 0.32/0.58 => ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym
% 0.32/0.58 thf(fact_4236_order__antisym,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_num @ Y3 @ X )
% 0.32/0.58 => ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym
% 0.32/0.58 thf(fact_4237_order__antisym,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_nat @ Y3 @ X )
% 0.32/0.58 => ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym
% 0.32/0.58 thf(fact_4238_order__antisym,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_int @ Y3 @ X )
% 0.32/0.58 => ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym
% 0.32/0.58 thf(fact_4239_order_Otrans,axiom,
% 0.32/0.58 ! [A: set_int,B: set_int,C: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ B @ C )
% 0.32/0.58 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.trans
% 0.32/0.58 thf(fact_4240_order_Otrans,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.trans
% 0.32/0.58 thf(fact_4241_order_Otrans,axiom,
% 0.32/0.58 ! [A: num,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.trans
% 0.32/0.58 thf(fact_4242_order_Otrans,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.58 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.trans
% 0.32/0.58 thf(fact_4243_order_Otrans,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.58 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.trans
% 0.32/0.58 thf(fact_4244_order__trans,axiom,
% 0.32/0.58 ! [X: set_int,Y3: set_int,Z: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_trans
% 0.32/0.58 thf(fact_4245_order__trans,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_trans
% 0.32/0.58 thf(fact_4246_order__trans,axiom,
% 0.32/0.58 ! [X: num,Y3: num,Z: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_eq_num @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_trans
% 0.32/0.58 thf(fact_4247_order__trans,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_trans
% 0.32/0.58 thf(fact_4248_order__trans,axiom,
% 0.32/0.58 ! [X: int,Y3: int,Z: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_trans
% 0.32/0.58 thf(fact_4249_linorder__wlog,axiom,
% 0.32/0.58 ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.32/0.58 ( ! [A5: rat,B3: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: rat,B3: rat] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_wlog
% 0.32/0.58 thf(fact_4250_linorder__wlog,axiom,
% 0.32/0.58 ! [P: num > num > $o,A: num,B: num] :
% 0.32/0.58 ( ! [A5: num,B3: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: num,B3: num] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_wlog
% 0.32/0.58 thf(fact_4251_linorder__wlog,axiom,
% 0.32/0.58 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.32/0.58 ( ! [A5: nat,B3: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: nat,B3: nat] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_wlog
% 0.32/0.58 thf(fact_4252_linorder__wlog,axiom,
% 0.32/0.58 ! [P: int > int > $o,A: int,B: int] :
% 0.32/0.58 ( ! [A5: int,B3: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: int,B3: int] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_wlog
% 0.32/0.58 thf(fact_4253_dual__order_Oeq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: set_int,B4: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ B4 @ A3 )
% 0.32/0.58 & ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.eq_iff
% 0.32/0.58 thf(fact_4254_dual__order_Oeq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: rat,B4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ B4 @ A3 )
% 0.32/0.58 & ( ord_less_eq_rat @ A3 @ B4 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.eq_iff
% 0.32/0.58 thf(fact_4255_dual__order_Oeq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: num,B4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ B4 @ A3 )
% 0.32/0.58 & ( ord_less_eq_num @ A3 @ B4 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.eq_iff
% 0.32/0.58 thf(fact_4256_dual__order_Oeq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ B4 @ A3 )
% 0.32/0.58 & ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.eq_iff
% 0.32/0.58 thf(fact_4257_dual__order_Oeq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: int,B4: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ B4 @ A3 )
% 0.32/0.58 & ( ord_less_eq_int @ A3 @ B4 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.eq_iff
% 0.32/0.58 thf(fact_4258_dual__order_Oantisym,axiom,
% 0.32/0.58 ! [B: set_int,A: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ A @ B )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.antisym
% 0.32/0.58 thf(fact_4259_dual__order_Oantisym,axiom,
% 0.32/0.58 ! [B: rat,A: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.antisym
% 0.32/0.58 thf(fact_4260_dual__order_Oantisym,axiom,
% 0.32/0.58 ! [B: num,A: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.antisym
% 0.32/0.58 thf(fact_4261_dual__order_Oantisym,axiom,
% 0.32/0.58 ! [B: nat,A: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.antisym
% 0.32/0.58 thf(fact_4262_dual__order_Oantisym,axiom,
% 0.32/0.58 ! [B: int,A: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.antisym
% 0.32/0.58 thf(fact_4263_dual__order_Otrans,axiom,
% 0.32/0.58 ! [B: set_int,A: set_int,C: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ C @ B )
% 0.32/0.58 => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.trans
% 0.32/0.58 thf(fact_4264_dual__order_Otrans,axiom,
% 0.32/0.58 ! [B: rat,A: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_rat @ C @ B )
% 0.32/0.58 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.trans
% 0.32/0.58 thf(fact_4265_dual__order_Otrans,axiom,
% 0.32/0.58 ! [B: num,A: num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_num @ C @ B )
% 0.32/0.58 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.trans
% 0.32/0.58 thf(fact_4266_dual__order_Otrans,axiom,
% 0.32/0.58 ! [B: nat,A: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_nat @ C @ B )
% 0.32/0.58 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.trans
% 0.32/0.58 thf(fact_4267_dual__order_Otrans,axiom,
% 0.32/0.58 ! [B: int,A: int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.58 => ( ( ord_less_eq_int @ C @ B )
% 0.32/0.58 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.trans
% 0.32/0.58 thf(fact_4268_antisym,axiom,
% 0.32/0.58 ! [A: set_int,B: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ B @ A )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym
% 0.32/0.58 thf(fact_4269_antisym,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym
% 0.32/0.58 thf(fact_4270_antisym,axiom,
% 0.32/0.58 ! [A: num,B: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ A )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym
% 0.32/0.58 thf(fact_4271_antisym,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym
% 0.32/0.58 thf(fact_4272_antisym,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_int @ B @ A )
% 0.32/0.58 => ( A = B ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym
% 0.32/0.58 thf(fact_4273_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: set_int,B4: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ A3 @ B4 )
% 0.32/0.58 & ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Orderings.order_eq_iff
% 0.32/0.58 thf(fact_4274_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: rat,B4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A3 @ B4 )
% 0.32/0.58 & ( ord_less_eq_rat @ B4 @ A3 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Orderings.order_eq_iff
% 0.32/0.58 thf(fact_4275_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: num,B4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ A3 @ B4 )
% 0.32/0.58 & ( ord_less_eq_num @ B4 @ A3 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Orderings.order_eq_iff
% 0.32/0.58 thf(fact_4276_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.32/0.58 & ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Orderings.order_eq_iff
% 0.32/0.58 thf(fact_4277_Orderings_Oorder__eq__iff,axiom,
% 0.32/0.58 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: int,B4: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ A3 @ B4 )
% 0.32/0.58 & ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % Orderings.order_eq_iff
% 0.32/0.58 thf(fact_4278_order__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4279_order__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4280_order__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4281_order__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.58 => ( ! [X3: int,Y4: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4282_order__subst1,axiom,
% 0.32/0.58 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4283_order__subst1,axiom,
% 0.32/0.58 ! [A: num,F: num > num,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4284_order__subst1,axiom,
% 0.32/0.58 ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4285_order__subst1,axiom,
% 0.32/0.58 ! [A: num,F: int > num,B: int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.58 => ( ! [X3: int,Y4: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4286_order__subst1,axiom,
% 0.32/0.58 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4287_order__subst1,axiom,
% 0.32/0.58 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst1
% 0.32/0.58 thf(fact_4288_order__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4289_order__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4290_order__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4291_order__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4292_order__subst2,axiom,
% 0.32/0.58 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4293_order__subst2,axiom,
% 0.32/0.58 ! [A: num,B: num,F: num > num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4294_order__subst2,axiom,
% 0.32/0.58 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4295_order__subst2,axiom,
% 0.32/0.58 ! [A: num,B: num,F: num > int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4296_order__subst2,axiom,
% 0.32/0.58 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4297_order__subst2,axiom,
% 0.32/0.58 ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.58 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_subst2
% 0.32/0.58 thf(fact_4298_order__eq__refl,axiom,
% 0.32/0.58 ! [X: set_int,Y3: set_int] :
% 0.32/0.58 ( ( X = Y3 )
% 0.32/0.58 => ( ord_less_eq_set_int @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_eq_refl
% 0.32/0.58 thf(fact_4299_order__eq__refl,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( X = Y3 )
% 0.32/0.58 => ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_eq_refl
% 0.32/0.58 thf(fact_4300_order__eq__refl,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( X = Y3 )
% 0.32/0.58 => ( ord_less_eq_num @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_eq_refl
% 0.32/0.58 thf(fact_4301_order__eq__refl,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( X = Y3 )
% 0.32/0.58 => ( ord_less_eq_nat @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_eq_refl
% 0.32/0.58 thf(fact_4302_order__eq__refl,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( X = Y3 )
% 0.32/0.58 => ( ord_less_eq_int @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_eq_refl
% 0.32/0.58 thf(fact_4303_linorder__linear,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.58 | ( ord_less_eq_rat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_linear
% 0.32/0.58 thf(fact_4304_linorder__linear,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.58 | ( ord_less_eq_num @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_linear
% 0.32/0.58 thf(fact_4305_linorder__linear,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.58 | ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_linear
% 0.32/0.58 thf(fact_4306_linorder__linear,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.58 | ( ord_less_eq_int @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_linear
% 0.32/0.58 thf(fact_4307_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4308_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4309_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4310_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4311_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4312_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: num,F: num > num,B: num,C: num] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4313_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4314_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: int,F: num > int,B: num,C: num] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4315_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4316_ord__eq__le__subst,axiom,
% 0.32/0.58 ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_le_subst
% 0.32/0.58 thf(fact_4317_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4318_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4319_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4320_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4321_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4322_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: num,B: num,F: num > num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4323_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4324_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: num,B: num,F: num > int,C: int] :
% 0.32/0.58 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4325_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4326_ord__le__eq__subst,axiom,
% 0.32/0.58 ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.32/0.58 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_le_eq_subst
% 0.32/0.58 thf(fact_4327_linorder__le__cases,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ~ ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.58 => ( ord_less_eq_rat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_le_cases
% 0.32/0.58 thf(fact_4328_linorder__le__cases,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ~ ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.58 => ( ord_less_eq_num @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_le_cases
% 0.32/0.58 thf(fact_4329_linorder__le__cases,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ~ ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.58 => ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_le_cases
% 0.32/0.58 thf(fact_4330_linorder__le__cases,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ~ ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.58 => ( ord_less_eq_int @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_le_cases
% 0.32/0.58 thf(fact_4331_order__antisym__conv,axiom,
% 0.32/0.58 ! [Y3: set_int,X: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ Y3 @ X )
% 0.32/0.58 => ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym_conv
% 0.32/0.58 thf(fact_4332_order__antisym__conv,axiom,
% 0.32/0.58 ! [Y3: rat,X: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ Y3 @ X )
% 0.32/0.58 => ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym_conv
% 0.32/0.58 thf(fact_4333_order__antisym__conv,axiom,
% 0.32/0.58 ! [Y3: num,X: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ Y3 @ X )
% 0.32/0.58 => ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym_conv
% 0.32/0.58 thf(fact_4334_order__antisym__conv,axiom,
% 0.32/0.58 ! [Y3: nat,X: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ Y3 @ X )
% 0.32/0.58 => ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym_conv
% 0.32/0.58 thf(fact_4335_order__antisym__conv,axiom,
% 0.32/0.58 ! [Y3: int,X: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ Y3 @ X )
% 0.32/0.58 => ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_antisym_conv
% 0.32/0.58 thf(fact_4336_mod__greater__zero__iff__not__dvd,axiom,
% 0.32/0.58 ! [M: nat,N: nat] :
% 0.32/0.58 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.58 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mod_greater_zero_iff_not_dvd
% 0.32/0.58 thf(fact_4337_list__update__same__conv,axiom,
% 0.32/0.58 ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
% 0.32/0.58 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.58 => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
% 0.32/0.58 = Xs2 )
% 0.32/0.58 = ( ( nth_VEBT_VEBT @ Xs2 @ I )
% 0.32/0.58 = X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % list_update_same_conv
% 0.32/0.58 thf(fact_4338_list__update__same__conv,axiom,
% 0.32/0.58 ! [I: nat,Xs2: list_o,X: $o] :
% 0.32/0.58 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.58 => ( ( ( list_update_o @ Xs2 @ I @ X )
% 0.32/0.58 = Xs2 )
% 0.32/0.58 = ( ( nth_o @ Xs2 @ I )
% 0.32/0.58 = X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % list_update_same_conv
% 0.32/0.58 thf(fact_4339_list__update__same__conv,axiom,
% 0.32/0.58 ! [I: nat,Xs2: list_nat,X: nat] :
% 0.32/0.58 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.58 => ( ( ( list_update_nat @ Xs2 @ I @ X )
% 0.32/0.58 = Xs2 )
% 0.32/0.58 = ( ( nth_nat @ Xs2 @ I )
% 0.32/0.58 = X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % list_update_same_conv
% 0.32/0.58 thf(fact_4340_list__update__same__conv,axiom,
% 0.32/0.58 ! [I: nat,Xs2: list_int,X: int] :
% 0.32/0.58 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.58 => ( ( ( list_update_int @ Xs2 @ I @ X )
% 0.32/0.58 = Xs2 )
% 0.32/0.58 = ( ( nth_int @ Xs2 @ I )
% 0.32/0.58 = X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % list_update_same_conv
% 0.32/0.58 thf(fact_4341_nth__list__update,axiom,
% 0.32/0.58 ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
% 0.32/0.58 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.32/0.58 => ( ( ( I = J )
% 0.32/0.58 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
% 0.32/0.58 = X ) )
% 0.32/0.58 & ( ( I != J )
% 0.32/0.58 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
% 0.32/0.58 = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nth_list_update
% 0.32/0.58 thf(fact_4342_nth__list__update,axiom,
% 0.32/0.58 ! [I: nat,Xs2: list_o,X: $o,J: nat] :
% 0.32/0.58 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 0.32/0.58 => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ J )
% 0.32/0.58 = ( ( ( I = J )
% 0.32/0.58 => X )
% 0.32/0.58 & ( ( I != J )
% 0.32/0.58 => ( nth_o @ Xs2 @ J ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nth_list_update
% 0.32/0.58 thf(fact_4343_nth__list__update,axiom,
% 0.32/0.58 ! [I: nat,Xs2: list_nat,J: nat,X: nat] :
% 0.32/0.58 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 0.32/0.58 => ( ( ( I = J )
% 0.32/0.58 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
% 0.32/0.58 = X ) )
% 0.32/0.58 & ( ( I != J )
% 0.32/0.58 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
% 0.32/0.58 = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nth_list_update
% 0.32/0.58 thf(fact_4344_nth__list__update,axiom,
% 0.32/0.58 ! [I: nat,Xs2: list_int,J: nat,X: int] :
% 0.32/0.58 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 0.32/0.58 => ( ( ( I = J )
% 0.32/0.58 => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
% 0.32/0.58 = X ) )
% 0.32/0.58 & ( ( I != J )
% 0.32/0.58 => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
% 0.32/0.58 = ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % nth_list_update
% 0.32/0.58 thf(fact_4345_lt__ex,axiom,
% 0.32/0.58 ! [X: real] :
% 0.32/0.58 ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% 0.32/0.58
% 0.32/0.58 % lt_ex
% 0.32/0.58 thf(fact_4346_lt__ex,axiom,
% 0.32/0.58 ! [X: rat] :
% 0.32/0.58 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).
% 0.32/0.58
% 0.32/0.58 % lt_ex
% 0.32/0.58 thf(fact_4347_lt__ex,axiom,
% 0.32/0.58 ! [X: int] :
% 0.32/0.58 ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% 0.32/0.58
% 0.32/0.58 % lt_ex
% 0.32/0.58 thf(fact_4348_gt__ex,axiom,
% 0.32/0.58 ! [X: real] :
% 0.32/0.58 ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 0.32/0.58
% 0.32/0.58 % gt_ex
% 0.32/0.58 thf(fact_4349_gt__ex,axiom,
% 0.32/0.58 ! [X: rat] :
% 0.32/0.58 ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 0.32/0.58
% 0.32/0.58 % gt_ex
% 0.32/0.58 thf(fact_4350_gt__ex,axiom,
% 0.32/0.58 ! [X: nat] :
% 0.32/0.58 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.32/0.58
% 0.32/0.58 % gt_ex
% 0.32/0.58 thf(fact_4351_gt__ex,axiom,
% 0.32/0.58 ! [X: int] :
% 0.32/0.58 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 0.32/0.58
% 0.32/0.58 % gt_ex
% 0.32/0.58 thf(fact_4352_dense,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ? [Z5: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Z5 )
% 0.32/0.58 & ( ord_less_real @ Z5 @ Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dense
% 0.32/0.58 thf(fact_4353_dense,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ? [Z5: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Z5 )
% 0.32/0.58 & ( ord_less_rat @ Z5 @ Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dense
% 0.32/0.58 thf(fact_4354_less__imp__neq,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % less_imp_neq
% 0.32/0.58 thf(fact_4355_less__imp__neq,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % less_imp_neq
% 0.32/0.58 thf(fact_4356_less__imp__neq,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % less_imp_neq
% 0.32/0.58 thf(fact_4357_less__imp__neq,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % less_imp_neq
% 0.32/0.58 thf(fact_4358_less__imp__neq,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % less_imp_neq
% 0.32/0.58 thf(fact_4359_order_Oasym,axiom,
% 0.32/0.58 ! [A: real,B: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.asym
% 0.32/0.58 thf(fact_4360_order_Oasym,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.asym
% 0.32/0.58 thf(fact_4361_order_Oasym,axiom,
% 0.32/0.58 ! [A: num,B: num] :
% 0.32/0.58 ( ( ord_less_num @ A @ B )
% 0.32/0.58 => ~ ( ord_less_num @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.asym
% 0.32/0.58 thf(fact_4362_order_Oasym,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( ord_less_nat @ A @ B )
% 0.32/0.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.asym
% 0.32/0.58 thf(fact_4363_order_Oasym,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( ord_less_int @ A @ B )
% 0.32/0.58 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.asym
% 0.32/0.58 thf(fact_4364_ord__eq__less__trans,axiom,
% 0.32/0.58 ! [A: real,B: real,C: real] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_trans
% 0.32/0.58 thf(fact_4365_ord__eq__less__trans,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_trans
% 0.32/0.58 thf(fact_4366_ord__eq__less__trans,axiom,
% 0.32/0.58 ! [A: num,B: num,C: num] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_num @ B @ C )
% 0.32/0.58 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_trans
% 0.32/0.58 thf(fact_4367_ord__eq__less__trans,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_nat @ B @ C )
% 0.32/0.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_trans
% 0.32/0.58 thf(fact_4368_ord__eq__less__trans,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( A = B )
% 0.32/0.58 => ( ( ord_less_int @ B @ C )
% 0.32/0.58 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_trans
% 0.32/0.58 thf(fact_4369_ord__less__eq__trans,axiom,
% 0.32/0.58 ! [A: real,B: real,C: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_trans
% 0.32/0.58 thf(fact_4370_ord__less__eq__trans,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_trans
% 0.32/0.58 thf(fact_4371_ord__less__eq__trans,axiom,
% 0.32/0.58 ! [A: num,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_num @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_trans
% 0.32/0.58 thf(fact_4372_ord__less__eq__trans,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_nat @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_trans
% 0.32/0.58 thf(fact_4373_ord__less__eq__trans,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( ord_less_int @ A @ B )
% 0.32/0.58 => ( ( B = C )
% 0.32/0.58 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_trans
% 0.32/0.58 thf(fact_4374_less__induct,axiom,
% 0.32/0.58 ! [P: nat > $o,A: nat] :
% 0.32/0.58 ( ! [X3: nat] :
% 0.32/0.58 ( ! [Y5: nat] :
% 0.32/0.58 ( ( ord_less_nat @ Y5 @ X3 )
% 0.32/0.58 => ( P @ Y5 ) )
% 0.32/0.58 => ( P @ X3 ) )
% 0.32/0.58 => ( P @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % less_induct
% 0.32/0.58 thf(fact_4375_antisym__conv3,axiom,
% 0.32/0.58 ! [Y3: real,X: real] :
% 0.32/0.58 ( ~ ( ord_less_real @ Y3 @ X )
% 0.32/0.58 => ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym_conv3
% 0.32/0.58 thf(fact_4376_antisym__conv3,axiom,
% 0.32/0.58 ! [Y3: rat,X: rat] :
% 0.32/0.58 ( ~ ( ord_less_rat @ Y3 @ X )
% 0.32/0.58 => ( ( ~ ( ord_less_rat @ X @ Y3 ) )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym_conv3
% 0.32/0.58 thf(fact_4377_antisym__conv3,axiom,
% 0.32/0.58 ! [Y3: num,X: num] :
% 0.32/0.58 ( ~ ( ord_less_num @ Y3 @ X )
% 0.32/0.58 => ( ( ~ ( ord_less_num @ X @ Y3 ) )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym_conv3
% 0.32/0.58 thf(fact_4378_antisym__conv3,axiom,
% 0.32/0.58 ! [Y3: nat,X: nat] :
% 0.32/0.58 ( ~ ( ord_less_nat @ Y3 @ X )
% 0.32/0.58 => ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym_conv3
% 0.32/0.58 thf(fact_4379_antisym__conv3,axiom,
% 0.32/0.58 ! [Y3: int,X: int] :
% 0.32/0.58 ( ~ ( ord_less_int @ Y3 @ X )
% 0.32/0.58 => ( ( ~ ( ord_less_int @ X @ Y3 ) )
% 0.32/0.58 = ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % antisym_conv3
% 0.32/0.58 thf(fact_4380_linorder__cases,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ~ ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ( ( X != Y3 )
% 0.32/0.58 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_cases
% 0.32/0.58 thf(fact_4381_linorder__cases,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ~ ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ( ( X != Y3 )
% 0.32/0.58 => ( ord_less_rat @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_cases
% 0.32/0.58 thf(fact_4382_linorder__cases,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ~ ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ( ( X != Y3 )
% 0.32/0.58 => ( ord_less_num @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_cases
% 0.32/0.58 thf(fact_4383_linorder__cases,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ~ ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ( ( X != Y3 )
% 0.32/0.58 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_cases
% 0.32/0.58 thf(fact_4384_linorder__cases,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ~ ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ( ( X != Y3 )
% 0.32/0.58 => ( ord_less_int @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_cases
% 0.32/0.58 thf(fact_4385_dual__order_Oasym,axiom,
% 0.32/0.58 ! [B: real,A: real] :
% 0.32/0.58 ( ( ord_less_real @ B @ A )
% 0.32/0.58 => ~ ( ord_less_real @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.asym
% 0.32/0.58 thf(fact_4386_dual__order_Oasym,axiom,
% 0.32/0.58 ! [B: rat,A: rat] :
% 0.32/0.58 ( ( ord_less_rat @ B @ A )
% 0.32/0.58 => ~ ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.asym
% 0.32/0.58 thf(fact_4387_dual__order_Oasym,axiom,
% 0.32/0.58 ! [B: num,A: num] :
% 0.32/0.58 ( ( ord_less_num @ B @ A )
% 0.32/0.58 => ~ ( ord_less_num @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.asym
% 0.32/0.58 thf(fact_4388_dual__order_Oasym,axiom,
% 0.32/0.58 ! [B: nat,A: nat] :
% 0.32/0.58 ( ( ord_less_nat @ B @ A )
% 0.32/0.58 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.asym
% 0.32/0.58 thf(fact_4389_dual__order_Oasym,axiom,
% 0.32/0.58 ! [B: int,A: int] :
% 0.32/0.58 ( ( ord_less_int @ B @ A )
% 0.32/0.58 => ~ ( ord_less_int @ A @ B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.asym
% 0.32/0.58 thf(fact_4390_dual__order_Oirrefl,axiom,
% 0.32/0.58 ! [A: real] :
% 0.32/0.58 ~ ( ord_less_real @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.irrefl
% 0.32/0.58 thf(fact_4391_dual__order_Oirrefl,axiom,
% 0.32/0.58 ! [A: rat] :
% 0.32/0.58 ~ ( ord_less_rat @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.irrefl
% 0.32/0.58 thf(fact_4392_dual__order_Oirrefl,axiom,
% 0.32/0.58 ! [A: num] :
% 0.32/0.58 ~ ( ord_less_num @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.irrefl
% 0.32/0.58 thf(fact_4393_dual__order_Oirrefl,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ~ ( ord_less_nat @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.irrefl
% 0.32/0.58 thf(fact_4394_dual__order_Oirrefl,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ~ ( ord_less_int @ A @ A ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.irrefl
% 0.32/0.58 thf(fact_4395_exists__least__iff,axiom,
% 0.32/0.58 ( ( ^ [P2: nat > $o] :
% 0.32/0.58 ? [X4: nat] : ( P2 @ X4 ) )
% 0.32/0.58 = ( ^ [P3: nat > $o] :
% 0.32/0.58 ? [N2: nat] :
% 0.32/0.58 ( ( P3 @ N2 )
% 0.32/0.58 & ! [M3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ M3 @ N2 )
% 0.32/0.58 => ~ ( P3 @ M3 ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % exists_least_iff
% 0.32/0.58 thf(fact_4396_linorder__less__wlog,axiom,
% 0.32/0.58 ! [P: real > real > $o,A: real,B: real] :
% 0.32/0.58 ( ! [A5: real,B3: real] :
% 0.32/0.58 ( ( ord_less_real @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: real] : ( P @ A5 @ A5 )
% 0.32/0.58 => ( ! [A5: real,B3: real] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_wlog
% 0.32/0.58 thf(fact_4397_linorder__less__wlog,axiom,
% 0.32/0.58 ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.32/0.58 ( ! [A5: rat,B3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: rat] : ( P @ A5 @ A5 )
% 0.32/0.58 => ( ! [A5: rat,B3: rat] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_wlog
% 0.32/0.58 thf(fact_4398_linorder__less__wlog,axiom,
% 0.32/0.58 ! [P: num > num > $o,A: num,B: num] :
% 0.32/0.58 ( ! [A5: num,B3: num] :
% 0.32/0.58 ( ( ord_less_num @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: num] : ( P @ A5 @ A5 )
% 0.32/0.58 => ( ! [A5: num,B3: num] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_wlog
% 0.32/0.58 thf(fact_4399_linorder__less__wlog,axiom,
% 0.32/0.58 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.32/0.58 ( ! [A5: nat,B3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: nat] : ( P @ A5 @ A5 )
% 0.32/0.58 => ( ! [A5: nat,B3: nat] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_wlog
% 0.32/0.58 thf(fact_4400_linorder__less__wlog,axiom,
% 0.32/0.58 ! [P: int > int > $o,A: int,B: int] :
% 0.32/0.58 ( ! [A5: int,B3: int] :
% 0.32/0.58 ( ( ord_less_int @ A5 @ B3 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( ! [A5: int] : ( P @ A5 @ A5 )
% 0.32/0.58 => ( ! [A5: int,B3: int] :
% 0.32/0.58 ( ( P @ B3 @ A5 )
% 0.32/0.58 => ( P @ A5 @ B3 ) )
% 0.32/0.58 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_wlog
% 0.32/0.58 thf(fact_4401_order_Ostrict__trans,axiom,
% 0.32/0.58 ! [A: real,B: real,C: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_trans
% 0.32/0.58 thf(fact_4402_order_Ostrict__trans,axiom,
% 0.32/0.58 ! [A: rat,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_trans
% 0.32/0.58 thf(fact_4403_order_Ostrict__trans,axiom,
% 0.32/0.58 ! [A: num,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_num @ A @ B )
% 0.32/0.58 => ( ( ord_less_num @ B @ C )
% 0.32/0.58 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_trans
% 0.32/0.58 thf(fact_4404_order_Ostrict__trans,axiom,
% 0.32/0.58 ! [A: nat,B: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_nat @ A @ B )
% 0.32/0.58 => ( ( ord_less_nat @ B @ C )
% 0.32/0.58 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_trans
% 0.32/0.58 thf(fact_4405_order_Ostrict__trans,axiom,
% 0.32/0.58 ! [A: int,B: int,C: int] :
% 0.32/0.58 ( ( ord_less_int @ A @ B )
% 0.32/0.58 => ( ( ord_less_int @ B @ C )
% 0.32/0.58 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_trans
% 0.32/0.58 thf(fact_4406_not__less__iff__gr__or__eq,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 0.32/0.58 = ( ( ord_less_real @ Y3 @ X )
% 0.32/0.58 | ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_less_iff_gr_or_eq
% 0.32/0.58 thf(fact_4407_not__less__iff__gr__or__eq,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ~ ( ord_less_rat @ X @ Y3 ) )
% 0.32/0.58 = ( ( ord_less_rat @ Y3 @ X )
% 0.32/0.58 | ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_less_iff_gr_or_eq
% 0.32/0.58 thf(fact_4408_not__less__iff__gr__or__eq,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ~ ( ord_less_num @ X @ Y3 ) )
% 0.32/0.58 = ( ( ord_less_num @ Y3 @ X )
% 0.32/0.58 | ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_less_iff_gr_or_eq
% 0.32/0.58 thf(fact_4409_not__less__iff__gr__or__eq,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 0.32/0.58 = ( ( ord_less_nat @ Y3 @ X )
% 0.32/0.58 | ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_less_iff_gr_or_eq
% 0.32/0.58 thf(fact_4410_not__less__iff__gr__or__eq,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ~ ( ord_less_int @ X @ Y3 ) )
% 0.32/0.58 = ( ( ord_less_int @ Y3 @ X )
% 0.32/0.58 | ( X = Y3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % not_less_iff_gr_or_eq
% 0.32/0.58 thf(fact_4411_dual__order_Ostrict__trans,axiom,
% 0.32/0.58 ! [B: real,A: real,C: real] :
% 0.32/0.58 ( ( ord_less_real @ B @ A )
% 0.32/0.58 => ( ( ord_less_real @ C @ B )
% 0.32/0.58 => ( ord_less_real @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_trans
% 0.32/0.58 thf(fact_4412_dual__order_Ostrict__trans,axiom,
% 0.32/0.58 ! [B: rat,A: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_rat @ B @ A )
% 0.32/0.58 => ( ( ord_less_rat @ C @ B )
% 0.32/0.58 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_trans
% 0.32/0.58 thf(fact_4413_dual__order_Ostrict__trans,axiom,
% 0.32/0.58 ! [B: num,A: num,C: num] :
% 0.32/0.58 ( ( ord_less_num @ B @ A )
% 0.32/0.58 => ( ( ord_less_num @ C @ B )
% 0.32/0.58 => ( ord_less_num @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_trans
% 0.32/0.58 thf(fact_4414_dual__order_Ostrict__trans,axiom,
% 0.32/0.58 ! [B: nat,A: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_nat @ B @ A )
% 0.32/0.58 => ( ( ord_less_nat @ C @ B )
% 0.32/0.58 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_trans
% 0.32/0.58 thf(fact_4415_dual__order_Ostrict__trans,axiom,
% 0.32/0.58 ! [B: int,A: int,C: int] :
% 0.32/0.58 ( ( ord_less_int @ B @ A )
% 0.32/0.58 => ( ( ord_less_int @ C @ B )
% 0.32/0.58 => ( ord_less_int @ C @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_trans
% 0.32/0.58 thf(fact_4416_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [A: real,B: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4417_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4418_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [A: num,B: num] :
% 0.32/0.58 ( ( ord_less_num @ A @ B )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4419_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( ord_less_nat @ A @ B )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4420_order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( ord_less_int @ A @ B )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4421_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [B: real,A: real] :
% 0.32/0.58 ( ( ord_less_real @ B @ A )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4422_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [B: rat,A: rat] :
% 0.32/0.58 ( ( ord_less_rat @ B @ A )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4423_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [B: num,A: num] :
% 0.32/0.58 ( ( ord_less_num @ B @ A )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4424_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [B: nat,A: nat] :
% 0.32/0.58 ( ( ord_less_nat @ B @ A )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4425_dual__order_Ostrict__implies__not__eq,axiom,
% 0.32/0.58 ! [B: int,A: int] :
% 0.32/0.58 ( ( ord_less_int @ B @ A )
% 0.32/0.58 => ( A != B ) ) ).
% 0.32/0.58
% 0.32/0.58 % dual_order.strict_implies_not_eq
% 0.32/0.58 thf(fact_4426_linorder__neqE,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 => ( ~ ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neqE
% 0.32/0.58 thf(fact_4427_linorder__neqE,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 => ( ~ ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ( ord_less_rat @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neqE
% 0.32/0.58 thf(fact_4428_linorder__neqE,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 => ( ~ ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ( ord_less_num @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neqE
% 0.32/0.58 thf(fact_4429_linorder__neqE,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 => ( ~ ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neqE
% 0.32/0.58 thf(fact_4430_linorder__neqE,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 => ( ~ ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ( ord_less_int @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neqE
% 0.32/0.58 thf(fact_4431_order__less__asym,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym
% 0.32/0.58 thf(fact_4432_order__less__asym,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_rat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym
% 0.32/0.58 thf(fact_4433_order__less__asym,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_num @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym
% 0.32/0.58 thf(fact_4434_order__less__asym,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym
% 0.32/0.58 thf(fact_4435_order__less__asym,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_int @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym
% 0.32/0.58 thf(fact_4436_linorder__neq__iff,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 = ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 | ( ord_less_real @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neq_iff
% 0.32/0.58 thf(fact_4437_linorder__neq__iff,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 = ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 | ( ord_less_rat @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neq_iff
% 0.32/0.58 thf(fact_4438_linorder__neq__iff,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 = ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 | ( ord_less_num @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neq_iff
% 0.32/0.58 thf(fact_4439_linorder__neq__iff,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 = ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 | ( ord_less_nat @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neq_iff
% 0.32/0.58 thf(fact_4440_linorder__neq__iff,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( X != Y3 )
% 0.32/0.58 = ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 | ( ord_less_int @ Y3 @ X ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_neq_iff
% 0.32/0.58 thf(fact_4441_order__less__asym_H,axiom,
% 0.32/0.58 ! [A: real,B: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ~ ( ord_less_real @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym'
% 0.32/0.58 thf(fact_4442_order__less__asym_H,axiom,
% 0.32/0.58 ! [A: rat,B: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym'
% 0.32/0.58 thf(fact_4443_order__less__asym_H,axiom,
% 0.32/0.58 ! [A: num,B: num] :
% 0.32/0.58 ( ( ord_less_num @ A @ B )
% 0.32/0.58 => ~ ( ord_less_num @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym'
% 0.32/0.58 thf(fact_4444_order__less__asym_H,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( ord_less_nat @ A @ B )
% 0.32/0.58 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym'
% 0.32/0.58 thf(fact_4445_order__less__asym_H,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( ord_less_int @ A @ B )
% 0.32/0.58 => ~ ( ord_less_int @ B @ A ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_asym'
% 0.32/0.58 thf(fact_4446_order__less__trans,axiom,
% 0.32/0.58 ! [X: real,Y3: real,Z: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_real @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_trans
% 0.32/0.58 thf(fact_4447_order__less__trans,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_rat @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_trans
% 0.32/0.58 thf(fact_4448_order__less__trans,axiom,
% 0.32/0.58 ! [X: num,Y3: num,Z: num] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_num @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_trans
% 0.32/0.58 thf(fact_4449_order__less__trans,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_nat @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_trans
% 0.32/0.58 thf(fact_4450_order__less__trans,axiom,
% 0.32/0.58 ! [X: int,Y3: int,Z: int] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_int @ Y3 @ Z )
% 0.32/0.58 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_trans
% 0.32/0.58 thf(fact_4451_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4452_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4453_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: num,F: real > num,B: real,C: real] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4454_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: nat,F: real > nat,B: real,C: real] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4455_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: int,F: real > int,B: real,C: real] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4456_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4457_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4458_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4459_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4460_ord__eq__less__subst,axiom,
% 0.32/0.58 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.32/0.58 ( ( A
% 0.32/0.58 = ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_eq_less_subst
% 0.32/0.58 thf(fact_4461_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4462_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4463_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > num,C: num] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4464_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4465_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > int,C: int] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4466_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4467_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4468_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4469_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4470_ord__less__eq__subst,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ( F @ B )
% 0.32/0.58 = C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % ord_less_eq_subst
% 0.32/0.58 thf(fact_4471_order__less__irrefl,axiom,
% 0.32/0.58 ! [X: real] :
% 0.32/0.58 ~ ( ord_less_real @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_irrefl
% 0.32/0.58 thf(fact_4472_order__less__irrefl,axiom,
% 0.32/0.58 ! [X: rat] :
% 0.32/0.58 ~ ( ord_less_rat @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_irrefl
% 0.32/0.58 thf(fact_4473_order__less__irrefl,axiom,
% 0.32/0.58 ! [X: num] :
% 0.32/0.58 ~ ( ord_less_num @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_irrefl
% 0.32/0.58 thf(fact_4474_order__less__irrefl,axiom,
% 0.32/0.58 ! [X: nat] :
% 0.32/0.58 ~ ( ord_less_nat @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_irrefl
% 0.32/0.58 thf(fact_4475_order__less__irrefl,axiom,
% 0.32/0.58 ! [X: int] :
% 0.32/0.58 ~ ( ord_less_int @ X @ X ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_irrefl
% 0.32/0.58 thf(fact_4476_order__less__subst1,axiom,
% 0.32/0.58 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4477_order__less__subst1,axiom,
% 0.32/0.58 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4478_order__less__subst1,axiom,
% 0.32/0.58 ! [A: real,F: num > real,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4479_order__less__subst1,axiom,
% 0.32/0.58 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_nat @ B @ C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4480_order__less__subst1,axiom,
% 0.32/0.58 ! [A: real,F: int > real,B: int,C: int] :
% 0.32/0.58 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_int @ B @ C )
% 0.32/0.58 => ( ! [X3: int,Y4: int] :
% 0.32/0.58 ( ( ord_less_int @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4481_order__less__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.32/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_real @ B @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4482_order__less__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_rat @ B @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4483_order__less__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_num @ B @ C )
% 0.32/0.58 => ( ! [X3: num,Y4: num] :
% 0.32/0.58 ( ( ord_less_num @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4484_order__less__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_nat @ B @ C )
% 0.32/0.58 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4485_order__less__subst1,axiom,
% 0.32/0.58 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.32/0.58 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.58 => ( ( ord_less_int @ B @ C )
% 0.32/0.58 => ( ! [X3: int,Y4: int] :
% 0.32/0.58 ( ( ord_less_int @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst1
% 0.32/0.58 thf(fact_4486_order__less__subst2,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4487_order__less__subst2,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4488_order__less__subst2,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > num,C: num] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4489_order__less__subst2,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > nat,C: nat] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4490_order__less__subst2,axiom,
% 0.32/0.58 ! [A: real,B: real,F: real > int,C: int] :
% 0.32/0.58 ( ( ord_less_real @ A @ B )
% 0.32/0.58 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: real,Y4: real] :
% 0.32/0.58 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4491_order__less__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4492_order__less__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4493_order__less__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4494_order__less__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4495_order__less__subst2,axiom,
% 0.32/0.58 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.58 ( ( ord_less_rat @ A @ B )
% 0.32/0.58 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.32/0.58 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.58 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.58 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_subst2
% 0.32/0.58 thf(fact_4496_order__less__not__sym,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_not_sym
% 0.32/0.58 thf(fact_4497_order__less__not__sym,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_rat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_not_sym
% 0.32/0.58 thf(fact_4498_order__less__not__sym,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_num @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_not_sym
% 0.32/0.58 thf(fact_4499_order__less__not__sym,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_not_sym
% 0.32/0.58 thf(fact_4500_order__less__not__sym,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_int @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_not_sym
% 0.32/0.58 thf(fact_4501_order__less__imp__triv,axiom,
% 0.32/0.58 ! [X: real,Y3: real,P: $o] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_real @ Y3 @ X )
% 0.32/0.58 => P ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_triv
% 0.32/0.58 thf(fact_4502_order__less__imp__triv,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat,P: $o] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_rat @ Y3 @ X )
% 0.32/0.58 => P ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_triv
% 0.32/0.58 thf(fact_4503_order__less__imp__triv,axiom,
% 0.32/0.58 ! [X: num,Y3: num,P: $o] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_num @ Y3 @ X )
% 0.32/0.58 => P ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_triv
% 0.32/0.58 thf(fact_4504_order__less__imp__triv,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat,P: $o] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_nat @ Y3 @ X )
% 0.32/0.58 => P ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_triv
% 0.32/0.58 thf(fact_4505_order__less__imp__triv,axiom,
% 0.32/0.58 ! [X: int,Y3: int,P: $o] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ( ( ord_less_int @ Y3 @ X )
% 0.32/0.58 => P ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_triv
% 0.32/0.58 thf(fact_4506_linorder__less__linear,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 | ( X = Y3 )
% 0.32/0.58 | ( ord_less_real @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_linear
% 0.32/0.58 thf(fact_4507_linorder__less__linear,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 | ( X = Y3 )
% 0.32/0.58 | ( ord_less_rat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_linear
% 0.32/0.58 thf(fact_4508_linorder__less__linear,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 | ( X = Y3 )
% 0.32/0.58 | ( ord_less_num @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_linear
% 0.32/0.58 thf(fact_4509_linorder__less__linear,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 | ( X = Y3 )
% 0.32/0.58 | ( ord_less_nat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_linear
% 0.32/0.58 thf(fact_4510_linorder__less__linear,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 | ( X = Y3 )
% 0.32/0.58 | ( ord_less_int @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % linorder_less_linear
% 0.32/0.58 thf(fact_4511_order__less__imp__not__eq,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq
% 0.32/0.58 thf(fact_4512_order__less__imp__not__eq,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq
% 0.32/0.58 thf(fact_4513_order__less__imp__not__eq,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq
% 0.32/0.58 thf(fact_4514_order__less__imp__not__eq,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq
% 0.32/0.58 thf(fact_4515_order__less__imp__not__eq,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ( X != Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq
% 0.32/0.58 thf(fact_4516_order__less__imp__not__eq2,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ( Y3 != X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq2
% 0.32/0.58 thf(fact_4517_order__less__imp__not__eq2,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ( Y3 != X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq2
% 0.32/0.58 thf(fact_4518_order__less__imp__not__eq2,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ( Y3 != X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq2
% 0.32/0.58 thf(fact_4519_order__less__imp__not__eq2,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ( Y3 != X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq2
% 0.32/0.58 thf(fact_4520_order__less__imp__not__eq2,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ( Y3 != X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_eq2
% 0.32/0.58 thf(fact_4521_order__less__imp__not__less,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_less
% 0.32/0.58 thf(fact_4522_order__less__imp__not__less,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_rat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_less
% 0.32/0.58 thf(fact_4523_order__less__imp__not__less,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_num @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_less
% 0.32/0.58 thf(fact_4524_order__less__imp__not__less,axiom,
% 0.32/0.58 ! [X: nat,Y3: nat] :
% 0.32/0.58 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_less
% 0.32/0.58 thf(fact_4525_order__less__imp__not__less,axiom,
% 0.32/0.58 ! [X: int,Y3: int] :
% 0.32/0.58 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.58 => ~ ( ord_less_int @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % order_less_imp_not_less
% 0.32/0.58 thf(fact_4526_mod__eq__dvd__iff__nat,axiom,
% 0.32/0.58 ! [N: nat,M: nat,Q2: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.58 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 0.32/0.58 = ( modulo_modulo_nat @ N @ Q2 ) )
% 0.32/0.58 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mod_eq_dvd_iff_nat
% 0.32/0.58 thf(fact_4527_even__zero,axiom,
% 0.32/0.58 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.32/0.58
% 0.32/0.58 % even_zero
% 0.32/0.58 thf(fact_4528_even__zero,axiom,
% 0.32/0.58 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.32/0.58
% 0.32/0.58 % even_zero
% 0.32/0.58 thf(fact_4529_even__zero,axiom,
% 0.32/0.58 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.32/0.58
% 0.32/0.58 % even_zero
% 0.32/0.58 thf(fact_4530_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult_cancel_right
% 0.32/0.58 thf(fact_4531_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( A != zero_zero_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 0.32/0.58 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult_cancel_right
% 0.32/0.58 thf(fact_4532_is__unit__div__mult__cancel__right,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( A != zero_zero_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 0.32/0.58 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult_cancel_right
% 0.32/0.58 thf(fact_4533_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult_cancel_left
% 0.32/0.58 thf(fact_4534_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ( A != zero_zero_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.32/0.58 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 0.32/0.58 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult_cancel_left
% 0.32/0.58 thf(fact_4535_is__unit__div__mult__cancel__left,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ( A != zero_zero_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.32/0.58 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 0.32/0.58 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unit_div_mult_cancel_left
% 0.32/0.58 thf(fact_4536_is__unitE,axiom,
% 0.32/0.58 ! [A: code_integer,C: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.32/0.58 => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.58 => ! [B3: code_integer] :
% 0.32/0.58 ( ( B3 != zero_z3403309356797280102nteger )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
% 0.32/0.58 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.32/0.58 = B3 )
% 0.32/0.58 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
% 0.32/0.58 = A )
% 0.32/0.58 => ( ( ( times_3573771949741848930nteger @ A @ B3 )
% 0.32/0.58 = one_one_Code_integer )
% 0.32/0.58 => ( ( divide6298287555418463151nteger @ C @ A )
% 0.32/0.58 != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unitE
% 0.32/0.58 thf(fact_4537_is__unitE,axiom,
% 0.32/0.58 ! [A: nat,C: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.32/0.58 => ~ ( ( A != zero_zero_nat )
% 0.32/0.58 => ! [B3: nat] :
% 0.32/0.58 ( ( B3 != zero_zero_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
% 0.32/0.58 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.32/0.58 = B3 )
% 0.32/0.58 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
% 0.32/0.58 = A )
% 0.32/0.58 => ( ( ( times_times_nat @ A @ B3 )
% 0.32/0.58 = one_one_nat )
% 0.32/0.58 => ( ( divide_divide_nat @ C @ A )
% 0.32/0.58 != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unitE
% 0.32/0.58 thf(fact_4538_is__unitE,axiom,
% 0.32/0.58 ! [A: int,C: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ A @ one_one_int )
% 0.32/0.58 => ~ ( ( A != zero_zero_int )
% 0.32/0.58 => ! [B3: int] :
% 0.32/0.58 ( ( B3 != zero_zero_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ B3 @ one_one_int )
% 0.32/0.58 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.32/0.58 = B3 )
% 0.32/0.58 => ( ( ( divide_divide_int @ one_one_int @ B3 )
% 0.32/0.58 = A )
% 0.32/0.58 => ( ( ( times_times_int @ A @ B3 )
% 0.32/0.58 = one_one_int )
% 0.32/0.58 => ( ( divide_divide_int @ C @ A )
% 0.32/0.58 != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % is_unitE
% 0.32/0.58 thf(fact_4539_evenE,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ~ ! [B3: code_integer] :
% 0.32/0.58 ( A
% 0.32/0.58 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % evenE
% 0.32/0.58 thf(fact_4540_evenE,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ~ ! [B3: nat] :
% 0.32/0.58 ( A
% 0.32/0.58 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % evenE
% 0.32/0.58 thf(fact_4541_evenE,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ~ ! [B3: int] :
% 0.32/0.58 ( A
% 0.32/0.58 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % evenE
% 0.32/0.58 thf(fact_4542_odd__one,axiom,
% 0.32/0.58 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 0.32/0.58
% 0.32/0.58 % odd_one
% 0.32/0.58 thf(fact_4543_odd__one,axiom,
% 0.32/0.58 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 0.32/0.58
% 0.32/0.58 % odd_one
% 0.32/0.58 thf(fact_4544_odd__one,axiom,
% 0.32/0.58 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 0.32/0.58
% 0.32/0.58 % odd_one
% 0.32/0.58 thf(fact_4545_odd__even__add,axiom,
% 0.32/0.58 ! [A: code_integer,B: code_integer] :
% 0.32/0.58 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_even_add
% 0.32/0.58 thf(fact_4546_odd__even__add,axiom,
% 0.32/0.58 ! [A: nat,B: nat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.58 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_even_add
% 0.32/0.58 thf(fact_4547_odd__even__add,axiom,
% 0.32/0.58 ! [A: int,B: int] :
% 0.32/0.58 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 0.32/0.58 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_even_add
% 0.32/0.58 thf(fact_4548_bit__eq__rec,axiom,
% 0.32/0.58 ( ( ^ [Y6: code_integer,Z4: code_integer] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: code_integer,B4: code_integer] :
% 0.32/0.58 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
% 0.32/0.58 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
% 0.32/0.58 & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % bit_eq_rec
% 0.32/0.58 thf(fact_4549_bit__eq__rec,axiom,
% 0.32/0.58 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.58 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
% 0.32/0.58 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
% 0.32/0.58 & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % bit_eq_rec
% 0.32/0.58 thf(fact_4550_bit__eq__rec,axiom,
% 0.32/0.58 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.58 = ( ^ [A3: int,B4: int] :
% 0.32/0.58 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
% 0.32/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
% 0.32/0.58 & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % bit_eq_rec
% 0.32/0.58 thf(fact_4551_dvd__power__iff,axiom,
% 0.32/0.58 ! [X: code_integer,M: nat,N: nat] :
% 0.32/0.58 ( ( X != zero_z3403309356797280102nteger )
% 0.32/0.58 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 0.32/0.58 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 0.32/0.58 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power_iff
% 0.32/0.58 thf(fact_4552_dvd__power__iff,axiom,
% 0.32/0.58 ! [X: nat,M: nat,N: nat] :
% 0.32/0.58 ( ( X != zero_zero_nat )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 0.32/0.58 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power_iff
% 0.32/0.58 thf(fact_4553_dvd__power__iff,axiom,
% 0.32/0.58 ! [X: int,M: nat,N: nat] :
% 0.32/0.58 ( ( X != zero_zero_int )
% 0.32/0.58 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 0.32/0.58 = ( ( dvd_dvd_int @ X @ one_one_int )
% 0.32/0.58 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power_iff
% 0.32/0.58 thf(fact_4554_dvd__power,axiom,
% 0.32/0.58 ! [N: nat,X: code_integer] :
% 0.32/0.58 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.58 | ( X = one_one_Code_integer ) )
% 0.32/0.58 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power
% 0.32/0.58 thf(fact_4555_dvd__power,axiom,
% 0.32/0.58 ! [N: nat,X: rat] :
% 0.32/0.58 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.58 | ( X = one_one_rat ) )
% 0.32/0.58 => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power
% 0.32/0.58 thf(fact_4556_dvd__power,axiom,
% 0.32/0.58 ! [N: nat,X: nat] :
% 0.32/0.58 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.58 | ( X = one_one_nat ) )
% 0.32/0.58 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power
% 0.32/0.58 thf(fact_4557_dvd__power,axiom,
% 0.32/0.58 ! [N: nat,X: real] :
% 0.32/0.58 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.58 | ( X = one_one_real ) )
% 0.32/0.58 => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power
% 0.32/0.58 thf(fact_4558_dvd__power,axiom,
% 0.32/0.58 ! [N: nat,X: int] :
% 0.32/0.58 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.58 | ( X = one_one_int ) )
% 0.32/0.58 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power
% 0.32/0.58 thf(fact_4559_dvd__power,axiom,
% 0.32/0.58 ! [N: nat,X: complex] :
% 0.32/0.58 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.58 | ( X = one_one_complex ) )
% 0.32/0.58 => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power
% 0.32/0.58 thf(fact_4560_even__even__mod__4__iff,axiom,
% 0.32/0.58 ! [N: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_even_mod_4_iff
% 0.32/0.58 thf(fact_4561_dvd__mult__cancel1,axiom,
% 0.32/0.58 ! [M: nat,N: nat] :
% 0.32/0.58 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.32/0.58 = ( N = one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel1
% 0.32/0.58 thf(fact_4562_dvd__mult__cancel2,axiom,
% 0.32/0.58 ! [M: nat,N: nat] :
% 0.32/0.58 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.32/0.58 = ( N = one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_mult_cancel2
% 0.32/0.58 thf(fact_4563_dvd__minus__add,axiom,
% 0.32/0.58 ! [Q2: nat,N: nat,R: nat,M: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ Q2 @ N )
% 0.32/0.58 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
% 0.32/0.58 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 0.32/0.58 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_minus_add
% 0.32/0.58 thf(fact_4564_power__dvd__imp__le,axiom,
% 0.32/0.58 ! [I: nat,M: nat,N: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.32/0.58 => ( ( ord_less_nat @ one_one_nat @ I )
% 0.32/0.58 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_dvd_imp_le
% 0.32/0.58 thf(fact_4565_mod__nat__eqI,axiom,
% 0.32/0.58 ! [R: nat,N: nat,M: nat] :
% 0.32/0.58 ( ( ord_less_nat @ R @ N )
% 0.32/0.58 => ( ( ord_less_eq_nat @ R @ M )
% 0.32/0.58 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
% 0.32/0.58 => ( ( modulo_modulo_nat @ M @ N )
% 0.32/0.58 = R ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mod_nat_eqI
% 0.32/0.58 thf(fact_4566_mod__int__pos__iff,axiom,
% 0.32/0.58 ! [K: int,L2: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
% 0.32/0.58 = ( ( dvd_dvd_int @ L2 @ K )
% 0.32/0.58 | ( ( L2 = zero_zero_int )
% 0.32/0.58 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.32/0.58 | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mod_int_pos_iff
% 0.32/0.58 thf(fact_4567_bset_I9_J,axiom,
% 0.32/0.58 ! [D: int,D4: int,B5: set_int,T: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ D @ D4 )
% 0.32/0.58 => ! [X5: int] :
% 0.32/0.58 ( ! [Xa3: int] :
% 0.32/0.58 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.58 => ! [Xb3: int] :
% 0.32/0.58 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.58 => ( X5
% 0.32/0.58 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.58 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.58 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % bset(9)
% 0.32/0.58 thf(fact_4568_bset_I10_J,axiom,
% 0.32/0.58 ! [D: int,D4: int,B5: set_int,T: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ D @ D4 )
% 0.32/0.58 => ! [X5: int] :
% 0.32/0.58 ( ! [Xa3: int] :
% 0.32/0.58 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.58 => ! [Xb3: int] :
% 0.32/0.58 ( ( member_int @ Xb3 @ B5 )
% 0.32/0.58 => ( X5
% 0.32/0.58 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.58 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.58 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % bset(10)
% 0.32/0.58 thf(fact_4569_aset_I9_J,axiom,
% 0.32/0.58 ! [D: int,D4: int,A2: set_int,T: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ D @ D4 )
% 0.32/0.58 => ! [X5: int] :
% 0.32/0.58 ( ! [Xa3: int] :
% 0.32/0.58 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.58 => ! [Xb3: int] :
% 0.32/0.58 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.58 => ( X5
% 0.32/0.58 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.58 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.58 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % aset(9)
% 0.32/0.58 thf(fact_4570_aset_I10_J,axiom,
% 0.32/0.58 ! [D: int,D4: int,A2: set_int,T: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ D @ D4 )
% 0.32/0.58 => ! [X5: int] :
% 0.32/0.58 ( ! [Xa3: int] :
% 0.32/0.58 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.32/0.58 => ! [Xb3: int] :
% 0.32/0.58 ( ( member_int @ Xb3 @ A2 )
% 0.32/0.58 => ( X5
% 0.32/0.58 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 0.32/0.58 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 0.32/0.58 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % aset(10)
% 0.32/0.58 thf(fact_4571_even__two__times__div__two,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_two_times_div_two
% 0.32/0.58 thf(fact_4572_even__two__times__div__two,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_two_times_div_two
% 0.32/0.58 thf(fact_4573_even__two__times__div__two,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.32/0.58 = A ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_two_times_div_two
% 0.32/0.58 thf(fact_4574_even__iff__mod__2__eq__zero,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = zero_zero_nat ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_iff_mod_2_eq_zero
% 0.32/0.58 thf(fact_4575_even__iff__mod__2__eq__zero,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 = zero_zero_int ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_iff_mod_2_eq_zero
% 0.32/0.58 thf(fact_4576_even__iff__mod__2__eq__zero,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_iff_mod_2_eq_zero
% 0.32/0.58 thf(fact_4577_odd__iff__mod__2__eq__one,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.58 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = one_one_nat ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_iff_mod_2_eq_one
% 0.32/0.58 thf(fact_4578_odd__iff__mod__2__eq__one,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.58 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 = one_one_int ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_iff_mod_2_eq_one
% 0.32/0.58 thf(fact_4579_odd__iff__mod__2__eq__one,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.58 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 = one_one_Code_integer ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_iff_mod_2_eq_one
% 0.32/0.58 thf(fact_4580_power__mono__odd,axiom,
% 0.32/0.58 ! [N: nat,A: real,B: real] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.58 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_mono_odd
% 0.32/0.58 thf(fact_4581_power__mono__odd,axiom,
% 0.32/0.58 ! [N: nat,A: rat,B: rat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.58 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_mono_odd
% 0.32/0.58 thf(fact_4582_power__mono__odd,axiom,
% 0.32/0.58 ! [N: nat,A: int,B: int] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.58 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % power_mono_odd
% 0.32/0.58 thf(fact_4583_odd__pos,axiom,
% 0.32/0.58 ! [N: nat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.58
% 0.32/0.58 % odd_pos
% 0.32/0.58 thf(fact_4584_dvd__power__iff__le,axiom,
% 0.32/0.58 ! [K: nat,M: nat,N: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.58 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.32/0.58 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % dvd_power_iff_le
% 0.32/0.58 thf(fact_4585_even__unset__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_unset_bit_iff
% 0.32/0.58 thf(fact_4586_even__unset__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_unset_bit_iff
% 0.32/0.58 thf(fact_4587_even__unset__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 | ( M = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_unset_bit_iff
% 0.32/0.58 thf(fact_4588_even__set__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_set_bit_iff
% 0.32/0.58 thf(fact_4589_even__set__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_set_bit_iff
% 0.32/0.58 thf(fact_4590_even__set__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 & ( M != zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_set_bit_iff
% 0.32/0.58 thf(fact_4591_even__flip__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: code_integer] :
% 0.32/0.58 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_flip_bit_iff
% 0.32/0.58 thf(fact_4592_even__flip__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_flip_bit_iff
% 0.32/0.58 thf(fact_4593_even__flip__bit__iff,axiom,
% 0.32/0.58 ! [M: nat,A: nat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 != ( M = zero_zero_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_flip_bit_iff
% 0.32/0.58 thf(fact_4594_even__diff__iff,axiom,
% 0.32/0.58 ! [K: int,L2: int] :
% 0.32/0.58 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
% 0.32/0.58 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % even_diff_iff
% 0.32/0.58 thf(fact_4595_oddE,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ~ ! [B3: code_integer] :
% 0.32/0.58 ( A
% 0.32/0.58 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % oddE
% 0.32/0.58 thf(fact_4596_oddE,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ~ ! [B3: nat] :
% 0.32/0.58 ( A
% 0.32/0.58 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % oddE
% 0.32/0.58 thf(fact_4597_oddE,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ~ ! [B3: int] :
% 0.32/0.58 ( A
% 0.32/0.58 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % oddE
% 0.32/0.58 thf(fact_4598_parity__cases,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 != zero_zero_nat ) )
% 0.32/0.58 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 != one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % parity_cases
% 0.32/0.58 thf(fact_4599_parity__cases,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 != zero_zero_int ) )
% 0.32/0.58 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 != one_one_int ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % parity_cases
% 0.32/0.58 thf(fact_4600_parity__cases,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 != zero_z3403309356797280102nteger ) )
% 0.32/0.58 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 != one_one_Code_integer ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % parity_cases
% 0.32/0.58 thf(fact_4601_mod2__eq__if,axiom,
% 0.32/0.58 ! [A: nat] :
% 0.32/0.58 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = zero_zero_nat ) )
% 0.32/0.58 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.58 = one_one_nat ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mod2_eq_if
% 0.32/0.58 thf(fact_4602_mod2__eq__if,axiom,
% 0.32/0.58 ! [A: int] :
% 0.32/0.58 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 = zero_zero_int ) )
% 0.32/0.58 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.58 = one_one_int ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mod2_eq_if
% 0.32/0.58 thf(fact_4603_mod2__eq__if,axiom,
% 0.32/0.58 ! [A: code_integer] :
% 0.32/0.58 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 = zero_z3403309356797280102nteger ) )
% 0.32/0.58 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.58 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.58 = one_one_Code_integer ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % mod2_eq_if
% 0.32/0.58 thf(fact_4604_zero__le__even__power,axiom,
% 0.32/0.58 ! [N: nat,A: real] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_even_power
% 0.32/0.58 thf(fact_4605_zero__le__even__power,axiom,
% 0.32/0.58 ! [N: nat,A: rat] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_even_power
% 0.32/0.58 thf(fact_4606_zero__le__even__power,axiom,
% 0.32/0.58 ! [N: nat,A: int] :
% 0.32/0.58 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_even_power
% 0.32/0.58 thf(fact_4607_zero__le__odd__power,axiom,
% 0.32/0.58 ! [N: nat,A: real] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.32/0.58 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_odd_power
% 0.32/0.58 thf(fact_4608_zero__le__odd__power,axiom,
% 0.32/0.58 ! [N: nat,A: rat] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.32/0.58 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_odd_power
% 0.32/0.58 thf(fact_4609_zero__le__odd__power,axiom,
% 0.32/0.58 ! [N: nat,A: int] :
% 0.32/0.58 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.32/0.58 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_odd_power
% 0.32/0.58 thf(fact_4610_zero__le__power__eq,axiom,
% 0.32/0.58 ! [A: real,N: nat] :
% 0.32/0.58 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_power_eq
% 0.32/0.58 thf(fact_4611_zero__le__power__eq,axiom,
% 0.32/0.58 ! [A: rat,N: nat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_power_eq
% 0.32/0.58 thf(fact_4612_zero__le__power__eq,axiom,
% 0.32/0.58 ! [A: int,N: nat] :
% 0.32/0.58 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.32/0.58 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.58 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.58
% 0.32/0.58 % zero_le_power_eq
% 0.32/0.58 thf(fact_4613_leD,axiom,
% 0.32/0.58 ! [Y3: real,X: real] :
% 0.32/0.58 ( ( ord_less_eq_real @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_real @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % leD
% 0.32/0.58 thf(fact_4614_leD,axiom,
% 0.32/0.58 ! [Y3: set_int,X: set_int] :
% 0.32/0.58 ( ( ord_less_eq_set_int @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_set_int @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % leD
% 0.32/0.58 thf(fact_4615_leD,axiom,
% 0.32/0.58 ! [Y3: rat,X: rat] :
% 0.32/0.58 ( ( ord_less_eq_rat @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_rat @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % leD
% 0.32/0.58 thf(fact_4616_leD,axiom,
% 0.32/0.58 ! [Y3: num,X: num] :
% 0.32/0.58 ( ( ord_less_eq_num @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_num @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % leD
% 0.32/0.58 thf(fact_4617_leD,axiom,
% 0.32/0.58 ! [Y3: nat,X: nat] :
% 0.32/0.58 ( ( ord_less_eq_nat @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_nat @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % leD
% 0.32/0.58 thf(fact_4618_leD,axiom,
% 0.32/0.58 ! [Y3: int,X: int] :
% 0.32/0.58 ( ( ord_less_eq_int @ Y3 @ X )
% 0.32/0.58 => ~ ( ord_less_int @ X @ Y3 ) ) ).
% 0.32/0.58
% 0.32/0.58 % leD
% 0.32/0.58 thf(fact_4619_leI,axiom,
% 0.32/0.58 ! [X: real,Y3: real] :
% 0.32/0.58 ( ~ ( ord_less_real @ X @ Y3 )
% 0.32/0.58 => ( ord_less_eq_real @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % leI
% 0.32/0.58 thf(fact_4620_leI,axiom,
% 0.32/0.58 ! [X: rat,Y3: rat] :
% 0.32/0.58 ( ~ ( ord_less_rat @ X @ Y3 )
% 0.32/0.58 => ( ord_less_eq_rat @ Y3 @ X ) ) ).
% 0.32/0.58
% 0.32/0.58 % leI
% 0.32/0.58 thf(fact_4621_leI,axiom,
% 0.32/0.58 ! [X: num,Y3: num] :
% 0.32/0.58 ( ~ ( ord_less_num @ X @ Y3 )
% 0.32/0.58 => ( ord_less_eq_num @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % leI
% 0.32/0.59 thf(fact_4622_leI,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat] :
% 0.32/0.59 ( ~ ( ord_less_nat @ X @ Y3 )
% 0.32/0.59 => ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % leI
% 0.32/0.59 thf(fact_4623_leI,axiom,
% 0.32/0.59 ! [X: int,Y3: int] :
% 0.32/0.59 ( ~ ( ord_less_int @ X @ Y3 )
% 0.32/0.59 => ( ord_less_eq_int @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % leI
% 0.32/0.59 thf(fact_4624_nless__le,axiom,
% 0.32/0.59 ! [A: real,B: real] :
% 0.32/0.59 ( ( ~ ( ord_less_real @ A @ B ) )
% 0.32/0.59 = ( ~ ( ord_less_eq_real @ A @ B )
% 0.32/0.59 | ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nless_le
% 0.32/0.59 thf(fact_4625_nless__le,axiom,
% 0.32/0.59 ! [A: set_int,B: set_int] :
% 0.32/0.59 ( ( ~ ( ord_less_set_int @ A @ B ) )
% 0.32/0.59 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.32/0.59 | ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nless_le
% 0.32/0.59 thf(fact_4626_nless__le,axiom,
% 0.32/0.59 ! [A: rat,B: rat] :
% 0.32/0.59 ( ( ~ ( ord_less_rat @ A @ B ) )
% 0.32/0.59 = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 | ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nless_le
% 0.32/0.59 thf(fact_4627_nless__le,axiom,
% 0.32/0.59 ! [A: num,B: num] :
% 0.32/0.59 ( ( ~ ( ord_less_num @ A @ B ) )
% 0.32/0.59 = ( ~ ( ord_less_eq_num @ A @ B )
% 0.32/0.59 | ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nless_le
% 0.32/0.59 thf(fact_4628_nless__le,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( ~ ( ord_less_nat @ A @ B ) )
% 0.32/0.59 = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 | ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nless_le
% 0.32/0.59 thf(fact_4629_nless__le,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( ~ ( ord_less_int @ A @ B ) )
% 0.32/0.59 = ( ~ ( ord_less_eq_int @ A @ B )
% 0.32/0.59 | ( A = B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % nless_le
% 0.32/0.59 thf(fact_4630_antisym__conv1,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ~ ( ord_less_real @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv1
% 0.32/0.59 thf(fact_4631_antisym__conv1,axiom,
% 0.32/0.59 ! [X: set_int,Y3: set_int] :
% 0.32/0.59 ( ~ ( ord_less_set_int @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv1
% 0.32/0.59 thf(fact_4632_antisym__conv1,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat] :
% 0.32/0.59 ( ~ ( ord_less_rat @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv1
% 0.32/0.59 thf(fact_4633_antisym__conv1,axiom,
% 0.32/0.59 ! [X: num,Y3: num] :
% 0.32/0.59 ( ~ ( ord_less_num @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv1
% 0.32/0.59 thf(fact_4634_antisym__conv1,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat] :
% 0.32/0.59 ( ~ ( ord_less_nat @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv1
% 0.32/0.59 thf(fact_4635_antisym__conv1,axiom,
% 0.32/0.59 ! [X: int,Y3: int] :
% 0.32/0.59 ( ~ ( ord_less_int @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv1
% 0.32/0.59 thf(fact_4636_antisym__conv2,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.59 => ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv2
% 0.32/0.59 thf(fact_4637_antisym__conv2,axiom,
% 0.32/0.59 ! [X: set_int,Y3: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.59 => ( ( ~ ( ord_less_set_int @ X @ Y3 ) )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv2
% 0.32/0.59 thf(fact_4638_antisym__conv2,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.59 => ( ( ~ ( ord_less_rat @ X @ Y3 ) )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv2
% 0.32/0.59 thf(fact_4639_antisym__conv2,axiom,
% 0.32/0.59 ! [X: num,Y3: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.59 => ( ( ~ ( ord_less_num @ X @ Y3 ) )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv2
% 0.32/0.59 thf(fact_4640_antisym__conv2,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.59 => ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv2
% 0.32/0.59 thf(fact_4641_antisym__conv2,axiom,
% 0.32/0.59 ! [X: int,Y3: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.59 => ( ( ~ ( ord_less_int @ X @ Y3 ) )
% 0.32/0.59 = ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % antisym_conv2
% 0.32/0.59 thf(fact_4642_dense__ge,axiom,
% 0.32/0.59 ! [Z: real,Y3: real] :
% 0.32/0.59 ( ! [X3: real] :
% 0.32/0.59 ( ( ord_less_real @ Z @ X3 )
% 0.32/0.59 => ( ord_less_eq_real @ Y3 @ X3 ) )
% 0.32/0.59 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.32/0.59
% 0.32/0.59 % dense_ge
% 0.32/0.59 thf(fact_4643_dense__ge,axiom,
% 0.32/0.59 ! [Z: rat,Y3: rat] :
% 0.32/0.59 ( ! [X3: rat] :
% 0.32/0.59 ( ( ord_less_rat @ Z @ X3 )
% 0.32/0.59 => ( ord_less_eq_rat @ Y3 @ X3 ) )
% 0.32/0.59 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.32/0.59
% 0.32/0.59 % dense_ge
% 0.32/0.59 thf(fact_4644_dense__le,axiom,
% 0.32/0.59 ! [Y3: real,Z: real] :
% 0.32/0.59 ( ! [X3: real] :
% 0.32/0.59 ( ( ord_less_real @ X3 @ Y3 )
% 0.32/0.59 => ( ord_less_eq_real @ X3 @ Z ) )
% 0.32/0.59 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.32/0.59
% 0.32/0.59 % dense_le
% 0.32/0.59 thf(fact_4645_dense__le,axiom,
% 0.32/0.59 ! [Y3: rat,Z: rat] :
% 0.32/0.59 ( ! [X3: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X3 @ Y3 )
% 0.32/0.59 => ( ord_less_eq_rat @ X3 @ Z ) )
% 0.32/0.59 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.32/0.59
% 0.32/0.59 % dense_le
% 0.32/0.59 thf(fact_4646_less__le__not__le,axiom,
% 0.32/0.59 ( ord_less_real
% 0.32/0.59 = ( ^ [X2: real,Y: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ X2 @ Y )
% 0.32/0.59 & ~ ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % less_le_not_le
% 0.32/0.59 thf(fact_4647_less__le__not__le,axiom,
% 0.32/0.59 ( ord_less_set_int
% 0.32/0.59 = ( ^ [X2: set_int,Y: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ X2 @ Y )
% 0.32/0.59 & ~ ( ord_less_eq_set_int @ Y @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % less_le_not_le
% 0.32/0.59 thf(fact_4648_less__le__not__le,axiom,
% 0.32/0.59 ( ord_less_rat
% 0.32/0.59 = ( ^ [X2: rat,Y: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X2 @ Y )
% 0.32/0.59 & ~ ( ord_less_eq_rat @ Y @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % less_le_not_le
% 0.32/0.59 thf(fact_4649_less__le__not__le,axiom,
% 0.32/0.59 ( ord_less_num
% 0.32/0.59 = ( ^ [X2: num,Y: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X2 @ Y )
% 0.32/0.59 & ~ ( ord_less_eq_num @ Y @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % less_le_not_le
% 0.32/0.59 thf(fact_4650_less__le__not__le,axiom,
% 0.32/0.59 ( ord_less_nat
% 0.32/0.59 = ( ^ [X2: nat,Y: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/0.59 & ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % less_le_not_le
% 0.32/0.59 thf(fact_4651_less__le__not__le,axiom,
% 0.32/0.59 ( ord_less_int
% 0.32/0.59 = ( ^ [X2: int,Y: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X2 @ Y )
% 0.32/0.59 & ~ ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % less_le_not_le
% 0.32/0.59 thf(fact_4652_not__le__imp__less,axiom,
% 0.32/0.59 ! [Y3: real,X: real] :
% 0.32/0.59 ( ~ ( ord_less_eq_real @ Y3 @ X )
% 0.32/0.59 => ( ord_less_real @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % not_le_imp_less
% 0.32/0.59 thf(fact_4653_not__le__imp__less,axiom,
% 0.32/0.59 ! [Y3: rat,X: rat] :
% 0.32/0.59 ( ~ ( ord_less_eq_rat @ Y3 @ X )
% 0.32/0.59 => ( ord_less_rat @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % not_le_imp_less
% 0.32/0.59 thf(fact_4654_not__le__imp__less,axiom,
% 0.32/0.59 ! [Y3: num,X: num] :
% 0.32/0.59 ( ~ ( ord_less_eq_num @ Y3 @ X )
% 0.32/0.59 => ( ord_less_num @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % not_le_imp_less
% 0.32/0.59 thf(fact_4655_not__le__imp__less,axiom,
% 0.32/0.59 ! [Y3: nat,X: nat] :
% 0.32/0.59 ( ~ ( ord_less_eq_nat @ Y3 @ X )
% 0.32/0.59 => ( ord_less_nat @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % not_le_imp_less
% 0.32/0.59 thf(fact_4656_not__le__imp__less,axiom,
% 0.32/0.59 ! [Y3: int,X: int] :
% 0.32/0.59 ( ~ ( ord_less_eq_int @ Y3 @ X )
% 0.32/0.59 => ( ord_less_int @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % not_le_imp_less
% 0.32/0.59 thf(fact_4657_order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_real
% 0.32/0.59 = ( ^ [A3: real,B4: real] :
% 0.32/0.59 ( ( ord_less_real @ A3 @ B4 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.order_iff_strict
% 0.32/0.59 thf(fact_4658_order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_set_int
% 0.32/0.59 = ( ^ [A3: set_int,B4: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ A3 @ B4 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.order_iff_strict
% 0.32/0.59 thf(fact_4659_order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_rat
% 0.32/0.59 = ( ^ [A3: rat,B4: rat] :
% 0.32/0.59 ( ( ord_less_rat @ A3 @ B4 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.order_iff_strict
% 0.32/0.59 thf(fact_4660_order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_num
% 0.32/0.59 = ( ^ [A3: num,B4: num] :
% 0.32/0.59 ( ( ord_less_num @ A3 @ B4 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.order_iff_strict
% 0.32/0.59 thf(fact_4661_order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_nat
% 0.32/0.59 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.59 ( ( ord_less_nat @ A3 @ B4 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.order_iff_strict
% 0.32/0.59 thf(fact_4662_order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_int
% 0.32/0.59 = ( ^ [A3: int,B4: int] :
% 0.32/0.59 ( ( ord_less_int @ A3 @ B4 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.order_iff_strict
% 0.32/0.59 thf(fact_4663_order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_real
% 0.32/0.59 = ( ^ [A3: real,B4: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ A3 @ B4 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_order
% 0.32/0.59 thf(fact_4664_order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_set_int
% 0.32/0.59 = ( ^ [A3: set_int,B4: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ A3 @ B4 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_order
% 0.32/0.59 thf(fact_4665_order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_rat
% 0.32/0.59 = ( ^ [A3: rat,B4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A3 @ B4 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_order
% 0.32/0.59 thf(fact_4666_order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_num
% 0.32/0.59 = ( ^ [A3: num,B4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A3 @ B4 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_order
% 0.32/0.59 thf(fact_4667_order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_nat
% 0.32/0.59 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_order
% 0.32/0.59 thf(fact_4668_order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_int
% 0.32/0.59 = ( ^ [A3: int,B4: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A3 @ B4 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_order
% 0.32/0.59 thf(fact_4669_order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [A: real,B: real,C: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.59 => ( ( ord_less_real @ B @ C )
% 0.32/0.59 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans1
% 0.32/0.59 thf(fact_4670_order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [A: set_int,B: set_int,C: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ A @ B )
% 0.32/0.59 => ( ( ord_less_set_int @ B @ C )
% 0.32/0.59 => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans1
% 0.32/0.59 thf(fact_4671_order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_rat @ B @ C )
% 0.32/0.59 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans1
% 0.32/0.59 thf(fact_4672_order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [A: num,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_num @ B @ C )
% 0.32/0.59 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans1
% 0.32/0.59 thf(fact_4673_order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_nat @ B @ C )
% 0.32/0.59 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans1
% 0.32/0.59 thf(fact_4674_order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.59 => ( ( ord_less_int @ B @ C )
% 0.32/0.59 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans1
% 0.32/0.59 thf(fact_4675_order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [A: real,B: real,C: real] :
% 0.32/0.59 ( ( ord_less_real @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_real @ B @ C )
% 0.32/0.59 => ( ord_less_real @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans2
% 0.32/0.59 thf(fact_4676_order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [A: set_int,B: set_int,C: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_set_int @ B @ C )
% 0.32/0.59 => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans2
% 0.32/0.59 thf(fact_4677_order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ord_less_rat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans2
% 0.32/0.59 thf(fact_4678_order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [A: num,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ord_less_num @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans2
% 0.32/0.59 thf(fact_4679_order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ B @ C )
% 0.32/0.59 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans2
% 0.32/0.59 thf(fact_4680_order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int] :
% 0.32/0.59 ( ( ord_less_int @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_int @ B @ C )
% 0.32/0.59 => ( ord_less_int @ A @ C ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_trans2
% 0.32/0.59 thf(fact_4681_order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_real
% 0.32/0.59 = ( ^ [A3: real,B4: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ A3 @ B4 )
% 0.32/0.59 & ~ ( ord_less_eq_real @ B4 @ A3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_not
% 0.32/0.59 thf(fact_4682_order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_set_int
% 0.32/0.59 = ( ^ [A3: set_int,B4: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ A3 @ B4 )
% 0.32/0.59 & ~ ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_not
% 0.32/0.59 thf(fact_4683_order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_rat
% 0.32/0.59 = ( ^ [A3: rat,B4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A3 @ B4 )
% 0.32/0.59 & ~ ( ord_less_eq_rat @ B4 @ A3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_not
% 0.32/0.59 thf(fact_4684_order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_num
% 0.32/0.59 = ( ^ [A3: num,B4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A3 @ B4 )
% 0.32/0.59 & ~ ( ord_less_eq_num @ B4 @ A3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_not
% 0.32/0.59 thf(fact_4685_order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_nat
% 0.32/0.59 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A3 @ B4 )
% 0.32/0.59 & ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_not
% 0.32/0.59 thf(fact_4686_order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_int
% 0.32/0.59 = ( ^ [A3: int,B4: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A3 @ B4 )
% 0.32/0.59 & ~ ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_iff_not
% 0.32/0.59 thf(fact_4687_dense__ge__bounded,axiom,
% 0.32/0.59 ! [Z: real,X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_real @ Z @ X )
% 0.32/0.59 => ( ! [W2: real] :
% 0.32/0.59 ( ( ord_less_real @ Z @ W2 )
% 0.32/0.59 => ( ( ord_less_real @ W2 @ X )
% 0.32/0.59 => ( ord_less_eq_real @ Y3 @ W2 ) ) )
% 0.32/0.59 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dense_ge_bounded
% 0.32/0.59 thf(fact_4688_dense__ge__bounded,axiom,
% 0.32/0.59 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.59 ( ( ord_less_rat @ Z @ X )
% 0.32/0.59 => ( ! [W2: rat] :
% 0.32/0.59 ( ( ord_less_rat @ Z @ W2 )
% 0.32/0.59 => ( ( ord_less_rat @ W2 @ X )
% 0.32/0.59 => ( ord_less_eq_rat @ Y3 @ W2 ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dense_ge_bounded
% 0.32/0.59 thf(fact_4689_dense__le__bounded,axiom,
% 0.32/0.59 ! [X: real,Y3: real,Z: real] :
% 0.32/0.59 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.59 => ( ! [W2: real] :
% 0.32/0.59 ( ( ord_less_real @ X @ W2 )
% 0.32/0.59 => ( ( ord_less_real @ W2 @ Y3 )
% 0.32/0.59 => ( ord_less_eq_real @ W2 @ Z ) ) )
% 0.32/0.59 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dense_le_bounded
% 0.32/0.59 thf(fact_4690_dense__le__bounded,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.59 => ( ! [W2: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X @ W2 )
% 0.32/0.59 => ( ( ord_less_rat @ W2 @ Y3 )
% 0.32/0.59 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 0.32/0.59 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dense_le_bounded
% 0.32/0.59 thf(fact_4691_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_real
% 0.32/0.59 = ( ^ [B4: real,A3: real] :
% 0.32/0.59 ( ( ord_less_real @ B4 @ A3 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.order_iff_strict
% 0.32/0.59 thf(fact_4692_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_set_int
% 0.32/0.59 = ( ^ [B4: set_int,A3: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ B4 @ A3 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.order_iff_strict
% 0.32/0.59 thf(fact_4693_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_rat
% 0.32/0.59 = ( ^ [B4: rat,A3: rat] :
% 0.32/0.59 ( ( ord_less_rat @ B4 @ A3 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.order_iff_strict
% 0.32/0.59 thf(fact_4694_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_num
% 0.32/0.59 = ( ^ [B4: num,A3: num] :
% 0.32/0.59 ( ( ord_less_num @ B4 @ A3 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.order_iff_strict
% 0.32/0.59 thf(fact_4695_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_nat
% 0.32/0.59 = ( ^ [B4: nat,A3: nat] :
% 0.32/0.59 ( ( ord_less_nat @ B4 @ A3 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.order_iff_strict
% 0.32/0.59 thf(fact_4696_dual__order_Oorder__iff__strict,axiom,
% 0.32/0.59 ( ord_less_eq_int
% 0.32/0.59 = ( ^ [B4: int,A3: int] :
% 0.32/0.59 ( ( ord_less_int @ B4 @ A3 )
% 0.32/0.59 | ( A3 = B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.order_iff_strict
% 0.32/0.59 thf(fact_4697_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_real
% 0.32/0.59 = ( ^ [B4: real,A3: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ B4 @ A3 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_order
% 0.32/0.59 thf(fact_4698_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_set_int
% 0.32/0.59 = ( ^ [B4: set_int,A3: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ B4 @ A3 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_order
% 0.32/0.59 thf(fact_4699_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_rat
% 0.32/0.59 = ( ^ [B4: rat,A3: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ B4 @ A3 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_order
% 0.32/0.59 thf(fact_4700_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_num
% 0.32/0.59 = ( ^ [B4: num,A3: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ B4 @ A3 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_order
% 0.32/0.59 thf(fact_4701_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_nat
% 0.32/0.59 = ( ^ [B4: nat,A3: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ B4 @ A3 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_order
% 0.32/0.59 thf(fact_4702_dual__order_Ostrict__iff__order,axiom,
% 0.32/0.59 ( ord_less_int
% 0.32/0.59 = ( ^ [B4: int,A3: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ B4 @ A3 )
% 0.32/0.59 & ( A3 != B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_order
% 0.32/0.59 thf(fact_4703_dual__order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [B: real,A: real,C: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ B @ A )
% 0.32/0.59 => ( ( ord_less_real @ C @ B )
% 0.32/0.59 => ( ord_less_real @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans1
% 0.32/0.59 thf(fact_4704_dual__order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [B: set_int,A: set_int,C: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ B @ A )
% 0.32/0.59 => ( ( ord_less_set_int @ C @ B )
% 0.32/0.59 => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans1
% 0.32/0.59 thf(fact_4705_dual__order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [B: rat,A: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ B @ A )
% 0.32/0.59 => ( ( ord_less_rat @ C @ B )
% 0.32/0.59 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans1
% 0.32/0.59 thf(fact_4706_dual__order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [B: num,A: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ B @ A )
% 0.32/0.59 => ( ( ord_less_num @ C @ B )
% 0.32/0.59 => ( ord_less_num @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans1
% 0.32/0.59 thf(fact_4707_dual__order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ B @ A )
% 0.32/0.59 => ( ( ord_less_nat @ C @ B )
% 0.32/0.59 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans1
% 0.32/0.59 thf(fact_4708_dual__order_Ostrict__trans1,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ B @ A )
% 0.32/0.59 => ( ( ord_less_int @ C @ B )
% 0.32/0.59 => ( ord_less_int @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans1
% 0.32/0.59 thf(fact_4709_dual__order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [B: real,A: real,C: real] :
% 0.32/0.59 ( ( ord_less_real @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_real @ C @ B )
% 0.32/0.59 => ( ord_less_real @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans2
% 0.32/0.59 thf(fact_4710_dual__order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [B: set_int,A: set_int,C: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_set_int @ C @ B )
% 0.32/0.59 => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans2
% 0.32/0.59 thf(fact_4711_dual__order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [B: rat,A: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_rat @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_rat @ C @ B )
% 0.32/0.59 => ( ord_less_rat @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans2
% 0.32/0.59 thf(fact_4712_dual__order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [B: num,A: num,C: num] :
% 0.32/0.59 ( ( ord_less_num @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_num @ C @ B )
% 0.32/0.59 => ( ord_less_num @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans2
% 0.32/0.59 thf(fact_4713_dual__order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [B: nat,A: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_nat @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_nat @ C @ B )
% 0.32/0.59 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans2
% 0.32/0.59 thf(fact_4714_dual__order_Ostrict__trans2,axiom,
% 0.32/0.59 ! [B: int,A: int,C: int] :
% 0.32/0.59 ( ( ord_less_int @ B @ A )
% 0.32/0.59 => ( ( ord_less_eq_int @ C @ B )
% 0.32/0.59 => ( ord_less_int @ C @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_trans2
% 0.32/0.59 thf(fact_4715_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_real
% 0.32/0.59 = ( ^ [B4: real,A3: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ B4 @ A3 )
% 0.32/0.59 & ~ ( ord_less_eq_real @ A3 @ B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_not
% 0.32/0.59 thf(fact_4716_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_set_int
% 0.32/0.59 = ( ^ [B4: set_int,A3: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ B4 @ A3 )
% 0.32/0.59 & ~ ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_not
% 0.32/0.59 thf(fact_4717_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_rat
% 0.32/0.59 = ( ^ [B4: rat,A3: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ B4 @ A3 )
% 0.32/0.59 & ~ ( ord_less_eq_rat @ A3 @ B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_not
% 0.32/0.59 thf(fact_4718_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_num
% 0.32/0.59 = ( ^ [B4: num,A3: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ B4 @ A3 )
% 0.32/0.59 & ~ ( ord_less_eq_num @ A3 @ B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_not
% 0.32/0.59 thf(fact_4719_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_nat
% 0.32/0.59 = ( ^ [B4: nat,A3: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ B4 @ A3 )
% 0.32/0.59 & ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_not
% 0.32/0.59 thf(fact_4720_dual__order_Ostrict__iff__not,axiom,
% 0.32/0.59 ( ord_less_int
% 0.32/0.59 = ( ^ [B4: int,A3: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ B4 @ A3 )
% 0.32/0.59 & ~ ( ord_less_eq_int @ A3 @ B4 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_iff_not
% 0.32/0.59 thf(fact_4721_order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [A: real,B: real] :
% 0.32/0.59 ( ( ord_less_real @ A @ B )
% 0.32/0.59 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_implies_order
% 0.32/0.59 thf(fact_4722_order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [A: set_int,B: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ A @ B )
% 0.32/0.59 => ( ord_less_eq_set_int @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_implies_order
% 0.32/0.59 thf(fact_4723_order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [A: rat,B: rat] :
% 0.32/0.59 ( ( ord_less_rat @ A @ B )
% 0.32/0.59 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_implies_order
% 0.32/0.59 thf(fact_4724_order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [A: num,B: num] :
% 0.32/0.59 ( ( ord_less_num @ A @ B )
% 0.32/0.59 => ( ord_less_eq_num @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_implies_order
% 0.32/0.59 thf(fact_4725_order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( ord_less_nat @ A @ B )
% 0.32/0.59 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_implies_order
% 0.32/0.59 thf(fact_4726_order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( ord_less_int @ A @ B )
% 0.32/0.59 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.59
% 0.32/0.59 % order.strict_implies_order
% 0.32/0.59 thf(fact_4727_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [B: real,A: real] :
% 0.32/0.59 ( ( ord_less_real @ B @ A )
% 0.32/0.59 => ( ord_less_eq_real @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_implies_order
% 0.32/0.59 thf(fact_4728_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [B: set_int,A: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ B @ A )
% 0.32/0.59 => ( ord_less_eq_set_int @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_implies_order
% 0.32/0.59 thf(fact_4729_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [B: rat,A: rat] :
% 0.32/0.59 ( ( ord_less_rat @ B @ A )
% 0.32/0.59 => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_implies_order
% 0.32/0.59 thf(fact_4730_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [B: num,A: num] :
% 0.32/0.59 ( ( ord_less_num @ B @ A )
% 0.32/0.59 => ( ord_less_eq_num @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_implies_order
% 0.32/0.59 thf(fact_4731_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [B: nat,A: nat] :
% 0.32/0.59 ( ( ord_less_nat @ B @ A )
% 0.32/0.59 => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_implies_order
% 0.32/0.59 thf(fact_4732_dual__order_Ostrict__implies__order,axiom,
% 0.32/0.59 ! [B: int,A: int] :
% 0.32/0.59 ( ( ord_less_int @ B @ A )
% 0.32/0.59 => ( ord_less_eq_int @ B @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % dual_order.strict_implies_order
% 0.32/0.59 thf(fact_4733_order__le__less,axiom,
% 0.32/0.59 ( ord_less_eq_real
% 0.32/0.59 = ( ^ [X2: real,Y: real] :
% 0.32/0.59 ( ( ord_less_real @ X2 @ Y )
% 0.32/0.59 | ( X2 = Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less
% 0.32/0.59 thf(fact_4734_order__le__less,axiom,
% 0.32/0.59 ( ord_less_eq_set_int
% 0.32/0.59 = ( ^ [X2: set_int,Y: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ X2 @ Y )
% 0.32/0.59 | ( X2 = Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less
% 0.32/0.59 thf(fact_4735_order__le__less,axiom,
% 0.32/0.59 ( ord_less_eq_rat
% 0.32/0.59 = ( ^ [X2: rat,Y: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X2 @ Y )
% 0.32/0.59 | ( X2 = Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less
% 0.32/0.59 thf(fact_4736_order__le__less,axiom,
% 0.32/0.59 ( ord_less_eq_num
% 0.32/0.59 = ( ^ [X2: num,Y: num] :
% 0.32/0.59 ( ( ord_less_num @ X2 @ Y )
% 0.32/0.59 | ( X2 = Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less
% 0.32/0.59 thf(fact_4737_order__le__less,axiom,
% 0.32/0.59 ( ord_less_eq_nat
% 0.32/0.59 = ( ^ [X2: nat,Y: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X2 @ Y )
% 0.32/0.59 | ( X2 = Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less
% 0.32/0.59 thf(fact_4738_order__le__less,axiom,
% 0.32/0.59 ( ord_less_eq_int
% 0.32/0.59 = ( ^ [X2: int,Y: int] :
% 0.32/0.59 ( ( ord_less_int @ X2 @ Y )
% 0.32/0.59 | ( X2 = Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less
% 0.32/0.59 thf(fact_4739_order__less__le,axiom,
% 0.32/0.59 ( ord_less_real
% 0.32/0.59 = ( ^ [X2: real,Y: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ X2 @ Y )
% 0.32/0.59 & ( X2 != Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le
% 0.32/0.59 thf(fact_4740_order__less__le,axiom,
% 0.32/0.59 ( ord_less_set_int
% 0.32/0.59 = ( ^ [X2: set_int,Y: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ X2 @ Y )
% 0.32/0.59 & ( X2 != Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le
% 0.32/0.59 thf(fact_4741_order__less__le,axiom,
% 0.32/0.59 ( ord_less_rat
% 0.32/0.59 = ( ^ [X2: rat,Y: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X2 @ Y )
% 0.32/0.59 & ( X2 != Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le
% 0.32/0.59 thf(fact_4742_order__less__le,axiom,
% 0.32/0.59 ( ord_less_num
% 0.32/0.59 = ( ^ [X2: num,Y: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X2 @ Y )
% 0.32/0.59 & ( X2 != Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le
% 0.32/0.59 thf(fact_4743_order__less__le,axiom,
% 0.32/0.59 ( ord_less_nat
% 0.32/0.59 = ( ^ [X2: nat,Y: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X2 @ Y )
% 0.32/0.59 & ( X2 != Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le
% 0.32/0.59 thf(fact_4744_order__less__le,axiom,
% 0.32/0.59 ( ord_less_int
% 0.32/0.59 = ( ^ [X2: int,Y: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X2 @ Y )
% 0.32/0.59 & ( X2 != Y ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le
% 0.32/0.59 thf(fact_4745_linorder__not__le,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_real @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_real @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_le
% 0.32/0.59 thf(fact_4746_linorder__not__le,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_rat @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_rat @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_le
% 0.32/0.59 thf(fact_4747_linorder__not__le,axiom,
% 0.32/0.59 ! [X: num,Y3: num] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_num @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_num @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_le
% 0.32/0.59 thf(fact_4748_linorder__not__le,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_nat @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_nat @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_le
% 0.32/0.59 thf(fact_4749_linorder__not__le,axiom,
% 0.32/0.59 ! [X: int,Y3: int] :
% 0.32/0.59 ( ( ~ ( ord_less_eq_int @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_int @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_le
% 0.32/0.59 thf(fact_4750_linorder__not__less,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_eq_real @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_less
% 0.32/0.59 thf(fact_4751_linorder__not__less,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat] :
% 0.32/0.59 ( ( ~ ( ord_less_rat @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_eq_rat @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_less
% 0.32/0.59 thf(fact_4752_linorder__not__less,axiom,
% 0.32/0.59 ! [X: num,Y3: num] :
% 0.32/0.59 ( ( ~ ( ord_less_num @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_eq_num @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_less
% 0.32/0.59 thf(fact_4753_linorder__not__less,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat] :
% 0.32/0.59 ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_less
% 0.32/0.59 thf(fact_4754_linorder__not__less,axiom,
% 0.32/0.59 ! [X: int,Y3: int] :
% 0.32/0.59 ( ( ~ ( ord_less_int @ X @ Y3 ) )
% 0.32/0.59 = ( ord_less_eq_int @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_not_less
% 0.32/0.59 thf(fact_4755_order__less__imp__le,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.59 => ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_imp_le
% 0.32/0.59 thf(fact_4756_order__less__imp__le,axiom,
% 0.32/0.59 ! [X: set_int,Y3: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ X @ Y3 )
% 0.32/0.59 => ( ord_less_eq_set_int @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_imp_le
% 0.32/0.59 thf(fact_4757_order__less__imp__le,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.59 => ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_imp_le
% 0.32/0.59 thf(fact_4758_order__less__imp__le,axiom,
% 0.32/0.59 ! [X: num,Y3: num] :
% 0.32/0.59 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.59 => ( ord_less_eq_num @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_imp_le
% 0.32/0.59 thf(fact_4759_order__less__imp__le,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.59 => ( ord_less_eq_nat @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_imp_le
% 0.32/0.59 thf(fact_4760_order__less__imp__le,axiom,
% 0.32/0.59 ! [X: int,Y3: int] :
% 0.32/0.59 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.59 => ( ord_less_eq_int @ X @ Y3 ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_imp_le
% 0.32/0.59 thf(fact_4761_order__le__neq__trans,axiom,
% 0.32/0.59 ! [A: real,B: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.59 => ( ( A != B )
% 0.32/0.59 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_neq_trans
% 0.32/0.59 thf(fact_4762_order__le__neq__trans,axiom,
% 0.32/0.59 ! [A: set_int,B: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ A @ B )
% 0.32/0.59 => ( ( A != B )
% 0.32/0.59 => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_neq_trans
% 0.32/0.59 thf(fact_4763_order__le__neq__trans,axiom,
% 0.32/0.59 ! [A: rat,B: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( A != B )
% 0.32/0.59 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_neq_trans
% 0.32/0.59 thf(fact_4764_order__le__neq__trans,axiom,
% 0.32/0.59 ! [A: num,B: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( A != B )
% 0.32/0.59 => ( ord_less_num @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_neq_trans
% 0.32/0.59 thf(fact_4765_order__le__neq__trans,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ( A != B )
% 0.32/0.59 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_neq_trans
% 0.32/0.59 thf(fact_4766_order__le__neq__trans,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.59 => ( ( A != B )
% 0.32/0.59 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_neq_trans
% 0.32/0.59 thf(fact_4767_order__neq__le__trans,axiom,
% 0.32/0.59 ! [A: real,B: real] :
% 0.32/0.59 ( ( A != B )
% 0.32/0.59 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.59 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_neq_le_trans
% 0.32/0.59 thf(fact_4768_order__neq__le__trans,axiom,
% 0.32/0.59 ! [A: set_int,B: set_int] :
% 0.32/0.59 ( ( A != B )
% 0.32/0.59 => ( ( ord_less_eq_set_int @ A @ B )
% 0.32/0.59 => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_neq_le_trans
% 0.32/0.59 thf(fact_4769_order__neq__le__trans,axiom,
% 0.32/0.59 ! [A: rat,B: rat] :
% 0.32/0.59 ( ( A != B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ord_less_rat @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_neq_le_trans
% 0.32/0.59 thf(fact_4770_order__neq__le__trans,axiom,
% 0.32/0.59 ! [A: num,B: num] :
% 0.32/0.59 ( ( A != B )
% 0.32/0.59 => ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ord_less_num @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_neq_le_trans
% 0.32/0.59 thf(fact_4771_order__neq__le__trans,axiom,
% 0.32/0.59 ! [A: nat,B: nat] :
% 0.32/0.59 ( ( A != B )
% 0.32/0.59 => ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.59 => ( ord_less_nat @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_neq_le_trans
% 0.32/0.59 thf(fact_4772_order__neq__le__trans,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( A != B )
% 0.32/0.59 => ( ( ord_less_eq_int @ A @ B )
% 0.32/0.59 => ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_neq_le_trans
% 0.32/0.59 thf(fact_4773_order__le__less__trans,axiom,
% 0.32/0.59 ! [X: real,Y3: real,Z: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_real @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_trans
% 0.32/0.59 thf(fact_4774_order__le__less__trans,axiom,
% 0.32/0.59 ! [X: set_int,Y3: set_int,Z: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_set_int @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_set_int @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_trans
% 0.32/0.59 thf(fact_4775_order__le__less__trans,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_rat @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_trans
% 0.32/0.59 thf(fact_4776_order__le__less__trans,axiom,
% 0.32/0.59 ! [X: num,Y3: num,Z: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_num @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_trans
% 0.32/0.59 thf(fact_4777_order__le__less__trans,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_nat @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_trans
% 0.32/0.59 thf(fact_4778_order__le__less__trans,axiom,
% 0.32/0.59 ! [X: int,Y3: int,Z: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_int @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_trans
% 0.32/0.59 thf(fact_4779_order__less__le__trans,axiom,
% 0.32/0.59 ! [X: real,Y3: real,Z: real] :
% 0.32/0.59 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_real @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_real @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_trans
% 0.32/0.59 thf(fact_4780_order__less__le__trans,axiom,
% 0.32/0.59 ! [X: set_int,Y3: set_int,Z: set_int] :
% 0.32/0.59 ( ( ord_less_set_int @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_set_int @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_trans
% 0.32/0.59 thf(fact_4781_order__less__le__trans,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat,Z: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_rat @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_trans
% 0.32/0.59 thf(fact_4782_order__less__le__trans,axiom,
% 0.32/0.59 ! [X: num,Y3: num,Z: num] :
% 0.32/0.59 ( ( ord_less_num @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_num @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_trans
% 0.32/0.59 thf(fact_4783_order__less__le__trans,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_nat @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_trans
% 0.32/0.59 thf(fact_4784_order__less__le__trans,axiom,
% 0.32/0.59 ! [X: int,Y3: int,Z: int] :
% 0.32/0.59 ( ( ord_less_int @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.32/0.59 => ( ord_less_int @ X @ Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_trans
% 0.32/0.59 thf(fact_4785_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: real,F: real > real,B: real,C: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_real @ B @ C )
% 0.32/0.59 => ( ! [X3: real,Y4: real] :
% 0.32/0.59 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4786_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_rat @ B @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4787_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: real,F: num > real,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_num @ B @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4788_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_nat @ B @ C )
% 0.32/0.59 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4789_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: real,F: int > real,B: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_int @ B @ C )
% 0.32/0.59 => ( ! [X3: int,Y4: int] :
% 0.32/0.59 ( ( ord_less_int @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4790_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: real > rat,B: real,C: real] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_real @ B @ C )
% 0.32/0.59 => ( ! [X3: real,Y4: real] :
% 0.32/0.59 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4791_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_rat @ B @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4792_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_num @ B @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4793_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_nat @ B @ C )
% 0.32/0.59 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4794_order__le__less__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: int > rat,B: int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_int @ B @ C )
% 0.32/0.59 => ( ! [X3: int,Y4: int] :
% 0.32/0.59 ( ( ord_less_int @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst1
% 0.32/0.59 thf(fact_4795_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4796_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4797_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4798_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4799_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4800_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > real,C: real] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4801_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4802_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > num,C: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4803_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > nat,C: nat] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4804_order__le__less__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > int,C: int] :
% 0.32/0.59 ( ( ord_less_eq_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_less_subst2
% 0.32/0.59 thf(fact_4805_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4806_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4807_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_num @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4808_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4809_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.32/0.59 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_rat @ B @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4810_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: real,F: num > real,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_real @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4811_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: rat,F: num > rat,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4812_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: num,F: num > num,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_num @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4813_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: nat,F: num > nat,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4814_order__less__le__subst1,axiom,
% 0.32/0.59 ! [A: int,F: num > int,B: num,C: num] :
% 0.32/0.59 ( ( ord_less_int @ A @ ( F @ B ) )
% 0.32/0.59 => ( ( ord_less_eq_num @ B @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst1
% 0.32/0.59 thf(fact_4815_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: real,B: real,F: real > real,C: real] :
% 0.32/0.59 ( ( ord_less_real @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: real,Y4: real] :
% 0.32/0.59 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4816_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.32/0.59 ( ( ord_less_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4817_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > real,C: real] :
% 0.32/0.59 ( ( ord_less_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4818_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.32/0.59 ( ( ord_less_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4819_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: int,B: int,F: int > real,C: real] :
% 0.32/0.59 ( ( ord_less_int @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: int,Y4: int] :
% 0.32/0.59 ( ( ord_less_int @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4820_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: real,B: real,F: real > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_real @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: real,Y4: real] :
% 0.32/0.59 ( ( ord_less_real @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4821_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_rat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: rat,Y4: rat] :
% 0.32/0.59 ( ( ord_less_rat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4822_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: num,B: num,F: num > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_num @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: num,Y4: num] :
% 0.32/0.59 ( ( ord_less_num @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4823_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_nat @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: nat,Y4: nat] :
% 0.32/0.59 ( ( ord_less_nat @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4824_order__less__le__subst2,axiom,
% 0.32/0.59 ! [A: int,B: int,F: int > rat,C: rat] :
% 0.32/0.59 ( ( ord_less_int @ A @ B )
% 0.32/0.59 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.32/0.59 => ( ! [X3: int,Y4: int] :
% 0.32/0.59 ( ( ord_less_int @ X3 @ Y4 )
% 0.32/0.59 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.32/0.59 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_less_le_subst2
% 0.32/0.59 thf(fact_4825_linorder__le__less__linear,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.59 | ( ord_less_real @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_le_less_linear
% 0.32/0.59 thf(fact_4826_linorder__le__less__linear,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.59 | ( ord_less_rat @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_le_less_linear
% 0.32/0.59 thf(fact_4827_linorder__le__less__linear,axiom,
% 0.32/0.59 ! [X: num,Y3: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.59 | ( ord_less_num @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_le_less_linear
% 0.32/0.59 thf(fact_4828_linorder__le__less__linear,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.59 | ( ord_less_nat @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_le_less_linear
% 0.32/0.59 thf(fact_4829_linorder__le__less__linear,axiom,
% 0.32/0.59 ! [X: int,Y3: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.59 | ( ord_less_int @ Y3 @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % linorder_le_less_linear
% 0.32/0.59 thf(fact_4830_order__le__imp__less__or__eq,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_real @ X @ Y3 )
% 0.32/0.59 | ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_imp_less_or_eq
% 0.32/0.59 thf(fact_4831_order__le__imp__less__or__eq,axiom,
% 0.32/0.59 ! [X: set_int,Y3: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_set_int @ X @ Y3 )
% 0.32/0.59 | ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_imp_less_or_eq
% 0.32/0.59 thf(fact_4832_order__le__imp__less__or__eq,axiom,
% 0.32/0.59 ! [X: rat,Y3: rat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_rat @ X @ Y3 )
% 0.32/0.59 | ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_imp_less_or_eq
% 0.32/0.59 thf(fact_4833_order__le__imp__less__or__eq,axiom,
% 0.32/0.59 ! [X: num,Y3: num] :
% 0.32/0.59 ( ( ord_less_eq_num @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_num @ X @ Y3 )
% 0.32/0.59 | ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_imp_less_or_eq
% 0.32/0.59 thf(fact_4834_order__le__imp__less__or__eq,axiom,
% 0.32/0.59 ! [X: nat,Y3: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.59 | ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_imp_less_or_eq
% 0.32/0.59 thf(fact_4835_order__le__imp__less__or__eq,axiom,
% 0.32/0.59 ! [X: int,Y3: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.59 => ( ( ord_less_int @ X @ Y3 )
% 0.32/0.59 | ( X = Y3 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % order_le_imp_less_or_eq
% 0.32/0.59 thf(fact_4836_bot_Oextremum,axiom,
% 0.32/0.59 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum
% 0.32/0.59 thf(fact_4837_bot_Oextremum,axiom,
% 0.32/0.59 ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum
% 0.32/0.59 thf(fact_4838_bot_Oextremum,axiom,
% 0.32/0.59 ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum
% 0.32/0.59 thf(fact_4839_bot_Oextremum,axiom,
% 0.32/0.59 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum
% 0.32/0.59 thf(fact_4840_bot_Oextremum,axiom,
% 0.32/0.59 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum
% 0.32/0.59 thf(fact_4841_bot_Oextremum__unique,axiom,
% 0.32/0.59 ! [A: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.32/0.59 = ( A = bot_bot_set_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_unique
% 0.32/0.59 thf(fact_4842_bot_Oextremum__unique,axiom,
% 0.32/0.59 ! [A: extended_enat] :
% 0.32/0.59 ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 0.32/0.59 = ( A = bot_bo4199563552545308370d_enat ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_unique
% 0.32/0.59 thf(fact_4843_bot_Oextremum__unique,axiom,
% 0.32/0.59 ! [A: set_real] :
% 0.32/0.59 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.32/0.59 = ( A = bot_bot_set_real ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_unique
% 0.32/0.59 thf(fact_4844_bot_Oextremum__unique,axiom,
% 0.32/0.59 ! [A: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.32/0.59 = ( A = bot_bot_set_int ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_unique
% 0.32/0.59 thf(fact_4845_bot_Oextremum__unique,axiom,
% 0.32/0.59 ! [A: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.32/0.59 = ( A = bot_bot_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_unique
% 0.32/0.59 thf(fact_4846_bot_Oextremum__uniqueI,axiom,
% 0.32/0.59 ! [A: set_nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.32/0.59 => ( A = bot_bot_set_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_uniqueI
% 0.32/0.59 thf(fact_4847_bot_Oextremum__uniqueI,axiom,
% 0.32/0.59 ! [A: extended_enat] :
% 0.32/0.59 ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 0.32/0.59 => ( A = bot_bo4199563552545308370d_enat ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_uniqueI
% 0.32/0.59 thf(fact_4848_bot_Oextremum__uniqueI,axiom,
% 0.32/0.59 ! [A: set_real] :
% 0.32/0.59 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.32/0.59 => ( A = bot_bot_set_real ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_uniqueI
% 0.32/0.59 thf(fact_4849_bot_Oextremum__uniqueI,axiom,
% 0.32/0.59 ! [A: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.32/0.59 => ( A = bot_bot_set_int ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_uniqueI
% 0.32/0.59 thf(fact_4850_bot_Oextremum__uniqueI,axiom,
% 0.32/0.59 ! [A: nat] :
% 0.32/0.59 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.32/0.59 => ( A = bot_bot_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_uniqueI
% 0.32/0.59 thf(fact_4851_bot_Oextremum__strict,axiom,
% 0.32/0.59 ! [A: set_nat] :
% 0.32/0.59 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_strict
% 0.32/0.59 thf(fact_4852_bot_Oextremum__strict,axiom,
% 0.32/0.59 ! [A: extended_enat] :
% 0.32/0.59 ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_strict
% 0.32/0.59 thf(fact_4853_bot_Oextremum__strict,axiom,
% 0.32/0.59 ! [A: set_int] :
% 0.32/0.59 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_strict
% 0.32/0.59 thf(fact_4854_bot_Oextremum__strict,axiom,
% 0.32/0.59 ! [A: set_real] :
% 0.32/0.59 ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_strict
% 0.32/0.59 thf(fact_4855_bot_Oextremum__strict,axiom,
% 0.32/0.59 ! [A: nat] :
% 0.32/0.59 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.32/0.59
% 0.32/0.59 % bot.extremum_strict
% 0.32/0.59 thf(fact_4856_bot_Onot__eq__extremum,axiom,
% 0.32/0.59 ! [A: set_nat] :
% 0.32/0.59 ( ( A != bot_bot_set_nat )
% 0.32/0.59 = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.not_eq_extremum
% 0.32/0.59 thf(fact_4857_bot_Onot__eq__extremum,axiom,
% 0.32/0.59 ! [A: extended_enat] :
% 0.32/0.59 ( ( A != bot_bo4199563552545308370d_enat )
% 0.32/0.59 = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.not_eq_extremum
% 0.32/0.59 thf(fact_4858_bot_Onot__eq__extremum,axiom,
% 0.32/0.59 ! [A: set_int] :
% 0.32/0.59 ( ( A != bot_bot_set_int )
% 0.32/0.59 = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.not_eq_extremum
% 0.32/0.59 thf(fact_4859_bot_Onot__eq__extremum,axiom,
% 0.32/0.59 ! [A: set_real] :
% 0.32/0.59 ( ( A != bot_bot_set_real )
% 0.32/0.59 = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.not_eq_extremum
% 0.32/0.59 thf(fact_4860_bot_Onot__eq__extremum,axiom,
% 0.32/0.59 ! [A: nat] :
% 0.32/0.59 ( ( A != bot_bot_nat )
% 0.32/0.59 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % bot.not_eq_extremum
% 0.32/0.59 thf(fact_4861_zero__less__power__eq,axiom,
% 0.32/0.59 ! [A: real,N: nat] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 0.32/0.59 = ( ( N = zero_zero_nat )
% 0.32/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( A != zero_zero_real ) )
% 0.32/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_power_eq
% 0.32/0.59 thf(fact_4862_zero__less__power__eq,axiom,
% 0.32/0.59 ! [A: rat,N: nat] :
% 0.32/0.59 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 0.32/0.59 = ( ( N = zero_zero_nat )
% 0.32/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( A != zero_zero_rat ) )
% 0.32/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_power_eq
% 0.32/0.59 thf(fact_4863_zero__less__power__eq,axiom,
% 0.32/0.59 ! [A: int,N: nat] :
% 0.32/0.59 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 0.32/0.59 = ( ( N = zero_zero_nat )
% 0.32/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( A != zero_zero_int ) )
% 0.32/0.59 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_power_eq
% 0.32/0.59 thf(fact_4864_Euclid__induct,axiom,
% 0.32/0.59 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.32/0.59 ( ! [A5: nat,B3: nat] :
% 0.32/0.59 ( ( P @ A5 @ B3 )
% 0.32/0.59 = ( P @ B3 @ A5 ) )
% 0.32/0.59 => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
% 0.32/0.59 => ( ! [A5: nat,B3: nat] :
% 0.32/0.59 ( ( P @ A5 @ B3 )
% 0.32/0.59 => ( P @ A5 @ ( plus_plus_nat @ A5 @ B3 ) ) )
% 0.32/0.59 => ( P @ A @ B ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Euclid_induct
% 0.32/0.59 thf(fact_4865_even__mask__div__iff_H,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mask_div_iff'
% 0.32/0.59 thf(fact_4866_even__mask__div__iff_H,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mask_div_iff'
% 0.32/0.59 thf(fact_4867_even__mask__div__iff_H,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.59 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mask_div_iff'
% 0.32/0.59 thf(fact_4868_vebt__insert_Osimps_I5_J,axiom,
% 0.32/0.59 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 0.32/0.59 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 0.32/0.59 = ( if_VEBT_VEBT
% 0.32/0.59 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.32/0.59 & ~ ( ( X = Mi )
% 0.32/0.59 | ( X = Ma ) ) )
% 0.32/0.59 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 0.32/0.59 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % vebt_insert.simps(5)
% 0.32/0.59 thf(fact_4869_power__le__zero__eq,axiom,
% 0.32/0.59 ! [A: real,N: nat] :
% 0.32/0.59 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 0.32/0.59 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.59 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.32/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( A = zero_zero_real ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % power_le_zero_eq
% 0.32/0.59 thf(fact_4870_power__le__zero__eq,axiom,
% 0.32/0.59 ! [A: rat,N: nat] :
% 0.32/0.59 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 0.32/0.59 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.59 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.32/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % power_le_zero_eq
% 0.32/0.59 thf(fact_4871_power__le__zero__eq,axiom,
% 0.32/0.59 ! [A: int,N: nat] :
% 0.32/0.59 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 0.32/0.59 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.59 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.32/0.59 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 & ( A = zero_zero_int ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % power_le_zero_eq
% 0.32/0.59 thf(fact_4872_option_Osize__gen_I1_J,axiom,
% 0.32/0.59 ! [X: nat > nat] :
% 0.32/0.59 ( ( size_option_nat @ X @ none_nat )
% 0.32/0.59 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % option.size_gen(1)
% 0.32/0.59 thf(fact_4873_option_Osize__gen_I1_J,axiom,
% 0.32/0.59 ! [X: product_prod_nat_nat > nat] :
% 0.32/0.59 ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
% 0.32/0.59 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % option.size_gen(1)
% 0.32/0.59 thf(fact_4874_option_Osize__gen_I1_J,axiom,
% 0.32/0.59 ! [X: num > nat] :
% 0.32/0.59 ( ( size_option_num @ X @ none_num )
% 0.32/0.59 = ( suc @ zero_zero_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % option.size_gen(1)
% 0.32/0.59 thf(fact_4875_even__mod__4__div__2,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.59 = ( suc @ zero_zero_nat ) )
% 0.32/0.59 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mod_4_div_2
% 0.32/0.59 thf(fact_4876_even__mask__div__iff,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.59 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 = zero_z3403309356797280102nteger )
% 0.32/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mask_div_iff
% 0.32/0.59 thf(fact_4877_even__mask__div__iff,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.59 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 = zero_zero_nat )
% 0.32/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mask_div_iff
% 0.32/0.59 thf(fact_4878_even__mask__div__iff,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.59 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 = zero_zero_int )
% 0.32/0.59 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mask_div_iff
% 0.32/0.59 thf(fact_4879_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.59 ! [A: code_integer,M: nat,N: nat] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.59 = ( ( ord_less_nat @ N @ M )
% 0.32/0.59 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 = zero_z3403309356797280102nteger )
% 0.32/0.59 | ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.59 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mult_exp_div_exp_iff
% 0.32/0.59 thf(fact_4880_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.59 ! [A: nat,M: nat,N: nat] :
% 0.32/0.59 ( ( 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 ) ) )
% 0.32/0.59 = ( ( ord_less_nat @ N @ M )
% 0.32/0.59 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 = zero_zero_nat )
% 0.32/0.59 | ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.59 & ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mult_exp_div_exp_iff
% 0.32/0.59 thf(fact_4881_even__mult__exp__div__exp__iff,axiom,
% 0.32/0.59 ! [A: int,M: nat,N: nat] :
% 0.32/0.59 ( ( 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 ) ) )
% 0.32/0.59 = ( ( ord_less_nat @ N @ M )
% 0.32/0.59 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.59 = zero_zero_int )
% 0.32/0.59 | ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.59 & ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_mult_exp_div_exp_iff
% 0.32/0.59 thf(fact_4882_vebt__insert_Oelims,axiom,
% 0.32/0.59 ! [X: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
% 0.32/0.59 ( ( ( vEBT_vebt_insert @ X @ Xa )
% 0.32/0.59 = Y3 )
% 0.32/0.59 => ( ! [A5: $o,B3: $o] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.59 => ~ ( ( ( Xa = zero_zero_nat )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Leaf @ $true @ B3 ) ) )
% 0.32/0.59 & ( ( Xa != zero_zero_nat )
% 0.32/0.59 => ( ( ( Xa = one_one_nat )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Leaf @ A5 @ $true ) ) )
% 0.32/0.59 & ( ( Xa != one_one_nat )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Leaf @ A5 @ B3 ) ) ) ) ) ) )
% 0.32/0.59 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
% 0.32/0.59 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
% 0.32/0.59 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.32/0.59 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 != ( if_VEBT_VEBT
% 0.32/0.59 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.59 & ~ ( ( Xa = Mi2 )
% 0.32/0.59 | ( Xa = Ma2 ) ) )
% 0.32/0.59 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.32/0.59 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % vebt_insert.elims
% 0.32/0.59 thf(fact_4883_vebt__insert_Opelims,axiom,
% 0.32/0.59 ! [X: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
% 0.32/0.59 ( ( ( vEBT_vebt_insert @ X @ Xa )
% 0.32/0.59 = Y3 )
% 0.32/0.59 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.59 => ( ! [A5: $o,B3: $o] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.59 => ( ( ( ( Xa = zero_zero_nat )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Leaf @ $true @ B3 ) ) )
% 0.32/0.59 & ( ( Xa != zero_zero_nat )
% 0.32/0.59 => ( ( ( Xa = one_one_nat )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Leaf @ A5 @ $true ) ) )
% 0.32/0.59 & ( ( Xa != one_one_nat )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Leaf @ A5 @ B3 ) ) ) ) ) )
% 0.32/0.59 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) ) ) )
% 0.32/0.59 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.32/0.59 => ( ( Y3
% 0.32/0.59 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 0.32/0.59 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa ) ) ) )
% 0.32/0.59 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.32/0.59 => ( ( Y3
% 0.32/0.59 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 0.32/0.59 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
% 0.32/0.59 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.59 => ( ( Y3
% 0.32/0.59 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.59 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
% 0.32/0.59 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 0.32/0.59 => ( ( Y3
% 0.32/0.59 = ( if_VEBT_VEBT
% 0.32/0.59 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.32/0.59 & ~ ( ( Xa = Mi2 )
% 0.32/0.59 | ( Xa = Ma2 ) ) )
% 0.32/0.59 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.32/0.59 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
% 0.32/0.59 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % vebt_insert.pelims
% 0.32/0.59 thf(fact_4884_triangle__def,axiom,
% 0.32/0.59 ( nat_triangle
% 0.32/0.59 = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % triangle_def
% 0.32/0.59 thf(fact_4885_vebt__buildup_Oelims,axiom,
% 0.32/0.59 ! [X: nat,Y3: vEBT_VEBT] :
% 0.32/0.59 ( ( ( vEBT_vebt_buildup @ X )
% 0.32/0.59 = Y3 )
% 0.32/0.59 => ( ( ( X = zero_zero_nat )
% 0.32/0.59 => ( Y3
% 0.32/0.59 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.32/0.59 => ( ( ( X
% 0.32/0.59 = ( suc @ zero_zero_nat ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.32/0.59 => ~ ! [Va3: nat] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( suc @ ( suc @ Va3 ) ) )
% 0.32/0.59 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.32/0.59 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % vebt_buildup.elims
% 0.32/0.59 thf(fact_4886_flip__bit__0,axiom,
% 0.32/0.59 ! [A: code_integer] :
% 0.32/0.59 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.32/0.59 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % flip_bit_0
% 0.32/0.59 thf(fact_4887_flip__bit__0,axiom,
% 0.32/0.59 ! [A: int] :
% 0.32/0.59 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.32/0.59 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % flip_bit_0
% 0.32/0.59 thf(fact_4888_flip__bit__0,axiom,
% 0.32/0.59 ! [A: nat] :
% 0.32/0.59 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.32/0.59 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % flip_bit_0
% 0.32/0.59 thf(fact_4889_signed__take__bit__Suc,axiom,
% 0.32/0.59 ! [N: nat,A: code_integer] :
% 0.32/0.59 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 0.32/0.59 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_Suc
% 0.32/0.59 thf(fact_4890_signed__take__bit__Suc,axiom,
% 0.32/0.59 ! [N: nat,A: int] :
% 0.32/0.59 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 0.32/0.59 = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_Suc
% 0.32/0.59 thf(fact_4891_set__decode__Suc,axiom,
% 0.32/0.59 ! [N: nat,X: nat] :
% 0.32/0.59 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 0.32/0.59 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % set_decode_Suc
% 0.32/0.59 thf(fact_4892_add__scale__eq__noteq,axiom,
% 0.32/0.59 ! [R: complex,A: complex,B: complex,C: complex,D: complex] :
% 0.32/0.59 ( ( R != zero_zero_complex )
% 0.32/0.59 => ( ( ( A = B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 => ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
% 0.32/0.59 != ( plus_plus_complex @ B @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_scale_eq_noteq
% 0.32/0.59 thf(fact_4893_add__scale__eq__noteq,axiom,
% 0.32/0.59 ! [R: real,A: real,B: real,C: real,D: real] :
% 0.32/0.59 ( ( R != zero_zero_real )
% 0.32/0.59 => ( ( ( A = B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
% 0.32/0.59 != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_scale_eq_noteq
% 0.32/0.59 thf(fact_4894_add__scale__eq__noteq,axiom,
% 0.32/0.59 ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.59 ( ( R != zero_zero_rat )
% 0.32/0.59 => ( ( ( A = B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
% 0.32/0.59 != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_scale_eq_noteq
% 0.32/0.59 thf(fact_4895_add__scale__eq__noteq,axiom,
% 0.32/0.59 ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.59 ( ( R != zero_zero_nat )
% 0.32/0.59 => ( ( ( A = B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
% 0.32/0.59 != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_scale_eq_noteq
% 0.32/0.59 thf(fact_4896_add__scale__eq__noteq,axiom,
% 0.32/0.59 ! [R: int,A: int,B: int,C: int,D: int] :
% 0.32/0.59 ( ( R != zero_zero_int )
% 0.32/0.59 => ( ( ( A = B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
% 0.32/0.59 != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_scale_eq_noteq
% 0.32/0.59 thf(fact_4897_artanh__def,axiom,
% 0.32/0.59 ( artanh_real
% 0.32/0.59 = ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % artanh_def
% 0.32/0.59 thf(fact_4898_intind,axiom,
% 0.32/0.59 ! [I: nat,N: nat,P: nat > $o,X: nat] :
% 0.32/0.59 ( ( ord_less_nat @ I @ N )
% 0.32/0.59 => ( ( P @ X )
% 0.32/0.59 => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % intind
% 0.32/0.59 thf(fact_4899_intind,axiom,
% 0.32/0.59 ! [I: nat,N: nat,P: int > $o,X: int] :
% 0.32/0.59 ( ( ord_less_nat @ I @ N )
% 0.32/0.59 => ( ( P @ X )
% 0.32/0.59 => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % intind
% 0.32/0.59 thf(fact_4900_intind,axiom,
% 0.32/0.59 ! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 0.32/0.59 ( ( ord_less_nat @ I @ N )
% 0.32/0.59 => ( ( P @ X )
% 0.32/0.59 => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % intind
% 0.32/0.59 thf(fact_4901_max__enat__simps_I2_J,axiom,
% 0.32/0.59 ! [Q2: extended_enat] :
% 0.32/0.59 ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 0.32/0.59 = Q2 ) ).
% 0.32/0.59
% 0.32/0.59 % max_enat_simps(2)
% 0.32/0.59 thf(fact_4902_max__enat__simps_I3_J,axiom,
% 0.32/0.59 ! [Q2: extended_enat] :
% 0.32/0.59 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 0.32/0.59 = Q2 ) ).
% 0.32/0.59
% 0.32/0.59 % max_enat_simps(3)
% 0.32/0.59 thf(fact_4903_ln__inj__iff,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.59 => ( ( ( ln_ln_real @ X )
% 0.32/0.59 = ( ln_ln_real @ Y3 ) )
% 0.32/0.59 = ( X = Y3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_inj_iff
% 0.32/0.59 thf(fact_4904_ln__less__cancel__iff,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.59 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) )
% 0.32/0.59 = ( ord_less_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_less_cancel_iff
% 0.32/0.59 thf(fact_4905_of__bool__less__eq__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.32/0.59 = ( P
% 0.32/0.59 => Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_iff
% 0.32/0.59 thf(fact_4906_of__bool__less__eq__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.32/0.59 = ( P
% 0.32/0.59 => Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_iff
% 0.32/0.59 thf(fact_4907_of__bool__less__eq__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.32/0.59 = ( P
% 0.32/0.59 => Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_iff
% 0.32/0.59 thf(fact_4908_of__bool__less__eq__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.32/0.59 = ( P
% 0.32/0.59 => Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_iff
% 0.32/0.59 thf(fact_4909_of__bool__eq_I1_J,axiom,
% 0.32/0.59 ( ( zero_n1201886186963655149omplex @ $false )
% 0.32/0.59 = zero_zero_complex ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(1)
% 0.32/0.59 thf(fact_4910_of__bool__eq_I1_J,axiom,
% 0.32/0.59 ( ( zero_n3304061248610475627l_real @ $false )
% 0.32/0.59 = zero_zero_real ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(1)
% 0.32/0.59 thf(fact_4911_of__bool__eq_I1_J,axiom,
% 0.32/0.59 ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.32/0.59 = zero_zero_rat ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(1)
% 0.32/0.59 thf(fact_4912_of__bool__eq_I1_J,axiom,
% 0.32/0.59 ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.32/0.59 = zero_zero_nat ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(1)
% 0.32/0.59 thf(fact_4913_of__bool__eq_I1_J,axiom,
% 0.32/0.59 ( ( zero_n2684676970156552555ol_int @ $false )
% 0.32/0.59 = zero_zero_int ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(1)
% 0.32/0.59 thf(fact_4914_of__bool__eq_I1_J,axiom,
% 0.32/0.59 ( ( zero_n356916108424825756nteger @ $false )
% 0.32/0.59 = zero_z3403309356797280102nteger ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(1)
% 0.32/0.59 thf(fact_4915_of__bool__eq__0__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.32/0.59 = zero_zero_complex )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_0_iff
% 0.32/0.59 thf(fact_4916_of__bool__eq__0__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.32/0.59 = zero_zero_real )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_0_iff
% 0.32/0.59 thf(fact_4917_of__bool__eq__0__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.32/0.59 = zero_zero_rat )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_0_iff
% 0.32/0.59 thf(fact_4918_of__bool__eq__0__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.32/0.59 = zero_zero_nat )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_0_iff
% 0.32/0.59 thf(fact_4919_of__bool__eq__0__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.32/0.59 = zero_zero_int )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_0_iff
% 0.32/0.59 thf(fact_4920_of__bool__eq__0__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n356916108424825756nteger @ P )
% 0.32/0.59 = zero_z3403309356797280102nteger )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_0_iff
% 0.32/0.59 thf(fact_4921_of__bool__less__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.32/0.59 = ( ~ P
% 0.32/0.59 & Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_iff
% 0.32/0.59 thf(fact_4922_of__bool__less__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.32/0.59 = ( ~ P
% 0.32/0.59 & Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_iff
% 0.32/0.59 thf(fact_4923_of__bool__less__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.32/0.59 = ( ~ P
% 0.32/0.59 & Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_iff
% 0.32/0.59 thf(fact_4924_of__bool__less__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.32/0.59 = ( ~ P
% 0.32/0.59 & Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_iff
% 0.32/0.59 thf(fact_4925_of__bool__less__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 0.32/0.59 = ( ~ P
% 0.32/0.59 & Q ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_iff
% 0.32/0.59 thf(fact_4926_of__bool__eq__1__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n1201886186963655149omplex @ P )
% 0.32/0.59 = one_one_complex )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_1_iff
% 0.32/0.59 thf(fact_4927_of__bool__eq__1__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n3304061248610475627l_real @ P )
% 0.32/0.59 = one_one_real )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_1_iff
% 0.32/0.59 thf(fact_4928_of__bool__eq__1__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.32/0.59 = one_one_rat )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_1_iff
% 0.32/0.59 thf(fact_4929_of__bool__eq__1__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.32/0.59 = one_one_nat )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_1_iff
% 0.32/0.59 thf(fact_4930_of__bool__eq__1__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.32/0.59 = one_one_int )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_1_iff
% 0.32/0.59 thf(fact_4931_of__bool__eq__1__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ( zero_n356916108424825756nteger @ P )
% 0.32/0.59 = one_one_Code_integer )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_1_iff
% 0.32/0.59 thf(fact_4932_of__bool__eq_I2_J,axiom,
% 0.32/0.59 ( ( zero_n1201886186963655149omplex @ $true )
% 0.32/0.59 = one_one_complex ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(2)
% 0.32/0.59 thf(fact_4933_of__bool__eq_I2_J,axiom,
% 0.32/0.59 ( ( zero_n3304061248610475627l_real @ $true )
% 0.32/0.59 = one_one_real ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(2)
% 0.32/0.59 thf(fact_4934_of__bool__eq_I2_J,axiom,
% 0.32/0.59 ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.32/0.59 = one_one_rat ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(2)
% 0.32/0.59 thf(fact_4935_of__bool__eq_I2_J,axiom,
% 0.32/0.59 ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.32/0.59 = one_one_nat ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(2)
% 0.32/0.59 thf(fact_4936_of__bool__eq_I2_J,axiom,
% 0.32/0.59 ( ( zero_n2684676970156552555ol_int @ $true )
% 0.32/0.59 = one_one_int ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(2)
% 0.32/0.59 thf(fact_4937_of__bool__eq_I2_J,axiom,
% 0.32/0.59 ( ( zero_n356916108424825756nteger @ $true )
% 0.32/0.59 = one_one_Code_integer ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq(2)
% 0.32/0.59 thf(fact_4938_signed__take__bit__of__0,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.32/0.59 = zero_zero_int ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_of_0
% 0.32/0.59 thf(fact_4939_of__bool__or__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( zero_n2687167440665602831ol_nat
% 0.32/0.59 @ ( P
% 0.32/0.59 | Q ) )
% 0.32/0.59 = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_or_iff
% 0.32/0.59 thf(fact_4940_of__bool__or__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( zero_n2684676970156552555ol_int
% 0.32/0.59 @ ( P
% 0.32/0.59 | Q ) )
% 0.32/0.59 = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_or_iff
% 0.32/0.59 thf(fact_4941_of__bool__or__iff,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( zero_n356916108424825756nteger
% 0.32/0.59 @ ( P
% 0.32/0.59 | Q ) )
% 0.32/0.59 = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_or_iff
% 0.32/0.59 thf(fact_4942_ln__le__cancel__iff,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.59 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) )
% 0.32/0.59 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_le_cancel_iff
% 0.32/0.59 thf(fact_4943_zero__less__of__bool__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_of_bool_iff
% 0.32/0.59 thf(fact_4944_zero__less__of__bool__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_of_bool_iff
% 0.32/0.59 thf(fact_4945_zero__less__of__bool__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_of_bool_iff
% 0.32/0.59 thf(fact_4946_zero__less__of__bool__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_of_bool_iff
% 0.32/0.59 thf(fact_4947_zero__less__of__bool__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 0.32/0.59 = P ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_of_bool_iff
% 0.32/0.59 thf(fact_4948_ln__eq__zero__iff,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ( ln_ln_real @ X )
% 0.32/0.59 = zero_zero_real )
% 0.32/0.59 = ( X = one_one_real ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_eq_zero_iff
% 0.32/0.59 thf(fact_4949_ln__gt__zero__iff,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.32/0.59 = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_gt_zero_iff
% 0.32/0.59 thf(fact_4950_ln__less__zero__iff,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.32/0.59 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_less_zero_iff
% 0.32/0.59 thf(fact_4951_of__bool__less__one__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_one_iff
% 0.32/0.59 thf(fact_4952_of__bool__less__one__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_one_iff
% 0.32/0.59 thf(fact_4953_of__bool__less__one__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_one_iff
% 0.32/0.59 thf(fact_4954_of__bool__less__one__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_one_iff
% 0.32/0.59 thf(fact_4955_of__bool__less__one__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 0.32/0.59 = ( ~ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_one_iff
% 0.32/0.59 thf(fact_4956_ln__one,axiom,
% 0.32/0.59 ( ( ln_ln_real @ one_one_real )
% 0.32/0.59 = zero_zero_real ) ).
% 0.32/0.59
% 0.32/0.59 % ln_one
% 0.32/0.59 thf(fact_4957_of__bool__not__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.32/0.59 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_not_iff
% 0.32/0.59 thf(fact_4958_of__bool__not__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.32/0.59 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_not_iff
% 0.32/0.59 thf(fact_4959_of__bool__not__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.32/0.59 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_not_iff
% 0.32/0.59 thf(fact_4960_of__bool__not__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.32/0.59 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_not_iff
% 0.32/0.59 thf(fact_4961_of__bool__not__iff,axiom,
% 0.32/0.59 ! [P: $o] :
% 0.32/0.59 ( ( zero_n356916108424825756nteger @ ~ P )
% 0.32/0.59 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_not_iff
% 0.32/0.59 thf(fact_4962_Suc__0__mod__eq,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.59 = ( zero_n2687167440665602831ol_nat
% 0.32/0.59 @ ( N
% 0.32/0.59 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Suc_0_mod_eq
% 0.32/0.59 thf(fact_4963_signed__take__bit__Suc__1,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 0.32/0.59 = one_one_int ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_Suc_1
% 0.32/0.59 thf(fact_4964_signed__take__bit__numeral__of__1,axiom,
% 0.32/0.59 ! [K: num] :
% 0.32/0.59 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 0.32/0.59 = one_one_int ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_numeral_of_1
% 0.32/0.59 thf(fact_4965_nth__replicate,axiom,
% 0.32/0.59 ! [I: nat,N: nat,X: nat] :
% 0.32/0.59 ( ( ord_less_nat @ I @ N )
% 0.32/0.59 => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
% 0.32/0.59 = X ) ) ).
% 0.32/0.59
% 0.32/0.59 % nth_replicate
% 0.32/0.59 thf(fact_4966_nth__replicate,axiom,
% 0.32/0.59 ! [I: nat,N: nat,X: int] :
% 0.32/0.59 ( ( ord_less_nat @ I @ N )
% 0.32/0.59 => ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
% 0.32/0.59 = X ) ) ).
% 0.32/0.59
% 0.32/0.59 % nth_replicate
% 0.32/0.59 thf(fact_4967_nth__replicate,axiom,
% 0.32/0.59 ! [I: nat,N: nat,X: vEBT_VEBT] :
% 0.32/0.59 ( ( ord_less_nat @ I @ N )
% 0.32/0.59 => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I )
% 0.32/0.59 = X ) ) ).
% 0.32/0.59
% 0.32/0.59 % nth_replicate
% 0.32/0.59 thf(fact_4968_triangle__Suc,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( nat_triangle @ ( suc @ N ) )
% 0.32/0.59 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % triangle_Suc
% 0.32/0.59 thf(fact_4969_ln__ge__zero__iff,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.32/0.59 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_ge_zero_iff
% 0.32/0.59 thf(fact_4970_ln__le__zero__iff,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 0.32/0.59 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_le_zero_iff
% 0.32/0.59 thf(fact_4971_signed__take__bit__Suc__bit0,axiom,
% 0.32/0.59 ! [N: nat,K: num] :
% 0.32/0.59 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.32/0.59 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_Suc_bit0
% 0.32/0.59 thf(fact_4972_odd__of__bool__self,axiom,
% 0.32/0.59 ! [P6: $o] :
% 0.32/0.59 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P6 ) ) )
% 0.32/0.59 = P6 ) ).
% 0.32/0.59
% 0.32/0.59 % odd_of_bool_self
% 0.32/0.59 thf(fact_4973_odd__of__bool__self,axiom,
% 0.32/0.59 ! [P6: $o] :
% 0.32/0.59 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P6 ) ) )
% 0.32/0.59 = P6 ) ).
% 0.32/0.59
% 0.32/0.59 % odd_of_bool_self
% 0.32/0.59 thf(fact_4974_odd__of__bool__self,axiom,
% 0.32/0.59 ! [P6: $o] :
% 0.32/0.59 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P6 ) ) )
% 0.32/0.59 = P6 ) ).
% 0.32/0.59
% 0.32/0.59 % odd_of_bool_self
% 0.32/0.59 thf(fact_4975_of__bool__half__eq__0,axiom,
% 0.32/0.59 ! [B: $o] :
% 0.32/0.59 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.59 = zero_zero_nat ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_half_eq_0
% 0.32/0.59 thf(fact_4976_of__bool__half__eq__0,axiom,
% 0.32/0.59 ! [B: $o] :
% 0.32/0.59 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.59 = zero_zero_int ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_half_eq_0
% 0.32/0.59 thf(fact_4977_of__bool__half__eq__0,axiom,
% 0.32/0.59 ! [B: $o] :
% 0.32/0.59 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.59 = zero_z3403309356797280102nteger ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_half_eq_0
% 0.32/0.59 thf(fact_4978_set__decode__0,axiom,
% 0.32/0.59 ! [X: nat] :
% 0.32/0.59 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 0.32/0.59 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % set_decode_0
% 0.32/0.59 thf(fact_4979_one__div__2__pow__eq,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % one_div_2_pow_eq
% 0.32/0.59 thf(fact_4980_one__div__2__pow__eq,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % one_div_2_pow_eq
% 0.32/0.59 thf(fact_4981_one__div__2__pow__eq,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % one_div_2_pow_eq
% 0.32/0.59 thf(fact_4982_bits__1__div__exp,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bits_1_div_exp
% 0.32/0.59 thf(fact_4983_bits__1__div__exp,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bits_1_div_exp
% 0.32/0.59 thf(fact_4984_bits__1__div__exp,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bits_1_div_exp
% 0.32/0.59 thf(fact_4985_one__mod__2__pow__eq,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % one_mod_2_pow_eq
% 0.32/0.59 thf(fact_4986_one__mod__2__pow__eq,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % one_mod_2_pow_eq
% 0.32/0.59 thf(fact_4987_one__mod__2__pow__eq,axiom,
% 0.32/0.59 ! [N: nat] :
% 0.32/0.59 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % one_mod_2_pow_eq
% 0.32/0.59 thf(fact_4988_dvd__antisym,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( dvd_dvd_nat @ M @ N )
% 0.32/0.59 => ( ( dvd_dvd_nat @ N @ M )
% 0.32/0.59 => ( M = N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % dvd_antisym
% 0.32/0.59 thf(fact_4989_of__bool__eq__iff,axiom,
% 0.32/0.59 ! [P6: $o,Q2: $o] :
% 0.32/0.59 ( ( ( zero_n2687167440665602831ol_nat @ P6 )
% 0.32/0.59 = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 0.32/0.59 = ( P6 = Q2 ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_iff
% 0.32/0.59 thf(fact_4990_of__bool__eq__iff,axiom,
% 0.32/0.59 ! [P6: $o,Q2: $o] :
% 0.32/0.59 ( ( ( zero_n2684676970156552555ol_int @ P6 )
% 0.32/0.59 = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 0.32/0.59 = ( P6 = Q2 ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_iff
% 0.32/0.59 thf(fact_4991_of__bool__eq__iff,axiom,
% 0.32/0.59 ! [P6: $o,Q2: $o] :
% 0.32/0.59 ( ( ( zero_n356916108424825756nteger @ P6 )
% 0.32/0.59 = ( zero_n356916108424825756nteger @ Q2 ) )
% 0.32/0.59 = ( P6 = Q2 ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_eq_iff
% 0.32/0.59 thf(fact_4992_of__bool__conj,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( zero_n3304061248610475627l_real
% 0.32/0.59 @ ( P
% 0.32/0.59 & Q ) )
% 0.32/0.59 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_conj
% 0.32/0.59 thf(fact_4993_of__bool__conj,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( zero_n2052037380579107095ol_rat
% 0.32/0.59 @ ( P
% 0.32/0.59 & Q ) )
% 0.32/0.59 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_conj
% 0.32/0.59 thf(fact_4994_of__bool__conj,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( zero_n2687167440665602831ol_nat
% 0.32/0.59 @ ( P
% 0.32/0.59 & Q ) )
% 0.32/0.59 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_conj
% 0.32/0.59 thf(fact_4995_of__bool__conj,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( zero_n2684676970156552555ol_int
% 0.32/0.59 @ ( P
% 0.32/0.59 & Q ) )
% 0.32/0.59 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_conj
% 0.32/0.59 thf(fact_4996_of__bool__conj,axiom,
% 0.32/0.59 ! [P: $o,Q: $o] :
% 0.32/0.59 ( ( zero_n356916108424825756nteger
% 0.32/0.59 @ ( P
% 0.32/0.59 & Q ) )
% 0.32/0.59 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_conj
% 0.32/0.59 thf(fact_4997_signed__take__bit__mult,axiom,
% 0.32/0.59 ! [N: nat,K: int,L2: int] :
% 0.32/0.59 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 0.32/0.59 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_mult
% 0.32/0.59 thf(fact_4998_signed__take__bit__add,axiom,
% 0.32/0.59 ! [N: nat,K: int,L2: int] :
% 0.32/0.59 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 0.32/0.59 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_add
% 0.32/0.59 thf(fact_4999_signed__take__bit__diff,axiom,
% 0.32/0.59 ! [N: nat,K: int,L2: int] :
% 0.32/0.59 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 0.32/0.59 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_diff
% 0.32/0.59 thf(fact_5000_ln__less__self,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_less_self
% 0.32/0.59 thf(fact_5001_zero__less__eq__of__bool,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_eq_of_bool
% 0.32/0.59 thf(fact_5002_zero__less__eq__of__bool,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_eq_of_bool
% 0.32/0.59 thf(fact_5003_zero__less__eq__of__bool,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_eq_of_bool
% 0.32/0.59 thf(fact_5004_zero__less__eq__of__bool,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_eq_of_bool
% 0.32/0.59 thf(fact_5005_zero__less__eq__of__bool,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 0.32/0.59
% 0.32/0.59 % zero_less_eq_of_bool
% 0.32/0.59 thf(fact_5006_of__bool__less__eq__one,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_one
% 0.32/0.59 thf(fact_5007_of__bool__less__eq__one,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_one
% 0.32/0.59 thf(fact_5008_of__bool__less__eq__one,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_one
% 0.32/0.59 thf(fact_5009_of__bool__less__eq__one,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_one
% 0.32/0.59 thf(fact_5010_of__bool__less__eq__one,axiom,
% 0.32/0.59 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_less_eq_one
% 0.32/0.59 thf(fact_5011_of__bool__def,axiom,
% 0.32/0.59 ( zero_n1201886186963655149omplex
% 0.32/0.59 = ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_def
% 0.32/0.59 thf(fact_5012_of__bool__def,axiom,
% 0.32/0.59 ( zero_n3304061248610475627l_real
% 0.32/0.59 = ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_def
% 0.32/0.59 thf(fact_5013_of__bool__def,axiom,
% 0.32/0.59 ( zero_n2052037380579107095ol_rat
% 0.32/0.59 = ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_def
% 0.32/0.59 thf(fact_5014_of__bool__def,axiom,
% 0.32/0.59 ( zero_n2687167440665602831ol_nat
% 0.32/0.59 = ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_def
% 0.32/0.59 thf(fact_5015_of__bool__def,axiom,
% 0.32/0.59 ( zero_n2684676970156552555ol_int
% 0.32/0.59 = ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_def
% 0.32/0.59 thf(fact_5016_of__bool__def,axiom,
% 0.32/0.59 ( zero_n356916108424825756nteger
% 0.32/0.59 = ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_def
% 0.32/0.59 thf(fact_5017_split__of__bool,axiom,
% 0.32/0.59 ! [P: complex > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
% 0.32/0.59 = ( ( P6
% 0.32/0.59 => ( P @ one_one_complex ) )
% 0.32/0.59 & ( ~ P6
% 0.32/0.59 => ( P @ zero_zero_complex ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool
% 0.32/0.59 thf(fact_5018_split__of__bool,axiom,
% 0.32/0.59 ! [P: real > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
% 0.32/0.59 = ( ( P6
% 0.32/0.59 => ( P @ one_one_real ) )
% 0.32/0.59 & ( ~ P6
% 0.32/0.59 => ( P @ zero_zero_real ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool
% 0.32/0.59 thf(fact_5019_split__of__bool,axiom,
% 0.32/0.59 ! [P: rat > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
% 0.32/0.59 = ( ( P6
% 0.32/0.59 => ( P @ one_one_rat ) )
% 0.32/0.59 & ( ~ P6
% 0.32/0.59 => ( P @ zero_zero_rat ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool
% 0.32/0.59 thf(fact_5020_split__of__bool,axiom,
% 0.32/0.59 ! [P: nat > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
% 0.32/0.59 = ( ( P6
% 0.32/0.59 => ( P @ one_one_nat ) )
% 0.32/0.59 & ( ~ P6
% 0.32/0.59 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool
% 0.32/0.59 thf(fact_5021_split__of__bool,axiom,
% 0.32/0.59 ! [P: int > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
% 0.32/0.59 = ( ( P6
% 0.32/0.59 => ( P @ one_one_int ) )
% 0.32/0.59 & ( ~ P6
% 0.32/0.59 => ( P @ zero_zero_int ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool
% 0.32/0.59 thf(fact_5022_split__of__bool,axiom,
% 0.32/0.59 ! [P: code_integer > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
% 0.32/0.59 = ( ( P6
% 0.32/0.59 => ( P @ one_one_Code_integer ) )
% 0.32/0.59 & ( ~ P6
% 0.32/0.59 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool
% 0.32/0.59 thf(fact_5023_split__of__bool__asm,axiom,
% 0.32/0.59 ! [P: complex > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
% 0.32/0.59 = ( ~ ( ( P6
% 0.32/0.59 & ~ ( P @ one_one_complex ) )
% 0.32/0.59 | ( ~ P6
% 0.32/0.59 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool_asm
% 0.32/0.59 thf(fact_5024_split__of__bool__asm,axiom,
% 0.32/0.59 ! [P: real > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
% 0.32/0.59 = ( ~ ( ( P6
% 0.32/0.59 & ~ ( P @ one_one_real ) )
% 0.32/0.59 | ( ~ P6
% 0.32/0.59 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool_asm
% 0.32/0.59 thf(fact_5025_split__of__bool__asm,axiom,
% 0.32/0.59 ! [P: rat > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
% 0.32/0.59 = ( ~ ( ( P6
% 0.32/0.59 & ~ ( P @ one_one_rat ) )
% 0.32/0.59 | ( ~ P6
% 0.32/0.59 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool_asm
% 0.32/0.59 thf(fact_5026_split__of__bool__asm,axiom,
% 0.32/0.59 ! [P: nat > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
% 0.32/0.59 = ( ~ ( ( P6
% 0.32/0.59 & ~ ( P @ one_one_nat ) )
% 0.32/0.59 | ( ~ P6
% 0.32/0.59 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool_asm
% 0.32/0.59 thf(fact_5027_split__of__bool__asm,axiom,
% 0.32/0.59 ! [P: int > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
% 0.32/0.59 = ( ~ ( ( P6
% 0.32/0.59 & ~ ( P @ one_one_int ) )
% 0.32/0.59 | ( ~ P6
% 0.32/0.59 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool_asm
% 0.32/0.59 thf(fact_5028_split__of__bool__asm,axiom,
% 0.32/0.59 ! [P: code_integer > $o,P6: $o] :
% 0.32/0.59 ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
% 0.32/0.59 = ( ~ ( ( P6
% 0.32/0.59 & ~ ( P @ one_one_Code_integer ) )
% 0.32/0.59 | ( ~ P6
% 0.32/0.59 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % split_of_bool_asm
% 0.32/0.59 thf(fact_5029_ln__bound,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_bound
% 0.32/0.59 thf(fact_5030_ln__gt__zero,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.59 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_gt_zero
% 0.32/0.59 thf(fact_5031_ln__less__zero,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.59 => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_less_zero
% 0.32/0.59 thf(fact_5032_ln__gt__zero__imp__gt__one,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_gt_zero_imp_gt_one
% 0.32/0.59 thf(fact_5033_ln__ge__zero,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_ge_zero
% 0.32/0.59 thf(fact_5034_ln__ge__zero__imp__ge__one,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_ge_zero_imp_ge_one
% 0.32/0.59 thf(fact_5035_ln__add__one__self__le__self,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_add_one_self_le_self
% 0.32/0.59 thf(fact_5036_ln__mult,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.59 => ( ( ln_ln_real @ ( times_times_real @ X @ Y3 ) )
% 0.32/0.59 = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_mult
% 0.32/0.59 thf(fact_5037_ln__eq__minus__one,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ( ln_ln_real @ X )
% 0.32/0.59 = ( minus_minus_real @ X @ one_one_real ) )
% 0.32/0.59 => ( X = one_one_real ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_eq_minus_one
% 0.32/0.59 thf(fact_5038_ln__div,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.59 => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y3 ) )
% 0.32/0.59 = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_div
% 0.32/0.59 thf(fact_5039_subset__decode__imp__le,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.32/0.59 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.59
% 0.32/0.59 % subset_decode_imp_le
% 0.32/0.59 thf(fact_5040_ln__le__minus__one,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_le_minus_one
% 0.32/0.59 thf(fact_5041_ln__diff__le,axiom,
% 0.32/0.59 ! [X: real,Y3: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.59 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y3 ) @ Y3 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_diff_le
% 0.32/0.59 thf(fact_5042_of__bool__odd__eq__mod__2,axiom,
% 0.32/0.59 ! [A: nat] :
% 0.32/0.59 ( ( zero_n2687167440665602831ol_nat
% 0.32/0.59 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.59 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_odd_eq_mod_2
% 0.32/0.59 thf(fact_5043_of__bool__odd__eq__mod__2,axiom,
% 0.32/0.59 ! [A: int] :
% 0.32/0.59 ( ( zero_n2684676970156552555ol_int
% 0.32/0.59 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.59 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_odd_eq_mod_2
% 0.32/0.59 thf(fact_5044_of__bool__odd__eq__mod__2,axiom,
% 0.32/0.59 ! [A: code_integer] :
% 0.32/0.59 ( ( zero_n356916108424825756nteger
% 0.32/0.59 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.59 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % of_bool_odd_eq_mod_2
% 0.32/0.59 thf(fact_5045_signed__take__bit__int__less__exp,axiom,
% 0.32/0.59 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_int_less_exp
% 0.32/0.59 thf(fact_5046_even__signed__take__bit__iff,axiom,
% 0.32/0.59 ! [M: nat,A: code_integer] :
% 0.32/0.59 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 0.32/0.59 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_signed_take_bit_iff
% 0.32/0.59 thf(fact_5047_even__signed__take__bit__iff,axiom,
% 0.32/0.59 ! [M: nat,A: int] :
% 0.32/0.59 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 0.32/0.59 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % even_signed_take_bit_iff
% 0.32/0.59 thf(fact_5048_bits__induct,axiom,
% 0.32/0.59 ! [P: nat > $o,A: nat] :
% 0.32/0.59 ( ! [A5: nat] :
% 0.32/0.59 ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.59 = A5 )
% 0.32/0.59 => ( P @ A5 ) )
% 0.32/0.59 => ( ! [A5: nat,B3: $o] :
% 0.32/0.59 ( ( P @ A5 )
% 0.32/0.59 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.59 = A5 )
% 0.32/0.59 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.32/0.59 => ( P @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bits_induct
% 0.32/0.59 thf(fact_5049_bits__induct,axiom,
% 0.32/0.59 ! [P: int > $o,A: int] :
% 0.32/0.59 ( ! [A5: int] :
% 0.32/0.59 ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.59 = A5 )
% 0.32/0.59 => ( P @ A5 ) )
% 0.32/0.59 => ( ! [A5: int,B3: $o] :
% 0.32/0.59 ( ( P @ A5 )
% 0.32/0.59 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.59 = A5 )
% 0.32/0.59 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.32/0.59 => ( P @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bits_induct
% 0.32/0.59 thf(fact_5050_bits__induct,axiom,
% 0.32/0.59 ! [P: code_integer > $o,A: code_integer] :
% 0.32/0.59 ( ! [A5: code_integer] :
% 0.32/0.59 ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.59 = A5 )
% 0.32/0.59 => ( P @ A5 ) )
% 0.32/0.59 => ( ! [A5: code_integer,B3: $o] :
% 0.32/0.59 ( ( P @ A5 )
% 0.32/0.59 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.59 = A5 )
% 0.32/0.59 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 0.32/0.59 => ( P @ A ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % bits_induct
% 0.32/0.59 thf(fact_5051_signed__take__bit__int__less__self__iff,axiom,
% 0.32/0.59 ! [N: nat,K: int] :
% 0.32/0.59 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.32/0.59 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_int_less_self_iff
% 0.32/0.59 thf(fact_5052_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.32/0.59 ! [K: int,N: nat] :
% 0.32/0.59 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.32/0.59 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_int_greater_eq_self_iff
% 0.32/0.59 thf(fact_5053_add__0__iff,axiom,
% 0.32/0.59 ! [B: complex,A: complex] :
% 0.32/0.59 ( ( B
% 0.32/0.59 = ( plus_plus_complex @ B @ A ) )
% 0.32/0.59 = ( A = zero_zero_complex ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_0_iff
% 0.32/0.59 thf(fact_5054_add__0__iff,axiom,
% 0.32/0.59 ! [B: real,A: real] :
% 0.32/0.59 ( ( B
% 0.32/0.59 = ( plus_plus_real @ B @ A ) )
% 0.32/0.59 = ( A = zero_zero_real ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_0_iff
% 0.32/0.59 thf(fact_5055_add__0__iff,axiom,
% 0.32/0.59 ! [B: rat,A: rat] :
% 0.32/0.59 ( ( B
% 0.32/0.59 = ( plus_plus_rat @ B @ A ) )
% 0.32/0.59 = ( A = zero_zero_rat ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_0_iff
% 0.32/0.59 thf(fact_5056_add__0__iff,axiom,
% 0.32/0.59 ! [B: nat,A: nat] :
% 0.32/0.59 ( ( B
% 0.32/0.59 = ( plus_plus_nat @ B @ A ) )
% 0.32/0.59 = ( A = zero_zero_nat ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_0_iff
% 0.32/0.59 thf(fact_5057_add__0__iff,axiom,
% 0.32/0.59 ! [B: int,A: int] :
% 0.32/0.59 ( ( B
% 0.32/0.59 = ( plus_plus_int @ B @ A ) )
% 0.32/0.59 = ( A = zero_zero_int ) ) ).
% 0.32/0.59
% 0.32/0.59 % add_0_iff
% 0.32/0.59 thf(fact_5058_exp__mod__exp,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % exp_mod_exp
% 0.32/0.59 thf(fact_5059_exp__mod__exp,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % exp_mod_exp
% 0.32/0.59 thf(fact_5060_exp__mod__exp,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % exp_mod_exp
% 0.32/0.59 thf(fact_5061_crossproduct__noteq,axiom,
% 0.32/0.59 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.59 ( ( ( A != B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 0.32/0.59 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % crossproduct_noteq
% 0.32/0.59 thf(fact_5062_crossproduct__noteq,axiom,
% 0.32/0.59 ! [A: rat,B: rat,C: rat,D: rat] :
% 0.32/0.59 ( ( ( A != B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 0.32/0.59 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % crossproduct_noteq
% 0.32/0.59 thf(fact_5063_crossproduct__noteq,axiom,
% 0.32/0.59 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.32/0.59 ( ( ( A != B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 0.32/0.59 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % crossproduct_noteq
% 0.32/0.59 thf(fact_5064_crossproduct__noteq,axiom,
% 0.32/0.59 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.59 ( ( ( A != B )
% 0.32/0.59 & ( C != D ) )
% 0.32/0.59 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 0.32/0.59 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % crossproduct_noteq
% 0.32/0.59 thf(fact_5065_crossproduct__eq,axiom,
% 0.32/0.59 ! [W: real,Y3: real,X: real,Z: real] :
% 0.32/0.59 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X @ Z ) )
% 0.32/0.59 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y3 ) ) )
% 0.32/0.59 = ( ( W = X )
% 0.32/0.59 | ( Y3 = Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % crossproduct_eq
% 0.32/0.59 thf(fact_5066_crossproduct__eq,axiom,
% 0.32/0.59 ! [W: rat,Y3: rat,X: rat,Z: rat] :
% 0.32/0.59 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X @ Z ) )
% 0.32/0.59 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y3 ) ) )
% 0.32/0.59 = ( ( W = X )
% 0.32/0.59 | ( Y3 = Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % crossproduct_eq
% 0.32/0.59 thf(fact_5067_crossproduct__eq,axiom,
% 0.32/0.59 ! [W: nat,Y3: nat,X: nat,Z: nat] :
% 0.32/0.59 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X @ Z ) )
% 0.32/0.59 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y3 ) ) )
% 0.32/0.59 = ( ( W = X )
% 0.32/0.59 | ( Y3 = Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % crossproduct_eq
% 0.32/0.59 thf(fact_5068_crossproduct__eq,axiom,
% 0.32/0.59 ! [W: int,Y3: int,X: int,Z: int] :
% 0.32/0.59 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X @ Z ) )
% 0.32/0.59 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y3 ) ) )
% 0.32/0.59 = ( ( W = X )
% 0.32/0.59 | ( Y3 = Z ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % crossproduct_eq
% 0.32/0.59 thf(fact_5069_signed__take__bit__int__less__eq,axiom,
% 0.32/0.59 ! [N: nat,K: int] :
% 0.32/0.59 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.32/0.59 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_int_less_eq
% 0.32/0.59 thf(fact_5070_ln__one__plus__pos__lower__bound,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.59 => ( 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 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_one_plus_pos_lower_bound
% 0.32/0.59 thf(fact_5071_set__decode__def,axiom,
% 0.32/0.59 ( nat_set_decode
% 0.32/0.59 = ( ^ [X2: nat] :
% 0.32/0.59 ( collect_nat
% 0.32/0.59 @ ^ [N2: nat] :
% 0.32/0.59 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % set_decode_def
% 0.32/0.59 thf(fact_5072_exp__div__exp__eq,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( times_times_nat
% 0.32/0.59 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.59 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.32/0.59 != zero_zero_nat )
% 0.32/0.59 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.32/0.59 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % exp_div_exp_eq
% 0.32/0.59 thf(fact_5073_exp__div__exp__eq,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( times_times_int
% 0.32/0.59 @ ( zero_n2684676970156552555ol_int
% 0.32/0.59 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.32/0.59 != zero_zero_int )
% 0.32/0.59 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.32/0.59 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % exp_div_exp_eq
% 0.32/0.59 thf(fact_5074_exp__div__exp__eq,axiom,
% 0.32/0.59 ! [M: nat,N: nat] :
% 0.32/0.59 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.59 = ( times_3573771949741848930nteger
% 0.32/0.59 @ ( zero_n356916108424825756nteger
% 0.32/0.59 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.32/0.59 != zero_z3403309356797280102nteger )
% 0.32/0.59 & ( ord_less_eq_nat @ N @ M ) ) )
% 0.32/0.59 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % exp_div_exp_eq
% 0.32/0.59 thf(fact_5075_vebt__buildup_Osimps_I3_J,axiom,
% 0.32/0.59 ! [Va: nat] :
% 0.32/0.59 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.32/0.59 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.32/0.59 = ( 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 ) ) ) ) ) ) )
% 0.32/0.59 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.32/0.59 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.32/0.59 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % vebt_buildup.simps(3)
% 0.32/0.59 thf(fact_5076_ln__2__less__1,axiom,
% 0.32/0.59 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.32/0.59
% 0.32/0.59 % ln_2_less_1
% 0.32/0.59 thf(fact_5077_tanh__ln__real,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 0.32/0.59 = ( 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 ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % tanh_ln_real
% 0.32/0.59 thf(fact_5078_Divides_Oadjust__div__eq,axiom,
% 0.32/0.59 ! [Q2: int,R: int] :
% 0.32/0.59 ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.59 = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Divides.adjust_div_eq
% 0.32/0.59 thf(fact_5079_ln__one__minus__pos__lower__bound,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.59 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.59 => ( 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 ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % ln_one_minus_pos_lower_bound
% 0.32/0.59 thf(fact_5080_signed__take__bit__rec,axiom,
% 0.32/0.59 ( bit_ri6519982836138164636nteger
% 0.32/0.59 = ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = 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 @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_rec
% 0.32/0.59 thf(fact_5081_signed__take__bit__rec,axiom,
% 0.32/0.59 ( bit_ri631733984087533419it_int
% 0.32/0.59 = ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = 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 @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % signed_take_bit_rec
% 0.32/0.59 thf(fact_5082_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.32/0.59 ! [X: real] :
% 0.32/0.59 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.59 => ( 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 ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % abs_ln_one_plus_x_minus_x_bound
% 0.32/0.59 thf(fact_5083_vebt__buildup_Opelims,axiom,
% 0.32/0.59 ! [X: nat,Y3: vEBT_VEBT] :
% 0.32/0.59 ( ( ( vEBT_vebt_buildup @ X )
% 0.32/0.59 = Y3 )
% 0.32/0.59 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 0.32/0.59 => ( ( ( X = zero_zero_nat )
% 0.32/0.59 => ( ( Y3
% 0.32/0.59 = ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.59 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.32/0.59 => ( ( ( X
% 0.32/0.59 = ( suc @ zero_zero_nat ) )
% 0.32/0.59 => ( ( Y3
% 0.32/0.59 = ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.59 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.32/0.59 => ~ ! [Va3: nat] :
% 0.32/0.59 ( ( X
% 0.32/0.59 = ( suc @ ( suc @ Va3 ) ) )
% 0.32/0.59 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.32/0.59 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 0.32/0.59 => ( Y3
% 0.32/0.59 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.32/0.59 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % vebt_buildup.pelims
% 0.32/0.59 thf(fact_5084_verit__minus__simplify_I4_J,axiom,
% 0.32/0.59 ! [B: real] :
% 0.32/0.59 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 0.32/0.59 = B ) ).
% 0.32/0.59
% 0.32/0.59 % verit_minus_simplify(4)
% 0.32/0.59 thf(fact_5085_verit__minus__simplify_I4_J,axiom,
% 0.32/0.59 ! [B: int] :
% 0.32/0.59 ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
% 0.32/0.59 = B ) ).
% 0.32/0.59
% 0.32/0.59 % verit_minus_simplify(4)
% 0.32/0.59 thf(fact_5086_verit__minus__simplify_I4_J,axiom,
% 0.32/0.59 ! [B: complex] :
% 0.32/0.59 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.59 = B ) ).
% 0.32/0.59
% 0.32/0.59 % verit_minus_simplify(4)
% 0.32/0.59 thf(fact_5087_verit__minus__simplify_I4_J,axiom,
% 0.32/0.59 ! [B: code_integer] :
% 0.32/0.59 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.59 = B ) ).
% 0.32/0.59
% 0.32/0.59 % verit_minus_simplify(4)
% 0.32/0.59 thf(fact_5088_verit__minus__simplify_I4_J,axiom,
% 0.32/0.59 ! [B: rat] :
% 0.32/0.59 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
% 0.32/0.59 = B ) ).
% 0.32/0.59
% 0.32/0.59 % verit_minus_simplify(4)
% 0.32/0.59 thf(fact_5089_neg__equal__iff__equal,axiom,
% 0.32/0.59 ! [A: real,B: real] :
% 0.32/0.59 ( ( ( uminus_uminus_real @ A )
% 0.32/0.59 = ( uminus_uminus_real @ B ) )
% 0.32/0.59 = ( A = B ) ) ).
% 0.32/0.59
% 0.32/0.59 % neg_equal_iff_equal
% 0.32/0.59 thf(fact_5090_neg__equal__iff__equal,axiom,
% 0.32/0.59 ! [A: int,B: int] :
% 0.32/0.59 ( ( ( uminus_uminus_int @ A )
% 0.32/0.59 = ( uminus_uminus_int @ B ) )
% 0.32/0.59 = ( A = B ) ) ).
% 0.32/0.59
% 0.32/0.59 % neg_equal_iff_equal
% 0.32/0.59 thf(fact_5091_neg__equal__iff__equal,axiom,
% 0.32/0.59 ! [A: complex,B: complex] :
% 0.32/0.59 ( ( ( uminus1482373934393186551omplex @ A )
% 0.32/0.59 = ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.59 = ( A = B ) ) ).
% 0.32/0.59
% 0.32/0.59 % neg_equal_iff_equal
% 0.32/0.59 thf(fact_5092_neg__equal__iff__equal,axiom,
% 0.32/0.59 ! [A: code_integer,B: code_integer] :
% 0.32/0.59 ( ( ( uminus1351360451143612070nteger @ A )
% 0.32/0.59 = ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.59 = ( A = B ) ) ).
% 0.32/0.59
% 0.32/0.59 % neg_equal_iff_equal
% 0.32/0.59 thf(fact_5093_neg__equal__iff__equal,axiom,
% 0.32/0.59 ! [A: rat,B: rat] :
% 0.32/0.59 ( ( ( uminus_uminus_rat @ A )
% 0.32/0.59 = ( uminus_uminus_rat @ B ) )
% 0.32/0.59 = ( A = B ) ) ).
% 0.32/0.59
% 0.32/0.59 % neg_equal_iff_equal
% 0.32/0.59 thf(fact_5094_add_Oinverse__inverse,axiom,
% 0.32/0.59 ! [A: real] :
% 0.32/0.59 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.32/0.59 = A ) ).
% 0.32/0.59
% 0.32/0.59 % add.inverse_inverse
% 0.32/0.59 thf(fact_5095_add_Oinverse__inverse,axiom,
% 0.32/0.59 ! [A: int] :
% 0.32/0.59 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.32/0.59 = A ) ).
% 0.32/0.59
% 0.32/0.59 % add.inverse_inverse
% 0.32/0.59 thf(fact_5096_add_Oinverse__inverse,axiom,
% 0.32/0.59 ! [A: complex] :
% 0.32/0.59 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 0.32/0.59 = A ) ).
% 0.32/0.59
% 0.32/0.59 % add.inverse_inverse
% 0.32/0.59 thf(fact_5097_add_Oinverse__inverse,axiom,
% 0.32/0.59 ! [A: code_integer] :
% 0.32/0.59 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.59 = A ) ).
% 0.32/0.59
% 0.32/0.59 % add.inverse_inverse
% 0.32/0.59 thf(fact_5098_add_Oinverse__inverse,axiom,
% 0.32/0.59 ! [A: rat] :
% 0.32/0.59 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.32/0.59 = A ) ).
% 0.32/0.59
% 0.32/0.59 % add.inverse_inverse
% 0.32/0.59 thf(fact_5099_Compl__anti__mono,axiom,
% 0.32/0.59 ! [A2: set_int,B5: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.59 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.32/0.59
% 0.32/0.59 % Compl_anti_mono
% 0.32/0.59 thf(fact_5100_Compl__subset__Compl__iff,axiom,
% 0.32/0.59 ! [A2: set_int,B5: set_int] :
% 0.32/0.59 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B5 ) )
% 0.32/0.59 = ( ord_less_eq_set_int @ B5 @ A2 ) ) ).
% 0.32/0.59
% 0.32/0.59 % Compl_subset_Compl_iff
% 0.32/0.59 thf(fact_5101_abs__abs,axiom,
% 0.32/0.59 ! [A: real] :
% 0.32/0.59 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.32/0.59 = ( abs_abs_real @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % abs_abs
% 0.32/0.59 thf(fact_5102_abs__abs,axiom,
% 0.32/0.59 ! [A: int] :
% 0.32/0.59 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.32/0.59 = ( abs_abs_int @ A ) ) ).
% 0.32/0.59
% 0.32/0.59 % abs_abs
% 0.32/0.59 thf(fact_5103_abs__abs,axiom,
% 0.32/0.59 ! [A: code_integer] :
% 0.32/0.59 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.32/0.60 = ( abs_abs_Code_integer @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_abs
% 0.32/0.60 thf(fact_5104_abs__abs,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.32/0.60 = ( abs_abs_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_abs
% 0.32/0.60 thf(fact_5105_abs__idempotent,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.32/0.60 = ( abs_abs_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_idempotent
% 0.32/0.60 thf(fact_5106_abs__idempotent,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.32/0.60 = ( abs_abs_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_idempotent
% 0.32/0.60 thf(fact_5107_abs__idempotent,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.32/0.60 = ( abs_abs_Code_integer @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_idempotent
% 0.32/0.60 thf(fact_5108_abs__idempotent,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.32/0.60 = ( abs_abs_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_idempotent
% 0.32/0.60 thf(fact_5109_compl__le__compl__iff,axiom,
% 0.32/0.60 ! [X: set_int,Y3: set_int] :
% 0.32/0.60 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 0.32/0.60 = ( ord_less_eq_set_int @ Y3 @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % compl_le_compl_iff
% 0.32/0.60 thf(fact_5110_neg__le__iff__le,axiom,
% 0.32/0.60 ! [B: real,A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_le_iff_le
% 0.32/0.60 thf(fact_5111_neg__le__iff__le,axiom,
% 0.32/0.60 ! [B: code_integer,A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_le_iff_le
% 0.32/0.60 thf(fact_5112_neg__le__iff__le,axiom,
% 0.32/0.60 ! [B: rat,A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_le_iff_le
% 0.32/0.60 thf(fact_5113_neg__le__iff__le,axiom,
% 0.32/0.60 ! [B: int,A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_le_iff_le
% 0.32/0.60 thf(fact_5114_neg__equal__zero,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ( uminus_uminus_real @ A )
% 0.32/0.60 = A )
% 0.32/0.60 = ( A = zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_zero
% 0.32/0.60 thf(fact_5115_neg__equal__zero,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ( uminus_uminus_int @ A )
% 0.32/0.60 = A )
% 0.32/0.60 = ( A = zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_zero
% 0.32/0.60 thf(fact_5116_neg__equal__zero,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ( uminus1351360451143612070nteger @ A )
% 0.32/0.60 = A )
% 0.32/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_zero
% 0.32/0.60 thf(fact_5117_neg__equal__zero,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ( uminus_uminus_rat @ A )
% 0.32/0.60 = A )
% 0.32/0.60 = ( A = zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_zero
% 0.32/0.60 thf(fact_5118_equal__neg__zero,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( A = zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % equal_neg_zero
% 0.32/0.60 thf(fact_5119_equal__neg__zero,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( A = zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % equal_neg_zero
% 0.32/0.60 thf(fact_5120_equal__neg__zero,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % equal_neg_zero
% 0.32/0.60 thf(fact_5121_equal__neg__zero,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( A = zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % equal_neg_zero
% 0.32/0.60 thf(fact_5122_neg__equal__0__iff__equal,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ( uminus_uminus_real @ A )
% 0.32/0.60 = zero_zero_real )
% 0.32/0.60 = ( A = zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_0_iff_equal
% 0.32/0.60 thf(fact_5123_neg__equal__0__iff__equal,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ( uminus_uminus_int @ A )
% 0.32/0.60 = zero_zero_int )
% 0.32/0.60 = ( A = zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_0_iff_equal
% 0.32/0.60 thf(fact_5124_neg__equal__0__iff__equal,axiom,
% 0.32/0.60 ! [A: complex] :
% 0.32/0.60 ( ( ( uminus1482373934393186551omplex @ A )
% 0.32/0.60 = zero_zero_complex )
% 0.32/0.60 = ( A = zero_zero_complex ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_0_iff_equal
% 0.32/0.60 thf(fact_5125_neg__equal__0__iff__equal,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ( uminus1351360451143612070nteger @ A )
% 0.32/0.60 = zero_z3403309356797280102nteger )
% 0.32/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_0_iff_equal
% 0.32/0.60 thf(fact_5126_neg__equal__0__iff__equal,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ( uminus_uminus_rat @ A )
% 0.32/0.60 = zero_zero_rat )
% 0.32/0.60 = ( A = zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_equal_0_iff_equal
% 0.32/0.60 thf(fact_5127_neg__0__equal__iff__equal,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( zero_zero_real
% 0.32/0.60 = ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( zero_zero_real = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_equal_iff_equal
% 0.32/0.60 thf(fact_5128_neg__0__equal__iff__equal,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( zero_zero_int
% 0.32/0.60 = ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( zero_zero_int = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_equal_iff_equal
% 0.32/0.60 thf(fact_5129_neg__0__equal__iff__equal,axiom,
% 0.32/0.60 ! [A: complex] :
% 0.32/0.60 ( ( zero_zero_complex
% 0.32/0.60 = ( uminus1482373934393186551omplex @ A ) )
% 0.32/0.60 = ( zero_zero_complex = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_equal_iff_equal
% 0.32/0.60 thf(fact_5130_neg__0__equal__iff__equal,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( zero_z3403309356797280102nteger
% 0.32/0.60 = ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_equal_iff_equal
% 0.32/0.60 thf(fact_5131_neg__0__equal__iff__equal,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( zero_zero_rat
% 0.32/0.60 = ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( zero_zero_rat = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_equal_iff_equal
% 0.32/0.60 thf(fact_5132_add_Oinverse__neutral,axiom,
% 0.32/0.60 ( ( uminus_uminus_real @ zero_zero_real )
% 0.32/0.60 = zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % add.inverse_neutral
% 0.32/0.60 thf(fact_5133_add_Oinverse__neutral,axiom,
% 0.32/0.60 ( ( uminus_uminus_int @ zero_zero_int )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % add.inverse_neutral
% 0.32/0.60 thf(fact_5134_add_Oinverse__neutral,axiom,
% 0.32/0.60 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.32/0.60 = zero_zero_complex ) ).
% 0.32/0.60
% 0.32/0.60 % add.inverse_neutral
% 0.32/0.60 thf(fact_5135_add_Oinverse__neutral,axiom,
% 0.32/0.60 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % add.inverse_neutral
% 0.32/0.60 thf(fact_5136_add_Oinverse__neutral,axiom,
% 0.32/0.60 ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.32/0.60 = zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % add.inverse_neutral
% 0.32/0.60 thf(fact_5137_neg__less__iff__less,axiom,
% 0.32/0.60 ! [B: real,A: real] :
% 0.32/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( ord_less_real @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_iff_less
% 0.32/0.60 thf(fact_5138_neg__less__iff__less,axiom,
% 0.32/0.60 ! [B: int,A: int] :
% 0.32/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( ord_less_int @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_iff_less
% 0.32/0.60 thf(fact_5139_neg__less__iff__less,axiom,
% 0.32/0.60 ! [B: code_integer,A: code_integer] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_iff_less
% 0.32/0.60 thf(fact_5140_neg__less__iff__less,axiom,
% 0.32/0.60 ! [B: rat,A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( ord_less_rat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_iff_less
% 0.32/0.60 thf(fact_5141_neg__numeral__eq__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( M = N ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_eq_iff
% 0.32/0.60 thf(fact_5142_neg__numeral__eq__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( M = N ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_eq_iff
% 0.32/0.60 thf(fact_5143_neg__numeral__eq__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.32/0.60 = ( M = N ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_eq_iff
% 0.32/0.60 thf(fact_5144_neg__numeral__eq__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( M = N ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_eq_iff
% 0.32/0.60 thf(fact_5145_neg__numeral__eq__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( M = N ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_eq_iff
% 0.32/0.60 thf(fact_5146_mult__minus__left,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.60 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_left
% 0.32/0.60 thf(fact_5147_mult__minus__left,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.60 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_left
% 0.32/0.60 thf(fact_5148_mult__minus__left,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_left
% 0.32/0.60 thf(fact_5149_mult__minus__left,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_left
% 0.32/0.60 thf(fact_5150_mult__minus__left,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.60 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_left
% 0.32/0.60 thf(fact_5151_minus__mult__minus,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.32/0.60 = ( times_times_real @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_mult_minus
% 0.32/0.60 thf(fact_5152_minus__mult__minus,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.32/0.60 = ( times_times_int @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_mult_minus
% 0.32/0.60 thf(fact_5153_minus__mult__minus,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.60 = ( times_times_complex @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_mult_minus
% 0.32/0.60 thf(fact_5154_minus__mult__minus,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.60 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_mult_minus
% 0.32/0.60 thf(fact_5155_minus__mult__minus,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.32/0.60 = ( times_times_rat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_mult_minus
% 0.32/0.60 thf(fact_5156_mult__minus__right,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_right
% 0.32/0.60 thf(fact_5157_mult__minus__right,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_right
% 0.32/0.60 thf(fact_5158_mult__minus__right,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_right
% 0.32/0.60 thf(fact_5159_mult__minus__right,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_right
% 0.32/0.60 thf(fact_5160_mult__minus__right,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus_right
% 0.32/0.60 thf(fact_5161_add__minus__cancel,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % add_minus_cancel
% 0.32/0.60 thf(fact_5162_add__minus__cancel,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % add_minus_cancel
% 0.32/0.60 thf(fact_5163_add__minus__cancel,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % add_minus_cancel
% 0.32/0.60 thf(fact_5164_add__minus__cancel,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % add_minus_cancel
% 0.32/0.60 thf(fact_5165_add__minus__cancel,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % add_minus_cancel
% 0.32/0.60 thf(fact_5166_minus__add__cancel,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_cancel
% 0.32/0.60 thf(fact_5167_minus__add__cancel,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_cancel
% 0.32/0.60 thf(fact_5168_minus__add__cancel,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_cancel
% 0.32/0.60 thf(fact_5169_minus__add__cancel,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_cancel
% 0.32/0.60 thf(fact_5170_minus__add__cancel,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.60 = B ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_cancel
% 0.32/0.60 thf(fact_5171_minus__add__distrib,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.32/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_distrib
% 0.32/0.60 thf(fact_5172_minus__add__distrib,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.32/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_distrib
% 0.32/0.60 thf(fact_5173_minus__add__distrib,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_distrib
% 0.32/0.60 thf(fact_5174_minus__add__distrib,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_distrib
% 0.32/0.60 thf(fact_5175_minus__add__distrib,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_add_distrib
% 0.32/0.60 thf(fact_5176_minus__diff__eq,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.32/0.60 = ( minus_minus_real @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_diff_eq
% 0.32/0.60 thf(fact_5177_minus__diff__eq,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.32/0.60 = ( minus_minus_int @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_diff_eq
% 0.32/0.60 thf(fact_5178_minus__diff__eq,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 0.32/0.60 = ( minus_minus_complex @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_diff_eq
% 0.32/0.60 thf(fact_5179_minus__diff__eq,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.32/0.60 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_diff_eq
% 0.32/0.60 thf(fact_5180_minus__diff__eq,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.60 = ( minus_minus_rat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_diff_eq
% 0.32/0.60 thf(fact_5181_div__minus__minus,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.32/0.60 = ( divide_divide_int @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % div_minus_minus
% 0.32/0.60 thf(fact_5182_div__minus__minus,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.60 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % div_minus_minus
% 0.32/0.60 thf(fact_5183_abs__0,axiom,
% 0.32/0.60 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0
% 0.32/0.60 thf(fact_5184_abs__0,axiom,
% 0.32/0.60 ( ( abs_abs_complex @ zero_zero_complex )
% 0.32/0.60 = zero_zero_complex ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0
% 0.32/0.60 thf(fact_5185_abs__0,axiom,
% 0.32/0.60 ( ( abs_abs_real @ zero_zero_real )
% 0.32/0.60 = zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0
% 0.32/0.60 thf(fact_5186_abs__0,axiom,
% 0.32/0.60 ( ( abs_abs_rat @ zero_zero_rat )
% 0.32/0.60 = zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0
% 0.32/0.60 thf(fact_5187_abs__0,axiom,
% 0.32/0.60 ( ( abs_abs_int @ zero_zero_int )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0
% 0.32/0.60 thf(fact_5188_abs__zero,axiom,
% 0.32/0.60 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % abs_zero
% 0.32/0.60 thf(fact_5189_abs__zero,axiom,
% 0.32/0.60 ( ( abs_abs_real @ zero_zero_real )
% 0.32/0.60 = zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % abs_zero
% 0.32/0.60 thf(fact_5190_abs__zero,axiom,
% 0.32/0.60 ( ( abs_abs_rat @ zero_zero_rat )
% 0.32/0.60 = zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % abs_zero
% 0.32/0.60 thf(fact_5191_abs__zero,axiom,
% 0.32/0.60 ( ( abs_abs_int @ zero_zero_int )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % abs_zero
% 0.32/0.60 thf(fact_5192_abs__eq__0,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ( abs_abs_Code_integer @ A )
% 0.32/0.60 = zero_z3403309356797280102nteger )
% 0.32/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_0
% 0.32/0.60 thf(fact_5193_abs__eq__0,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ( abs_abs_real @ A )
% 0.32/0.60 = zero_zero_real )
% 0.32/0.60 = ( A = zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_0
% 0.32/0.60 thf(fact_5194_abs__eq__0,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ( abs_abs_rat @ A )
% 0.32/0.60 = zero_zero_rat )
% 0.32/0.60 = ( A = zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_0
% 0.32/0.60 thf(fact_5195_abs__eq__0,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ( abs_abs_int @ A )
% 0.32/0.60 = zero_zero_int )
% 0.32/0.60 = ( A = zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_0
% 0.32/0.60 thf(fact_5196_abs__0__eq,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( zero_z3403309356797280102nteger
% 0.32/0.60 = ( abs_abs_Code_integer @ A ) )
% 0.32/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0_eq
% 0.32/0.60 thf(fact_5197_abs__0__eq,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( zero_zero_real
% 0.32/0.60 = ( abs_abs_real @ A ) )
% 0.32/0.60 = ( A = zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0_eq
% 0.32/0.60 thf(fact_5198_abs__0__eq,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( zero_zero_rat
% 0.32/0.60 = ( abs_abs_rat @ A ) )
% 0.32/0.60 = ( A = zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0_eq
% 0.32/0.60 thf(fact_5199_abs__0__eq,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( zero_zero_int
% 0.32/0.60 = ( abs_abs_int @ A ) )
% 0.32/0.60 = ( A = zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_0_eq
% 0.32/0.60 thf(fact_5200_minus__dvd__iff,axiom,
% 0.32/0.60 ! [X: real,Y3: real] :
% 0.32/0.60 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y3 )
% 0.32/0.60 = ( dvd_dvd_real @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_dvd_iff
% 0.32/0.60 thf(fact_5201_minus__dvd__iff,axiom,
% 0.32/0.60 ! [X: int,Y3: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y3 )
% 0.32/0.60 = ( dvd_dvd_int @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_dvd_iff
% 0.32/0.60 thf(fact_5202_minus__dvd__iff,axiom,
% 0.32/0.60 ! [X: complex,Y3: complex] :
% 0.32/0.60 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y3 )
% 0.32/0.60 = ( dvd_dvd_complex @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_dvd_iff
% 0.32/0.60 thf(fact_5203_minus__dvd__iff,axiom,
% 0.32/0.60 ! [X: code_integer,Y3: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y3 )
% 0.32/0.60 = ( dvd_dvd_Code_integer @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_dvd_iff
% 0.32/0.60 thf(fact_5204_minus__dvd__iff,axiom,
% 0.32/0.60 ! [X: rat,Y3: rat] :
% 0.32/0.60 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y3 )
% 0.32/0.60 = ( dvd_dvd_rat @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_dvd_iff
% 0.32/0.60 thf(fact_5205_dvd__minus__iff,axiom,
% 0.32/0.60 ! [X: real,Y3: real] :
% 0.32/0.60 ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y3 ) )
% 0.32/0.60 = ( dvd_dvd_real @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_minus_iff
% 0.32/0.60 thf(fact_5206_dvd__minus__iff,axiom,
% 0.32/0.60 ! [X: int,Y3: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y3 ) )
% 0.32/0.60 = ( dvd_dvd_int @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_minus_iff
% 0.32/0.60 thf(fact_5207_dvd__minus__iff,axiom,
% 0.32/0.60 ! [X: complex,Y3: complex] :
% 0.32/0.60 ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y3 ) )
% 0.32/0.60 = ( dvd_dvd_complex @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_minus_iff
% 0.32/0.60 thf(fact_5208_dvd__minus__iff,axiom,
% 0.32/0.60 ! [X: code_integer,Y3: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y3 ) )
% 0.32/0.60 = ( dvd_dvd_Code_integer @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_minus_iff
% 0.32/0.60 thf(fact_5209_dvd__minus__iff,axiom,
% 0.32/0.60 ! [X: rat,Y3: rat] :
% 0.32/0.60 ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y3 ) )
% 0.32/0.60 = ( dvd_dvd_rat @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_minus_iff
% 0.32/0.60 thf(fact_5210_abs__numeral,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 0.32/0.60 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_numeral
% 0.32/0.60 thf(fact_5211_abs__numeral,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 0.32/0.60 = ( numeral_numeral_real @ N ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_numeral
% 0.32/0.60 thf(fact_5212_abs__numeral,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 0.32/0.60 = ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_numeral
% 0.32/0.60 thf(fact_5213_abs__numeral,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 0.32/0.60 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_numeral
% 0.32/0.60 thf(fact_5214_abs__mult__self__eq,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.32/0.60 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_mult_self_eq
% 0.32/0.60 thf(fact_5215_abs__mult__self__eq,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.32/0.60 = ( times_times_real @ A @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_mult_self_eq
% 0.32/0.60 thf(fact_5216_abs__mult__self__eq,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.32/0.60 = ( times_times_rat @ A @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_mult_self_eq
% 0.32/0.60 thf(fact_5217_abs__mult__self__eq,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 0.32/0.60 = ( times_times_int @ A @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_mult_self_eq
% 0.32/0.60 thf(fact_5218_abs__1,axiom,
% 0.32/0.60 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.32/0.60 = one_one_Code_integer ) ).
% 0.32/0.60
% 0.32/0.60 % abs_1
% 0.32/0.60 thf(fact_5219_abs__1,axiom,
% 0.32/0.60 ( ( abs_abs_complex @ one_one_complex )
% 0.32/0.60 = one_one_complex ) ).
% 0.32/0.60
% 0.32/0.60 % abs_1
% 0.32/0.60 thf(fact_5220_abs__1,axiom,
% 0.32/0.60 ( ( abs_abs_real @ one_one_real )
% 0.32/0.60 = one_one_real ) ).
% 0.32/0.60
% 0.32/0.60 % abs_1
% 0.32/0.60 thf(fact_5221_abs__1,axiom,
% 0.32/0.60 ( ( abs_abs_rat @ one_one_rat )
% 0.32/0.60 = one_one_rat ) ).
% 0.32/0.60
% 0.32/0.60 % abs_1
% 0.32/0.60 thf(fact_5222_abs__1,axiom,
% 0.32/0.60 ( ( abs_abs_int @ one_one_int )
% 0.32/0.60 = one_one_int ) ).
% 0.32/0.60
% 0.32/0.60 % abs_1
% 0.32/0.60 thf(fact_5223_abs__add__abs,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 0.32/0.60 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_add_abs
% 0.32/0.60 thf(fact_5224_abs__add__abs,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.32/0.60 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_add_abs
% 0.32/0.60 thf(fact_5225_abs__add__abs,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 0.32/0.60 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_add_abs
% 0.32/0.60 thf(fact_5226_abs__add__abs,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 0.32/0.60 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_add_abs
% 0.32/0.60 thf(fact_5227_abs__divide,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.60 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_divide
% 0.32/0.60 thf(fact_5228_abs__divide,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.60 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_divide
% 0.32/0.60 thf(fact_5229_abs__divide,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.60 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_divide
% 0.32/0.60 thf(fact_5230_abs__minus,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( abs_abs_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus
% 0.32/0.60 thf(fact_5231_abs__minus,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( abs_abs_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus
% 0.32/0.60 thf(fact_5232_abs__minus,axiom,
% 0.32/0.60 ! [A: complex] :
% 0.32/0.60 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 0.32/0.60 = ( abs_abs_complex @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus
% 0.32/0.60 thf(fact_5233_abs__minus,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( abs_abs_Code_integer @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus
% 0.32/0.60 thf(fact_5234_abs__minus,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( abs_abs_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus
% 0.32/0.60 thf(fact_5235_abs__minus__cancel,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( abs_abs_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus_cancel
% 0.32/0.60 thf(fact_5236_abs__minus__cancel,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( abs_abs_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus_cancel
% 0.32/0.60 thf(fact_5237_abs__minus__cancel,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( abs_abs_Code_integer @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus_cancel
% 0.32/0.60 thf(fact_5238_abs__minus__cancel,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( abs_abs_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus_cancel
% 0.32/0.60 thf(fact_5239_mod__minus__minus,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod_minus_minus
% 0.32/0.60 thf(fact_5240_mod__minus__minus,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mod_minus_minus
% 0.32/0.60 thf(fact_5241_dvd__abs__iff,axiom,
% 0.32/0.60 ! [M: real,K: real] :
% 0.32/0.60 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 0.32/0.60 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_abs_iff
% 0.32/0.60 thf(fact_5242_dvd__abs__iff,axiom,
% 0.32/0.60 ! [M: int,K: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 0.32/0.60 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_abs_iff
% 0.32/0.60 thf(fact_5243_dvd__abs__iff,axiom,
% 0.32/0.60 ! [M: code_integer,K: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 0.32/0.60 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_abs_iff
% 0.32/0.60 thf(fact_5244_dvd__abs__iff,axiom,
% 0.32/0.60 ! [M: rat,K: rat] :
% 0.32/0.60 ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 0.32/0.60 = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.32/0.60
% 0.32/0.60 % dvd_abs_iff
% 0.32/0.60 thf(fact_5245_abs__dvd__iff,axiom,
% 0.32/0.60 ! [M: real,K: real] :
% 0.32/0.60 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 0.32/0.60 = ( dvd_dvd_real @ M @ K ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_dvd_iff
% 0.32/0.60 thf(fact_5246_abs__dvd__iff,axiom,
% 0.32/0.60 ! [M: int,K: int] :
% 0.32/0.60 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 0.32/0.60 = ( dvd_dvd_int @ M @ K ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_dvd_iff
% 0.32/0.60 thf(fact_5247_abs__dvd__iff,axiom,
% 0.32/0.60 ! [M: code_integer,K: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 0.32/0.60 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_dvd_iff
% 0.32/0.60 thf(fact_5248_abs__dvd__iff,axiom,
% 0.32/0.60 ! [M: rat,K: rat] :
% 0.32/0.60 ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 0.32/0.60 = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_dvd_iff
% 0.32/0.60 thf(fact_5249_abs__bool__eq,axiom,
% 0.32/0.60 ! [P: $o] :
% 0.32/0.60 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.32/0.60 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_bool_eq
% 0.32/0.60 thf(fact_5250_abs__bool__eq,axiom,
% 0.32/0.60 ! [P: $o] :
% 0.32/0.60 ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.32/0.60 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_bool_eq
% 0.32/0.60 thf(fact_5251_abs__bool__eq,axiom,
% 0.32/0.60 ! [P: $o] :
% 0.32/0.60 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.32/0.60 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_bool_eq
% 0.32/0.60 thf(fact_5252_abs__bool__eq,axiom,
% 0.32/0.60 ! [P: $o] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 0.32/0.60 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_bool_eq
% 0.32/0.60 thf(fact_5253_tanh__real__less__iff,axiom,
% 0.32/0.60 ! [X: real,Y3: real] :
% 0.32/0.60 ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y3 ) )
% 0.32/0.60 = ( ord_less_real @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % tanh_real_less_iff
% 0.32/0.60 thf(fact_5254_tanh__real__le__iff,axiom,
% 0.32/0.60 ! [X: real,Y3: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y3 ) )
% 0.32/0.60 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % tanh_real_le_iff
% 0.32/0.60 thf(fact_5255_neg__less__eq__nonneg,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.32/0.60 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_eq_nonneg
% 0.32/0.60 thf(fact_5256_neg__less__eq__nonneg,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.32/0.60 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_eq_nonneg
% 0.32/0.60 thf(fact_5257_neg__less__eq__nonneg,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.32/0.60 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_eq_nonneg
% 0.32/0.60 thf(fact_5258_neg__less__eq__nonneg,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.32/0.60 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_eq_nonneg
% 0.32/0.60 thf(fact_5259_less__eq__neg__nonpos,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_eq_neg_nonpos
% 0.32/0.60 thf(fact_5260_less__eq__neg__nonpos,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_eq_neg_nonpos
% 0.32/0.60 thf(fact_5261_less__eq__neg__nonpos,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_eq_neg_nonpos
% 0.32/0.60 thf(fact_5262_less__eq__neg__nonpos,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_eq_neg_nonpos
% 0.32/0.60 thf(fact_5263_neg__le__0__iff__le,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.32/0.60 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_le_0_iff_le
% 0.32/0.60 thf(fact_5264_neg__le__0__iff__le,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.32/0.60 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_le_0_iff_le
% 0.32/0.60 thf(fact_5265_neg__le__0__iff__le,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.32/0.60 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_le_0_iff_le
% 0.32/0.60 thf(fact_5266_neg__le__0__iff__le,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.32/0.60 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_le_0_iff_le
% 0.32/0.60 thf(fact_5267_neg__0__le__iff__le,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_le_iff_le
% 0.32/0.60 thf(fact_5268_neg__0__le__iff__le,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_le_iff_le
% 0.32/0.60 thf(fact_5269_neg__0__le__iff__le,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_le_iff_le
% 0.32/0.60 thf(fact_5270_neg__0__le__iff__le,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_le_iff_le
% 0.32/0.60 thf(fact_5271_less__neg__neg,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_neg_neg
% 0.32/0.60 thf(fact_5272_less__neg__neg,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_neg_neg
% 0.32/0.60 thf(fact_5273_less__neg__neg,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_neg_neg
% 0.32/0.60 thf(fact_5274_less__neg__neg,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_neg_neg
% 0.32/0.60 thf(fact_5275_neg__less__pos,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.32/0.60 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_pos
% 0.32/0.60 thf(fact_5276_neg__less__pos,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.32/0.60 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_pos
% 0.32/0.60 thf(fact_5277_neg__less__pos,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.32/0.60 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_pos
% 0.32/0.60 thf(fact_5278_neg__less__pos,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.32/0.60 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_pos
% 0.32/0.60 thf(fact_5279_neg__0__less__iff__less,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_less_iff_less
% 0.32/0.60 thf(fact_5280_neg__0__less__iff__less,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_less_iff_less
% 0.32/0.60 thf(fact_5281_neg__0__less__iff__less,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_less_iff_less
% 0.32/0.60 thf(fact_5282_neg__0__less__iff__less,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_0_less_iff_less
% 0.32/0.60 thf(fact_5283_neg__less__0__iff__less,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.32/0.60 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_0_iff_less
% 0.32/0.60 thf(fact_5284_neg__less__0__iff__less,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.32/0.60 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_0_iff_less
% 0.32/0.60 thf(fact_5285_neg__less__0__iff__less,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.32/0.60 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_0_iff_less
% 0.32/0.60 thf(fact_5286_neg__less__0__iff__less,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.32/0.60 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_less_0_iff_less
% 0.32/0.60 thf(fact_5287_add_Oright__inverse,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % add.right_inverse
% 0.32/0.60 thf(fact_5288_add_Oright__inverse,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % add.right_inverse
% 0.32/0.60 thf(fact_5289_add_Oright__inverse,axiom,
% 0.32/0.60 ! [A: complex] :
% 0.32/0.60 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.32/0.60 = zero_zero_complex ) ).
% 0.32/0.60
% 0.32/0.60 % add.right_inverse
% 0.32/0.60 thf(fact_5290_add_Oright__inverse,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % add.right_inverse
% 0.32/0.60 thf(fact_5291_add_Oright__inverse,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.32/0.60 = zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % add.right_inverse
% 0.32/0.60 thf(fact_5292_ab__left__minus,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.32/0.60 = zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % ab_left_minus
% 0.32/0.60 thf(fact_5293_ab__left__minus,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % ab_left_minus
% 0.32/0.60 thf(fact_5294_ab__left__minus,axiom,
% 0.32/0.60 ! [A: complex] :
% 0.32/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.32/0.60 = zero_zero_complex ) ).
% 0.32/0.60
% 0.32/0.60 % ab_left_minus
% 0.32/0.60 thf(fact_5295_ab__left__minus,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % ab_left_minus
% 0.32/0.60 thf(fact_5296_ab__left__minus,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.32/0.60 = zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % ab_left_minus
% 0.32/0.60 thf(fact_5297_diff__0,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( minus_minus_real @ zero_zero_real @ A )
% 0.32/0.60 = ( uminus_uminus_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_0
% 0.32/0.60 thf(fact_5298_diff__0,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( minus_minus_int @ zero_zero_int @ A )
% 0.32/0.60 = ( uminus_uminus_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_0
% 0.32/0.60 thf(fact_5299_diff__0,axiom,
% 0.32/0.60 ! [A: complex] :
% 0.32/0.60 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_0
% 0.32/0.60 thf(fact_5300_diff__0,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_0
% 0.32/0.60 thf(fact_5301_diff__0,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.32/0.60 = ( uminus_uminus_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_0
% 0.32/0.60 thf(fact_5302_verit__minus__simplify_I3_J,axiom,
% 0.32/0.60 ! [B: real] :
% 0.32/0.60 ( ( minus_minus_real @ zero_zero_real @ B )
% 0.32/0.60 = ( uminus_uminus_real @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_minus_simplify(3)
% 0.32/0.60 thf(fact_5303_verit__minus__simplify_I3_J,axiom,
% 0.32/0.60 ! [B: int] :
% 0.32/0.60 ( ( minus_minus_int @ zero_zero_int @ B )
% 0.32/0.60 = ( uminus_uminus_int @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_minus_simplify(3)
% 0.32/0.60 thf(fact_5304_verit__minus__simplify_I3_J,axiom,
% 0.32/0.60 ! [B: complex] :
% 0.32/0.60 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_minus_simplify(3)
% 0.32/0.60 thf(fact_5305_verit__minus__simplify_I3_J,axiom,
% 0.32/0.60 ! [B: code_integer] :
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_minus_simplify(3)
% 0.32/0.60 thf(fact_5306_verit__minus__simplify_I3_J,axiom,
% 0.32/0.60 ! [B: rat] :
% 0.32/0.60 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 0.32/0.60 = ( uminus_uminus_rat @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_minus_simplify(3)
% 0.32/0.60 thf(fact_5307_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5308_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5309_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5310_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5311_add__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5312_mult__minus1__right,axiom,
% 0.32/0.60 ! [Z: real] :
% 0.32/0.60 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = ( uminus_uminus_real @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1_right
% 0.32/0.60 thf(fact_5313_mult__minus1__right,axiom,
% 0.32/0.60 ! [Z: int] :
% 0.32/0.60 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = ( uminus_uminus_int @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1_right
% 0.32/0.60 thf(fact_5314_mult__minus1__right,axiom,
% 0.32/0.60 ! [Z: complex] :
% 0.32/0.60 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1_right
% 0.32/0.60 thf(fact_5315_mult__minus1__right,axiom,
% 0.32/0.60 ! [Z: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1_right
% 0.32/0.60 thf(fact_5316_mult__minus1__right,axiom,
% 0.32/0.60 ! [Z: rat] :
% 0.32/0.60 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = ( uminus_uminus_rat @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1_right
% 0.32/0.60 thf(fact_5317_mult__minus1,axiom,
% 0.32/0.60 ! [Z: real] :
% 0.32/0.60 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 0.32/0.60 = ( uminus_uminus_real @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1
% 0.32/0.60 thf(fact_5318_mult__minus1,axiom,
% 0.32/0.60 ! [Z: int] :
% 0.32/0.60 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 0.32/0.60 = ( uminus_uminus_int @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1
% 0.32/0.60 thf(fact_5319_mult__minus1,axiom,
% 0.32/0.60 ! [Z: complex] :
% 0.32/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1
% 0.32/0.60 thf(fact_5320_mult__minus1,axiom,
% 0.32/0.60 ! [Z: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1
% 0.32/0.60 thf(fact_5321_mult__minus1,axiom,
% 0.32/0.60 ! [Z: rat] :
% 0.32/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 0.32/0.60 = ( uminus_uminus_rat @ Z ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_minus1
% 0.32/0.60 thf(fact_5322_abs__le__zero__iff,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.32/0.60 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_zero_iff
% 0.32/0.60 thf(fact_5323_abs__le__zero__iff,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.32/0.60 = ( A = zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_zero_iff
% 0.32/0.60 thf(fact_5324_abs__le__zero__iff,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.32/0.60 = ( A = zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_zero_iff
% 0.32/0.60 thf(fact_5325_abs__le__zero__iff,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.32/0.60 = ( A = zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_zero_iff
% 0.32/0.60 thf(fact_5326_abs__le__self__iff,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.32/0.60 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_self_iff
% 0.32/0.60 thf(fact_5327_abs__le__self__iff,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.32/0.60 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_self_iff
% 0.32/0.60 thf(fact_5328_abs__le__self__iff,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.32/0.60 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_self_iff
% 0.32/0.60 thf(fact_5329_abs__le__self__iff,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.32/0.60 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_self_iff
% 0.32/0.60 thf(fact_5330_abs__of__nonneg,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.60 => ( ( abs_abs_Code_integer @ A )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_nonneg
% 0.32/0.60 thf(fact_5331_abs__of__nonneg,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.60 => ( ( abs_abs_real @ A )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_nonneg
% 0.32/0.60 thf(fact_5332_abs__of__nonneg,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.60 => ( ( abs_abs_rat @ A )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_nonneg
% 0.32/0.60 thf(fact_5333_abs__of__nonneg,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.60 => ( ( abs_abs_int @ A )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_nonneg
% 0.32/0.60 thf(fact_5334_zero__less__abs__iff,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.32/0.60 = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_less_abs_iff
% 0.32/0.60 thf(fact_5335_zero__less__abs__iff,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.32/0.60 = ( A != zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_less_abs_iff
% 0.32/0.60 thf(fact_5336_zero__less__abs__iff,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.32/0.60 = ( A != zero_zero_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_less_abs_iff
% 0.32/0.60 thf(fact_5337_zero__less__abs__iff,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.32/0.60 = ( A != zero_zero_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_less_abs_iff
% 0.32/0.60 thf(fact_5338_diff__minus__eq__add,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.60 = ( plus_plus_real @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_minus_eq_add
% 0.32/0.60 thf(fact_5339_diff__minus__eq__add,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.60 = ( plus_plus_int @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_minus_eq_add
% 0.32/0.60 thf(fact_5340_diff__minus__eq__add,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.60 = ( plus_plus_complex @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_minus_eq_add
% 0.32/0.60 thf(fact_5341_diff__minus__eq__add,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.60 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_minus_eq_add
% 0.32/0.60 thf(fact_5342_diff__minus__eq__add,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.60 = ( plus_plus_rat @ A @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_minus_eq_add
% 0.32/0.60 thf(fact_5343_uminus__add__conv__diff,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.60 = ( minus_minus_real @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % uminus_add_conv_diff
% 0.32/0.60 thf(fact_5344_uminus__add__conv__diff,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.60 = ( minus_minus_int @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % uminus_add_conv_diff
% 0.32/0.60 thf(fact_5345_uminus__add__conv__diff,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.32/0.60 = ( minus_minus_complex @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % uminus_add_conv_diff
% 0.32/0.60 thf(fact_5346_uminus__add__conv__diff,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.60 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % uminus_add_conv_diff
% 0.32/0.60 thf(fact_5347_uminus__add__conv__diff,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.60 = ( minus_minus_rat @ B @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % uminus_add_conv_diff
% 0.32/0.60 thf(fact_5348_div__minus1__right,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = ( uminus_uminus_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % div_minus1_right
% 0.32/0.60 thf(fact_5349_div__minus1__right,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % div_minus1_right
% 0.32/0.60 thf(fact_5350_divide__minus1,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = ( uminus_uminus_real @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_minus1
% 0.32/0.60 thf(fact_5351_divide__minus1,axiom,
% 0.32/0.60 ! [X: complex] :
% 0.32/0.60 ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_minus1
% 0.32/0.60 thf(fact_5352_divide__minus1,axiom,
% 0.32/0.60 ! [X: rat] :
% 0.32/0.60 ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = ( uminus_uminus_rat @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_minus1
% 0.32/0.60 thf(fact_5353_abs__neg__numeral,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_real @ N ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_neg_numeral
% 0.32/0.60 thf(fact_5354_abs__neg__numeral,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_neg_numeral
% 0.32/0.60 thf(fact_5355_abs__neg__numeral,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( numera6620942414471956472nteger @ N ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_neg_numeral
% 0.32/0.60 thf(fact_5356_abs__neg__numeral,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_neg_numeral
% 0.32/0.60 thf(fact_5357_abs__neg__one,axiom,
% 0.32/0.60 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = one_one_real ) ).
% 0.32/0.60
% 0.32/0.60 % abs_neg_one
% 0.32/0.60 thf(fact_5358_abs__neg__one,axiom,
% 0.32/0.60 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = one_one_int ) ).
% 0.32/0.60
% 0.32/0.60 % abs_neg_one
% 0.32/0.60 thf(fact_5359_abs__neg__one,axiom,
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = one_one_Code_integer ) ).
% 0.32/0.60
% 0.32/0.60 % abs_neg_one
% 0.32/0.60 thf(fact_5360_abs__neg__one,axiom,
% 0.32/0.60 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = one_one_rat ) ).
% 0.32/0.60
% 0.32/0.60 % abs_neg_one
% 0.32/0.60 thf(fact_5361_minus__mod__self1,axiom,
% 0.32/0.60 ! [B: int,A: int] :
% 0.32/0.60 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 0.32/0.60 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_mod_self1
% 0.32/0.60 thf(fact_5362_minus__mod__self1,axiom,
% 0.32/0.60 ! [B: code_integer,A: code_integer] :
% 0.32/0.60 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 0.32/0.60 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_mod_self1
% 0.32/0.60 thf(fact_5363_abs__power__minus,axiom,
% 0.32/0.60 ! [A: real,N: nat] :
% 0.32/0.60 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.32/0.60 = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_power_minus
% 0.32/0.60 thf(fact_5364_abs__power__minus,axiom,
% 0.32/0.60 ! [A: int,N: nat] :
% 0.32/0.60 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.32/0.60 = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_power_minus
% 0.32/0.60 thf(fact_5365_abs__power__minus,axiom,
% 0.32/0.60 ! [A: code_integer,N: nat] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.32/0.60 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_power_minus
% 0.32/0.60 thf(fact_5366_abs__power__minus,axiom,
% 0.32/0.60 ! [A: rat,N: nat] :
% 0.32/0.60 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.32/0.60 = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_power_minus
% 0.32/0.60 thf(fact_5367_signed__take__bit__of__minus__1,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.60
% 0.32/0.60 % signed_take_bit_of_minus_1
% 0.32/0.60 thf(fact_5368_signed__take__bit__of__minus__1,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % signed_take_bit_of_minus_1
% 0.32/0.60 thf(fact_5369_real__add__minus__iff,axiom,
% 0.32/0.60 ! [X: real,A: real] :
% 0.32/0.60 ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 0.32/0.60 = zero_zero_real )
% 0.32/0.60 = ( X = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % real_add_minus_iff
% 0.32/0.60 thf(fact_5370_tanh__real__pos__iff,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.32/0.60 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % tanh_real_pos_iff
% 0.32/0.60 thf(fact_5371_tanh__real__neg__iff,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.32/0.60 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % tanh_real_neg_iff
% 0.32/0.60 thf(fact_5372_tanh__real__nonneg__iff,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
% 0.32/0.60 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.32/0.60
% 0.32/0.60 % tanh_real_nonneg_iff
% 0.32/0.60 thf(fact_5373_tanh__real__nonpos__iff,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
% 0.32/0.60 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % tanh_real_nonpos_iff
% 0.32/0.60 thf(fact_5374_dbl__simps_I1_J,axiom,
% 0.32/0.60 ! [K: num] :
% 0.32/0.60 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(1)
% 0.32/0.60 thf(fact_5375_dbl__simps_I1_J,axiom,
% 0.32/0.60 ! [K: num] :
% 0.32/0.60 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(1)
% 0.32/0.60 thf(fact_5376_dbl__simps_I1_J,axiom,
% 0.32/0.60 ! [K: num] :
% 0.32/0.60 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(1)
% 0.32/0.60 thf(fact_5377_dbl__simps_I1_J,axiom,
% 0.32/0.60 ! [K: num] :
% 0.32/0.60 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(1)
% 0.32/0.60 thf(fact_5378_dbl__simps_I1_J,axiom,
% 0.32/0.60 ! [K: num] :
% 0.32/0.60 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(1)
% 0.32/0.60 thf(fact_5379_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.60 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(7)
% 0.32/0.60 thf(fact_5380_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.60 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(7)
% 0.32/0.60 thf(fact_5381_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.60 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.60 = zero_zero_complex ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(7)
% 0.32/0.60 thf(fact_5382_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(7)
% 0.32/0.60 thf(fact_5383_add__neg__numeral__special_I7_J,axiom,
% 0.32/0.60 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(7)
% 0.32/0.60 thf(fact_5384_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.32/0.60 = zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(8)
% 0.32/0.60 thf(fact_5385_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(8)
% 0.32/0.60 thf(fact_5386_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.32/0.60 = zero_zero_complex ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(8)
% 0.32/0.60 thf(fact_5387_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(8)
% 0.32/0.60 thf(fact_5388_add__neg__numeral__special_I8_J,axiom,
% 0.32/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.32/0.60 = zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(8)
% 0.32/0.60 thf(fact_5389_diff__numeral__special_I12_J,axiom,
% 0.32/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(12)
% 0.32/0.60 thf(fact_5390_diff__numeral__special_I12_J,axiom,
% 0.32/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(12)
% 0.32/0.60 thf(fact_5391_diff__numeral__special_I12_J,axiom,
% 0.32/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.60 = zero_zero_complex ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(12)
% 0.32/0.60 thf(fact_5392_diff__numeral__special_I12_J,axiom,
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(12)
% 0.32/0.60 thf(fact_5393_diff__numeral__special_I12_J,axiom,
% 0.32/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(12)
% 0.32/0.60 thf(fact_5394_numeral__eq__neg__one__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 0.32/0.60 = ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % numeral_eq_neg_one_iff
% 0.32/0.60 thf(fact_5395_numeral__eq__neg__one__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 0.32/0.60 = ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % numeral_eq_neg_one_iff
% 0.32/0.60 thf(fact_5396_numeral__eq__neg__one__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % numeral_eq_neg_one_iff
% 0.32/0.60 thf(fact_5397_numeral__eq__neg__one__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % numeral_eq_neg_one_iff
% 0.32/0.60 thf(fact_5398_numeral__eq__neg__one__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 0.32/0.60 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % numeral_eq_neg_one_iff
% 0.32/0.60 thf(fact_5399_neg__one__eq__numeral__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_real @ one_one_real )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_eq_numeral_iff
% 0.32/0.60 thf(fact_5400_neg__one__eq__numeral__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_int @ one_one_int )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_eq_numeral_iff
% 0.32/0.60 thf(fact_5401_neg__one__eq__numeral__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_eq_numeral_iff
% 0.32/0.60 thf(fact_5402_neg__one__eq__numeral__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_eq_numeral_iff
% 0.32/0.60 thf(fact_5403_neg__one__eq__numeral__iff,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( ( uminus_uminus_rat @ one_one_rat )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( N = one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_eq_numeral_iff
% 0.32/0.60 thf(fact_5404_divide__le__0__abs__iff,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.32/0.60 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.60 | ( B = zero_zero_real ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_le_0_abs_iff
% 0.32/0.60 thf(fact_5405_divide__le__0__abs__iff,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 0.32/0.60 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.60 | ( B = zero_zero_rat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_le_0_abs_iff
% 0.32/0.60 thf(fact_5406_zero__le__divide__abs__iff,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.32/0.60 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.60 | ( B = zero_zero_real ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_divide_abs_iff
% 0.32/0.60 thf(fact_5407_zero__le__divide__abs__iff,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 0.32/0.60 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.60 | ( B = zero_zero_rat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_le_divide_abs_iff
% 0.32/0.60 thf(fact_5408_abs__of__nonpos,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.60 => ( ( abs_abs_real @ A )
% 0.32/0.60 = ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_nonpos
% 0.32/0.60 thf(fact_5409_abs__of__nonpos,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.32/0.60 => ( ( abs_abs_Code_integer @ A )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_nonpos
% 0.32/0.60 thf(fact_5410_abs__of__nonpos,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.32/0.60 => ( ( abs_abs_rat @ A )
% 0.32/0.60 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_nonpos
% 0.32/0.60 thf(fact_5411_abs__of__nonpos,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.32/0.60 => ( ( abs_abs_int @ A )
% 0.32/0.60 = ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_nonpos
% 0.32/0.60 thf(fact_5412_left__minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat,A: real] :
% 0.32/0.60 ( ( 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 ) )
% 0.32/0.60 = A ) ).
% 0.32/0.60
% 0.32/0.60 % left_minus_one_mult_self
% 0.32/0.60 thf(fact_5413_left__minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat,A: int] :
% 0.32/0.60 ( ( 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 ) )
% 0.32/0.60 = A ) ).
% 0.32/0.60
% 0.32/0.60 % left_minus_one_mult_self
% 0.32/0.60 thf(fact_5414_left__minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat,A: complex] :
% 0.32/0.60 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 0.32/0.60 = A ) ).
% 0.32/0.60
% 0.32/0.60 % left_minus_one_mult_self
% 0.32/0.60 thf(fact_5415_left__minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat,A: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 0.32/0.60 = A ) ).
% 0.32/0.60
% 0.32/0.60 % left_minus_one_mult_self
% 0.32/0.60 thf(fact_5416_left__minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat,A: rat] :
% 0.32/0.60 ( ( 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 ) )
% 0.32/0.60 = A ) ).
% 0.32/0.60
% 0.32/0.60 % left_minus_one_mult_self
% 0.32/0.60 thf(fact_5417_minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 0.32/0.60 = one_one_real ) ).
% 0.32/0.60
% 0.32/0.60 % minus_one_mult_self
% 0.32/0.60 thf(fact_5418_minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 0.32/0.60 = one_one_int ) ).
% 0.32/0.60
% 0.32/0.60 % minus_one_mult_self
% 0.32/0.60 thf(fact_5419_minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 0.32/0.60 = one_one_complex ) ).
% 0.32/0.60
% 0.32/0.60 % minus_one_mult_self
% 0.32/0.60 thf(fact_5420_minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 0.32/0.60 = one_one_Code_integer ) ).
% 0.32/0.60
% 0.32/0.60 % minus_one_mult_self
% 0.32/0.60 thf(fact_5421_minus__one__mult__self,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 0.32/0.60 = one_one_rat ) ).
% 0.32/0.60
% 0.32/0.60 % minus_one_mult_self
% 0.32/0.60 thf(fact_5422_mod__minus1__right,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % mod_minus1_right
% 0.32/0.60 thf(fact_5423_mod__minus1__right,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % mod_minus1_right
% 0.32/0.60 thf(fact_5424_max__number__of_I4_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(4)
% 0.32/0.60 thf(fact_5425_max__number__of_I4_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.32/0.60 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(4)
% 0.32/0.60 thf(fact_5426_max__number__of_I4_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(4)
% 0.32/0.60 thf(fact_5427_max__number__of_I4_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(4)
% 0.32/0.60 thf(fact_5428_max__number__of_I3_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.60 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.60 = ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.60 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(3)
% 0.32/0.60 thf(fact_5429_max__number__of_I3_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.32/0.60 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.32/0.60 = ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.32/0.60 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(3)
% 0.32/0.60 thf(fact_5430_max__number__of_I3_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.60 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.60 = ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.60 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(3)
% 0.32/0.60 thf(fact_5431_max__number__of_I3_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.60 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.60 = ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.60 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(3)
% 0.32/0.60 thf(fact_5432_max__number__of_I2_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.32/0.60 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(2)
% 0.32/0.60 thf(fact_5433_max__number__of_I2_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.32/0.60 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.32/0.60 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(2)
% 0.32/0.60 thf(fact_5434_max__number__of_I2_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.32/0.60 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(2)
% 0.32/0.60 thf(fact_5435_max__number__of_I2_J,axiom,
% 0.32/0.60 ! [U: num,V: num] :
% 0.32/0.60 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.32/0.60 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.32/0.60 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % max_number_of(2)
% 0.32/0.60 thf(fact_5436_semiring__norm_I168_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: real] :
% 0.32/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(168)
% 0.32/0.60 thf(fact_5437_semiring__norm_I168_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: int] :
% 0.32/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(168)
% 0.32/0.60 thf(fact_5438_semiring__norm_I168_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: complex] :
% 0.32/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(168)
% 0.32/0.60 thf(fact_5439_semiring__norm_I168_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: code_integer] :
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(168)
% 0.32/0.60 thf(fact_5440_semiring__norm_I168_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: rat] :
% 0.32/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(168)
% 0.32/0.60 thf(fact_5441_diff__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(2)
% 0.32/0.60 thf(fact_5442_diff__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(2)
% 0.32/0.60 thf(fact_5443_diff__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.32/0.60 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(2)
% 0.32/0.60 thf(fact_5444_diff__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(2)
% 0.32/0.60 thf(fact_5445_diff__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(2)
% 0.32/0.60 thf(fact_5446_diff__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(3)
% 0.32/0.60 thf(fact_5447_diff__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(3)
% 0.32/0.60 thf(fact_5448_diff__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(3)
% 0.32/0.60 thf(fact_5449_diff__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(3)
% 0.32/0.60 thf(fact_5450_diff__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_simps(3)
% 0.32/0.60 thf(fact_5451_semiring__norm_I172_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: real] :
% 0.32/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(172)
% 0.32/0.60 thf(fact_5452_semiring__norm_I172_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: int] :
% 0.32/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(172)
% 0.32/0.60 thf(fact_5453_semiring__norm_I172_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: complex] :
% 0.32/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(172)
% 0.32/0.60 thf(fact_5454_semiring__norm_I172_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(172)
% 0.32/0.60 thf(fact_5455_semiring__norm_I172_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: rat] :
% 0.32/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(172)
% 0.32/0.60 thf(fact_5456_semiring__norm_I171_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: real] :
% 0.32/0.60 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(171)
% 0.32/0.60 thf(fact_5457_semiring__norm_I171_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: int] :
% 0.32/0.60 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(171)
% 0.32/0.60 thf(fact_5458_semiring__norm_I171_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: complex] :
% 0.32/0.60 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(171)
% 0.32/0.60 thf(fact_5459_semiring__norm_I171_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(171)
% 0.32/0.60 thf(fact_5460_semiring__norm_I171_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: rat] :
% 0.32/0.60 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 0.32/0.60 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(171)
% 0.32/0.60 thf(fact_5461_semiring__norm_I170_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: real] :
% 0.32/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
% 0.32/0.60 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(170)
% 0.32/0.60 thf(fact_5462_semiring__norm_I170_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: int] :
% 0.32/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
% 0.32/0.60 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(170)
% 0.32/0.60 thf(fact_5463_semiring__norm_I170_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: complex] :
% 0.32/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
% 0.32/0.60 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(170)
% 0.32/0.60 thf(fact_5464_semiring__norm_I170_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: code_integer] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
% 0.32/0.60 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(170)
% 0.32/0.60 thf(fact_5465_semiring__norm_I170_J,axiom,
% 0.32/0.60 ! [V: num,W: num,Y3: rat] :
% 0.32/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
% 0.32/0.60 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 0.32/0.60
% 0.32/0.60 % semiring_norm(170)
% 0.32/0.60 thf(fact_5466_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5467_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5468_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5469_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5470_mult__neg__numeral__simps_I3_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(3)
% 0.32/0.60 thf(fact_5471_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(2)
% 0.32/0.60 thf(fact_5472_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(2)
% 0.32/0.60 thf(fact_5473_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(2)
% 0.32/0.60 thf(fact_5474_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(2)
% 0.32/0.60 thf(fact_5475_mult__neg__numeral__simps_I2_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(2)
% 0.32/0.60 thf(fact_5476_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(1)
% 0.32/0.60 thf(fact_5477_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(1)
% 0.32/0.60 thf(fact_5478_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.32/0.60 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(1)
% 0.32/0.60 thf(fact_5479_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(1)
% 0.32/0.60 thf(fact_5480_mult__neg__numeral__simps_I1_J,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % mult_neg_numeral_simps(1)
% 0.32/0.60 thf(fact_5481_artanh__minus__real,axiom,
% 0.32/0.60 ! [X: real] :
% 0.32/0.60 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.60 => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % artanh_minus_real
% 0.32/0.60 thf(fact_5482_neg__numeral__le__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_le_iff
% 0.32/0.60 thf(fact_5483_neg__numeral__le__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_le_iff
% 0.32/0.60 thf(fact_5484_neg__numeral__le__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_le_iff
% 0.32/0.60 thf(fact_5485_neg__numeral__le__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_le_iff
% 0.32/0.60 thf(fact_5486_neg__numeral__less__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( ord_less_num @ N @ M ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_less_iff
% 0.32/0.60 thf(fact_5487_neg__numeral__less__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( ord_less_num @ N @ M ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_less_iff
% 0.32/0.60 thf(fact_5488_neg__numeral__less__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( ord_less_num @ N @ M ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_less_iff
% 0.32/0.60 thf(fact_5489_neg__numeral__less__iff,axiom,
% 0.32/0.60 ! [M: num,N: num] :
% 0.32/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( ord_less_num @ N @ M ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_less_iff
% 0.32/0.60 thf(fact_5490_not__neg__one__le__neg__numeral__iff,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.32/0.60 = ( M != one ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_neg_one_le_neg_numeral_iff
% 0.32/0.60 thf(fact_5491_not__neg__one__le__neg__numeral__iff,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.32/0.60 = ( M != one ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_neg_one_le_neg_numeral_iff
% 0.32/0.60 thf(fact_5492_not__neg__one__le__neg__numeral__iff,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.32/0.60 = ( M != one ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_neg_one_le_neg_numeral_iff
% 0.32/0.60 thf(fact_5493_not__neg__one__le__neg__numeral__iff,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.32/0.60 = ( M != one ) ) ).
% 0.32/0.60
% 0.32/0.60 % not_neg_one_le_neg_numeral_iff
% 0.32/0.60 thf(fact_5494_le__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [A: real,B: real,W: num] :
% 0.32/0.60 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.32/0.60 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % le_divide_eq_numeral1(2)
% 0.32/0.60 thf(fact_5495_le__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [A: rat,B: rat,W: num] :
% 0.32/0.60 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.32/0.60 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % le_divide_eq_numeral1(2)
% 0.32/0.60 thf(fact_5496_divide__le__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [B: real,W: num,A: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.32/0.60 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_le_eq_numeral1(2)
% 0.32/0.60 thf(fact_5497_divide__le__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [B: rat,W: num,A: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.32/0.60 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_le_eq_numeral1(2)
% 0.32/0.60 thf(fact_5498_eq__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [A: real,B: real,W: num] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.32/0.60 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.60 != zero_zero_real )
% 0.32/0.60 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.60 = B ) )
% 0.32/0.60 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.60 = zero_zero_real )
% 0.32/0.60 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % eq_divide_eq_numeral1(2)
% 0.32/0.60 thf(fact_5499_eq__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [A: complex,B: complex,W: num] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 0.32/0.60 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.60 != zero_zero_complex )
% 0.32/0.60 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.60 = B ) )
% 0.32/0.60 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.60 = zero_zero_complex )
% 0.32/0.60 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % eq_divide_eq_numeral1(2)
% 0.32/0.60 thf(fact_5500_eq__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [A: rat,B: rat,W: num] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.32/0.60 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.60 != zero_zero_rat )
% 0.32/0.60 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.60 = B ) )
% 0.32/0.60 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.60 = zero_zero_rat )
% 0.32/0.60 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % eq_divide_eq_numeral1(2)
% 0.32/0.60 thf(fact_5501_divide__eq__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [B: real,W: num,A: real] :
% 0.32/0.60 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.60 = A )
% 0.32/0.60 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.60 != zero_zero_real )
% 0.32/0.60 => ( B
% 0.32/0.60 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 0.32/0.60 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.60 = zero_zero_real )
% 0.32/0.60 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_eq_eq_numeral1(2)
% 0.32/0.60 thf(fact_5502_divide__eq__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [B: complex,W: num,A: complex] :
% 0.32/0.60 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.60 = A )
% 0.32/0.60 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.60 != zero_zero_complex )
% 0.32/0.60 => ( B
% 0.32/0.60 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 0.32/0.60 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.60 = zero_zero_complex )
% 0.32/0.60 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_eq_eq_numeral1(2)
% 0.32/0.60 thf(fact_5503_divide__eq__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [B: rat,W: num,A: rat] :
% 0.32/0.60 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.60 = A )
% 0.32/0.60 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.60 != zero_zero_rat )
% 0.32/0.60 => ( B
% 0.32/0.60 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 0.32/0.60 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.60 = zero_zero_rat )
% 0.32/0.60 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_eq_eq_numeral1(2)
% 0.32/0.60 thf(fact_5504_neg__numeral__less__neg__one__iff,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = ( M != one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_less_neg_one_iff
% 0.32/0.60 thf(fact_5505_neg__numeral__less__neg__one__iff,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = ( M != one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_less_neg_one_iff
% 0.32/0.60 thf(fact_5506_neg__numeral__less__neg__one__iff,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = ( M != one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_less_neg_one_iff
% 0.32/0.60 thf(fact_5507_neg__numeral__less__neg__one__iff,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = ( M != one ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_numeral_less_neg_one_iff
% 0.32/0.60 thf(fact_5508_less__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [A: real,B: real,W: num] :
% 0.32/0.60 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 0.32/0.60 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_divide_eq_numeral1(2)
% 0.32/0.60 thf(fact_5509_less__divide__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [A: rat,B: rat,W: num] :
% 0.32/0.60 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 0.32/0.60 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % less_divide_eq_numeral1(2)
% 0.32/0.60 thf(fact_5510_divide__less__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [B: real,W: num,A: real] :
% 0.32/0.60 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 0.32/0.60 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_less_eq_numeral1(2)
% 0.32/0.60 thf(fact_5511_divide__less__eq__numeral1_I2_J,axiom,
% 0.32/0.60 ! [B: rat,W: num,A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 0.32/0.60 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % divide_less_eq_numeral1(2)
% 0.32/0.60 thf(fact_5512_power2__minus,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_minus
% 0.32/0.60 thf(fact_5513_power2__minus,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_minus
% 0.32/0.60 thf(fact_5514_power2__minus,axiom,
% 0.32/0.60 ! [A: complex] :
% 0.32/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_minus
% 0.32/0.60 thf(fact_5515_power2__minus,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_minus
% 0.32/0.60 thf(fact_5516_power2__minus,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_minus
% 0.32/0.60 thf(fact_5517_zero__less__power__abs__iff,axiom,
% 0.32/0.60 ! [A: code_integer,N: nat] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 0.32/0.60 = ( ( A != zero_z3403309356797280102nteger )
% 0.32/0.60 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_less_power_abs_iff
% 0.32/0.60 thf(fact_5518_zero__less__power__abs__iff,axiom,
% 0.32/0.60 ! [A: real,N: nat] :
% 0.32/0.60 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.32/0.60 = ( ( A != zero_zero_real )
% 0.32/0.60 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_less_power_abs_iff
% 0.32/0.60 thf(fact_5519_zero__less__power__abs__iff,axiom,
% 0.32/0.60 ! [A: rat,N: nat] :
% 0.32/0.60 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 0.32/0.60 = ( ( A != zero_zero_rat )
% 0.32/0.60 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_less_power_abs_iff
% 0.32/0.60 thf(fact_5520_zero__less__power__abs__iff,axiom,
% 0.32/0.60 ! [A: int,N: nat] :
% 0.32/0.60 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 0.32/0.60 = ( ( A != zero_zero_int )
% 0.32/0.60 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % zero_less_power_abs_iff
% 0.32/0.60 thf(fact_5521_power2__abs,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_abs
% 0.32/0.60 thf(fact_5522_power2__abs,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_abs
% 0.32/0.60 thf(fact_5523_power2__abs,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_abs
% 0.32/0.60 thf(fact_5524_power2__abs,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power2_abs
% 0.32/0.60 thf(fact_5525_abs__power2,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.60 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_power2
% 0.32/0.60 thf(fact_5526_abs__power2,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.60 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_power2
% 0.32/0.60 thf(fact_5527_abs__power2,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.60 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_power2
% 0.32/0.60 thf(fact_5528_abs__power2,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.60 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_power2
% 0.32/0.60 thf(fact_5529_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.60 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(9)
% 0.32/0.60 thf(fact_5530_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.60 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(9)
% 0.32/0.60 thf(fact_5531_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.60 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(9)
% 0.32/0.60 thf(fact_5532_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.60 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(9)
% 0.32/0.60 thf(fact_5533_add__neg__numeral__special_I9_J,axiom,
% 0.32/0.60 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % add_neg_numeral_special(9)
% 0.32/0.60 thf(fact_5534_diff__numeral__special_I10_J,axiom,
% 0.32/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(10)
% 0.32/0.60 thf(fact_5535_diff__numeral__special_I10_J,axiom,
% 0.32/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(10)
% 0.32/0.60 thf(fact_5536_diff__numeral__special_I10_J,axiom,
% 0.32/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(10)
% 0.32/0.60 thf(fact_5537_diff__numeral__special_I10_J,axiom,
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(10)
% 0.32/0.60 thf(fact_5538_diff__numeral__special_I10_J,axiom,
% 0.32/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(10)
% 0.32/0.60 thf(fact_5539_diff__numeral__special_I11_J,axiom,
% 0.32/0.60 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(11)
% 0.32/0.60 thf(fact_5540_diff__numeral__special_I11_J,axiom,
% 0.32/0.60 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(11)
% 0.32/0.60 thf(fact_5541_diff__numeral__special_I11_J,axiom,
% 0.32/0.60 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.60 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(11)
% 0.32/0.60 thf(fact_5542_diff__numeral__special_I11_J,axiom,
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(11)
% 0.32/0.60 thf(fact_5543_diff__numeral__special_I11_J,axiom,
% 0.32/0.60 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(11)
% 0.32/0.60 thf(fact_5544_minus__1__div__2__eq,axiom,
% 0.32/0.60 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_1_div_2_eq
% 0.32/0.60 thf(fact_5545_minus__1__div__2__eq,axiom,
% 0.32/0.60 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_1_div_2_eq
% 0.32/0.60 thf(fact_5546_bits__minus__1__mod__2__eq,axiom,
% 0.32/0.60 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_int ) ).
% 0.32/0.60
% 0.32/0.60 % bits_minus_1_mod_2_eq
% 0.32/0.60 thf(fact_5547_bits__minus__1__mod__2__eq,axiom,
% 0.32/0.60 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_Code_integer ) ).
% 0.32/0.60
% 0.32/0.60 % bits_minus_1_mod_2_eq
% 0.32/0.60 thf(fact_5548_minus__1__mod__2__eq,axiom,
% 0.32/0.60 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_int ) ).
% 0.32/0.60
% 0.32/0.60 % minus_1_mod_2_eq
% 0.32/0.60 thf(fact_5549_minus__1__mod__2__eq,axiom,
% 0.32/0.60 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.60 = one_one_Code_integer ) ).
% 0.32/0.60
% 0.32/0.60 % minus_1_mod_2_eq
% 0.32/0.60 thf(fact_5550_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [A: real,N: nat] :
% 0.32/0.60 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Power.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5551_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [A: int,N: nat] :
% 0.32/0.60 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Power.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5552_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [A: complex,N: nat] :
% 0.32/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Power.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5553_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [A: code_integer,N: nat] :
% 0.32/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Power.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5554_Power_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [A: rat,N: nat] :
% 0.32/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Power.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5555_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [N: nat,A: real] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.32/0.60 = ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Parity.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5556_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [N: nat,A: int] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.32/0.60 = ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Parity.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5557_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [N: nat,A: complex] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.32/0.60 = ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Parity.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5558_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [N: nat,A: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.32/0.60 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Parity.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5559_Parity_Oring__1__class_Opower__minus__even,axiom,
% 0.32/0.60 ! [N: nat,A: rat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.32/0.60 = ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % Parity.ring_1_class.power_minus_even
% 0.32/0.60 thf(fact_5560_power__minus__odd,axiom,
% 0.32/0.60 ! [N: nat,A: real] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.32/0.60 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus_odd
% 0.32/0.60 thf(fact_5561_power__minus__odd,axiom,
% 0.32/0.60 ! [N: nat,A: int] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.32/0.60 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus_odd
% 0.32/0.60 thf(fact_5562_power__minus__odd,axiom,
% 0.32/0.60 ! [N: nat,A: complex] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus_odd
% 0.32/0.60 thf(fact_5563_power__minus__odd,axiom,
% 0.32/0.60 ! [N: nat,A: code_integer] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus_odd
% 0.32/0.60 thf(fact_5564_power__minus__odd,axiom,
% 0.32/0.60 ! [N: nat,A: rat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.32/0.60 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus_odd
% 0.32/0.60 thf(fact_5565_power__even__abs__numeral,axiom,
% 0.32/0.60 ! [W: num,A: code_integer] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.60 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.60 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_even_abs_numeral
% 0.32/0.60 thf(fact_5566_power__even__abs__numeral,axiom,
% 0.32/0.60 ! [W: num,A: rat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.60 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.60 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_even_abs_numeral
% 0.32/0.60 thf(fact_5567_power__even__abs__numeral,axiom,
% 0.32/0.60 ! [W: num,A: real] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.60 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.60 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_even_abs_numeral
% 0.32/0.60 thf(fact_5568_power__even__abs__numeral,axiom,
% 0.32/0.60 ! [W: num,A: int] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.60 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 0.32/0.60 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % power_even_abs_numeral
% 0.32/0.60 thf(fact_5569_diff__numeral__special_I4_J,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(4)
% 0.32/0.60 thf(fact_5570_diff__numeral__special_I4_J,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(4)
% 0.32/0.60 thf(fact_5571_diff__numeral__special_I4_J,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(4)
% 0.32/0.60 thf(fact_5572_diff__numeral__special_I4_J,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(4)
% 0.32/0.60 thf(fact_5573_diff__numeral__special_I4_J,axiom,
% 0.32/0.60 ! [M: num] :
% 0.32/0.60 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(4)
% 0.32/0.60 thf(fact_5574_diff__numeral__special_I3_J,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(3)
% 0.32/0.60 thf(fact_5575_diff__numeral__special_I3_J,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(3)
% 0.32/0.60 thf(fact_5576_diff__numeral__special_I3_J,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.32/0.60 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(3)
% 0.32/0.60 thf(fact_5577_diff__numeral__special_I3_J,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.60 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(3)
% 0.32/0.60 thf(fact_5578_diff__numeral__special_I3_J,axiom,
% 0.32/0.60 ! [N: num] :
% 0.32/0.60 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.60 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % diff_numeral_special(3)
% 0.32/0.60 thf(fact_5579_signed__take__bit__Suc__minus__bit0,axiom,
% 0.32/0.60 ! [N: nat,K: num] :
% 0.32/0.60 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.32/0.60 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % signed_take_bit_Suc_minus_bit0
% 0.32/0.60 thf(fact_5580_dbl__simps_I4_J,axiom,
% 0.32/0.60 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.60 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(4)
% 0.32/0.60 thf(fact_5581_dbl__simps_I4_J,axiom,
% 0.32/0.60 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.60 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(4)
% 0.32/0.60 thf(fact_5582_dbl__simps_I4_J,axiom,
% 0.32/0.60 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(4)
% 0.32/0.60 thf(fact_5583_dbl__simps_I4_J,axiom,
% 0.32/0.60 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(4)
% 0.32/0.60 thf(fact_5584_dbl__simps_I4_J,axiom,
% 0.32/0.60 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.60 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % dbl_simps(4)
% 0.32/0.60 thf(fact_5585_power__minus1__even,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = one_one_real ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus1_even
% 0.32/0.60 thf(fact_5586_power__minus1__even,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = one_one_int ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus1_even
% 0.32/0.60 thf(fact_5587_power__minus1__even,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = one_one_complex ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus1_even
% 0.32/0.60 thf(fact_5588_power__minus1__even,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = one_one_Code_integer ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus1_even
% 0.32/0.60 thf(fact_5589_power__minus1__even,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.60 = one_one_rat ) ).
% 0.32/0.60
% 0.32/0.60 % power_minus1_even
% 0.32/0.60 thf(fact_5590_neg__one__even__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.32/0.60 = one_one_real ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_even_power
% 0.32/0.60 thf(fact_5591_neg__one__even__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.32/0.60 = one_one_int ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_even_power
% 0.32/0.60 thf(fact_5592_neg__one__even__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.32/0.60 = one_one_complex ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_even_power
% 0.32/0.60 thf(fact_5593_neg__one__even__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.32/0.60 = one_one_Code_integer ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_even_power
% 0.32/0.60 thf(fact_5594_neg__one__even__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.32/0.60 = one_one_rat ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_even_power
% 0.32/0.60 thf(fact_5595_neg__one__odd__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.32/0.60 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_odd_power
% 0.32/0.60 thf(fact_5596_neg__one__odd__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.32/0.60 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_odd_power
% 0.32/0.60 thf(fact_5597_neg__one__odd__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.32/0.60 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_odd_power
% 0.32/0.60 thf(fact_5598_neg__one__odd__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_odd_power
% 0.32/0.60 thf(fact_5599_neg__one__odd__power,axiom,
% 0.32/0.60 ! [N: nat] :
% 0.32/0.60 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.60 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.32/0.60 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % neg_one_odd_power
% 0.32/0.60 thf(fact_5600_signed__take__bit__0,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % signed_take_bit_0
% 0.32/0.60 thf(fact_5601_signed__take__bit__0,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.32/0.60 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % signed_take_bit_0
% 0.32/0.60 thf(fact_5602_abs__leI,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.60 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_leI
% 0.32/0.60 thf(fact_5603_abs__leI,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.32/0.60 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.60 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_leI
% 0.32/0.60 thf(fact_5604_abs__leI,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.60 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_leI
% 0.32/0.60 thf(fact_5605_abs__leI,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.60 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.60 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_leI
% 0.32/0.60 thf(fact_5606_abs__le__D2,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.32/0.60 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_D2
% 0.32/0.60 thf(fact_5607_abs__le__D2,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.32/0.60 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_D2
% 0.32/0.60 thf(fact_5608_abs__le__D2,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.32/0.60 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_D2
% 0.32/0.60 thf(fact_5609_abs__le__D2,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.32/0.60 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_D2
% 0.32/0.60 thf(fact_5610_abs__le__iff,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.32/0.60 = ( ( ord_less_eq_real @ A @ B )
% 0.32/0.60 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_iff
% 0.32/0.60 thf(fact_5611_abs__le__iff,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.32/0.60 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.32/0.60 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_iff
% 0.32/0.60 thf(fact_5612_abs__le__iff,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.32/0.60 = ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.60 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_iff
% 0.32/0.60 thf(fact_5613_abs__le__iff,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.32/0.60 = ( ( ord_less_eq_int @ A @ B )
% 0.32/0.60 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_le_iff
% 0.32/0.60 thf(fact_5614_abs__ge__minus__self,axiom,
% 0.32/0.60 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_ge_minus_self
% 0.32/0.60 thf(fact_5615_abs__ge__minus__self,axiom,
% 0.32/0.60 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_ge_minus_self
% 0.32/0.60 thf(fact_5616_abs__ge__minus__self,axiom,
% 0.32/0.60 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_ge_minus_self
% 0.32/0.60 thf(fact_5617_abs__ge__minus__self,axiom,
% 0.32/0.60 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_ge_minus_self
% 0.32/0.60 thf(fact_5618_abs__less__iff,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.32/0.60 = ( ( ord_less_real @ A @ B )
% 0.32/0.60 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_less_iff
% 0.32/0.60 thf(fact_5619_abs__less__iff,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 0.32/0.60 = ( ( ord_less_int @ A @ B )
% 0.32/0.60 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_less_iff
% 0.32/0.60 thf(fact_5620_abs__less__iff,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.32/0.60 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.32/0.60 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_less_iff
% 0.32/0.60 thf(fact_5621_abs__less__iff,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 0.32/0.60 = ( ( ord_less_rat @ A @ B )
% 0.32/0.60 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_less_iff
% 0.32/0.60 thf(fact_5622_signed__take__bit__minus,axiom,
% 0.32/0.60 ! [N: nat,K: int] :
% 0.32/0.60 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 0.32/0.60 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % signed_take_bit_minus
% 0.32/0.60 thf(fact_5623_verit__negate__coefficient_I3_J,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( uminus_uminus_real @ A )
% 0.32/0.60 = ( uminus_uminus_real @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_negate_coefficient(3)
% 0.32/0.60 thf(fact_5624_verit__negate__coefficient_I3_J,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( uminus_uminus_int @ A )
% 0.32/0.60 = ( uminus_uminus_int @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_negate_coefficient(3)
% 0.32/0.60 thf(fact_5625_verit__negate__coefficient_I3_J,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( uminus1351360451143612070nteger @ A )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_negate_coefficient(3)
% 0.32/0.60 thf(fact_5626_verit__negate__coefficient_I3_J,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( A = B )
% 0.32/0.60 => ( ( uminus_uminus_rat @ A )
% 0.32/0.60 = ( uminus_uminus_rat @ B ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % verit_negate_coefficient(3)
% 0.32/0.60 thf(fact_5627_abs__eq__iff,axiom,
% 0.32/0.60 ! [X: real,Y3: real] :
% 0.32/0.60 ( ( ( abs_abs_real @ X )
% 0.32/0.60 = ( abs_abs_real @ Y3 ) )
% 0.32/0.60 = ( ( X = Y3 )
% 0.32/0.60 | ( X
% 0.32/0.60 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_iff
% 0.32/0.60 thf(fact_5628_abs__eq__iff,axiom,
% 0.32/0.60 ! [X: int,Y3: int] :
% 0.32/0.60 ( ( ( abs_abs_int @ X )
% 0.32/0.60 = ( abs_abs_int @ Y3 ) )
% 0.32/0.60 = ( ( X = Y3 )
% 0.32/0.60 | ( X
% 0.32/0.60 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_iff
% 0.32/0.60 thf(fact_5629_abs__eq__iff,axiom,
% 0.32/0.60 ! [X: code_integer,Y3: code_integer] :
% 0.32/0.60 ( ( ( abs_abs_Code_integer @ X )
% 0.32/0.60 = ( abs_abs_Code_integer @ Y3 ) )
% 0.32/0.60 = ( ( X = Y3 )
% 0.32/0.60 | ( X
% 0.32/0.60 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_iff
% 0.32/0.60 thf(fact_5630_abs__eq__iff,axiom,
% 0.32/0.60 ! [X: rat,Y3: rat] :
% 0.32/0.60 ( ( ( abs_abs_rat @ X )
% 0.32/0.60 = ( abs_abs_rat @ Y3 ) )
% 0.32/0.60 = ( ( X = Y3 )
% 0.32/0.60 | ( X
% 0.32/0.60 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_iff
% 0.32/0.60 thf(fact_5631_minus__equation__iff,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ( uminus_uminus_real @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( uminus_uminus_real @ B )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_equation_iff
% 0.32/0.60 thf(fact_5632_minus__equation__iff,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ( uminus_uminus_int @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( uminus_uminus_int @ B )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_equation_iff
% 0.32/0.60 thf(fact_5633_minus__equation__iff,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( ( uminus1482373934393186551omplex @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( uminus1482373934393186551omplex @ B )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_equation_iff
% 0.32/0.60 thf(fact_5634_minus__equation__iff,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( ( uminus1351360451143612070nteger @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( uminus1351360451143612070nteger @ B )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_equation_iff
% 0.32/0.60 thf(fact_5635_minus__equation__iff,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ( uminus_uminus_rat @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( uminus_uminus_rat @ B )
% 0.32/0.60 = A ) ) ).
% 0.32/0.60
% 0.32/0.60 % minus_equation_iff
% 0.32/0.60 thf(fact_5636_equation__minus__iff,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus_uminus_real @ B ) )
% 0.32/0.60 = ( B
% 0.32/0.60 = ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % equation_minus_iff
% 0.32/0.60 thf(fact_5637_equation__minus__iff,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus_uminus_int @ B ) )
% 0.32/0.60 = ( B
% 0.32/0.60 = ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % equation_minus_iff
% 0.32/0.60 thf(fact_5638_equation__minus__iff,axiom,
% 0.32/0.60 ! [A: complex,B: complex] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.60 = ( B
% 0.32/0.60 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % equation_minus_iff
% 0.32/0.60 thf(fact_5639_equation__minus__iff,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.60 = ( B
% 0.32/0.60 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % equation_minus_iff
% 0.32/0.60 thf(fact_5640_equation__minus__iff,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( uminus_uminus_rat @ B ) )
% 0.32/0.60 = ( B
% 0.32/0.60 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % equation_minus_iff
% 0.32/0.60 thf(fact_5641_abs__minus__le__zero,axiom,
% 0.32/0.60 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus_le_zero
% 0.32/0.60 thf(fact_5642_abs__minus__le__zero,axiom,
% 0.32/0.60 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus_le_zero
% 0.32/0.60 thf(fact_5643_abs__minus__le__zero,axiom,
% 0.32/0.60 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus_le_zero
% 0.32/0.60 thf(fact_5644_abs__minus__le__zero,axiom,
% 0.32/0.60 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.32/0.60
% 0.32/0.60 % abs_minus_le_zero
% 0.32/0.60 thf(fact_5645_eq__abs__iff_H,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( abs_abs_real @ B ) )
% 0.32/0.60 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.60 & ( ( B = A )
% 0.32/0.60 | ( B
% 0.32/0.60 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % eq_abs_iff'
% 0.32/0.60 thf(fact_5646_eq__abs__iff_H,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( abs_abs_Code_integer @ B ) )
% 0.32/0.60 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.60 & ( ( B = A )
% 0.32/0.60 | ( B
% 0.32/0.60 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % eq_abs_iff'
% 0.32/0.60 thf(fact_5647_eq__abs__iff_H,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( abs_abs_rat @ B ) )
% 0.32/0.60 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.60 & ( ( B = A )
% 0.32/0.60 | ( B
% 0.32/0.60 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % eq_abs_iff'
% 0.32/0.60 thf(fact_5648_eq__abs__iff_H,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( A
% 0.32/0.60 = ( abs_abs_int @ B ) )
% 0.32/0.60 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.60 & ( ( B = A )
% 0.32/0.60 | ( B
% 0.32/0.60 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % eq_abs_iff'
% 0.32/0.60 thf(fact_5649_abs__eq__iff_H,axiom,
% 0.32/0.60 ! [A: real,B: real] :
% 0.32/0.60 ( ( ( abs_abs_real @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.60 & ( ( A = B )
% 0.32/0.60 | ( A
% 0.32/0.60 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_iff'
% 0.32/0.60 thf(fact_5650_abs__eq__iff_H,axiom,
% 0.32/0.60 ! [A: code_integer,B: code_integer] :
% 0.32/0.60 ( ( ( abs_abs_Code_integer @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.60 & ( ( A = B )
% 0.32/0.60 | ( A
% 0.32/0.60 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_iff'
% 0.32/0.60 thf(fact_5651_abs__eq__iff_H,axiom,
% 0.32/0.60 ! [A: rat,B: rat] :
% 0.32/0.60 ( ( ( abs_abs_rat @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.60 & ( ( A = B )
% 0.32/0.60 | ( A
% 0.32/0.60 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_iff'
% 0.32/0.60 thf(fact_5652_abs__eq__iff_H,axiom,
% 0.32/0.60 ! [A: int,B: int] :
% 0.32/0.60 ( ( ( abs_abs_int @ A )
% 0.32/0.60 = B )
% 0.32/0.60 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.60 & ( ( A = B )
% 0.32/0.60 | ( A
% 0.32/0.60 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_eq_iff'
% 0.32/0.60 thf(fact_5653_abs__if__raw,axiom,
% 0.32/0.60 ( abs_abs_real
% 0.32/0.60 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_if_raw
% 0.32/0.60 thf(fact_5654_abs__if__raw,axiom,
% 0.32/0.60 ( abs_abs_int
% 0.32/0.60 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_if_raw
% 0.32/0.60 thf(fact_5655_abs__if__raw,axiom,
% 0.32/0.60 ( abs_abs_Code_integer
% 0.32/0.60 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_if_raw
% 0.32/0.60 thf(fact_5656_abs__if__raw,axiom,
% 0.32/0.60 ( abs_abs_rat
% 0.32/0.60 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_if_raw
% 0.32/0.60 thf(fact_5657_abs__of__neg,axiom,
% 0.32/0.60 ! [A: real] :
% 0.32/0.60 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.60 => ( ( abs_abs_real @ A )
% 0.32/0.60 = ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_neg
% 0.32/0.60 thf(fact_5658_abs__of__neg,axiom,
% 0.32/0.60 ! [A: int] :
% 0.32/0.60 ( ( ord_less_int @ A @ zero_zero_int )
% 0.32/0.60 => ( ( abs_abs_int @ A )
% 0.32/0.60 = ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_neg
% 0.32/0.60 thf(fact_5659_abs__of__neg,axiom,
% 0.32/0.60 ! [A: code_integer] :
% 0.32/0.60 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.32/0.60 => ( ( abs_abs_Code_integer @ A )
% 0.32/0.60 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.60
% 0.32/0.60 % abs_of_neg
% 0.32/0.60 thf(fact_5660_abs__of__neg,axiom,
% 0.32/0.60 ! [A: rat] :
% 0.32/0.60 ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.32/0.60 => ( ( abs_abs_rat @ A )
% 0.32/0.60 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_of_neg
% 0.32/0.61 thf(fact_5661_abs__if,axiom,
% 0.32/0.61 ( abs_abs_real
% 0.32/0.61 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_if
% 0.32/0.61 thf(fact_5662_abs__if,axiom,
% 0.32/0.61 ( abs_abs_int
% 0.32/0.61 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_if
% 0.32/0.61 thf(fact_5663_abs__if,axiom,
% 0.32/0.61 ( abs_abs_Code_integer
% 0.32/0.61 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_if
% 0.32/0.61 thf(fact_5664_abs__if,axiom,
% 0.32/0.61 ( abs_abs_rat
% 0.32/0.61 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_if
% 0.32/0.61 thf(fact_5665_abs__real__def,axiom,
% 0.32/0.61 ( abs_abs_real
% 0.32/0.61 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_real_def
% 0.32/0.61 thf(fact_5666_abs__le__D1,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.32/0.61 => ( ord_less_eq_real @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_le_D1
% 0.32/0.61 thf(fact_5667_abs__le__D1,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.32/0.61 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_le_D1
% 0.32/0.61 thf(fact_5668_abs__le__D1,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.32/0.61 => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_le_D1
% 0.32/0.61 thf(fact_5669_abs__le__D1,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.32/0.61 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_le_D1
% 0.32/0.61 thf(fact_5670_abs__ge__self,axiom,
% 0.32/0.61 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_ge_self
% 0.32/0.61 thf(fact_5671_abs__ge__self,axiom,
% 0.32/0.61 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_ge_self
% 0.32/0.61 thf(fact_5672_abs__ge__self,axiom,
% 0.32/0.61 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_ge_self
% 0.32/0.61 thf(fact_5673_abs__ge__self,axiom,
% 0.32/0.61 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_ge_self
% 0.32/0.61 thf(fact_5674_abs__eq__0__iff,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ( abs_abs_Code_integer @ A )
% 0.32/0.61 = zero_z3403309356797280102nteger )
% 0.32/0.61 = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_0_iff
% 0.32/0.61 thf(fact_5675_abs__eq__0__iff,axiom,
% 0.32/0.61 ! [A: complex] :
% 0.32/0.61 ( ( ( abs_abs_complex @ A )
% 0.32/0.61 = zero_zero_complex )
% 0.32/0.61 = ( A = zero_zero_complex ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_0_iff
% 0.32/0.61 thf(fact_5676_abs__eq__0__iff,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ( abs_abs_real @ A )
% 0.32/0.61 = zero_zero_real )
% 0.32/0.61 = ( A = zero_zero_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_0_iff
% 0.32/0.61 thf(fact_5677_abs__eq__0__iff,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ( abs_abs_rat @ A )
% 0.32/0.61 = zero_zero_rat )
% 0.32/0.61 = ( A = zero_zero_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_0_iff
% 0.32/0.61 thf(fact_5678_abs__eq__0__iff,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ( abs_abs_int @ A )
% 0.32/0.61 = zero_zero_int )
% 0.32/0.61 = ( A = zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_0_iff
% 0.32/0.61 thf(fact_5679_abs__mult,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.32/0.61 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult
% 0.32/0.61 thf(fact_5680_abs__mult,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.32/0.61 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult
% 0.32/0.61 thf(fact_5681_abs__mult,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.61 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult
% 0.32/0.61 thf(fact_5682_abs__mult,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.32/0.61 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult
% 0.32/0.61 thf(fact_5683_abs__one,axiom,
% 0.32/0.61 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.32/0.61 = one_one_Code_integer ) ).
% 0.32/0.61
% 0.32/0.61 % abs_one
% 0.32/0.61 thf(fact_5684_abs__one,axiom,
% 0.32/0.61 ( ( abs_abs_real @ one_one_real )
% 0.32/0.61 = one_one_real ) ).
% 0.32/0.61
% 0.32/0.61 % abs_one
% 0.32/0.61 thf(fact_5685_abs__one,axiom,
% 0.32/0.61 ( ( abs_abs_rat @ one_one_rat )
% 0.32/0.61 = one_one_rat ) ).
% 0.32/0.61
% 0.32/0.61 % abs_one
% 0.32/0.61 thf(fact_5686_abs__one,axiom,
% 0.32/0.61 ( ( abs_abs_int @ one_one_int )
% 0.32/0.61 = one_one_int ) ).
% 0.32/0.61
% 0.32/0.61 % abs_one
% 0.32/0.61 thf(fact_5687_abs__minus__commute,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.32/0.61 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_minus_commute
% 0.32/0.61 thf(fact_5688_abs__minus__commute,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.32/0.61 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_minus_commute
% 0.32/0.61 thf(fact_5689_abs__minus__commute,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 0.32/0.61 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_minus_commute
% 0.32/0.61 thf(fact_5690_abs__minus__commute,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.32/0.61 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_minus_commute
% 0.32/0.61 thf(fact_5691_tanh__real__gt__neg1,axiom,
% 0.32/0.61 ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % tanh_real_gt_neg1
% 0.32/0.61 thf(fact_5692_power__abs,axiom,
% 0.32/0.61 ! [A: code_integer,N: nat] :
% 0.32/0.61 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 0.32/0.61 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_abs
% 0.32/0.61 thf(fact_5693_power__abs,axiom,
% 0.32/0.61 ! [A: rat,N: nat] :
% 0.32/0.61 ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 0.32/0.61 = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_abs
% 0.32/0.61 thf(fact_5694_power__abs,axiom,
% 0.32/0.61 ! [A: real,N: nat] :
% 0.32/0.61 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 0.32/0.61 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_abs
% 0.32/0.61 thf(fact_5695_power__abs,axiom,
% 0.32/0.61 ! [A: int,N: nat] :
% 0.32/0.61 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 0.32/0.61 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_abs
% 0.32/0.61 thf(fact_5696_dvd__if__abs__eq,axiom,
% 0.32/0.61 ! [L2: real,K: real] :
% 0.32/0.61 ( ( ( abs_abs_real @ L2 )
% 0.32/0.61 = ( abs_abs_real @ K ) )
% 0.32/0.61 => ( dvd_dvd_real @ L2 @ K ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_if_abs_eq
% 0.32/0.61 thf(fact_5697_dvd__if__abs__eq,axiom,
% 0.32/0.61 ! [L2: int,K: int] :
% 0.32/0.61 ( ( ( abs_abs_int @ L2 )
% 0.32/0.61 = ( abs_abs_int @ K ) )
% 0.32/0.61 => ( dvd_dvd_int @ L2 @ K ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_if_abs_eq
% 0.32/0.61 thf(fact_5698_dvd__if__abs__eq,axiom,
% 0.32/0.61 ! [L2: code_integer,K: code_integer] :
% 0.32/0.61 ( ( ( abs_abs_Code_integer @ L2 )
% 0.32/0.61 = ( abs_abs_Code_integer @ K ) )
% 0.32/0.61 => ( dvd_dvd_Code_integer @ L2 @ K ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_if_abs_eq
% 0.32/0.61 thf(fact_5699_dvd__if__abs__eq,axiom,
% 0.32/0.61 ! [L2: rat,K: rat] :
% 0.32/0.61 ( ( ( abs_abs_rat @ L2 )
% 0.32/0.61 = ( abs_abs_rat @ K ) )
% 0.32/0.61 => ( dvd_dvd_rat @ L2 @ K ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_if_abs_eq
% 0.32/0.61 thf(fact_5700_compl__mono,axiom,
% 0.32/0.61 ! [X: set_int,Y3: set_int] :
% 0.32/0.61 ( ( ord_less_eq_set_int @ X @ Y3 )
% 0.32/0.61 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % compl_mono
% 0.32/0.61 thf(fact_5701_compl__le__swap1,axiom,
% 0.32/0.61 ! [Y3: set_int,X: set_int] :
% 0.32/0.61 ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X ) )
% 0.32/0.61 => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % compl_le_swap1
% 0.32/0.61 thf(fact_5702_compl__le__swap2,axiom,
% 0.32/0.61 ! [Y3: set_int,X: set_int] :
% 0.32/0.61 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X )
% 0.32/0.61 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y3 ) ) ).
% 0.32/0.61
% 0.32/0.61 % compl_le_swap2
% 0.32/0.61 thf(fact_5703_le__minus__iff,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_iff
% 0.32/0.61 thf(fact_5704_le__minus__iff,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_iff
% 0.32/0.61 thf(fact_5705_le__minus__iff,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_iff
% 0.32/0.61 thf(fact_5706_le__minus__iff,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_iff
% 0.32/0.61 thf(fact_5707_minus__le__iff,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.61 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_le_iff
% 0.32/0.61 thf(fact_5708_minus__le__iff,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.61 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_le_iff
% 0.32/0.61 thf(fact_5709_minus__le__iff,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.61 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_le_iff
% 0.32/0.61 thf(fact_5710_minus__le__iff,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_le_iff
% 0.32/0.61 thf(fact_5711_le__imp__neg__le,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.61 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_imp_neg_le
% 0.32/0.61 thf(fact_5712_le__imp__neg__le,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.32/0.61 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_imp_neg_le
% 0.32/0.61 thf(fact_5713_le__imp__neg__le,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ B )
% 0.32/0.61 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_imp_neg_le
% 0.32/0.61 thf(fact_5714_le__imp__neg__le,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ A @ B )
% 0.32/0.61 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_imp_neg_le
% 0.32/0.61 thf(fact_5715_less__minus__iff,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_iff
% 0.32/0.61 thf(fact_5716_less__minus__iff,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_iff
% 0.32/0.61 thf(fact_5717_less__minus__iff,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_iff
% 0.32/0.61 thf(fact_5718_less__minus__iff,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_iff
% 0.32/0.61 thf(fact_5719_minus__less__iff,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.61 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_less_iff
% 0.32/0.61 thf(fact_5720_minus__less__iff,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_less_iff
% 0.32/0.61 thf(fact_5721_minus__less__iff,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.61 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_less_iff
% 0.32/0.61 thf(fact_5722_minus__less__iff,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.61 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_less_iff
% 0.32/0.61 thf(fact_5723_verit__negate__coefficient_I2_J,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ A @ B )
% 0.32/0.61 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_negate_coefficient(2)
% 0.32/0.61 thf(fact_5724_verit__negate__coefficient_I2_J,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ord_less_int @ A @ B )
% 0.32/0.61 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_negate_coefficient(2)
% 0.32/0.61 thf(fact_5725_verit__negate__coefficient_I2_J,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.32/0.61 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_negate_coefficient(2)
% 0.32/0.61 thf(fact_5726_verit__negate__coefficient_I2_J,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_rat @ A @ B )
% 0.32/0.61 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_negate_coefficient(2)
% 0.32/0.61 thf(fact_5727_numeral__neq__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( numeral_numeral_real @ M )
% 0.32/0.61 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_numeral
% 0.32/0.61 thf(fact_5728_numeral__neq__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( numeral_numeral_int @ M )
% 0.32/0.61 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_numeral
% 0.32/0.61 thf(fact_5729_numeral__neq__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( numera6690914467698888265omplex @ M )
% 0.32/0.61 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_numeral
% 0.32/0.61 thf(fact_5730_numeral__neq__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( numera6620942414471956472nteger @ M )
% 0.32/0.61 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_numeral
% 0.32/0.61 thf(fact_5731_numeral__neq__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( numeral_numeral_rat @ M )
% 0.32/0.61 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_numeral
% 0.32/0.61 thf(fact_5732_neg__numeral__neq__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 0.32/0.61 != ( numeral_numeral_real @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_neq_numeral
% 0.32/0.61 thf(fact_5733_neg__numeral__neq__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 0.32/0.61 != ( numeral_numeral_int @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_neq_numeral
% 0.32/0.61 thf(fact_5734_neg__numeral__neq__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 0.32/0.61 != ( numera6690914467698888265omplex @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_neq_numeral
% 0.32/0.61 thf(fact_5735_neg__numeral__neq__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 0.32/0.61 != ( numera6620942414471956472nteger @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_neq_numeral
% 0.32/0.61 thf(fact_5736_neg__numeral__neq__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 0.32/0.61 != ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_neq_numeral
% 0.32/0.61 thf(fact_5737_square__eq__iff,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ( times_times_real @ A @ A )
% 0.32/0.61 = ( times_times_real @ B @ B ) )
% 0.32/0.61 = ( ( A = B )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_iff
% 0.32/0.61 thf(fact_5738_square__eq__iff,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ( times_times_int @ A @ A )
% 0.32/0.61 = ( times_times_int @ B @ B ) )
% 0.32/0.61 = ( ( A = B )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus_uminus_int @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_iff
% 0.32/0.61 thf(fact_5739_square__eq__iff,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( ( times_times_complex @ A @ A )
% 0.32/0.61 = ( times_times_complex @ B @ B ) )
% 0.32/0.61 = ( ( A = B )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_iff
% 0.32/0.61 thf(fact_5740_square__eq__iff,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ( times_3573771949741848930nteger @ A @ A )
% 0.32/0.61 = ( times_3573771949741848930nteger @ B @ B ) )
% 0.32/0.61 = ( ( A = B )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_iff
% 0.32/0.61 thf(fact_5741_square__eq__iff,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ( times_times_rat @ A @ A )
% 0.32/0.61 = ( times_times_rat @ B @ B ) )
% 0.32/0.61 = ( ( A = B )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_iff
% 0.32/0.61 thf(fact_5742_minus__mult__commute,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.61 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_commute
% 0.32/0.61 thf(fact_5743_minus__mult__commute,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_commute
% 0.32/0.61 thf(fact_5744_minus__mult__commute,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.32/0.61 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_commute
% 0.32/0.61 thf(fact_5745_minus__mult__commute,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.61 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_commute
% 0.32/0.61 thf(fact_5746_minus__mult__commute,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.61 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mult_commute
% 0.32/0.61 thf(fact_5747_one__neq__neg__one,axiom,
% 0.32/0.61 ( one_one_real
% 0.32/0.61 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_one
% 0.32/0.61 thf(fact_5748_one__neq__neg__one,axiom,
% 0.32/0.61 ( one_one_int
% 0.32/0.61 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_one
% 0.32/0.61 thf(fact_5749_one__neq__neg__one,axiom,
% 0.32/0.61 ( one_one_complex
% 0.32/0.61 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_one
% 0.32/0.61 thf(fact_5750_one__neq__neg__one,axiom,
% 0.32/0.61 ( one_one_Code_integer
% 0.32/0.61 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_one
% 0.32/0.61 thf(fact_5751_one__neq__neg__one,axiom,
% 0.32/0.61 ( one_one_rat
% 0.32/0.61 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_one
% 0.32/0.61 thf(fact_5752_group__cancel_Oneg1,axiom,
% 0.32/0.61 ! [A2: real,K: real,A: real] :
% 0.32/0.61 ( ( A2
% 0.32/0.61 = ( plus_plus_real @ K @ A ) )
% 0.32/0.61 => ( ( uminus_uminus_real @ A2 )
% 0.32/0.61 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.neg1
% 0.32/0.61 thf(fact_5753_group__cancel_Oneg1,axiom,
% 0.32/0.61 ! [A2: int,K: int,A: int] :
% 0.32/0.61 ( ( A2
% 0.32/0.61 = ( plus_plus_int @ K @ A ) )
% 0.32/0.61 => ( ( uminus_uminus_int @ A2 )
% 0.32/0.61 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.neg1
% 0.32/0.61 thf(fact_5754_group__cancel_Oneg1,axiom,
% 0.32/0.61 ! [A2: complex,K: complex,A: complex] :
% 0.32/0.61 ( ( A2
% 0.32/0.61 = ( plus_plus_complex @ K @ A ) )
% 0.32/0.61 => ( ( uminus1482373934393186551omplex @ A2 )
% 0.32/0.61 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.neg1
% 0.32/0.61 thf(fact_5755_group__cancel_Oneg1,axiom,
% 0.32/0.61 ! [A2: code_integer,K: code_integer,A: code_integer] :
% 0.32/0.61 ( ( A2
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.32/0.61 => ( ( uminus1351360451143612070nteger @ A2 )
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.neg1
% 0.32/0.61 thf(fact_5756_group__cancel_Oneg1,axiom,
% 0.32/0.61 ! [A2: rat,K: rat,A: rat] :
% 0.32/0.61 ( ( A2
% 0.32/0.61 = ( plus_plus_rat @ K @ A ) )
% 0.32/0.61 => ( ( uminus_uminus_rat @ A2 )
% 0.32/0.61 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.neg1
% 0.32/0.61 thf(fact_5757_add_Oinverse__distrib__swap,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.32/0.61 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_distrib_swap
% 0.32/0.61 thf(fact_5758_add_Oinverse__distrib__swap,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.32/0.61 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_distrib_swap
% 0.32/0.61 thf(fact_5759_add_Oinverse__distrib__swap,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.61 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_distrib_swap
% 0.32/0.61 thf(fact_5760_add_Oinverse__distrib__swap,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_distrib_swap
% 0.32/0.61 thf(fact_5761_add_Oinverse__distrib__swap,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.61 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_distrib_swap
% 0.32/0.61 thf(fact_5762_is__num__normalize_I8_J,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.32/0.61 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % is_num_normalize(8)
% 0.32/0.61 thf(fact_5763_is__num__normalize_I8_J,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.32/0.61 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % is_num_normalize(8)
% 0.32/0.61 thf(fact_5764_is__num__normalize_I8_J,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 0.32/0.61 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % is_num_normalize(8)
% 0.32/0.61 thf(fact_5765_is__num__normalize_I8_J,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % is_num_normalize(8)
% 0.32/0.61 thf(fact_5766_is__num__normalize_I8_J,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.32/0.61 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % is_num_normalize(8)
% 0.32/0.61 thf(fact_5767_minus__diff__commute,axiom,
% 0.32/0.61 ! [B: real,A: real] :
% 0.32/0.61 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.32/0.61 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_commute
% 0.32/0.61 thf(fact_5768_minus__diff__commute,axiom,
% 0.32/0.61 ! [B: int,A: int] :
% 0.32/0.61 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.32/0.61 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_commute
% 0.32/0.61 thf(fact_5769_minus__diff__commute,axiom,
% 0.32/0.61 ! [B: complex,A: complex] :
% 0.32/0.61 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 0.32/0.61 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_commute
% 0.32/0.61 thf(fact_5770_minus__diff__commute,axiom,
% 0.32/0.61 ! [B: code_integer,A: code_integer] :
% 0.32/0.61 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 0.32/0.61 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_commute
% 0.32/0.61 thf(fact_5771_minus__diff__commute,axiom,
% 0.32/0.61 ! [B: rat,A: rat] :
% 0.32/0.61 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 0.32/0.61 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_commute
% 0.32/0.61 thf(fact_5772_minus__diff__minus,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_minus
% 0.32/0.61 thf(fact_5773_minus__diff__minus,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_minus
% 0.32/0.61 thf(fact_5774_minus__diff__minus,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_minus
% 0.32/0.61 thf(fact_5775_minus__diff__minus,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_minus
% 0.32/0.61 thf(fact_5776_minus__diff__minus,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_diff_minus
% 0.32/0.61 thf(fact_5777_div__minus__right,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % div_minus_right
% 0.32/0.61 thf(fact_5778_div__minus__right,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % div_minus_right
% 0.32/0.61 thf(fact_5779_minus__divide__left,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.61 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_left
% 0.32/0.61 thf(fact_5780_minus__divide__left,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.61 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_left
% 0.32/0.61 thf(fact_5781_minus__divide__left,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.61 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_left
% 0.32/0.61 thf(fact_5782_minus__divide__divide,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( divide_divide_real @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_divide
% 0.32/0.61 thf(fact_5783_minus__divide__divide,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.61 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_divide
% 0.32/0.61 thf(fact_5784_minus__divide__divide,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( divide_divide_rat @ A @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_divide
% 0.32/0.61 thf(fact_5785_minus__divide__right,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.61 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_right
% 0.32/0.61 thf(fact_5786_minus__divide__right,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.61 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_right
% 0.32/0.61 thf(fact_5787_minus__divide__right,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.61 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_right
% 0.32/0.61 thf(fact_5788_mod__minus__right,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mod_minus_right
% 0.32/0.61 thf(fact_5789_mod__minus__right,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mod_minus_right
% 0.32/0.61 thf(fact_5790_mod__minus__cong,axiom,
% 0.32/0.61 ! [A: int,B: int,A4: int] :
% 0.32/0.61 ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.61 = ( modulo_modulo_int @ A4 @ B ) )
% 0.32/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mod_minus_cong
% 0.32/0.61 thf(fact_5791_mod__minus__cong,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer,A4: code_integer] :
% 0.32/0.61 ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.32/0.61 = ( modulo364778990260209775nteger @ A4 @ B ) )
% 0.32/0.61 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.61 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % mod_minus_cong
% 0.32/0.61 thf(fact_5792_mod__minus__eq,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 0.32/0.61 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mod_minus_eq
% 0.32/0.61 thf(fact_5793_mod__minus__eq,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 0.32/0.61 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mod_minus_eq
% 0.32/0.61 thf(fact_5794_abs__ge__zero,axiom,
% 0.32/0.61 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_ge_zero
% 0.32/0.61 thf(fact_5795_abs__ge__zero,axiom,
% 0.32/0.61 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_ge_zero
% 0.32/0.61 thf(fact_5796_abs__ge__zero,axiom,
% 0.32/0.61 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_ge_zero
% 0.32/0.61 thf(fact_5797_abs__ge__zero,axiom,
% 0.32/0.61 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_ge_zero
% 0.32/0.61 thf(fact_5798_abs__of__pos,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.61 => ( ( abs_abs_Code_integer @ A )
% 0.32/0.61 = A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_of_pos
% 0.32/0.61 thf(fact_5799_abs__of__pos,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.61 => ( ( abs_abs_real @ A )
% 0.32/0.61 = A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_of_pos
% 0.32/0.61 thf(fact_5800_abs__of__pos,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.32/0.61 => ( ( abs_abs_rat @ A )
% 0.32/0.61 = A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_of_pos
% 0.32/0.61 thf(fact_5801_abs__of__pos,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.61 => ( ( abs_abs_int @ A )
% 0.32/0.61 = A ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_of_pos
% 0.32/0.61 thf(fact_5802_abs__not__less__zero,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.32/0.61
% 0.32/0.61 % abs_not_less_zero
% 0.32/0.61 thf(fact_5803_abs__not__less__zero,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.32/0.61
% 0.32/0.61 % abs_not_less_zero
% 0.32/0.61 thf(fact_5804_abs__not__less__zero,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.32/0.61
% 0.32/0.61 % abs_not_less_zero
% 0.32/0.61 thf(fact_5805_abs__not__less__zero,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % abs_not_less_zero
% 0.32/0.61 thf(fact_5806_abs__triangle__ineq,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq
% 0.32/0.61 thf(fact_5807_abs__triangle__ineq,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq
% 0.32/0.61 thf(fact_5808_abs__triangle__ineq,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq
% 0.32/0.61 thf(fact_5809_abs__triangle__ineq,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq
% 0.32/0.61 thf(fact_5810_abs__mult__less,axiom,
% 0.32/0.61 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.32/0.61 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 0.32/0.61 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult_less
% 0.32/0.61 thf(fact_5811_abs__mult__less,axiom,
% 0.32/0.61 ! [A: real,C: real,B: real,D: real] :
% 0.32/0.61 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.32/0.61 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 0.32/0.61 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult_less
% 0.32/0.61 thf(fact_5812_abs__mult__less,axiom,
% 0.32/0.61 ! [A: rat,C: rat,B: rat,D: rat] :
% 0.32/0.61 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.32/0.61 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 0.32/0.61 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult_less
% 0.32/0.61 thf(fact_5813_abs__mult__less,axiom,
% 0.32/0.61 ! [A: int,C: int,B: int,D: int] :
% 0.32/0.61 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.32/0.61 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 0.32/0.61 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult_less
% 0.32/0.61 thf(fact_5814_abs__triangle__ineq2__sym,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq2_sym
% 0.32/0.61 thf(fact_5815_abs__triangle__ineq2__sym,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq2_sym
% 0.32/0.61 thf(fact_5816_abs__triangle__ineq2__sym,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq2_sym
% 0.32/0.61 thf(fact_5817_abs__triangle__ineq2__sym,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq2_sym
% 0.32/0.61 thf(fact_5818_abs__triangle__ineq3,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq3
% 0.32/0.61 thf(fact_5819_abs__triangle__ineq3,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq3
% 0.32/0.61 thf(fact_5820_abs__triangle__ineq3,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq3
% 0.32/0.61 thf(fact_5821_abs__triangle__ineq3,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq3
% 0.32/0.61 thf(fact_5822_abs__triangle__ineq2,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq2
% 0.32/0.61 thf(fact_5823_abs__triangle__ineq2,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq2
% 0.32/0.61 thf(fact_5824_abs__triangle__ineq2,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq2
% 0.32/0.61 thf(fact_5825_abs__triangle__ineq2,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq2
% 0.32/0.61 thf(fact_5826_nonzero__abs__divide,axiom,
% 0.32/0.61 ! [B: real,A: real] :
% 0.32/0.61 ( ( B != zero_zero_real )
% 0.32/0.61 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.61 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_abs_divide
% 0.32/0.61 thf(fact_5827_nonzero__abs__divide,axiom,
% 0.32/0.61 ! [B: rat,A: rat] :
% 0.32/0.61 ( ( B != zero_zero_rat )
% 0.32/0.61 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.61 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_abs_divide
% 0.32/0.61 thf(fact_5828_neg__numeral__le__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_numeral
% 0.32/0.61 thf(fact_5829_neg__numeral__le__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_numeral
% 0.32/0.61 thf(fact_5830_neg__numeral__le__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_numeral
% 0.32/0.61 thf(fact_5831_neg__numeral__le__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_numeral
% 0.32/0.61 thf(fact_5832_not__numeral__le__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_le_neg_numeral
% 0.32/0.61 thf(fact_5833_not__numeral__le__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_le_neg_numeral
% 0.32/0.61 thf(fact_5834_not__numeral__le__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_le_neg_numeral
% 0.32/0.61 thf(fact_5835_not__numeral__le__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_le_neg_numeral
% 0.32/0.61 thf(fact_5836_zero__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( zero_zero_real
% 0.32/0.61 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_numeral
% 0.32/0.61 thf(fact_5837_zero__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( zero_zero_int
% 0.32/0.61 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_numeral
% 0.32/0.61 thf(fact_5838_zero__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( zero_zero_complex
% 0.32/0.61 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_numeral
% 0.32/0.61 thf(fact_5839_zero__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( zero_z3403309356797280102nteger
% 0.32/0.61 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_numeral
% 0.32/0.61 thf(fact_5840_zero__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( zero_zero_rat
% 0.32/0.61 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_numeral
% 0.32/0.61 thf(fact_5841_neg__numeral__less__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_numeral
% 0.32/0.61 thf(fact_5842_neg__numeral__less__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_numeral
% 0.32/0.61 thf(fact_5843_neg__numeral__less__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_numeral
% 0.32/0.61 thf(fact_5844_neg__numeral__less__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_numeral
% 0.32/0.61 thf(fact_5845_not__numeral__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_less_neg_numeral
% 0.32/0.61 thf(fact_5846_not__numeral__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_less_neg_numeral
% 0.32/0.61 thf(fact_5847_not__numeral__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_less_neg_numeral
% 0.32/0.61 thf(fact_5848_not__numeral__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num,N: num] :
% 0.32/0.61 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_less_neg_numeral
% 0.32/0.61 thf(fact_5849_le__minus__one__simps_I4_J,axiom,
% 0.32/0.61 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(4)
% 0.32/0.61 thf(fact_5850_le__minus__one__simps_I4_J,axiom,
% 0.32/0.61 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(4)
% 0.32/0.61 thf(fact_5851_le__minus__one__simps_I4_J,axiom,
% 0.32/0.61 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(4)
% 0.32/0.61 thf(fact_5852_le__minus__one__simps_I4_J,axiom,
% 0.32/0.61 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(4)
% 0.32/0.61 thf(fact_5853_le__minus__one__simps_I2_J,axiom,
% 0.32/0.61 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(2)
% 0.32/0.61 thf(fact_5854_le__minus__one__simps_I2_J,axiom,
% 0.32/0.61 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(2)
% 0.32/0.61 thf(fact_5855_le__minus__one__simps_I2_J,axiom,
% 0.32/0.61 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(2)
% 0.32/0.61 thf(fact_5856_le__minus__one__simps_I2_J,axiom,
% 0.32/0.61 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(2)
% 0.32/0.61 thf(fact_5857_zero__neq__neg__one,axiom,
% 0.32/0.61 ( zero_zero_real
% 0.32/0.61 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_one
% 0.32/0.61 thf(fact_5858_zero__neq__neg__one,axiom,
% 0.32/0.61 ( zero_zero_int
% 0.32/0.61 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_one
% 0.32/0.61 thf(fact_5859_zero__neq__neg__one,axiom,
% 0.32/0.61 ( zero_zero_complex
% 0.32/0.61 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_one
% 0.32/0.61 thf(fact_5860_zero__neq__neg__one,axiom,
% 0.32/0.61 ( zero_z3403309356797280102nteger
% 0.32/0.61 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_one
% 0.32/0.61 thf(fact_5861_zero__neq__neg__one,axiom,
% 0.32/0.61 ( zero_zero_rat
% 0.32/0.61 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_neq_neg_one
% 0.32/0.61 thf(fact_5862_add__eq__0__iff,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.61 = zero_zero_real )
% 0.32/0.61 = ( B
% 0.32/0.61 = ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_eq_0_iff
% 0.32/0.61 thf(fact_5863_add__eq__0__iff,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.61 = zero_zero_int )
% 0.32/0.61 = ( B
% 0.32/0.61 = ( uminus_uminus_int @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_eq_0_iff
% 0.32/0.61 thf(fact_5864_add__eq__0__iff,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( ( plus_plus_complex @ A @ B )
% 0.32/0.61 = zero_zero_complex )
% 0.32/0.61 = ( B
% 0.32/0.61 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_eq_0_iff
% 0.32/0.61 thf(fact_5865_add__eq__0__iff,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.32/0.61 = zero_z3403309356797280102nteger )
% 0.32/0.61 = ( B
% 0.32/0.61 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_eq_0_iff
% 0.32/0.61 thf(fact_5866_add__eq__0__iff,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.61 = zero_zero_rat )
% 0.32/0.61 = ( B
% 0.32/0.61 = ( uminus_uminus_rat @ A ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_eq_0_iff
% 0.32/0.61 thf(fact_5867_ab__group__add__class_Oab__left__minus,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.32/0.61 = zero_zero_real ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_left_minus
% 0.32/0.61 thf(fact_5868_ab__group__add__class_Oab__left__minus,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.32/0.61 = zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_left_minus
% 0.32/0.61 thf(fact_5869_ab__group__add__class_Oab__left__minus,axiom,
% 0.32/0.61 ! [A: complex] :
% 0.32/0.61 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.32/0.61 = zero_zero_complex ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_left_minus
% 0.32/0.61 thf(fact_5870_ab__group__add__class_Oab__left__minus,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.32/0.61 = zero_z3403309356797280102nteger ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_left_minus
% 0.32/0.61 thf(fact_5871_ab__group__add__class_Oab__left__minus,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.32/0.61 = zero_zero_rat ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_left_minus
% 0.32/0.61 thf(fact_5872_add_Oinverse__unique,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ( plus_plus_real @ A @ B )
% 0.32/0.61 = zero_zero_real )
% 0.32/0.61 => ( ( uminus_uminus_real @ A )
% 0.32/0.61 = B ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_unique
% 0.32/0.61 thf(fact_5873_add_Oinverse__unique,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ( plus_plus_int @ A @ B )
% 0.32/0.61 = zero_zero_int )
% 0.32/0.61 => ( ( uminus_uminus_int @ A )
% 0.32/0.61 = B ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_unique
% 0.32/0.61 thf(fact_5874_add_Oinverse__unique,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( ( plus_plus_complex @ A @ B )
% 0.32/0.61 = zero_zero_complex )
% 0.32/0.61 => ( ( uminus1482373934393186551omplex @ A )
% 0.32/0.61 = B ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_unique
% 0.32/0.61 thf(fact_5875_add_Oinverse__unique,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.32/0.61 = zero_z3403309356797280102nteger )
% 0.32/0.61 => ( ( uminus1351360451143612070nteger @ A )
% 0.32/0.61 = B ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_unique
% 0.32/0.61 thf(fact_5876_add_Oinverse__unique,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ( plus_plus_rat @ A @ B )
% 0.32/0.61 = zero_zero_rat )
% 0.32/0.61 => ( ( uminus_uminus_rat @ A )
% 0.32/0.61 = B ) ) ).
% 0.32/0.61
% 0.32/0.61 % add.inverse_unique
% 0.32/0.61 thf(fact_5877_eq__neg__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( A
% 0.32/0.61 = ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( ( plus_plus_real @ A @ B )
% 0.32/0.61 = zero_zero_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_neg_iff_add_eq_0
% 0.32/0.61 thf(fact_5878_eq__neg__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( A
% 0.32/0.61 = ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( ( plus_plus_int @ A @ B )
% 0.32/0.61 = zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_neg_iff_add_eq_0
% 0.32/0.61 thf(fact_5879_eq__neg__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( A
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.61 = ( ( plus_plus_complex @ A @ B )
% 0.32/0.61 = zero_zero_complex ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_neg_iff_add_eq_0
% 0.32/0.61 thf(fact_5880_eq__neg__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( A
% 0.32/0.61 = ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.32/0.61 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_neg_iff_add_eq_0
% 0.32/0.61 thf(fact_5881_eq__neg__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( A
% 0.32/0.61 = ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( ( plus_plus_rat @ A @ B )
% 0.32/0.61 = zero_zero_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_neg_iff_add_eq_0
% 0.32/0.61 thf(fact_5882_neg__eq__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ( uminus_uminus_real @ A )
% 0.32/0.61 = B )
% 0.32/0.61 = ( ( plus_plus_real @ A @ B )
% 0.32/0.61 = zero_zero_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_eq_iff_add_eq_0
% 0.32/0.61 thf(fact_5883_neg__eq__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ( uminus_uminus_int @ A )
% 0.32/0.61 = B )
% 0.32/0.61 = ( ( plus_plus_int @ A @ B )
% 0.32/0.61 = zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_eq_iff_add_eq_0
% 0.32/0.61 thf(fact_5884_neg__eq__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( ( uminus1482373934393186551omplex @ A )
% 0.32/0.61 = B )
% 0.32/0.61 = ( ( plus_plus_complex @ A @ B )
% 0.32/0.61 = zero_zero_complex ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_eq_iff_add_eq_0
% 0.32/0.61 thf(fact_5885_neg__eq__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ( uminus1351360451143612070nteger @ A )
% 0.32/0.61 = B )
% 0.32/0.61 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.32/0.61 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_eq_iff_add_eq_0
% 0.32/0.61 thf(fact_5886_neg__eq__iff__add__eq__0,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ( uminus_uminus_rat @ A )
% 0.32/0.61 = B )
% 0.32/0.61 = ( ( plus_plus_rat @ A @ B )
% 0.32/0.61 = zero_zero_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_eq_iff_add_eq_0
% 0.32/0.61 thf(fact_5887_less__minus__one__simps_I2_J,axiom,
% 0.32/0.61 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(2)
% 0.32/0.61 thf(fact_5888_less__minus__one__simps_I2_J,axiom,
% 0.32/0.61 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(2)
% 0.32/0.61 thf(fact_5889_less__minus__one__simps_I2_J,axiom,
% 0.32/0.61 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(2)
% 0.32/0.61 thf(fact_5890_less__minus__one__simps_I2_J,axiom,
% 0.32/0.61 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(2)
% 0.32/0.61 thf(fact_5891_less__minus__one__simps_I4_J,axiom,
% 0.32/0.61 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(4)
% 0.32/0.61 thf(fact_5892_less__minus__one__simps_I4_J,axiom,
% 0.32/0.61 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(4)
% 0.32/0.61 thf(fact_5893_less__minus__one__simps_I4_J,axiom,
% 0.32/0.61 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(4)
% 0.32/0.61 thf(fact_5894_less__minus__one__simps_I4_J,axiom,
% 0.32/0.61 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(4)
% 0.32/0.61 thf(fact_5895_numeral__times__minus__swap,axiom,
% 0.32/0.61 ! [W: num,X: real] :
% 0.32/0.61 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 0.32/0.61 = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_times_minus_swap
% 0.32/0.61 thf(fact_5896_numeral__times__minus__swap,axiom,
% 0.32/0.61 ! [W: num,X: int] :
% 0.32/0.61 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 0.32/0.61 = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_times_minus_swap
% 0.32/0.61 thf(fact_5897_numeral__times__minus__swap,axiom,
% 0.32/0.61 ! [W: num,X: complex] :
% 0.32/0.61 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 0.32/0.61 = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_times_minus_swap
% 0.32/0.61 thf(fact_5898_numeral__times__minus__swap,axiom,
% 0.32/0.61 ! [W: num,X: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 0.32/0.61 = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_times_minus_swap
% 0.32/0.61 thf(fact_5899_numeral__times__minus__swap,axiom,
% 0.32/0.61 ! [W: num,X: rat] :
% 0.32/0.61 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 0.32/0.61 = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_times_minus_swap
% 0.32/0.61 thf(fact_5900_nonzero__minus__divide__divide,axiom,
% 0.32/0.61 ! [B: real,A: real] :
% 0.32/0.61 ( ( B != zero_zero_real )
% 0.32/0.61 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( divide_divide_real @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_minus_divide_divide
% 0.32/0.61 thf(fact_5901_nonzero__minus__divide__divide,axiom,
% 0.32/0.61 ! [B: complex,A: complex] :
% 0.32/0.61 ( ( B != zero_zero_complex )
% 0.32/0.61 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.61 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_minus_divide_divide
% 0.32/0.61 thf(fact_5902_nonzero__minus__divide__divide,axiom,
% 0.32/0.61 ! [B: rat,A: rat] :
% 0.32/0.61 ( ( B != zero_zero_rat )
% 0.32/0.61 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_minus_divide_divide
% 0.32/0.61 thf(fact_5903_nonzero__minus__divide__right,axiom,
% 0.32/0.61 ! [B: real,A: real] :
% 0.32/0.61 ( ( B != zero_zero_real )
% 0.32/0.61 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.61 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_minus_divide_right
% 0.32/0.61 thf(fact_5904_nonzero__minus__divide__right,axiom,
% 0.32/0.61 ! [B: complex,A: complex] :
% 0.32/0.61 ( ( B != zero_zero_complex )
% 0.32/0.61 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.61 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_minus_divide_right
% 0.32/0.61 thf(fact_5905_nonzero__minus__divide__right,axiom,
% 0.32/0.61 ! [B: rat,A: rat] :
% 0.32/0.61 ( ( B != zero_zero_rat )
% 0.32/0.61 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.61 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_minus_divide_right
% 0.32/0.61 thf(fact_5906_numeral__neq__neg__one,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( ( numeral_numeral_real @ N )
% 0.32/0.61 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_one
% 0.32/0.61 thf(fact_5907_numeral__neq__neg__one,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( ( numeral_numeral_int @ N )
% 0.32/0.61 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_one
% 0.32/0.61 thf(fact_5908_numeral__neq__neg__one,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( ( numera6690914467698888265omplex @ N )
% 0.32/0.61 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_one
% 0.32/0.61 thf(fact_5909_numeral__neq__neg__one,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( ( numera6620942414471956472nteger @ N )
% 0.32/0.61 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_one
% 0.32/0.61 thf(fact_5910_numeral__neq__neg__one,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( ( numeral_numeral_rat @ N )
% 0.32/0.61 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % numeral_neq_neg_one
% 0.32/0.61 thf(fact_5911_one__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( one_one_real
% 0.32/0.61 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_numeral
% 0.32/0.61 thf(fact_5912_one__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( one_one_int
% 0.32/0.61 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_numeral
% 0.32/0.61 thf(fact_5913_one__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( one_one_complex
% 0.32/0.61 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_numeral
% 0.32/0.61 thf(fact_5914_one__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( one_one_Code_integer
% 0.32/0.61 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_numeral
% 0.32/0.61 thf(fact_5915_one__neq__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ( one_one_rat
% 0.32/0.61 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % one_neq_neg_numeral
% 0.32/0.61 thf(fact_5916_square__eq__1__iff,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ( ( times_times_real @ X @ X )
% 0.32/0.61 = one_one_real )
% 0.32/0.61 = ( ( X = one_one_real )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_1_iff
% 0.32/0.61 thf(fact_5917_square__eq__1__iff,axiom,
% 0.32/0.61 ! [X: int] :
% 0.32/0.61 ( ( ( times_times_int @ X @ X )
% 0.32/0.61 = one_one_int )
% 0.32/0.61 = ( ( X = one_one_int )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_1_iff
% 0.32/0.61 thf(fact_5918_square__eq__1__iff,axiom,
% 0.32/0.61 ! [X: complex] :
% 0.32/0.61 ( ( ( times_times_complex @ X @ X )
% 0.32/0.61 = one_one_complex )
% 0.32/0.61 = ( ( X = one_one_complex )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_1_iff
% 0.32/0.61 thf(fact_5919_square__eq__1__iff,axiom,
% 0.32/0.61 ! [X: code_integer] :
% 0.32/0.61 ( ( ( times_3573771949741848930nteger @ X @ X )
% 0.32/0.61 = one_one_Code_integer )
% 0.32/0.61 = ( ( X = one_one_Code_integer )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_1_iff
% 0.32/0.61 thf(fact_5920_square__eq__1__iff,axiom,
% 0.32/0.61 ! [X: rat] :
% 0.32/0.61 ( ( ( times_times_rat @ X @ X )
% 0.32/0.61 = one_one_rat )
% 0.32/0.61 = ( ( X = one_one_rat )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % square_eq_1_iff
% 0.32/0.61 thf(fact_5921_group__cancel_Osub2,axiom,
% 0.32/0.61 ! [B5: real,K: real,B: real,A: real] :
% 0.32/0.61 ( ( B5
% 0.32/0.61 = ( plus_plus_real @ K @ B ) )
% 0.32/0.61 => ( ( minus_minus_real @ A @ B5 )
% 0.32/0.61 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.sub2
% 0.32/0.61 thf(fact_5922_group__cancel_Osub2,axiom,
% 0.32/0.61 ! [B5: int,K: int,B: int,A: int] :
% 0.32/0.61 ( ( B5
% 0.32/0.61 = ( plus_plus_int @ K @ B ) )
% 0.32/0.61 => ( ( minus_minus_int @ A @ B5 )
% 0.32/0.61 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.sub2
% 0.32/0.61 thf(fact_5923_group__cancel_Osub2,axiom,
% 0.32/0.61 ! [B5: complex,K: complex,B: complex,A: complex] :
% 0.32/0.61 ( ( B5
% 0.32/0.61 = ( plus_plus_complex @ K @ B ) )
% 0.32/0.61 => ( ( minus_minus_complex @ A @ B5 )
% 0.32/0.61 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.sub2
% 0.32/0.61 thf(fact_5924_group__cancel_Osub2,axiom,
% 0.32/0.61 ! [B5: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 0.32/0.61 ( ( B5
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 0.32/0.61 => ( ( minus_8373710615458151222nteger @ A @ B5 )
% 0.32/0.61 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.sub2
% 0.32/0.61 thf(fact_5925_group__cancel_Osub2,axiom,
% 0.32/0.61 ! [B5: rat,K: rat,B: rat,A: rat] :
% 0.32/0.61 ( ( B5
% 0.32/0.61 = ( plus_plus_rat @ K @ B ) )
% 0.32/0.61 => ( ( minus_minus_rat @ A @ B5 )
% 0.32/0.61 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % group_cancel.sub2
% 0.32/0.61 thf(fact_5926_diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_minus_real
% 0.32/0.61 = ( ^ [A3: real,B4: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_conv_add_uminus
% 0.32/0.61 thf(fact_5927_diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_minus_int
% 0.32/0.61 = ( ^ [A3: int,B4: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_conv_add_uminus
% 0.32/0.61 thf(fact_5928_diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_minus_complex
% 0.32/0.61 = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_conv_add_uminus
% 0.32/0.61 thf(fact_5929_diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_8373710615458151222nteger
% 0.32/0.61 = ( ^ [A3: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_conv_add_uminus
% 0.32/0.61 thf(fact_5930_diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_minus_rat
% 0.32/0.61 = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % diff_conv_add_uminus
% 0.32/0.61 thf(fact_5931_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_minus_real
% 0.32/0.61 = ( ^ [A3: real,B4: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.32/0.61 thf(fact_5932_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_minus_int
% 0.32/0.61 = ( ^ [A3: int,B4: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.32/0.61 thf(fact_5933_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_minus_complex
% 0.32/0.61 = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.32/0.61 thf(fact_5934_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_8373710615458151222nteger
% 0.32/0.61 = ( ^ [A3: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.32/0.61 thf(fact_5935_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.32/0.61 ( minus_minus_rat
% 0.32/0.61 = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.32/0.61 thf(fact_5936_dvd__neg__div,axiom,
% 0.32/0.61 ! [B: real,A: real] :
% 0.32/0.61 ( ( dvd_dvd_real @ B @ A )
% 0.32/0.61 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 0.32/0.61 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_neg_div
% 0.32/0.61 thf(fact_5937_dvd__neg__div,axiom,
% 0.32/0.61 ! [B: int,A: int] :
% 0.32/0.61 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.61 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_neg_div
% 0.32/0.61 thf(fact_5938_dvd__neg__div,axiom,
% 0.32/0.61 ! [B: complex,A: complex] :
% 0.32/0.61 ( ( dvd_dvd_complex @ B @ A )
% 0.32/0.61 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_neg_div
% 0.32/0.61 thf(fact_5939_dvd__neg__div,axiom,
% 0.32/0.61 ! [B: code_integer,A: code_integer] :
% 0.32/0.61 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.61 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_neg_div
% 0.32/0.61 thf(fact_5940_dvd__neg__div,axiom,
% 0.32/0.61 ! [B: rat,A: rat] :
% 0.32/0.61 ( ( dvd_dvd_rat @ B @ A )
% 0.32/0.61 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.32/0.61 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_neg_div
% 0.32/0.61 thf(fact_5941_dvd__div__neg,axiom,
% 0.32/0.61 ! [B: real,A: real] :
% 0.32/0.61 ( ( dvd_dvd_real @ B @ A )
% 0.32/0.61 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 0.32/0.61 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_div_neg
% 0.32/0.61 thf(fact_5942_dvd__div__neg,axiom,
% 0.32/0.61 ! [B: int,A: int] :
% 0.32/0.61 ( ( dvd_dvd_int @ B @ A )
% 0.32/0.61 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_div_neg
% 0.32/0.61 thf(fact_5943_dvd__div__neg,axiom,
% 0.32/0.61 ! [B: complex,A: complex] :
% 0.32/0.61 ( ( dvd_dvd_complex @ B @ A )
% 0.32/0.61 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_div_neg
% 0.32/0.61 thf(fact_5944_dvd__div__neg,axiom,
% 0.32/0.61 ! [B: code_integer,A: code_integer] :
% 0.32/0.61 ( ( dvd_dvd_Code_integer @ B @ A )
% 0.32/0.61 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_div_neg
% 0.32/0.61 thf(fact_5945_dvd__div__neg,axiom,
% 0.32/0.61 ! [B: rat,A: rat] :
% 0.32/0.61 ( ( dvd_dvd_rat @ B @ A )
% 0.32/0.61 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.32/0.61 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % dvd_div_neg
% 0.32/0.61 thf(fact_5946_subset__Compl__self__eq,axiom,
% 0.32/0.61 ! [A2: set_nat] :
% 0.32/0.61 ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.32/0.61 = ( A2 = bot_bot_set_nat ) ) ).
% 0.32/0.61
% 0.32/0.61 % subset_Compl_self_eq
% 0.32/0.61 thf(fact_5947_subset__Compl__self__eq,axiom,
% 0.32/0.61 ! [A2: set_real] :
% 0.32/0.61 ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 0.32/0.61 = ( A2 = bot_bot_set_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % subset_Compl_self_eq
% 0.32/0.61 thf(fact_5948_subset__Compl__self__eq,axiom,
% 0.32/0.61 ! [A2: set_int] :
% 0.32/0.61 ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.32/0.61 = ( A2 = bot_bot_set_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % subset_Compl_self_eq
% 0.32/0.61 thf(fact_5949_sin__bound__lemma,axiom,
% 0.32/0.61 ! [X: real,Y3: real,U: real,V: real] :
% 0.32/0.61 ( ( X = Y3 )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.32/0.61 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y3 ) ) @ V ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % sin_bound_lemma
% 0.32/0.61 thf(fact_5950_real__minus__mult__self__le,axiom,
% 0.32/0.61 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % real_minus_mult_self_le
% 0.32/0.61 thf(fact_5951_zmult__eq__1__iff,axiom,
% 0.32/0.61 ! [M: int,N: int] :
% 0.32/0.61 ( ( ( times_times_int @ M @ N )
% 0.32/0.61 = one_one_int )
% 0.32/0.61 = ( ( ( M = one_one_int )
% 0.32/0.61 & ( N = one_one_int ) )
% 0.32/0.61 | ( ( M
% 0.32/0.61 = ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.61 & ( N
% 0.32/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zmult_eq_1_iff
% 0.32/0.61 thf(fact_5952_pos__zmult__eq__1__iff__lemma,axiom,
% 0.32/0.61 ! [M: int,N: int] :
% 0.32/0.61 ( ( ( times_times_int @ M @ N )
% 0.32/0.61 = one_one_int )
% 0.32/0.61 => ( ( M = one_one_int )
% 0.32/0.61 | ( M
% 0.32/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_zmult_eq_1_iff_lemma
% 0.32/0.61 thf(fact_5953_minus__int__code_I2_J,axiom,
% 0.32/0.61 ! [L2: int] :
% 0.32/0.61 ( ( minus_minus_int @ zero_zero_int @ L2 )
% 0.32/0.61 = ( uminus_uminus_int @ L2 ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_int_code(2)
% 0.32/0.61 thf(fact_5954_zmod__zminus2__not__zero,axiom,
% 0.32/0.61 ! [K: int,L2: int] :
% 0.32/0.61 ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L2 ) )
% 0.32/0.61 != zero_zero_int )
% 0.32/0.61 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.61 != zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % zmod_zminus2_not_zero
% 0.32/0.61 thf(fact_5955_zmod__zminus1__not__zero,axiom,
% 0.32/0.61 ! [K: int,L2: int] :
% 0.32/0.61 ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 0.32/0.61 != zero_zero_int )
% 0.32/0.61 => ( ( modulo_modulo_int @ K @ L2 )
% 0.32/0.61 != zero_zero_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % zmod_zminus1_not_zero
% 0.32/0.61 thf(fact_5956_minus__real__def,axiom,
% 0.32/0.61 ( minus_minus_real
% 0.32/0.61 = ( ^ [X2: real,Y: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_real_def
% 0.32/0.61 thf(fact_5957_tanh__real__lt__1,axiom,
% 0.32/0.61 ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 0.32/0.61
% 0.32/0.61 % tanh_real_lt_1
% 0.32/0.61 thf(fact_5958_dense__eq0__I,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ! [E2: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.32/0.61 => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
% 0.32/0.61 => ( X = zero_zero_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % dense_eq0_I
% 0.32/0.61 thf(fact_5959_dense__eq0__I,axiom,
% 0.32/0.61 ! [X: rat] :
% 0.32/0.61 ( ! [E2: rat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.32/0.61 => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
% 0.32/0.61 => ( X = zero_zero_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % dense_eq0_I
% 0.32/0.61 thf(fact_5960_abs__mult__pos,axiom,
% 0.32/0.61 ! [X: code_integer,Y3: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 0.32/0.61 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X )
% 0.32/0.61 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult_pos
% 0.32/0.61 thf(fact_5961_abs__mult__pos,axiom,
% 0.32/0.61 ! [X: real,Y3: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.61 => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X )
% 0.32/0.61 = ( abs_abs_real @ ( times_times_real @ Y3 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult_pos
% 0.32/0.61 thf(fact_5962_abs__mult__pos,axiom,
% 0.32/0.61 ! [X: rat,Y3: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 0.32/0.61 => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X )
% 0.32/0.61 = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult_pos
% 0.32/0.61 thf(fact_5963_abs__mult__pos,axiom,
% 0.32/0.61 ! [X: int,Y3: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.61 => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X )
% 0.32/0.61 = ( abs_abs_int @ ( times_times_int @ Y3 @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_mult_pos
% 0.32/0.61 thf(fact_5964_abs__eq__mult,axiom,
% 0.32/0.61 ! [A: code_integer,B: code_integer] :
% 0.32/0.61 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.32/0.61 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.32/0.61 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.32/0.61 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.32/0.61 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.32/0.61 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_mult
% 0.32/0.61 thf(fact_5965_abs__eq__mult,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.61 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.32/0.61 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.32/0.61 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.32/0.61 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.32/0.61 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_mult
% 0.32/0.61 thf(fact_5966_abs__eq__mult,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.32/0.61 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.32/0.61 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.32/0.61 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.32/0.61 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.32/0.61 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_mult
% 0.32/0.61 thf(fact_5967_abs__eq__mult,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.32/0.61 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.32/0.61 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.61 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.32/0.61 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.32/0.61 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_eq_mult
% 0.32/0.61 thf(fact_5968_abs__div__pos,axiom,
% 0.32/0.61 ! [Y3: real,X: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.61 => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y3 )
% 0.32/0.61 = ( abs_abs_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_div_pos
% 0.32/0.61 thf(fact_5969_abs__div__pos,axiom,
% 0.32/0.61 ! [Y3: rat,X: rat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.32/0.61 => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y3 )
% 0.32/0.61 = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_div_pos
% 0.32/0.61 thf(fact_5970_zero__le__power__abs,axiom,
% 0.32/0.61 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_le_power_abs
% 0.32/0.61 thf(fact_5971_zero__le__power__abs,axiom,
% 0.32/0.61 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_le_power_abs
% 0.32/0.61 thf(fact_5972_zero__le__power__abs,axiom,
% 0.32/0.61 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_le_power_abs
% 0.32/0.61 thf(fact_5973_zero__le__power__abs,axiom,
% 0.32/0.61 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 0.32/0.61
% 0.32/0.61 % zero_le_power_abs
% 0.32/0.61 thf(fact_5974_abs__diff__triangle__ineq,axiom,
% 0.32/0.61 ! [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 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_triangle_ineq
% 0.32/0.61 thf(fact_5975_abs__diff__triangle__ineq,axiom,
% 0.32/0.61 ! [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 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_triangle_ineq
% 0.32/0.61 thf(fact_5976_abs__diff__triangle__ineq,axiom,
% 0.32/0.61 ! [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 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_triangle_ineq
% 0.32/0.61 thf(fact_5977_abs__diff__triangle__ineq,axiom,
% 0.32/0.61 ! [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 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_triangle_ineq
% 0.32/0.61 thf(fact_5978_abs__triangle__ineq4,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq4
% 0.32/0.61 thf(fact_5979_abs__triangle__ineq4,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq4
% 0.32/0.61 thf(fact_5980_abs__triangle__ineq4,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq4
% 0.32/0.61 thf(fact_5981_abs__triangle__ineq4,axiom,
% 0.32/0.61 ! [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 ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_triangle_ineq4
% 0.32/0.61 thf(fact_5982_abs__diff__le__iff,axiom,
% 0.32/0.61 ! [X: code_integer,A: code_integer,R: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 0.32/0.61 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 0.32/0.61 & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_le_iff
% 0.32/0.61 thf(fact_5983_abs__diff__le__iff,axiom,
% 0.32/0.61 ! [X: real,A: real,R: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 0.32/0.61 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 0.32/0.61 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_le_iff
% 0.32/0.61 thf(fact_5984_abs__diff__le__iff,axiom,
% 0.32/0.61 ! [X: rat,A: rat,R: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 0.32/0.61 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 0.32/0.61 & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_le_iff
% 0.32/0.61 thf(fact_5985_abs__diff__le__iff,axiom,
% 0.32/0.61 ! [X: int,A: int,R: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 0.32/0.61 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
% 0.32/0.61 & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_le_iff
% 0.32/0.61 thf(fact_5986_abs__diff__less__iff,axiom,
% 0.32/0.61 ! [X: code_integer,A: code_integer,R: code_integer] :
% 0.32/0.61 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 0.32/0.61 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 0.32/0.61 & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_less_iff
% 0.32/0.61 thf(fact_5987_abs__diff__less__iff,axiom,
% 0.32/0.61 ! [X: real,A: real,R: real] :
% 0.32/0.61 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 0.32/0.61 = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X )
% 0.32/0.61 & ( ord_less_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_less_iff
% 0.32/0.61 thf(fact_5988_abs__diff__less__iff,axiom,
% 0.32/0.61 ! [X: rat,A: rat,R: rat] :
% 0.32/0.61 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 0.32/0.61 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 0.32/0.61 & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_less_iff
% 0.32/0.61 thf(fact_5989_abs__diff__less__iff,axiom,
% 0.32/0.61 ! [X: int,A: int,R: int] :
% 0.32/0.61 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 0.32/0.61 = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
% 0.32/0.61 & ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_diff_less_iff
% 0.32/0.61 thf(fact_5990_neg__numeral__le__zero,axiom,
% 0.32/0.61 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_zero
% 0.32/0.61 thf(fact_5991_neg__numeral__le__zero,axiom,
% 0.32/0.61 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_zero
% 0.32/0.61 thf(fact_5992_neg__numeral__le__zero,axiom,
% 0.32/0.61 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_zero
% 0.32/0.61 thf(fact_5993_neg__numeral__le__zero,axiom,
% 0.32/0.61 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_zero
% 0.32/0.61 thf(fact_5994_not__zero__le__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_zero_le_neg_numeral
% 0.32/0.61 thf(fact_5995_not__zero__le__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_zero_le_neg_numeral
% 0.32/0.61 thf(fact_5996_not__zero__le__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_zero_le_neg_numeral
% 0.32/0.61 thf(fact_5997_not__zero__le__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_zero_le_neg_numeral
% 0.32/0.61 thf(fact_5998_not__zero__less__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_zero_less_neg_numeral
% 0.32/0.61 thf(fact_5999_not__zero__less__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_zero_less_neg_numeral
% 0.32/0.61 thf(fact_6000_not__zero__less__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_zero_less_neg_numeral
% 0.32/0.61 thf(fact_6001_not__zero__less__neg__numeral,axiom,
% 0.32/0.61 ! [N: num] :
% 0.32/0.61 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_zero_less_neg_numeral
% 0.32/0.61 thf(fact_6002_neg__numeral__less__zero,axiom,
% 0.32/0.61 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_zero
% 0.32/0.61 thf(fact_6003_neg__numeral__less__zero,axiom,
% 0.32/0.61 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_zero
% 0.32/0.61 thf(fact_6004_neg__numeral__less__zero,axiom,
% 0.32/0.61 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_zero
% 0.32/0.61 thf(fact_6005_neg__numeral__less__zero,axiom,
% 0.32/0.61 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_zero
% 0.32/0.61 thf(fact_6006_le__minus__one__simps_I1_J,axiom,
% 0.32/0.61 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(1)
% 0.32/0.61 thf(fact_6007_le__minus__one__simps_I1_J,axiom,
% 0.32/0.61 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(1)
% 0.32/0.61 thf(fact_6008_le__minus__one__simps_I1_J,axiom,
% 0.32/0.61 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(1)
% 0.32/0.61 thf(fact_6009_le__minus__one__simps_I1_J,axiom,
% 0.32/0.61 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(1)
% 0.32/0.61 thf(fact_6010_le__minus__one__simps_I3_J,axiom,
% 0.32/0.61 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(3)
% 0.32/0.61 thf(fact_6011_le__minus__one__simps_I3_J,axiom,
% 0.32/0.61 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(3)
% 0.32/0.61 thf(fact_6012_le__minus__one__simps_I3_J,axiom,
% 0.32/0.61 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(3)
% 0.32/0.61 thf(fact_6013_le__minus__one__simps_I3_J,axiom,
% 0.32/0.61 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_one_simps(3)
% 0.32/0.61 thf(fact_6014_less__minus__one__simps_I1_J,axiom,
% 0.32/0.61 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(1)
% 0.32/0.61 thf(fact_6015_less__minus__one__simps_I1_J,axiom,
% 0.32/0.61 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(1)
% 0.32/0.61 thf(fact_6016_less__minus__one__simps_I1_J,axiom,
% 0.32/0.61 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(1)
% 0.32/0.61 thf(fact_6017_less__minus__one__simps_I1_J,axiom,
% 0.32/0.61 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(1)
% 0.32/0.61 thf(fact_6018_less__minus__one__simps_I3_J,axiom,
% 0.32/0.61 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(3)
% 0.32/0.61 thf(fact_6019_less__minus__one__simps_I3_J,axiom,
% 0.32/0.61 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(3)
% 0.32/0.61 thf(fact_6020_less__minus__one__simps_I3_J,axiom,
% 0.32/0.61 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(3)
% 0.32/0.61 thf(fact_6021_less__minus__one__simps_I3_J,axiom,
% 0.32/0.61 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_one_simps(3)
% 0.32/0.61 thf(fact_6022_neg__numeral__le__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_one
% 0.32/0.61 thf(fact_6023_neg__numeral__le__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_one
% 0.32/0.61 thf(fact_6024_neg__numeral__le__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_one
% 0.32/0.61 thf(fact_6025_neg__numeral__le__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_one
% 0.32/0.61 thf(fact_6026_neg__one__le__numeral,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_le_numeral
% 0.32/0.61 thf(fact_6027_neg__one__le__numeral,axiom,
% 0.32/0.61 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_le_numeral
% 0.32/0.61 thf(fact_6028_neg__one__le__numeral,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_le_numeral
% 0.32/0.61 thf(fact_6029_neg__one__le__numeral,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_le_numeral
% 0.32/0.61 thf(fact_6030_neg__numeral__le__neg__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_neg_one
% 0.32/0.61 thf(fact_6031_neg__numeral__le__neg__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_neg_one
% 0.32/0.61 thf(fact_6032_neg__numeral__le__neg__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_neg_one
% 0.32/0.61 thf(fact_6033_neg__numeral__le__neg__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_le_neg_one
% 0.32/0.61 thf(fact_6034_not__numeral__le__neg__one,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_le_neg_one
% 0.32/0.61 thf(fact_6035_not__numeral__le__neg__one,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_le_neg_one
% 0.32/0.61 thf(fact_6036_not__numeral__le__neg__one,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_le_neg_one
% 0.32/0.61 thf(fact_6037_not__numeral__le__neg__one,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_le_neg_one
% 0.32/0.61 thf(fact_6038_not__one__le__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_one_le_neg_numeral
% 0.32/0.61 thf(fact_6039_not__one__le__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_one_le_neg_numeral
% 0.32/0.61 thf(fact_6040_not__one__le__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_one_le_neg_numeral
% 0.32/0.61 thf(fact_6041_not__one__le__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_one_le_neg_numeral
% 0.32/0.61 thf(fact_6042_not__neg__one__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_neg_one_less_neg_numeral
% 0.32/0.61 thf(fact_6043_not__neg__one__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_neg_one_less_neg_numeral
% 0.32/0.61 thf(fact_6044_not__neg__one__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_neg_one_less_neg_numeral
% 0.32/0.61 thf(fact_6045_not__neg__one__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_neg_one_less_neg_numeral
% 0.32/0.61 thf(fact_6046_not__one__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_one_less_neg_numeral
% 0.32/0.61 thf(fact_6047_not__one__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_one_less_neg_numeral
% 0.32/0.61 thf(fact_6048_not__one__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_one_less_neg_numeral
% 0.32/0.61 thf(fact_6049_not__one__less__neg__numeral,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_one_less_neg_numeral
% 0.32/0.61 thf(fact_6050_not__numeral__less__neg__one,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_less_neg_one
% 0.32/0.61 thf(fact_6051_not__numeral__less__neg__one,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_less_neg_one
% 0.32/0.61 thf(fact_6052_not__numeral__less__neg__one,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_less_neg_one
% 0.32/0.61 thf(fact_6053_not__numeral__less__neg__one,axiom,
% 0.32/0.61 ! [M: num] :
% 0.32/0.61 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % not_numeral_less_neg_one
% 0.32/0.61 thf(fact_6054_neg__one__less__numeral,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_less_numeral
% 0.32/0.61 thf(fact_6055_neg__one__less__numeral,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_less_numeral
% 0.32/0.61 thf(fact_6056_neg__one__less__numeral,axiom,
% 0.32/0.61 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_less_numeral
% 0.32/0.61 thf(fact_6057_neg__one__less__numeral,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_less_numeral
% 0.32/0.61 thf(fact_6058_neg__numeral__less__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_one
% 0.32/0.61 thf(fact_6059_neg__numeral__less__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_one
% 0.32/0.61 thf(fact_6060_neg__numeral__less__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_one
% 0.32/0.61 thf(fact_6061_neg__numeral__less__one,axiom,
% 0.32/0.61 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.32/0.61
% 0.32/0.61 % neg_numeral_less_one
% 0.32/0.61 thf(fact_6062_eq__minus__divide__eq,axiom,
% 0.32/0.61 ! [A: real,B: real,C: real] :
% 0.32/0.61 ( ( A
% 0.32/0.61 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.32/0.61 = ( ( ( C != zero_zero_real )
% 0.32/0.61 => ( ( times_times_real @ A @ C )
% 0.32/0.61 = ( uminus_uminus_real @ B ) ) )
% 0.32/0.61 & ( ( C = zero_zero_real )
% 0.32/0.61 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_minus_divide_eq
% 0.32/0.61 thf(fact_6063_eq__minus__divide__eq,axiom,
% 0.32/0.61 ! [A: complex,B: complex,C: complex] :
% 0.32/0.61 ( ( A
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 0.32/0.61 = ( ( ( C != zero_zero_complex )
% 0.32/0.61 => ( ( times_times_complex @ A @ C )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.32/0.61 & ( ( C = zero_zero_complex )
% 0.32/0.61 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_minus_divide_eq
% 0.32/0.61 thf(fact_6064_eq__minus__divide__eq,axiom,
% 0.32/0.61 ! [A: rat,B: rat,C: rat] :
% 0.32/0.61 ( ( A
% 0.32/0.61 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.32/0.61 = ( ( ( C != zero_zero_rat )
% 0.32/0.61 => ( ( times_times_rat @ A @ C )
% 0.32/0.61 = ( uminus_uminus_rat @ B ) ) )
% 0.32/0.61 & ( ( C = zero_zero_rat )
% 0.32/0.61 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_minus_divide_eq
% 0.32/0.61 thf(fact_6065_minus__divide__eq__eq,axiom,
% 0.32/0.61 ! [B: real,C: real,A: real] :
% 0.32/0.61 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 0.32/0.61 = A )
% 0.32/0.61 = ( ( ( C != zero_zero_real )
% 0.32/0.61 => ( ( uminus_uminus_real @ B )
% 0.32/0.61 = ( times_times_real @ A @ C ) ) )
% 0.32/0.61 & ( ( C = zero_zero_real )
% 0.32/0.61 => ( A = zero_zero_real ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_eq_eq
% 0.32/0.61 thf(fact_6066_minus__divide__eq__eq,axiom,
% 0.32/0.61 ! [B: complex,C: complex,A: complex] :
% 0.32/0.61 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.61 = A )
% 0.32/0.61 = ( ( ( C != zero_zero_complex )
% 0.32/0.61 => ( ( uminus1482373934393186551omplex @ B )
% 0.32/0.61 = ( times_times_complex @ A @ C ) ) )
% 0.32/0.61 & ( ( C = zero_zero_complex )
% 0.32/0.61 => ( A = zero_zero_complex ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_eq_eq
% 0.32/0.61 thf(fact_6067_minus__divide__eq__eq,axiom,
% 0.32/0.61 ! [B: rat,C: rat,A: rat] :
% 0.32/0.61 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.61 = A )
% 0.32/0.61 = ( ( ( C != zero_zero_rat )
% 0.32/0.61 => ( ( uminus_uminus_rat @ B )
% 0.32/0.61 = ( times_times_rat @ A @ C ) ) )
% 0.32/0.61 & ( ( C = zero_zero_rat )
% 0.32/0.61 => ( A = zero_zero_rat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_eq_eq
% 0.32/0.61 thf(fact_6068_nonzero__neg__divide__eq__eq,axiom,
% 0.32/0.61 ! [B: real,A: real,C: real] :
% 0.32/0.61 ( ( B != zero_zero_real )
% 0.32/0.61 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.61 = C )
% 0.32/0.61 = ( ( uminus_uminus_real @ A )
% 0.32/0.61 = ( times_times_real @ C @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_neg_divide_eq_eq
% 0.32/0.61 thf(fact_6069_nonzero__neg__divide__eq__eq,axiom,
% 0.32/0.61 ! [B: complex,A: complex,C: complex] :
% 0.32/0.61 ( ( B != zero_zero_complex )
% 0.32/0.61 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.61 = C )
% 0.32/0.61 = ( ( uminus1482373934393186551omplex @ A )
% 0.32/0.61 = ( times_times_complex @ C @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_neg_divide_eq_eq
% 0.32/0.61 thf(fact_6070_nonzero__neg__divide__eq__eq,axiom,
% 0.32/0.61 ! [B: rat,A: rat,C: rat] :
% 0.32/0.61 ( ( B != zero_zero_rat )
% 0.32/0.61 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.32/0.61 = C )
% 0.32/0.61 = ( ( uminus_uminus_rat @ A )
% 0.32/0.61 = ( times_times_rat @ C @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_neg_divide_eq_eq
% 0.32/0.61 thf(fact_6071_nonzero__neg__divide__eq__eq2,axiom,
% 0.32/0.61 ! [B: real,C: real,A: real] :
% 0.32/0.61 ( ( B != zero_zero_real )
% 0.32/0.61 => ( ( C
% 0.32/0.61 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 0.32/0.61 = ( ( times_times_real @ C @ B )
% 0.32/0.61 = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_neg_divide_eq_eq2
% 0.32/0.61 thf(fact_6072_nonzero__neg__divide__eq__eq2,axiom,
% 0.32/0.61 ! [B: complex,C: complex,A: complex] :
% 0.32/0.61 ( ( B != zero_zero_complex )
% 0.32/0.61 => ( ( C
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.32/0.61 = ( ( times_times_complex @ C @ B )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_neg_divide_eq_eq2
% 0.32/0.61 thf(fact_6073_nonzero__neg__divide__eq__eq2,axiom,
% 0.32/0.61 ! [B: rat,C: rat,A: rat] :
% 0.32/0.61 ( ( B != zero_zero_rat )
% 0.32/0.61 => ( ( C
% 0.32/0.61 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 0.32/0.61 = ( ( times_times_rat @ C @ B )
% 0.32/0.61 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % nonzero_neg_divide_eq_eq2
% 0.32/0.61 thf(fact_6074_mult__1s__ring__1_I1_J,axiom,
% 0.32/0.61 ! [B: real] :
% 0.32/0.61 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 0.32/0.61 = ( uminus_uminus_real @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(1)
% 0.32/0.61 thf(fact_6075_mult__1s__ring__1_I1_J,axiom,
% 0.32/0.61 ! [B: int] :
% 0.32/0.61 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 0.32/0.61 = ( uminus_uminus_int @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(1)
% 0.32/0.61 thf(fact_6076_mult__1s__ring__1_I1_J,axiom,
% 0.32/0.61 ! [B: complex] :
% 0.32/0.61 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(1)
% 0.32/0.61 thf(fact_6077_mult__1s__ring__1_I1_J,axiom,
% 0.32/0.61 ! [B: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(1)
% 0.32/0.61 thf(fact_6078_mult__1s__ring__1_I1_J,axiom,
% 0.32/0.61 ! [B: rat] :
% 0.32/0.61 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 0.32/0.61 = ( uminus_uminus_rat @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(1)
% 0.32/0.61 thf(fact_6079_mult__1s__ring__1_I2_J,axiom,
% 0.32/0.61 ! [B: real] :
% 0.32/0.61 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 0.32/0.61 = ( uminus_uminus_real @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(2)
% 0.32/0.61 thf(fact_6080_mult__1s__ring__1_I2_J,axiom,
% 0.32/0.61 ! [B: int] :
% 0.32/0.61 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 0.32/0.61 = ( uminus_uminus_int @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(2)
% 0.32/0.61 thf(fact_6081_mult__1s__ring__1_I2_J,axiom,
% 0.32/0.61 ! [B: complex] :
% 0.32/0.61 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(2)
% 0.32/0.61 thf(fact_6082_mult__1s__ring__1_I2_J,axiom,
% 0.32/0.61 ! [B: code_integer] :
% 0.32/0.61 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(2)
% 0.32/0.61 thf(fact_6083_mult__1s__ring__1_I2_J,axiom,
% 0.32/0.61 ! [B: rat] :
% 0.32/0.61 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 0.32/0.61 = ( uminus_uminus_rat @ B ) ) ).
% 0.32/0.61
% 0.32/0.61 % mult_1s_ring_1(2)
% 0.32/0.61 thf(fact_6084_divide__eq__minus__1__iff,axiom,
% 0.32/0.61 ! [A: real,B: real] :
% 0.32/0.61 ( ( ( divide_divide_real @ A @ B )
% 0.32/0.61 = ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.61 = ( ( B != zero_zero_real )
% 0.32/0.61 & ( A
% 0.32/0.61 = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_eq_minus_1_iff
% 0.32/0.61 thf(fact_6085_divide__eq__minus__1__iff,axiom,
% 0.32/0.61 ! [A: complex,B: complex] :
% 0.32/0.61 ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.61 = ( ( B != zero_zero_complex )
% 0.32/0.61 & ( A
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_eq_minus_1_iff
% 0.32/0.61 thf(fact_6086_divide__eq__minus__1__iff,axiom,
% 0.32/0.61 ! [A: rat,B: rat] :
% 0.32/0.61 ( ( ( divide_divide_rat @ A @ B )
% 0.32/0.61 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.61 = ( ( B != zero_zero_rat )
% 0.32/0.61 & ( A
% 0.32/0.61 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_eq_minus_1_iff
% 0.32/0.61 thf(fact_6087_uminus__numeral__One,axiom,
% 0.32/0.61 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 0.32/0.61 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_numeral_One
% 0.32/0.61 thf(fact_6088_uminus__numeral__One,axiom,
% 0.32/0.61 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 0.32/0.61 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_numeral_One
% 0.32/0.61 thf(fact_6089_uminus__numeral__One,axiom,
% 0.32/0.61 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_numeral_One
% 0.32/0.61 thf(fact_6090_uminus__numeral__One,axiom,
% 0.32/0.61 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_numeral_One
% 0.32/0.61 thf(fact_6091_uminus__numeral__One,axiom,
% 0.32/0.61 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 0.32/0.61 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_numeral_One
% 0.32/0.61 thf(fact_6092_power__minus,axiom,
% 0.32/0.61 ! [A: real,N: nat] :
% 0.32/0.61 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.32/0.61 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus
% 0.32/0.61 thf(fact_6093_power__minus,axiom,
% 0.32/0.61 ! [A: int,N: nat] :
% 0.32/0.61 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.32/0.61 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus
% 0.32/0.61 thf(fact_6094_power__minus,axiom,
% 0.32/0.61 ! [A: complex,N: nat] :
% 0.32/0.61 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.32/0.61 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus
% 0.32/0.61 thf(fact_6095_power__minus,axiom,
% 0.32/0.61 ! [A: code_integer,N: nat] :
% 0.32/0.61 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.32/0.61 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus
% 0.32/0.61 thf(fact_6096_power__minus,axiom,
% 0.32/0.61 ! [A: rat,N: nat] :
% 0.32/0.61 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.32/0.61 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus
% 0.32/0.61 thf(fact_6097_power__minus__Bit0,axiom,
% 0.32/0.61 ! [X: real,K: num] :
% 0.32/0.61 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.32/0.61 = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus_Bit0
% 0.32/0.61 thf(fact_6098_power__minus__Bit0,axiom,
% 0.32/0.61 ! [X: int,K: num] :
% 0.32/0.61 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.32/0.61 = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus_Bit0
% 0.32/0.61 thf(fact_6099_power__minus__Bit0,axiom,
% 0.32/0.61 ! [X: complex,K: num] :
% 0.32/0.61 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.32/0.61 = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus_Bit0
% 0.32/0.61 thf(fact_6100_power__minus__Bit0,axiom,
% 0.32/0.61 ! [X: code_integer,K: num] :
% 0.32/0.61 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.32/0.61 = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus_Bit0
% 0.32/0.61 thf(fact_6101_power__minus__Bit0,axiom,
% 0.32/0.61 ! [X: rat,K: num] :
% 0.32/0.61 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.32/0.61 = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_minus_Bit0
% 0.32/0.61 thf(fact_6102_lemma__interval__lt,axiom,
% 0.32/0.61 ! [A: real,X: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ A @ X )
% 0.32/0.61 => ( ( ord_less_real @ X @ B )
% 0.32/0.61 => ? [D3: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.61 & ! [Y5: real] :
% 0.32/0.61 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
% 0.32/0.61 => ( ( ord_less_real @ A @ Y5 )
% 0.32/0.61 & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % lemma_interval_lt
% 0.32/0.61 thf(fact_6103_real__0__less__add__iff,axiom,
% 0.32/0.61 ! [X: real,Y3: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y3 ) )
% 0.32/0.61 = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y3 ) ) ).
% 0.32/0.61
% 0.32/0.61 % real_0_less_add_iff
% 0.32/0.61 thf(fact_6104_real__add__less__0__iff,axiom,
% 0.32/0.61 ! [X: real,Y3: real] :
% 0.32/0.61 ( ( ord_less_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
% 0.32/0.61 = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % real_add_less_0_iff
% 0.32/0.61 thf(fact_6105_real__add__le__0__iff,axiom,
% 0.32/0.61 ! [X: real,Y3: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
% 0.32/0.61 = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % real_add_le_0_iff
% 0.32/0.61 thf(fact_6106_real__0__le__add__iff,axiom,
% 0.32/0.61 ! [X: real,Y3: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y3 ) )
% 0.32/0.61 = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y3 ) ) ).
% 0.32/0.61
% 0.32/0.61 % real_0_le_add_iff
% 0.32/0.61 thf(fact_6107_zmod__zminus2__eq__if,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.61 = zero_zero_int )
% 0.32/0.61 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = zero_zero_int ) )
% 0.32/0.61 & ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.61 != zero_zero_int )
% 0.32/0.61 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.61 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zmod_zminus2_eq_if
% 0.32/0.61 thf(fact_6108_zmod__zminus1__eq__if,axiom,
% 0.32/0.61 ! [A: int,B: int] :
% 0.32/0.61 ( ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.61 = zero_zero_int )
% 0.32/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = zero_zero_int ) )
% 0.32/0.61 & ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.61 != zero_zero_int )
% 0.32/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.61 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zmod_zminus1_eq_if
% 0.32/0.61 thf(fact_6109_abs__add__one__gt__zero,axiom,
% 0.32/0.61 ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_add_one_gt_zero
% 0.32/0.61 thf(fact_6110_abs__add__one__gt__zero,axiom,
% 0.32/0.61 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_add_one_gt_zero
% 0.32/0.61 thf(fact_6111_abs__add__one__gt__zero,axiom,
% 0.32/0.61 ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_add_one_gt_zero
% 0.32/0.61 thf(fact_6112_abs__add__one__gt__zero,axiom,
% 0.32/0.61 ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_add_one_gt_zero
% 0.32/0.61 thf(fact_6113_pos__minus__divide__less__eq,axiom,
% 0.32/0.61 ! [C: real,B: real,A: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.32/0.61 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_minus_divide_less_eq
% 0.32/0.61 thf(fact_6114_pos__minus__divide__less__eq,axiom,
% 0.32/0.61 ! [C: rat,B: rat,A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.32/0.61 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_minus_divide_less_eq
% 0.32/0.61 thf(fact_6115_pos__less__minus__divide__eq,axiom,
% 0.32/0.61 ! [C: real,A: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.32/0.61 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_less_minus_divide_eq
% 0.32/0.61 thf(fact_6116_pos__less__minus__divide__eq,axiom,
% 0.32/0.61 ! [C: rat,A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.32/0.61 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_less_minus_divide_eq
% 0.32/0.61 thf(fact_6117_neg__minus__divide__less__eq,axiom,
% 0.32/0.61 ! [C: real,B: real,A: real] :
% 0.32/0.61 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.32/0.61 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_minus_divide_less_eq
% 0.32/0.61 thf(fact_6118_neg__minus__divide__less__eq,axiom,
% 0.32/0.61 ! [C: rat,B: rat,A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.32/0.61 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_minus_divide_less_eq
% 0.32/0.61 thf(fact_6119_neg__less__minus__divide__eq,axiom,
% 0.32/0.61 ! [C: real,A: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.32/0.61 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_minus_divide_eq
% 0.32/0.61 thf(fact_6120_neg__less__minus__divide__eq,axiom,
% 0.32/0.61 ! [C: rat,A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.32/0.61 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_less_minus_divide_eq
% 0.32/0.61 thf(fact_6121_minus__divide__less__eq,axiom,
% 0.32/0.61 ! [B: real,C: real,A: real] :
% 0.32/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.32/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_less_eq
% 0.32/0.61 thf(fact_6122_minus__divide__less__eq,axiom,
% 0.32/0.61 ! [B: rat,C: rat,A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.32/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_less_eq
% 0.32/0.61 thf(fact_6123_less__minus__divide__eq,axiom,
% 0.32/0.61 ! [A: real,B: real,C: real] :
% 0.32/0.61 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.32/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_divide_eq
% 0.32/0.61 thf(fact_6124_less__minus__divide__eq,axiom,
% 0.32/0.61 ! [A: rat,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.32/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_minus_divide_eq
% 0.32/0.61 thf(fact_6125_eq__divide__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [W: num,B: real,C: real] :
% 0.32/0.61 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.61 = ( divide_divide_real @ B @ C ) )
% 0.32/0.61 = ( ( ( C != zero_zero_real )
% 0.32/0.61 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 0.32/0.61 = B ) )
% 0.32/0.61 & ( ( C = zero_zero_real )
% 0.32/0.61 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.61 = zero_zero_real ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_divide_eq_numeral(2)
% 0.32/0.61 thf(fact_6126_eq__divide__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [W: num,B: complex,C: complex] :
% 0.32/0.61 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.61 = ( divide1717551699836669952omplex @ B @ C ) )
% 0.32/0.61 = ( ( ( C != zero_zero_complex )
% 0.32/0.61 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 0.32/0.61 = B ) )
% 0.32/0.61 & ( ( C = zero_zero_complex )
% 0.32/0.61 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.61 = zero_zero_complex ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_divide_eq_numeral(2)
% 0.32/0.61 thf(fact_6127_eq__divide__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [W: num,B: rat,C: rat] :
% 0.32/0.61 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.61 = ( divide_divide_rat @ B @ C ) )
% 0.32/0.61 = ( ( ( C != zero_zero_rat )
% 0.32/0.61 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 0.32/0.61 = B ) )
% 0.32/0.61 & ( ( C = zero_zero_rat )
% 0.32/0.61 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.61 = zero_zero_rat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % eq_divide_eq_numeral(2)
% 0.32/0.61 thf(fact_6128_divide__eq__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [B: real,C: real,W: num] :
% 0.32/0.61 ( ( ( divide_divide_real @ B @ C )
% 0.32/0.61 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.61 = ( ( ( C != zero_zero_real )
% 0.32/0.61 => ( B
% 0.32/0.61 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.32/0.61 & ( ( C = zero_zero_real )
% 0.32/0.61 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.61 = zero_zero_real ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_eq_eq_numeral(2)
% 0.32/0.61 thf(fact_6129_divide__eq__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [B: complex,C: complex,W: num] :
% 0.32/0.61 ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.61 = ( ( ( C != zero_zero_complex )
% 0.32/0.61 => ( B
% 0.32/0.61 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 0.32/0.61 & ( ( C = zero_zero_complex )
% 0.32/0.61 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.61 = zero_zero_complex ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_eq_eq_numeral(2)
% 0.32/0.61 thf(fact_6130_divide__eq__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [B: rat,C: rat,W: num] :
% 0.32/0.61 ( ( ( divide_divide_rat @ B @ C )
% 0.32/0.61 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.61 = ( ( ( C != zero_zero_rat )
% 0.32/0.61 => ( B
% 0.32/0.61 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.32/0.61 & ( ( C = zero_zero_rat )
% 0.32/0.61 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 0.32/0.61 = zero_zero_rat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_eq_eq_numeral(2)
% 0.32/0.61 thf(fact_6131_minus__divide__add__eq__iff,axiom,
% 0.32/0.61 ! [Z: real,X: real,Y3: real] :
% 0.32/0.61 ( ( Z != zero_zero_real )
% 0.32/0.61 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y3 )
% 0.32/0.61 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_add_eq_iff
% 0.32/0.61 thf(fact_6132_minus__divide__add__eq__iff,axiom,
% 0.32/0.61 ! [Z: complex,X: complex,Y3: complex] :
% 0.32/0.61 ( ( Z != zero_zero_complex )
% 0.32/0.61 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y3 )
% 0.32/0.61 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_add_eq_iff
% 0.32/0.61 thf(fact_6133_minus__divide__add__eq__iff,axiom,
% 0.32/0.61 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.61 ( ( Z != zero_zero_rat )
% 0.32/0.61 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y3 )
% 0.32/0.61 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_add_eq_iff
% 0.32/0.61 thf(fact_6134_add__divide__eq__if__simps_I3_J,axiom,
% 0.32/0.61 ! [Z: real,A: real,B: real] :
% 0.32/0.61 ( ( ( Z = zero_zero_real )
% 0.32/0.61 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.32/0.61 = B ) )
% 0.32/0.61 & ( ( Z != zero_zero_real )
% 0.32/0.61 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.32/0.61 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(3)
% 0.32/0.61 thf(fact_6135_add__divide__eq__if__simps_I3_J,axiom,
% 0.32/0.61 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.61 ( ( ( Z = zero_zero_complex )
% 0.32/0.61 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.32/0.61 = B ) )
% 0.32/0.61 & ( ( Z != zero_zero_complex )
% 0.32/0.61 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.32/0.61 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(3)
% 0.32/0.61 thf(fact_6136_add__divide__eq__if__simps_I3_J,axiom,
% 0.32/0.61 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.61 ( ( ( Z = zero_zero_rat )
% 0.32/0.61 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.32/0.61 = B ) )
% 0.32/0.61 & ( ( Z != zero_zero_rat )
% 0.32/0.61 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.32/0.61 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(3)
% 0.32/0.61 thf(fact_6137_minus__divide__diff__eq__iff,axiom,
% 0.32/0.61 ! [Z: real,X: real,Y3: real] :
% 0.32/0.61 ( ( Z != zero_zero_real )
% 0.32/0.61 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y3 )
% 0.32/0.61 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_diff_eq_iff
% 0.32/0.61 thf(fact_6138_minus__divide__diff__eq__iff,axiom,
% 0.32/0.61 ! [Z: complex,X: complex,Y3: complex] :
% 0.32/0.61 ( ( Z != zero_zero_complex )
% 0.32/0.61 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y3 )
% 0.32/0.61 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_diff_eq_iff
% 0.32/0.61 thf(fact_6139_minus__divide__diff__eq__iff,axiom,
% 0.32/0.61 ! [Z: rat,X: rat,Y3: rat] :
% 0.32/0.61 ( ( Z != zero_zero_rat )
% 0.32/0.61 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y3 )
% 0.32/0.61 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_diff_eq_iff
% 0.32/0.61 thf(fact_6140_add__divide__eq__if__simps_I5_J,axiom,
% 0.32/0.61 ! [Z: real,A: real,B: real] :
% 0.32/0.61 ( ( ( Z = zero_zero_real )
% 0.32/0.61 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.32/0.61 = ( uminus_uminus_real @ B ) ) )
% 0.32/0.61 & ( ( Z != zero_zero_real )
% 0.32/0.61 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 0.32/0.61 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(5)
% 0.32/0.61 thf(fact_6141_add__divide__eq__if__simps_I5_J,axiom,
% 0.32/0.61 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.61 ( ( ( Z = zero_zero_complex )
% 0.32/0.61 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.32/0.61 & ( ( Z != zero_zero_complex )
% 0.32/0.61 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 0.32/0.61 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(5)
% 0.32/0.61 thf(fact_6142_add__divide__eq__if__simps_I5_J,axiom,
% 0.32/0.61 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.61 ( ( ( Z = zero_zero_rat )
% 0.32/0.61 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.32/0.61 = ( uminus_uminus_rat @ B ) ) )
% 0.32/0.61 & ( ( Z != zero_zero_rat )
% 0.32/0.61 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 0.32/0.61 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(5)
% 0.32/0.61 thf(fact_6143_add__divide__eq__if__simps_I6_J,axiom,
% 0.32/0.61 ! [Z: real,A: real,B: real] :
% 0.32/0.61 ( ( ( Z = zero_zero_real )
% 0.32/0.61 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.32/0.61 = ( uminus_uminus_real @ B ) ) )
% 0.32/0.61 & ( ( Z != zero_zero_real )
% 0.32/0.61 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 0.32/0.61 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(6)
% 0.32/0.61 thf(fact_6144_add__divide__eq__if__simps_I6_J,axiom,
% 0.32/0.61 ! [Z: complex,A: complex,B: complex] :
% 0.32/0.61 ( ( ( Z = zero_zero_complex )
% 0.32/0.61 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ B ) ) )
% 0.32/0.61 & ( ( Z != zero_zero_complex )
% 0.32/0.61 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 0.32/0.61 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(6)
% 0.32/0.61 thf(fact_6145_add__divide__eq__if__simps_I6_J,axiom,
% 0.32/0.61 ! [Z: rat,A: rat,B: rat] :
% 0.32/0.61 ( ( ( Z = zero_zero_rat )
% 0.32/0.61 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.32/0.61 = ( uminus_uminus_rat @ B ) ) )
% 0.32/0.61 & ( ( Z != zero_zero_rat )
% 0.32/0.61 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 0.32/0.61 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % add_divide_eq_if_simps(6)
% 0.32/0.61 thf(fact_6146_even__minus,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 0.32/0.61 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_minus
% 0.32/0.61 thf(fact_6147_even__minus,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.32/0.61 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 0.32/0.61
% 0.32/0.61 % even_minus
% 0.32/0.61 thf(fact_6148_lemma__interval,axiom,
% 0.32/0.61 ! [A: real,X: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ A @ X )
% 0.32/0.61 => ( ( ord_less_real @ X @ B )
% 0.32/0.61 => ? [D3: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.61 & ! [Y5: real] :
% 0.32/0.61 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
% 0.32/0.61 => ( ( ord_less_eq_real @ A @ Y5 )
% 0.32/0.61 & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % lemma_interval
% 0.32/0.61 thf(fact_6149_power2__eq__iff,axiom,
% 0.32/0.61 ! [X: real,Y3: real] :
% 0.32/0.61 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.61 = ( ( X = Y3 )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_iff
% 0.32/0.61 thf(fact_6150_power2__eq__iff,axiom,
% 0.32/0.61 ! [X: int,Y3: int] :
% 0.32/0.61 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.61 = ( ( X = Y3 )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_iff
% 0.32/0.61 thf(fact_6151_power2__eq__iff,axiom,
% 0.32/0.61 ! [X: complex,Y3: complex] :
% 0.32/0.61 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.61 = ( ( X = Y3 )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_iff
% 0.32/0.61 thf(fact_6152_power2__eq__iff,axiom,
% 0.32/0.61 ! [X: code_integer,Y3: code_integer] :
% 0.32/0.61 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.61 = ( ( X = Y3 )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_iff
% 0.32/0.61 thf(fact_6153_power2__eq__iff,axiom,
% 0.32/0.61 ! [X: rat,Y3: rat] :
% 0.32/0.61 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.61 = ( ( X = Y3 )
% 0.32/0.61 | ( X
% 0.32/0.61 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_iff
% 0.32/0.61 thf(fact_6154_ln__add__one__self__le__self2,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.61 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 0.32/0.61
% 0.32/0.61 % ln_add_one_self_le_self2
% 0.32/0.61 thf(fact_6155_verit__less__mono__div__int2,axiom,
% 0.32/0.61 ! [A2: int,B5: int,N: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ A2 @ B5 )
% 0.32/0.61 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.32/0.61 => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % verit_less_mono_div_int2
% 0.32/0.61 thf(fact_6156_div__eq__minus1,axiom,
% 0.32/0.61 ! [B: int] :
% 0.32/0.61 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.61 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.32/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % div_eq_minus1
% 0.32/0.61 thf(fact_6157_abs__le__square__iff,axiom,
% 0.32/0.61 ! [X: code_integer,Y3: code_integer] :
% 0.32/0.61 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y3 ) )
% 0.32/0.61 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_le_square_iff
% 0.32/0.61 thf(fact_6158_abs__le__square__iff,axiom,
% 0.32/0.61 ! [X: real,Y3: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y3 ) )
% 0.32/0.61 = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_le_square_iff
% 0.32/0.61 thf(fact_6159_abs__le__square__iff,axiom,
% 0.32/0.61 ! [X: rat,Y3: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y3 ) )
% 0.32/0.61 = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_le_square_iff
% 0.32/0.61 thf(fact_6160_abs__le__square__iff,axiom,
% 0.32/0.61 ! [X: int,Y3: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y3 ) )
% 0.32/0.61 = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_le_square_iff
% 0.32/0.61 thf(fact_6161_abs__square__eq__1,axiom,
% 0.32/0.61 ! [X: code_integer] :
% 0.32/0.61 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_Code_integer )
% 0.32/0.61 = ( ( abs_abs_Code_integer @ X )
% 0.32/0.61 = one_one_Code_integer ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_square_eq_1
% 0.32/0.61 thf(fact_6162_abs__square__eq__1,axiom,
% 0.32/0.61 ! [X: rat] :
% 0.32/0.61 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_rat )
% 0.32/0.61 = ( ( abs_abs_rat @ X )
% 0.32/0.61 = one_one_rat ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_square_eq_1
% 0.32/0.61 thf(fact_6163_abs__square__eq__1,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_real )
% 0.32/0.61 = ( ( abs_abs_real @ X )
% 0.32/0.61 = one_one_real ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_square_eq_1
% 0.32/0.61 thf(fact_6164_abs__square__eq__1,axiom,
% 0.32/0.61 ! [X: int] :
% 0.32/0.61 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_int )
% 0.32/0.61 = ( ( abs_abs_int @ X )
% 0.32/0.61 = one_one_int ) ) ).
% 0.32/0.61
% 0.32/0.61 % abs_square_eq_1
% 0.32/0.61 thf(fact_6165_power__even__abs,axiom,
% 0.32/0.61 ! [N: nat,A: code_integer] :
% 0.32/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 0.32/0.61 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_even_abs
% 0.32/0.61 thf(fact_6166_power__even__abs,axiom,
% 0.32/0.61 ! [N: nat,A: rat] :
% 0.32/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 0.32/0.61 = ( power_power_rat @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_even_abs
% 0.32/0.61 thf(fact_6167_power__even__abs,axiom,
% 0.32/0.61 ! [N: nat,A: real] :
% 0.32/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 0.32/0.61 = ( power_power_real @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_even_abs
% 0.32/0.61 thf(fact_6168_power__even__abs,axiom,
% 0.32/0.61 ! [N: nat,A: int] :
% 0.32/0.61 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 0.32/0.61 = ( power_power_int @ A @ N ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power_even_abs
% 0.32/0.61 thf(fact_6169_pos__minus__divide__le__eq,axiom,
% 0.32/0.61 ! [C: real,B: real,A: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.32/0.61 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_minus_divide_le_eq
% 0.32/0.61 thf(fact_6170_pos__minus__divide__le__eq,axiom,
% 0.32/0.61 ! [C: rat,B: rat,A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.32/0.61 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_minus_divide_le_eq
% 0.32/0.61 thf(fact_6171_pos__le__minus__divide__eq,axiom,
% 0.32/0.61 ! [C: real,A: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.32/0.61 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_le_minus_divide_eq
% 0.32/0.61 thf(fact_6172_pos__le__minus__divide__eq,axiom,
% 0.32/0.61 ! [C: rat,A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.32/0.61 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % pos_le_minus_divide_eq
% 0.32/0.61 thf(fact_6173_neg__minus__divide__le__eq,axiom,
% 0.32/0.61 ! [C: real,B: real,A: real] :
% 0.32/0.61 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.32/0.61 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_minus_divide_le_eq
% 0.32/0.61 thf(fact_6174_neg__minus__divide__le__eq,axiom,
% 0.32/0.61 ! [C: rat,B: rat,A: rat] :
% 0.32/0.61 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.32/0.61 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_minus_divide_le_eq
% 0.32/0.61 thf(fact_6175_neg__le__minus__divide__eq,axiom,
% 0.32/0.61 ! [C: real,A: real,B: real] :
% 0.32/0.61 ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.32/0.61 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_minus_divide_eq
% 0.32/0.61 thf(fact_6176_neg__le__minus__divide__eq,axiom,
% 0.32/0.61 ! [C: rat,A: rat,B: rat] :
% 0.32/0.61 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.32/0.61 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_le_minus_divide_eq
% 0.32/0.61 thf(fact_6177_minus__divide__le__eq,axiom,
% 0.32/0.61 ! [B: real,C: real,A: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.32/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_le_eq
% 0.32/0.61 thf(fact_6178_minus__divide__le__eq,axiom,
% 0.32/0.61 ! [B: rat,C: rat,A: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.32/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_divide_le_eq
% 0.32/0.61 thf(fact_6179_le__minus__divide__eq,axiom,
% 0.32/0.61 ! [A: real,B: real,C: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.32/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_divide_eq
% 0.32/0.61 thf(fact_6180_le__minus__divide__eq,axiom,
% 0.32/0.61 ! [A: rat,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.32/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % le_minus_divide_eq
% 0.32/0.61 thf(fact_6181_divide__less__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [B: real,C: real,W: num] :
% 0.32/0.61 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_less_eq_numeral(2)
% 0.32/0.61 thf(fact_6182_divide__less__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [B: rat,C: rat,W: num] :
% 0.32/0.61 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % divide_less_eq_numeral(2)
% 0.32/0.61 thf(fact_6183_less__divide__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [W: num,B: real,C: real] :
% 0.32/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.61 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.61 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.61 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_divide_eq_numeral(2)
% 0.32/0.61 thf(fact_6184_less__divide__eq__numeral_I2_J,axiom,
% 0.32/0.61 ! [W: num,B: rat,C: rat] :
% 0.32/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.61 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.61 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.32/0.61 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.61 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % less_divide_eq_numeral(2)
% 0.32/0.61 thf(fact_6185_power2__eq__1__iff,axiom,
% 0.32/0.61 ! [A: real] :
% 0.32/0.61 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_real )
% 0.32/0.61 = ( ( A = one_one_real )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_1_iff
% 0.32/0.61 thf(fact_6186_power2__eq__1__iff,axiom,
% 0.32/0.61 ! [A: int] :
% 0.32/0.61 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_int )
% 0.32/0.61 = ( ( A = one_one_int )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_1_iff
% 0.32/0.61 thf(fact_6187_power2__eq__1__iff,axiom,
% 0.32/0.61 ! [A: complex] :
% 0.32/0.61 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_complex )
% 0.32/0.61 = ( ( A = one_one_complex )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_1_iff
% 0.32/0.61 thf(fact_6188_power2__eq__1__iff,axiom,
% 0.32/0.61 ! [A: code_integer] :
% 0.32/0.61 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_Code_integer )
% 0.32/0.61 = ( ( A = one_one_Code_integer )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_1_iff
% 0.32/0.61 thf(fact_6189_power2__eq__1__iff,axiom,
% 0.32/0.61 ! [A: rat] :
% 0.32/0.61 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.61 = one_one_rat )
% 0.32/0.61 = ( ( A = one_one_rat )
% 0.32/0.61 | ( A
% 0.32/0.61 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % power2_eq_1_iff
% 0.32/0.61 thf(fact_6190_uminus__power__if,axiom,
% 0.32/0.61 ! [N: nat,A: real] :
% 0.32/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.32/0.61 = ( power_power_real @ A @ N ) ) )
% 0.32/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 0.32/0.61 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_power_if
% 0.32/0.61 thf(fact_6191_uminus__power__if,axiom,
% 0.32/0.61 ! [N: nat,A: int] :
% 0.32/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.32/0.61 = ( power_power_int @ A @ N ) ) )
% 0.32/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 0.32/0.61 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_power_if
% 0.32/0.61 thf(fact_6192_uminus__power__if,axiom,
% 0.32/0.61 ! [N: nat,A: complex] :
% 0.32/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.32/0.61 = ( power_power_complex @ A @ N ) ) )
% 0.32/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 0.32/0.61 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_power_if
% 0.32/0.61 thf(fact_6193_uminus__power__if,axiom,
% 0.32/0.61 ! [N: nat,A: code_integer] :
% 0.32/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.32/0.61 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 0.32/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 0.32/0.61 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_power_if
% 0.32/0.61 thf(fact_6194_uminus__power__if,axiom,
% 0.32/0.61 ! [N: nat,A: rat] :
% 0.32/0.61 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.32/0.61 = ( power_power_rat @ A @ N ) ) )
% 0.32/0.61 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.61 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 0.32/0.61 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % uminus_power_if
% 0.32/0.61 thf(fact_6195_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.32/0.61 ! [K: nat,N: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.61 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.32/0.61 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.32/0.61 thf(fact_6196_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.32/0.61 ! [K: nat,N: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.61 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.32/0.61 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.32/0.61 thf(fact_6197_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.32/0.61 ! [K: nat,N: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.61 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.32/0.61 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.32/0.61 thf(fact_6198_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.32/0.61 ! [K: nat,N: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.61 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.32/0.61 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.32/0.61 thf(fact_6199_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.32/0.61 ! [K: nat,N: nat] :
% 0.32/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.61 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.32/0.61 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % neg_one_power_add_eq_neg_one_power_diff
% 0.32/0.61 thf(fact_6200_realpow__square__minus__le,axiom,
% 0.32/0.61 ! [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 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % realpow_square_minus_le
% 0.32/0.61 thf(fact_6201_ln__one__minus__pos__upper__bound,axiom,
% 0.32/0.61 ! [X: real] :
% 0.32/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.61 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.61 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % ln_one_minus_pos_upper_bound
% 0.32/0.61 thf(fact_6202_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.32/0.61 ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_int_greater_eq_minus_exp
% 0.32/0.61 thf(fact_6203_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.32/0.61 ! [N: nat,K: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.32/0.61 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_int_less_eq_self_iff
% 0.32/0.61 thf(fact_6204_signed__take__bit__int__greater__self__iff,axiom,
% 0.32/0.61 ! [K: int,N: nat] :
% 0.32/0.61 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.32/0.61 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % signed_take_bit_int_greater_self_iff
% 0.32/0.61 thf(fact_6205_minus__mod__int__eq,axiom,
% 0.32/0.61 ! [L2: int,K: int] :
% 0.32/0.61 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 0.32/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 0.32/0.61 = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % minus_mod_int_eq
% 0.32/0.61 thf(fact_6206_zmod__minus1,axiom,
% 0.32/0.61 ! [B: int] :
% 0.32/0.61 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.61 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.32/0.61 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 0.32/0.61
% 0.32/0.61 % zmod_minus1
% 0.32/0.61 thf(fact_6207_zdiv__zminus1__eq__if,axiom,
% 0.32/0.61 ! [B: int,A: int] :
% 0.32/0.61 ( ( B != zero_zero_int )
% 0.32/0.62 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.62 = zero_zero_int )
% 0.32/0.62 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.62 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.32/0.62 & ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.62 != zero_zero_int )
% 0.32/0.62 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.32/0.62 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % zdiv_zminus1_eq_if
% 0.32/0.62 thf(fact_6208_zdiv__zminus2__eq__if,axiom,
% 0.32/0.62 ! [B: int,A: int] :
% 0.32/0.62 ( ( B != zero_zero_int )
% 0.32/0.62 => ( ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.62 = zero_zero_int )
% 0.32/0.62 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 0.32/0.62 & ( ( ( modulo_modulo_int @ A @ B )
% 0.32/0.62 != zero_zero_int )
% 0.32/0.62 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.32/0.62 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % zdiv_zminus2_eq_if
% 0.32/0.62 thf(fact_6209_abs__sqrt__wlog,axiom,
% 0.32/0.62 ! [P: code_integer > code_integer > $o,X: code_integer] :
% 0.32/0.62 ( ! [X3: code_integer] :
% 0.32/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 0.32/0.62 => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.62 => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_sqrt_wlog
% 0.32/0.62 thf(fact_6210_abs__sqrt__wlog,axiom,
% 0.32/0.62 ! [P: real > real > $o,X: real] :
% 0.32/0.62 ( ! [X3: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.32/0.62 => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.62 => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_sqrt_wlog
% 0.32/0.62 thf(fact_6211_abs__sqrt__wlog,axiom,
% 0.32/0.62 ! [P: rat > rat > $o,X: rat] :
% 0.32/0.62 ( ! [X3: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.32/0.62 => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.62 => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_sqrt_wlog
% 0.32/0.62 thf(fact_6212_abs__sqrt__wlog,axiom,
% 0.32/0.62 ! [P: int > int > $o,X: int] :
% 0.32/0.62 ( ! [X3: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.32/0.62 => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.62 => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_sqrt_wlog
% 0.32/0.62 thf(fact_6213_power2__le__iff__abs__le,axiom,
% 0.32/0.62 ! [Y3: code_integer,X: code_integer] :
% 0.32/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 0.32/0.62 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.62 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y3 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_le_iff_abs_le
% 0.32/0.62 thf(fact_6214_power2__le__iff__abs__le,axiom,
% 0.32/0.62 ! [Y3: real,X: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.62 => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.62 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y3 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_le_iff_abs_le
% 0.32/0.62 thf(fact_6215_power2__le__iff__abs__le,axiom,
% 0.32/0.62 ! [Y3: rat,X: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.32/0.62 => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.62 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y3 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_le_iff_abs_le
% 0.32/0.62 thf(fact_6216_power2__le__iff__abs__le,axiom,
% 0.32/0.62 ! [Y3: int,X: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.62 => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.62 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y3 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power2_le_iff_abs_le
% 0.32/0.62 thf(fact_6217_abs__square__le__1,axiom,
% 0.32/0.62 ! [X: code_integer] :
% 0.32/0.62 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.32/0.62 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_square_le_1
% 0.32/0.62 thf(fact_6218_abs__square__le__1,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.32/0.62 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_square_le_1
% 0.32/0.62 thf(fact_6219_abs__square__le__1,axiom,
% 0.32/0.62 ! [X: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.32/0.62 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_square_le_1
% 0.32/0.62 thf(fact_6220_abs__square__le__1,axiom,
% 0.32/0.62 ! [X: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.32/0.62 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_square_le_1
% 0.32/0.62 thf(fact_6221_abs__square__less__1,axiom,
% 0.32/0.62 ! [X: code_integer] :
% 0.32/0.62 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.32/0.62 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_square_less_1
% 0.32/0.62 thf(fact_6222_abs__square__less__1,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.32/0.62 = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_square_less_1
% 0.32/0.62 thf(fact_6223_abs__square__less__1,axiom,
% 0.32/0.62 ! [X: rat] :
% 0.32/0.62 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.32/0.62 = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_square_less_1
% 0.32/0.62 thf(fact_6224_abs__square__less__1,axiom,
% 0.32/0.62 ! [X: int] :
% 0.32/0.62 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.32/0.62 = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_square_less_1
% 0.32/0.62 thf(fact_6225_power__mono__even,axiom,
% 0.32/0.62 ! [N: nat,A: code_integer,B: code_integer] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 0.32/0.62 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_mono_even
% 0.32/0.62 thf(fact_6226_power__mono__even,axiom,
% 0.32/0.62 ! [N: nat,A: real,B: real] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_mono_even
% 0.32/0.62 thf(fact_6227_power__mono__even,axiom,
% 0.32/0.62 ! [N: nat,A: rat,B: rat] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_mono_even
% 0.32/0.62 thf(fact_6228_power__mono__even,axiom,
% 0.32/0.62 ! [N: nat,A: int,B: int] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 0.32/0.62 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_mono_even
% 0.32/0.62 thf(fact_6229_zminus1__lemma,axiom,
% 0.32/0.62 ! [A: int,B: int,Q2: int,R: int] :
% 0.32/0.62 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 0.32/0.62 => ( ( B != zero_zero_int )
% 0.32/0.62 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % zminus1_lemma
% 0.32/0.62 thf(fact_6230_divide__le__eq__numeral_I2_J,axiom,
% 0.32/0.62 ! [B: real,C: real,W: num] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.62 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.32/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.62 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.32/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_le_eq_numeral(2)
% 0.32/0.62 thf(fact_6231_divide__le__eq__numeral_I2_J,axiom,
% 0.32/0.62 ! [B: rat,C: rat,W: num] :
% 0.32/0.62 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 0.32/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.62 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.32/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.62 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.32/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % divide_le_eq_numeral(2)
% 0.32/0.62 thf(fact_6232_le__divide__eq__numeral_I2_J,axiom,
% 0.32/0.62 ! [W: num,B: real,C: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 0.32/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.62 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 0.32/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.62 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 0.32/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.32/0.62 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_divide_eq_numeral(2)
% 0.32/0.62 thf(fact_6233_le__divide__eq__numeral_I2_J,axiom,
% 0.32/0.62 ! [W: num,B: rat,C: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.32/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.62 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 0.32/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.32/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.62 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 0.32/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.32/0.62 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % le_divide_eq_numeral(2)
% 0.32/0.62 thf(fact_6234_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.62 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.62 => ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.32/0.62 thf(fact_6235_square__le__1,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.62 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.62 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_le_1
% 0.32/0.62 thf(fact_6236_square__le__1,axiom,
% 0.32/0.62 ! [X: code_integer] :
% 0.32/0.62 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 0.32/0.62 => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 0.32/0.62 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_le_1
% 0.32/0.62 thf(fact_6237_square__le__1,axiom,
% 0.32/0.62 ! [X: rat] :
% 0.32/0.62 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 0.32/0.62 => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 0.32/0.62 => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_le_1
% 0.32/0.62 thf(fact_6238_square__le__1,axiom,
% 0.32/0.62 ! [X: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 0.32/0.62 => ( ( ord_less_eq_int @ X @ one_one_int )
% 0.32/0.62 => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % square_le_1
% 0.32/0.62 thf(fact_6239_minus__power__mult__self,axiom,
% 0.32/0.62 ! [A: real,N: nat] :
% 0.32/0.62 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 0.32/0.62 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_power_mult_self
% 0.32/0.62 thf(fact_6240_minus__power__mult__self,axiom,
% 0.32/0.62 ! [A: int,N: nat] :
% 0.32/0.62 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 0.32/0.62 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_power_mult_self
% 0.32/0.62 thf(fact_6241_minus__power__mult__self,axiom,
% 0.32/0.62 ! [A: complex,N: nat] :
% 0.32/0.62 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 0.32/0.62 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_power_mult_self
% 0.32/0.62 thf(fact_6242_minus__power__mult__self,axiom,
% 0.32/0.62 ! [A: code_integer,N: nat] :
% 0.32/0.62 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 0.32/0.62 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_power_mult_self
% 0.32/0.62 thf(fact_6243_minus__power__mult__self,axiom,
% 0.32/0.62 ! [A: rat,N: nat] :
% 0.32/0.62 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 0.32/0.62 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_power_mult_self
% 0.32/0.62 thf(fact_6244_minus__one__power__iff,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.32/0.62 = one_one_real ) )
% 0.32/0.62 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 0.32/0.62 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_power_iff
% 0.32/0.62 thf(fact_6245_minus__one__power__iff,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.32/0.62 = one_one_int ) )
% 0.32/0.62 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 0.32/0.62 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_power_iff
% 0.32/0.62 thf(fact_6246_minus__one__power__iff,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.32/0.62 = one_one_complex ) )
% 0.32/0.62 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_power_iff
% 0.32/0.62 thf(fact_6247_minus__one__power__iff,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.32/0.62 = one_one_Code_integer ) )
% 0.32/0.62 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_power_iff
% 0.32/0.62 thf(fact_6248_minus__one__power__iff,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.32/0.62 = one_one_rat ) )
% 0.32/0.62 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.62 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 0.32/0.62 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_one_power_iff
% 0.32/0.62 thf(fact_6249_signed__take__bit__int__eq__self__iff,axiom,
% 0.32/0.62 ! [N: nat,K: int] :
% 0.32/0.62 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.32/0.62 = K )
% 0.32/0.62 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.32/0.62 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_int_eq_self_iff
% 0.32/0.62 thf(fact_6250_signed__take__bit__int__eq__self,axiom,
% 0.32/0.62 ! [N: nat,K: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.32/0.62 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.62 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.32/0.62 = K ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_int_eq_self
% 0.32/0.62 thf(fact_6251_minus__1__div__exp__eq__int,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.62 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % minus_1_div_exp_eq_int
% 0.32/0.62 thf(fact_6252_div__pos__neg__trivial,axiom,
% 0.32/0.62 ! [K: int,L2: int] :
% 0.32/0.62 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.62 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 0.32/0.62 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.62 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % div_pos_neg_trivial
% 0.32/0.62 thf(fact_6253_power__minus1__odd,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.62 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_odd
% 0.32/0.62 thf(fact_6254_power__minus1__odd,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.62 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_odd
% 0.32/0.62 thf(fact_6255_power__minus1__odd,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_odd
% 0.32/0.62 thf(fact_6256_power__minus1__odd,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_odd
% 0.32/0.62 thf(fact_6257_power__minus1__odd,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.62 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus1_odd
% 0.32/0.62 thf(fact_6258_int__bit__induct,axiom,
% 0.32/0.62 ! [P: int > $o,K: int] :
% 0.32/0.62 ( ( P @ zero_zero_int )
% 0.32/0.62 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 => ( ! [K3: int] :
% 0.32/0.62 ( ( P @ K3 )
% 0.32/0.62 => ( ( K3 != zero_zero_int )
% 0.32/0.62 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.62 => ( ! [K3: int] :
% 0.32/0.62 ( ( P @ K3 )
% 0.32/0.62 => ( ( K3
% 0.32/0.62 != ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.62 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 0.32/0.62 => ( P @ K ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % int_bit_induct
% 0.32/0.62 thf(fact_6259_eq__diff__eq_H,axiom,
% 0.32/0.62 ! [X: real,Y3: real,Z: real] :
% 0.32/0.62 ( ( X
% 0.32/0.62 = ( minus_minus_real @ Y3 @ Z ) )
% 0.32/0.62 = ( Y3
% 0.32/0.62 = ( plus_plus_real @ X @ Z ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % eq_diff_eq'
% 0.32/0.62 thf(fact_6260_signed__take__bit__int__greater__eq,axiom,
% 0.32/0.62 ! [K: int,N: nat] :
% 0.32/0.62 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.62 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_int_greater_eq
% 0.32/0.62 thf(fact_6261_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.62 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.62 => ( 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 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.32/0.62 thf(fact_6262_arctan__double,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.62 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 0.32/0.62 = ( 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 ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % arctan_double
% 0.32/0.62 thf(fact_6263_signed__take__bit__Suc__minus__bit1,axiom,
% 0.32/0.62 ! [N: nat,K: num] :
% 0.32/0.62 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.32/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_Suc_minus_bit1
% 0.32/0.62 thf(fact_6264_Sum__Icc__int,axiom,
% 0.32/0.62 ! [M: int,N: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ M @ N )
% 0.32/0.62 => ( ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [X2: int] : X2
% 0.32/0.62 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 0.32/0.62 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Sum_Icc_int
% 0.32/0.62 thf(fact_6265_divmod__step__def,axiom,
% 0.32/0.62 ( unique5026877609467782581ep_nat
% 0.32/0.62 = ( ^ [L: num] :
% 0.32/0.62 ( produc2626176000494625587at_nat
% 0.32/0.62 @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % divmod_step_def
% 0.32/0.62 thf(fact_6266_divmod__step__def,axiom,
% 0.32/0.62 ( unique5024387138958732305ep_int
% 0.32/0.62 = ( ^ [L: num] :
% 0.32/0.62 ( produc4245557441103728435nt_int
% 0.32/0.62 @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % divmod_step_def
% 0.32/0.62 thf(fact_6267_divmod__step__def,axiom,
% 0.32/0.62 ( unique4921790084139445826nteger
% 0.32/0.62 = ( ^ [L: num] :
% 0.32/0.62 ( produc6916734918728496179nteger
% 0.32/0.62 @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % divmod_step_def
% 0.32/0.62 thf(fact_6268_even__set__encode__iff,axiom,
% 0.32/0.62 ! [A2: set_nat] :
% 0.32/0.62 ( ( finite_finite_nat @ A2 )
% 0.32/0.62 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 0.32/0.62 = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % even_set_encode_iff
% 0.32/0.62 thf(fact_6269_signed__take__bit__Suc__bit1,axiom,
% 0.32/0.62 ! [N: nat,K: num] :
% 0.32/0.62 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.32/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_Suc_bit1
% 0.32/0.62 thf(fact_6270_take__bit__rec,axiom,
% 0.32/0.62 ( bit_se1745604003318907178nteger
% 0.32/0.62 = ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_rec
% 0.32/0.62 thf(fact_6271_take__bit__rec,axiom,
% 0.32/0.62 ( bit_se2923211474154528505it_int
% 0.32/0.62 = ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_rec
% 0.32/0.62 thf(fact_6272_take__bit__rec,axiom,
% 0.32/0.62 ( bit_se2925701944663578781it_nat
% 0.32/0.62 = ( ^ [N2: nat,A3: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_rec
% 0.32/0.62 thf(fact_6273_verit__eq__simplify_I9_J,axiom,
% 0.32/0.62 ! [X32: num,Y32: num] :
% 0.32/0.62 ( ( ( bit1 @ X32 )
% 0.32/0.62 = ( bit1 @ Y32 ) )
% 0.32/0.62 = ( X32 = Y32 ) ) ).
% 0.32/0.62
% 0.32/0.62 % verit_eq_simplify(9)
% 0.32/0.62 thf(fact_6274_semiring__norm_I90_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( ( bit1 @ M )
% 0.32/0.62 = ( bit1 @ N ) )
% 0.32/0.62 = ( M = N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(90)
% 0.32/0.62 thf(fact_6275_semiring__norm_I89_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( bit1 @ M )
% 0.32/0.62 != ( bit0 @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(89)
% 0.32/0.62 thf(fact_6276_semiring__norm_I88_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( bit0 @ M )
% 0.32/0.62 != ( bit1 @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(88)
% 0.32/0.62 thf(fact_6277_semiring__norm_I86_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( bit1 @ M )
% 0.32/0.62 != one ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(86)
% 0.32/0.62 thf(fact_6278_semiring__norm_I84_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( one
% 0.32/0.62 != ( bit1 @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(84)
% 0.32/0.62 thf(fact_6279_take__bit__of__0,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 0.32/0.62 = zero_zero_int ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_0
% 0.32/0.62 thf(fact_6280_take__bit__of__0,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 0.32/0.62 = zero_zero_nat ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_0
% 0.32/0.62 thf(fact_6281_concat__bit__of__zero__2,axiom,
% 0.32/0.62 ! [N: nat,K: int] :
% 0.32/0.62 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.32/0.62
% 0.32/0.62 % concat_bit_of_zero_2
% 0.32/0.62 thf(fact_6282_semiring__norm_I73_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.62 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(73)
% 0.32/0.62 thf(fact_6283_semiring__norm_I80_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.62 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(80)
% 0.32/0.62 thf(fact_6284_take__bit__0,axiom,
% 0.32/0.62 ! [A: int] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 0.32/0.62 = zero_zero_int ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_0
% 0.32/0.62 thf(fact_6285_take__bit__0,axiom,
% 0.32/0.62 ! [A: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 0.32/0.62 = zero_zero_nat ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_0
% 0.32/0.62 thf(fact_6286_take__bit__Suc__1,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 0.32/0.62 = one_one_int ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_Suc_1
% 0.32/0.62 thf(fact_6287_take__bit__Suc__1,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 0.32/0.62 = one_one_nat ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_Suc_1
% 0.32/0.62 thf(fact_6288_take__bit__numeral__1,axiom,
% 0.32/0.62 ! [L2: num] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ one_one_int )
% 0.32/0.62 = one_one_int ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_numeral_1
% 0.32/0.62 thf(fact_6289_take__bit__numeral__1,axiom,
% 0.32/0.62 ! [L2: num] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ one_one_nat )
% 0.32/0.62 = one_one_nat ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_numeral_1
% 0.32/0.62 thf(fact_6290_zabs__less__one__iff,axiom,
% 0.32/0.62 ! [Z: int] :
% 0.32/0.62 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.32/0.62 = ( Z = zero_zero_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % zabs_less_one_iff
% 0.32/0.62 thf(fact_6291_semiring__norm_I7_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.62 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(7)
% 0.32/0.62 thf(fact_6292_semiring__norm_I9_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.62 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(9)
% 0.32/0.62 thf(fact_6293_arctan__less__zero__iff,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 0.32/0.62 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % arctan_less_zero_iff
% 0.32/0.62 thf(fact_6294_zero__less__arctan__iff,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 0.32/0.62 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.32/0.62
% 0.32/0.62 % zero_less_arctan_iff
% 0.32/0.62 thf(fact_6295_zero__le__arctan__iff,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
% 0.32/0.62 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.32/0.62
% 0.32/0.62 % zero_le_arctan_iff
% 0.32/0.62 thf(fact_6296_arctan__le__zero__iff,axiom,
% 0.32/0.62 ! [X: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
% 0.32/0.62 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % arctan_le_zero_iff
% 0.32/0.62 thf(fact_6297_semiring__norm_I14_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.62 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(14)
% 0.32/0.62 thf(fact_6298_semiring__norm_I15_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.62 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(15)
% 0.32/0.62 thf(fact_6299_semiring__norm_I72_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.62 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(72)
% 0.32/0.62 thf(fact_6300_semiring__norm_I81_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.62 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(81)
% 0.32/0.62 thf(fact_6301_semiring__norm_I70_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(70)
% 0.32/0.62 thf(fact_6302_semiring__norm_I77_J,axiom,
% 0.32/0.62 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(77)
% 0.32/0.62 thf(fact_6303_sum__abs,axiom,
% 0.32/0.62 ! [F: int > int,A2: set_int] :
% 0.32/0.62 ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.32/0.62 @ ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_abs
% 0.32/0.62 thf(fact_6304_sum__abs,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat] :
% 0.32/0.62 ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.32/0.62 @ ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_abs
% 0.32/0.62 thf(fact_6305_take__bit__of__1__eq__0__iff,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.32/0.62 = zero_zero_int )
% 0.32/0.62 = ( N = zero_zero_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_1_eq_0_iff
% 0.32/0.62 thf(fact_6306_take__bit__of__1__eq__0__iff,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.32/0.62 = zero_zero_nat )
% 0.32/0.62 = ( N = zero_zero_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_1_eq_0_iff
% 0.32/0.62 thf(fact_6307_zdiv__numeral__Bit1,axiom,
% 0.32/0.62 ! [V: num,W: num] :
% 0.32/0.62 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.32/0.62 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % zdiv_numeral_Bit1
% 0.32/0.62 thf(fact_6308_semiring__norm_I10_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.62 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(10)
% 0.32/0.62 thf(fact_6309_semiring__norm_I8_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 0.32/0.62 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(8)
% 0.32/0.62 thf(fact_6310_semiring__norm_I5_J,axiom,
% 0.32/0.62 ! [M: num] :
% 0.32/0.62 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 0.32/0.62 = ( bit1 @ M ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(5)
% 0.32/0.62 thf(fact_6311_semiring__norm_I4_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 0.32/0.62 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(4)
% 0.32/0.62 thf(fact_6312_semiring__norm_I3_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 0.32/0.62 = ( bit1 @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(3)
% 0.32/0.62 thf(fact_6313_take__bit__of__Suc__0,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.62 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_Suc_0
% 0.32/0.62 thf(fact_6314_sum__abs__ge__zero,axiom,
% 0.32/0.62 ! [F: int > int,A2: set_int] :
% 0.32/0.62 ( ord_less_eq_int @ zero_zero_int
% 0.32/0.62 @ ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_abs_ge_zero
% 0.32/0.62 thf(fact_6315_sum__abs__ge__zero,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat] :
% 0.32/0.62 ( ord_less_eq_real @ zero_zero_real
% 0.32/0.62 @ ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_abs_ge_zero
% 0.32/0.62 thf(fact_6316_semiring__norm_I16_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.62 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(16)
% 0.32/0.62 thf(fact_6317_semiring__norm_I79_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.62 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(79)
% 0.32/0.62 thf(fact_6318_semiring__norm_I74_J,axiom,
% 0.32/0.62 ! [M: num,N: num] :
% 0.32/0.62 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.62 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.62
% 0.32/0.62 % semiring_norm(74)
% 0.32/0.62 thf(fact_6319_take__bit__of__1,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 0.32/0.62 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_1
% 0.32/0.62 thf(fact_6320_take__bit__of__1,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 0.32/0.62 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_1
% 0.32/0.62 thf(fact_6321_take__bit__of__1,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 0.32/0.62 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_1
% 0.32/0.62 thf(fact_6322_even__take__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,A: code_integer] :
% 0.32/0.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 0.32/0.62 = ( ( N = zero_zero_nat )
% 0.32/0.62 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % even_take_bit_eq
% 0.32/0.62 thf(fact_6323_even__take__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,A: int] :
% 0.32/0.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.32/0.62 = ( ( N = zero_zero_nat )
% 0.32/0.62 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % even_take_bit_eq
% 0.32/0.62 thf(fact_6324_even__take__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,A: nat] :
% 0.32/0.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 0.32/0.62 = ( ( N = zero_zero_nat )
% 0.32/0.62 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % even_take_bit_eq
% 0.32/0.62 thf(fact_6325_div__Suc__eq__div__add3,axiom,
% 0.32/0.62 ! [M: nat,N: nat] :
% 0.32/0.62 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.32/0.62 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % div_Suc_eq_div_add3
% 0.32/0.62 thf(fact_6326_Suc__div__eq__add3__div__numeral,axiom,
% 0.32/0.62 ! [M: nat,V: num] :
% 0.32/0.62 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.32/0.62 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Suc_div_eq_add3_div_numeral
% 0.32/0.62 thf(fact_6327_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.32/0.62 ! [M: nat,V: num] :
% 0.32/0.62 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.32/0.62 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % Suc_mod_eq_add3_mod_numeral
% 0.32/0.62 thf(fact_6328_mod__Suc__eq__mod__add3,axiom,
% 0.32/0.62 ! [M: nat,N: nat] :
% 0.32/0.62 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.32/0.62 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_Suc_eq_mod_add3
% 0.32/0.62 thf(fact_6329_take__bit__Suc__0,axiom,
% 0.32/0.62 ! [A: code_integer] :
% 0.32/0.62 ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 0.32/0.62 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_Suc_0
% 0.32/0.62 thf(fact_6330_take__bit__Suc__0,axiom,
% 0.32/0.62 ! [A: int] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 0.32/0.62 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_Suc_0
% 0.32/0.62 thf(fact_6331_take__bit__Suc__0,axiom,
% 0.32/0.62 ! [A: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 0.32/0.62 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_Suc_0
% 0.32/0.62 thf(fact_6332_take__bit__of__exp,axiom,
% 0.32/0.62 ! [M: nat,N: nat] :
% 0.32/0.62 ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.62 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_exp
% 0.32/0.62 thf(fact_6333_take__bit__of__exp,axiom,
% 0.32/0.62 ! [M: nat,N: nat] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.62 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_exp
% 0.32/0.62 thf(fact_6334_take__bit__of__exp,axiom,
% 0.32/0.62 ! [M: nat,N: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.62 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_exp
% 0.32/0.62 thf(fact_6335_take__bit__of__2,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_2
% 0.32/0.62 thf(fact_6336_take__bit__of__2,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_2
% 0.32/0.62 thf(fact_6337_take__bit__of__2,axiom,
% 0.32/0.62 ! [N: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.62 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_of_2
% 0.32/0.62 thf(fact_6338_zmod__numeral__Bit1,axiom,
% 0.32/0.62 ! [V: num,W: num] :
% 0.32/0.62 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 0.32/0.62 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % zmod_numeral_Bit1
% 0.32/0.62 thf(fact_6339_take__bit__minus,axiom,
% 0.32/0.62 ! [N: nat,K: int] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_minus
% 0.32/0.62 thf(fact_6340_take__bit__add,axiom,
% 0.32/0.62 ! [N: nat,A: int,B: int] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_add
% 0.32/0.62 thf(fact_6341_take__bit__add,axiom,
% 0.32/0.62 ! [N: nat,A: nat,B: nat] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
% 0.32/0.62 = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_add
% 0.32/0.62 thf(fact_6342_take__bit__tightened,axiom,
% 0.32/0.62 ! [N: nat,A: int,B: int,M: nat] :
% 0.32/0.62 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.32/0.62 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ M @ A )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_tightened
% 0.32/0.62 thf(fact_6343_take__bit__tightened,axiom,
% 0.32/0.62 ! [N: nat,A: nat,B: nat,M: nat] :
% 0.32/0.62 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.32/0.62 = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 0.32/0.62 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.62 => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 0.32/0.62 = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_tightened
% 0.32/0.62 thf(fact_6344_take__bit__tightened__less__eq__nat,axiom,
% 0.32/0.62 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.62 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_tightened_less_eq_nat
% 0.32/0.62 thf(fact_6345_take__bit__nat__less__eq__self,axiom,
% 0.32/0.62 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_nat_less_eq_self
% 0.32/0.62 thf(fact_6346_take__bit__mult,axiom,
% 0.32/0.62 ! [N: nat,K: int,L2: int] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_mult
% 0.32/0.62 thf(fact_6347_verit__eq__simplify_I14_J,axiom,
% 0.32/0.62 ! [X22: num,X32: num] :
% 0.32/0.62 ( ( bit0 @ X22 )
% 0.32/0.62 != ( bit1 @ X32 ) ) ).
% 0.32/0.62
% 0.32/0.62 % verit_eq_simplify(14)
% 0.32/0.62 thf(fact_6348_verit__eq__simplify_I12_J,axiom,
% 0.32/0.62 ! [X32: num] :
% 0.32/0.62 ( one
% 0.32/0.62 != ( bit1 @ X32 ) ) ).
% 0.32/0.62
% 0.32/0.62 % verit_eq_simplify(12)
% 0.32/0.62 thf(fact_6349_take__bit__diff,axiom,
% 0.32/0.62 ! [N: nat,K: int,L2: int] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_diff
% 0.32/0.62 thf(fact_6350_arctan__monotone,axiom,
% 0.32/0.62 ! [X: real,Y3: real] :
% 0.32/0.62 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.62 => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y3 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % arctan_monotone
% 0.32/0.62 thf(fact_6351_arctan__less__iff,axiom,
% 0.32/0.62 ! [X: real,Y3: real] :
% 0.32/0.62 ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y3 ) )
% 0.32/0.62 = ( ord_less_real @ X @ Y3 ) ) ).
% 0.32/0.62
% 0.32/0.62 % arctan_less_iff
% 0.32/0.62 thf(fact_6352_arctan__monotone_H,axiom,
% 0.32/0.62 ! [X: real,Y3: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.62 => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y3 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % arctan_monotone'
% 0.32/0.62 thf(fact_6353_arctan__le__iff,axiom,
% 0.32/0.62 ! [X: real,Y3: real] :
% 0.32/0.62 ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y3 ) )
% 0.32/0.62 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.62
% 0.32/0.62 % arctan_le_iff
% 0.32/0.62 thf(fact_6354_concat__bit__eq__iff,axiom,
% 0.32/0.62 ! [N: nat,K: int,L2: int,R: int,S: int] :
% 0.32/0.62 ( ( ( bit_concat_bit @ N @ K @ L2 )
% 0.32/0.62 = ( bit_concat_bit @ N @ R @ S ) )
% 0.32/0.62 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ R ) )
% 0.32/0.62 & ( L2 = S ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % concat_bit_eq_iff
% 0.32/0.62 thf(fact_6355_concat__bit__take__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,B: int] :
% 0.32/0.62 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
% 0.32/0.62 = ( bit_concat_bit @ N @ B ) ) ).
% 0.32/0.62
% 0.32/0.62 % concat_bit_take_bit_eq
% 0.32/0.62 thf(fact_6356_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 0.32/0.62 ( ! [I2: nat] :
% 0.32/0.62 ( ( member_nat @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6357_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_real,F: real > rat,G: real > rat] :
% 0.32/0.62 ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6358_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_int,F: int > rat,G: int > rat] :
% 0.32/0.62 ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6359_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_complex,F: complex > rat,G: complex > rat] :
% 0.32/0.62 ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6360_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_real,F: real > nat,G: real > nat] :
% 0.32/0.62 ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6361_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_int,F: int > nat,G: int > nat] :
% 0.32/0.62 ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6362_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_complex,F: complex > nat,G: complex > nat] :
% 0.32/0.62 ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6363_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_nat,F: nat > int,G: nat > int] :
% 0.32/0.62 ( ! [I2: nat] :
% 0.32/0.62 ( ( member_nat @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6364_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_real,F: real > int,G: real > int] :
% 0.32/0.62 ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6365_sum__mono,axiom,
% 0.32/0.62 ! [K5: set_complex,F: complex > int,G: complex > int] :
% 0.32/0.62 ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ K5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono
% 0.32/0.62 thf(fact_6366_sum__product,axiom,
% 0.32/0.62 ! [F: int > int,A2: set_int,G: int > int,B5: set_int] :
% 0.32/0.62 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [I4: int] :
% 0.32/0.62 ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.32/0.62 @ B5 )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_product
% 0.32/0.62 thf(fact_6367_sum__product,axiom,
% 0.32/0.62 ! [F: complex > complex,A2: set_complex,G: complex > complex,B5: set_complex] :
% 0.32/0.62 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B5 ) )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [I4: complex] :
% 0.32/0.62 ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.32/0.62 @ B5 )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_product
% 0.32/0.62 thf(fact_6368_sum__product,axiom,
% 0.32/0.62 ! [F: nat > nat,A2: set_nat,G: nat > nat,B5: set_nat] :
% 0.32/0.62 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [I4: nat] :
% 0.32/0.62 ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.32/0.62 @ B5 )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_product
% 0.32/0.62 thf(fact_6369_sum__product,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat,G: nat > real,B5: set_nat] :
% 0.32/0.62 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [I4: nat] :
% 0.32/0.62 ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
% 0.32/0.62 @ B5 )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_product
% 0.32/0.62 thf(fact_6370_sum__distrib__right,axiom,
% 0.32/0.62 ! [F: int > int,A2: set_int,R: int] :
% 0.32/0.62 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_right
% 0.32/0.62 thf(fact_6371_sum__distrib__right,axiom,
% 0.32/0.62 ! [F: complex > complex,A2: set_complex,R: complex] :
% 0.32/0.62 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_right
% 0.32/0.62 thf(fact_6372_sum__distrib__right,axiom,
% 0.32/0.62 ! [F: nat > nat,A2: set_nat,R: nat] :
% 0.32/0.62 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_right
% 0.32/0.62 thf(fact_6373_sum__distrib__right,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat,R: real] :
% 0.32/0.62 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_right
% 0.32/0.62 thf(fact_6374_sum__distrib__left,axiom,
% 0.32/0.62 ! [R: int,F: int > int,A2: set_int] :
% 0.32/0.62 ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 0.32/0.62 = ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [N2: int] : ( times_times_int @ R @ ( F @ N2 ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_left
% 0.32/0.62 thf(fact_6375_sum__distrib__left,axiom,
% 0.32/0.62 ! [R: complex,F: complex > complex,A2: set_complex] :
% 0.32/0.62 ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [N2: complex] : ( times_times_complex @ R @ ( F @ N2 ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_left
% 0.32/0.62 thf(fact_6376_sum__distrib__left,axiom,
% 0.32/0.62 ! [R: nat,F: nat > nat,A2: set_nat] :
% 0.32/0.62 ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.32/0.62 = ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_left
% 0.32/0.62 thf(fact_6377_sum__distrib__left,axiom,
% 0.32/0.62 ! [R: real,F: nat > real,A2: set_nat] :
% 0.32/0.62 ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [N2: nat] : ( times_times_real @ R @ ( F @ N2 ) )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_distrib_left
% 0.32/0.62 thf(fact_6378_sum_Odistrib,axiom,
% 0.32/0.62 ! [G: int > int,H2: int > int,A2: set_int] :
% 0.32/0.62 ( ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.distrib
% 0.32/0.62 thf(fact_6379_sum_Odistrib,axiom,
% 0.32/0.62 ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
% 0.32/0.62 ( ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.distrib
% 0.32/0.62 thf(fact_6380_sum_Odistrib,axiom,
% 0.32/0.62 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 0.32/0.62 ( ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.distrib
% 0.32/0.62 thf(fact_6381_sum_Odistrib,axiom,
% 0.32/0.62 ! [G: nat > real,H2: nat > real,A2: set_nat] :
% 0.32/0.62 ( ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.distrib
% 0.32/0.62 thf(fact_6382_sum__subtractf,axiom,
% 0.32/0.62 ! [F: int > int,G: int > int,A2: set_int] :
% 0.32/0.62 ( ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [X2: int] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_subtractf
% 0.32/0.62 thf(fact_6383_sum__subtractf,axiom,
% 0.32/0.62 ! [F: complex > complex,G: complex > complex,A2: set_complex] :
% 0.32/0.62 ( ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [X2: complex] : ( minus_minus_complex @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_subtractf
% 0.32/0.62 thf(fact_6384_sum__subtractf,axiom,
% 0.32/0.62 ! [F: nat > real,G: nat > real,A2: set_nat] :
% 0.32/0.62 ( ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.62 @ A2 )
% 0.32/0.62 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_subtractf
% 0.32/0.62 thf(fact_6385_sum__divide__distrib,axiom,
% 0.32/0.62 ! [F: complex > complex,A2: set_complex,R: complex] :
% 0.32/0.62 ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups7754918857620584856omplex
% 0.32/0.62 @ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_divide_distrib
% 0.32/0.62 thf(fact_6386_sum__divide__distrib,axiom,
% 0.32/0.62 ! [F: nat > real,A2: set_nat,R: real] :
% 0.32/0.62 ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
% 0.32/0.62 = ( groups6591440286371151544t_real
% 0.32/0.62 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R )
% 0.32/0.62 @ A2 ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_divide_distrib
% 0.32/0.62 thf(fact_6387_mod__sum__eq,axiom,
% 0.32/0.62 ! [F: int > int,A: int,A2: set_int] :
% 0.32/0.62 ( ( modulo_modulo_int
% 0.32/0.62 @ ( groups4538972089207619220nt_int
% 0.32/0.62 @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 0.32/0.62 @ A2 )
% 0.32/0.62 @ A )
% 0.32/0.62 = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_sum_eq
% 0.32/0.62 thf(fact_6388_mod__sum__eq,axiom,
% 0.32/0.62 ! [F: nat > nat,A: nat,A2: set_nat] :
% 0.32/0.62 ( ( modulo_modulo_nat
% 0.32/0.62 @ ( groups3542108847815614940at_nat
% 0.32/0.62 @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 0.32/0.62 @ A2 )
% 0.32/0.62 @ A )
% 0.32/0.62 = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % mod_sum_eq
% 0.32/0.62 thf(fact_6389_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > real] :
% 0.32/0.62 ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6390_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > real] :
% 0.32/0.62 ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6391_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > real] :
% 0.32/0.62 ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6392_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_nat,F: nat > rat] :
% 0.32/0.62 ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6393_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > rat] :
% 0.32/0.62 ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6394_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > rat] :
% 0.32/0.62 ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6395_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > rat] :
% 0.32/0.62 ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6396_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > nat] :
% 0.32/0.62 ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6397_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > nat] :
% 0.32/0.62 ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6398_sum__nonpos,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > nat] :
% 0.32/0.62 ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
% 0.32/0.62 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonpos
% 0.32/0.62 thf(fact_6399_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > real] :
% 0.32/0.62 ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6400_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > real] :
% 0.32/0.62 ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6401_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > real] :
% 0.32/0.62 ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6402_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_nat,F: nat > rat] :
% 0.32/0.62 ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6403_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > rat] :
% 0.32/0.62 ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6404_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > rat] :
% 0.32/0.62 ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6405_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > rat] :
% 0.32/0.62 ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6406_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > nat] :
% 0.32/0.62 ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6407_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > nat] :
% 0.32/0.62 ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6408_sum__nonneg,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > nat] :
% 0.32/0.62 ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg
% 0.32/0.62 thf(fact_6409_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
% 0.32/0.62 ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 0.32/0.62 = ( groups1300246762558778688al_rat @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_real @ I @ I5 )
% 0.32/0.62 => ( ( finite_finite_real @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6410_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
% 0.32/0.62 ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 0.32/0.62 = ( groups2906978787729119204at_rat @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: nat] :
% 0.32/0.62 ( ( member_nat @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_nat @ I @ I5 )
% 0.32/0.62 => ( ( finite_finite_nat @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6411_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
% 0.32/0.62 ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 0.32/0.62 = ( groups3906332499630173760nt_rat @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_int @ I @ I5 )
% 0.32/0.62 => ( ( finite_finite_int @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6412_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
% 0.32/0.62 ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 0.32/0.62 = ( groups5058264527183730370ex_rat @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_complex @ I @ I5 )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6413_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
% 0.32/0.62 ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 0.32/0.62 = ( groups1935376822645274424al_nat @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_real @ I @ I5 )
% 0.32/0.62 => ( ( finite_finite_real @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6414_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
% 0.32/0.62 ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 0.32/0.62 = ( groups4541462559716669496nt_nat @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_int @ I @ I5 )
% 0.32/0.62 => ( ( finite_finite_int @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6415_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
% 0.32/0.62 ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 0.32/0.62 = ( groups5693394587270226106ex_nat @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_complex @ I @ I5 )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6416_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: real > int,I5: set_real,G: real > int,I: real] :
% 0.32/0.62 ( ( ( groups1932886352136224148al_int @ F @ I5 )
% 0.32/0.62 = ( groups1932886352136224148al_int @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_real @ I @ I5 )
% 0.32/0.62 => ( ( finite_finite_real @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6417_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
% 0.32/0.62 ( ( ( groups3539618377306564664at_int @ F @ I5 )
% 0.32/0.62 = ( groups3539618377306564664at_int @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: nat] :
% 0.32/0.62 ( ( member_nat @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_nat @ I @ I5 )
% 0.32/0.62 => ( ( finite_finite_nat @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6418_sum__mono__inv,axiom,
% 0.32/0.62 ! [F: complex > int,I5: set_complex,G: complex > int,I: complex] :
% 0.32/0.62 ( ( ( groups5690904116761175830ex_int @ F @ I5 )
% 0.32/0.62 = ( groups5690904116761175830ex_int @ G @ I5 ) )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.32/0.62 => ( ( member_complex @ I @ I5 )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = ( G @ I ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_mono_inv
% 0.32/0.62 thf(fact_6419_take__bit__tightened__less__eq__int,axiom,
% 0.32/0.62 ! [M: nat,N: nat,K: int] :
% 0.32/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.62 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_tightened_less_eq_int
% 0.32/0.62 thf(fact_6420_take__bit__int__less__eq__self__iff,axiom,
% 0.32/0.62 ! [N: nat,K: int] :
% 0.32/0.62 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.32/0.62 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_int_less_eq_self_iff
% 0.32/0.62 thf(fact_6421_take__bit__nonnegative,axiom,
% 0.32/0.62 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_nonnegative
% 0.32/0.62 thf(fact_6422_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,A: int,B: int] :
% 0.32/0.62 ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.32/0.62 = ( bit_ri631733984087533419it_int @ N @ B ) )
% 0.32/0.62 = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_eq_iff_take_bit_eq
% 0.32/0.62 thf(fact_6423_take__bit__int__greater__self__iff,axiom,
% 0.32/0.62 ! [K: int,N: nat] :
% 0.32/0.62 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.32/0.62 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_int_greater_self_iff
% 0.32/0.62 thf(fact_6424_not__take__bit__negative,axiom,
% 0.32/0.62 ! [N: nat,K: int] :
% 0.32/0.62 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.32/0.62
% 0.32/0.62 % not_take_bit_negative
% 0.32/0.62 thf(fact_6425_signed__take__bit__take__bit,axiom,
% 0.32/0.62 ! [M: nat,N: nat,A: int] :
% 0.32/0.62 ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 0.32/0.62 = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 0.32/0.62
% 0.32/0.62 % signed_take_bit_take_bit
% 0.32/0.62 thf(fact_6426_xor__num_Ocases,axiom,
% 0.32/0.62 ! [X: product_prod_num_num] :
% 0.32/0.62 ( ( X
% 0.32/0.62 != ( product_Pair_num_num @ one @ one ) )
% 0.32/0.62 => ( ! [N4: num] :
% 0.32/0.62 ( X
% 0.32/0.62 != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
% 0.32/0.62 => ( ! [N4: num] :
% 0.32/0.62 ( X
% 0.32/0.62 != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
% 0.32/0.62 => ( ! [M4: num] :
% 0.32/0.62 ( X
% 0.32/0.62 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 0.32/0.62 => ( ! [M4: num,N4: num] :
% 0.32/0.62 ( X
% 0.32/0.62 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) )
% 0.32/0.62 => ( ! [M4: num,N4: num] :
% 0.32/0.62 ( X
% 0.32/0.62 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) )
% 0.32/0.62 => ( ! [M4: num] :
% 0.32/0.62 ( X
% 0.32/0.62 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 0.32/0.62 => ( ! [M4: num,N4: num] :
% 0.32/0.62 ( X
% 0.32/0.62 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) )
% 0.32/0.62 => ~ ! [M4: num,N4: num] :
% 0.32/0.62 ( X
% 0.32/0.62 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % xor_num.cases
% 0.32/0.62 thf(fact_6427_num_Oexhaust,axiom,
% 0.32/0.62 ! [Y3: num] :
% 0.32/0.62 ( ( Y3 != one )
% 0.32/0.62 => ( ! [X23: num] :
% 0.32/0.62 ( Y3
% 0.32/0.62 != ( bit0 @ X23 ) )
% 0.32/0.62 => ~ ! [X33: num] :
% 0.32/0.62 ( Y3
% 0.32/0.62 != ( bit1 @ X33 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % num.exhaust
% 0.32/0.62 thf(fact_6428_zabs__def,axiom,
% 0.32/0.62 ( abs_abs_int
% 0.32/0.62 = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % zabs_def
% 0.32/0.62 thf(fact_6429_take__bit__unset__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,M: nat,A: int] :
% 0.32/0.62 ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.32/0.62 = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_unset_bit_eq
% 0.32/0.62 thf(fact_6430_take__bit__unset__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,M: nat,A: nat] :
% 0.32/0.62 ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.32/0.62 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.32/0.62 = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_unset_bit_eq
% 0.32/0.62 thf(fact_6431_take__bit__set__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,M: nat,A: int] :
% 0.32/0.62 ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.32/0.62 = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_set_bit_eq
% 0.32/0.62 thf(fact_6432_take__bit__set__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,M: nat,A: nat] :
% 0.32/0.62 ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.32/0.62 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.32/0.62 = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_set_bit_eq
% 0.32/0.62 thf(fact_6433_take__bit__flip__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,M: nat,A: int] :
% 0.32/0.62 ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.32/0.62 = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_flip_bit_eq
% 0.32/0.62 thf(fact_6434_take__bit__flip__bit__eq,axiom,
% 0.32/0.62 ! [N: nat,M: nat,A: nat] :
% 0.32/0.62 ( ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.32/0.62 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 0.32/0.62 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.32/0.62 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.32/0.62 = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_flip_bit_eq
% 0.32/0.62 thf(fact_6435_abs__zmult__eq__1,axiom,
% 0.32/0.62 ! [M: int,N: int] :
% 0.32/0.62 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 0.32/0.62 = one_one_int )
% 0.32/0.62 => ( ( abs_abs_int @ M )
% 0.32/0.62 = one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_zmult_eq_1
% 0.32/0.62 thf(fact_6436_abs__div,axiom,
% 0.32/0.62 ! [Y3: int,X: int] :
% 0.32/0.62 ( ( dvd_dvd_int @ Y3 @ X )
% 0.32/0.62 => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y3 ) )
% 0.32/0.62 = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y3 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_div
% 0.32/0.62 thf(fact_6437_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > real] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 0.32/0.62 = zero_zero_real )
% 0.32/0.62 = ( ! [X2: real] :
% 0.32/0.62 ( ( member_real @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6438_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 0.32/0.62 = zero_zero_real )
% 0.32/0.62 = ( ! [X2: int] :
% 0.32/0.62 ( ( member_int @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6439_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 0.32/0.62 = zero_zero_real )
% 0.32/0.62 = ( ! [X2: complex] :
% 0.32/0.62 ( ( member_complex @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_real ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6440_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > rat] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 = ( ! [X2: real] :
% 0.32/0.62 ( ( member_real @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6441_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_nat,F: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ A2 )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 = ( ! [X2: nat] :
% 0.32/0.62 ( ( member_nat @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6442_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 = ( ! [X2: int] :
% 0.32/0.62 ( ( member_int @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6443_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > rat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 = ( ! [X2: complex] :
% 0.32/0.62 ( ( member_complex @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6444_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > nat] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 0.32/0.62 = zero_zero_nat )
% 0.32/0.62 = ( ! [X2: real] :
% 0.32/0.62 ( ( member_real @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6445_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > nat] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.32/0.62 = zero_zero_nat )
% 0.32/0.62 = ( ! [X2: int] :
% 0.32/0.62 ( ( member_int @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6446_sum__nonneg__eq__0__iff,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > nat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.62 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.32/0.62 = zero_zero_nat )
% 0.32/0.62 = ( ! [X2: complex] :
% 0.32/0.62 ( ( member_complex @ X2 @ A2 )
% 0.32/0.62 => ( ( F @ X2 )
% 0.32/0.62 = zero_zero_nat ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_eq_0_iff
% 0.32/0.62 thf(fact_6447_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( finite_finite_int @ T )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S )
% 0.32/0.62 => ? [Xa2: int] :
% 0.32/0.62 ( ( member_int @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6448_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ T )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S )
% 0.32/0.62 => ? [Xa2: complex] :
% 0.32/0.62 ( ( member_complex @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6449_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ( finite_finite_int @ T )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ S )
% 0.32/0.62 => ? [Xa2: int] :
% 0.32/0.62 ( ( member_int @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6450_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ T )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ S )
% 0.32/0.62 => ? [Xa2: complex] :
% 0.32/0.62 ( ( member_complex @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6451_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ( finite_finite_nat @ T )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ S )
% 0.32/0.62 => ? [Xa2: nat] :
% 0.32/0.62 ( ( member_nat @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6452_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ( finite_finite_int @ T )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ S )
% 0.32/0.62 => ? [Xa2: int] :
% 0.32/0.62 ( ( member_int @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6453_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ T )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ S )
% 0.32/0.62 => ? [Xa2: complex] :
% 0.32/0.62 ( ( member_complex @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6454_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( finite_finite_nat @ T )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S )
% 0.32/0.62 => ? [Xa2: nat] :
% 0.32/0.62 ( ( member_nat @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6455_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( finite_finite_int @ T )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S )
% 0.32/0.62 => ? [Xa2: int] :
% 0.32/0.62 ( ( member_int @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6456_sum__le__included,axiom,
% 0.32/0.62 ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ T )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ T )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S )
% 0.32/0.62 => ? [Xa2: complex] :
% 0.32/0.62 ( ( member_complex @ Xa2 @ T )
% 0.32/0.62 & ( ( I @ Xa2 )
% 0.32/0.62 = X3 )
% 0.32/0.62 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 0.32/0.62 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_le_included
% 0.32/0.62 thf(fact_6457_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > real,G: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: int] :
% 0.32/0.62 ( ( member_int @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6458_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: complex] :
% 0.32/0.62 ( ( member_complex @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6459_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ A2 )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: nat] :
% 0.32/0.62 ( ( member_nat @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6460_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: int] :
% 0.32/0.62 ( ( member_int @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6461_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: complex] :
% 0.32/0.62 ( ( member_complex @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6462_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: int] :
% 0.32/0.62 ( ( member_int @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6463_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: complex] :
% 0.32/0.62 ( ( member_complex @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6464_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_nat,F: nat > int,G: nat > int] :
% 0.32/0.62 ( ( finite_finite_nat @ A2 )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: nat] :
% 0.32/0.62 ( ( member_nat @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6465_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > int,G: complex > int] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: complex] :
% 0.32/0.62 ( ( member_complex @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6466_sum__strict__mono__ex1,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > int,G: int > int] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ? [X5: int] :
% 0.32/0.62 ( ( member_int @ X5 @ A2 )
% 0.32/0.62 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 0.32/0.62 => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono_ex1
% 0.32/0.62 thf(fact_6467_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.32/0.62 ( ( R4 @ zero_zero_complex @ zero_zero_complex )
% 0.32/0.62 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite_finite_nat @ S2 )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6468_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 0.32/0.62 ( ( R4 @ zero_zero_complex @ zero_zero_complex )
% 0.32/0.62 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite_finite_int @ S2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6469_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 0.32/0.62 ( ( R4 @ zero_zero_real @ zero_zero_real )
% 0.32/0.62 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite_finite_int @ S2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6470_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.32/0.62 ( ( R4 @ zero_zero_real @ zero_zero_real )
% 0.32/0.62 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6471_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 0.32/0.62 ( ( R4 @ zero_zero_rat @ zero_zero_rat )
% 0.32/0.62 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite_finite_nat @ S2 )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6472_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 0.32/0.62 ( ( R4 @ zero_zero_rat @ zero_zero_rat )
% 0.32/0.62 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite_finite_int @ S2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6473_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.32/0.62 ( ( R4 @ zero_zero_rat @ zero_zero_rat )
% 0.32/0.62 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6474_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 0.32/0.62 ( ( R4 @ zero_zero_nat @ zero_zero_nat )
% 0.32/0.62 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite_finite_int @ S2 )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6475_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 0.32/0.62 ( ( R4 @ zero_zero_nat @ zero_zero_nat )
% 0.32/0.62 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6476_sum_Orelated,axiom,
% 0.32/0.62 ! [R4: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
% 0.32/0.62 ( ( R4 @ zero_zero_int @ zero_zero_int )
% 0.32/0.62 => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
% 0.32/0.62 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.62 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.62 => ( R4 @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y22 ) ) )
% 0.32/0.62 => ( ( finite_finite_nat @ S2 )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ S2 )
% 0.32/0.62 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( R4 @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.related
% 0.32/0.62 thf(fact_6477_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_complex )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6478_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > real,G: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_int )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6479_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > real,G: real > real] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_real )
% 0.32/0.62 => ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6480_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_complex )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6481_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_nat )
% 0.32/0.62 => ( ! [X3: nat] :
% 0.32/0.62 ( ( member_nat @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6482_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_int )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6483_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > rat,G: real > rat] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_real )
% 0.32/0.62 => ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6484_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_complex )
% 0.32/0.62 => ( ! [X3: complex] :
% 0.32/0.62 ( ( member_complex @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6485_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_int )
% 0.32/0.62 => ( ! [X3: int] :
% 0.32/0.62 ( ( member_int @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6486_sum__strict__mono,axiom,
% 0.32/0.62 ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( A2 != bot_bot_set_real )
% 0.32/0.62 => ( ! [X3: real] :
% 0.32/0.62 ( ( member_real @ X3 @ A2 )
% 0.32/0.62 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.62 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_strict_mono
% 0.32/0.62 thf(fact_6487_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 0.32/0.62 ( ( finite_finite_real @ S4 )
% 0.32/0.62 => ( ( finite_finite_real @ T4 )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.62 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_complex ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_complex ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups5754745047067104278omplex @ G @ S2 )
% 0.32/0.62 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6488_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 0.32/0.62 ( ( finite_finite_real @ S4 )
% 0.32/0.62 => ( ( finite_finite_int @ T4 )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.62 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_complex ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_complex ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups5754745047067104278omplex @ G @ S2 )
% 0.32/0.62 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6489_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 0.32/0.62 ( ( finite_finite_int @ S4 )
% 0.32/0.62 => ( ( finite_finite_real @ T4 )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.62 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_complex ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_complex ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups3049146728041665814omplex @ G @ S2 )
% 0.32/0.62 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6490_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 0.32/0.62 ( ( finite_finite_int @ S4 )
% 0.32/0.62 => ( ( finite_finite_int @ T4 )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.62 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_complex ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_complex ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups3049146728041665814omplex @ G @ S2 )
% 0.32/0.62 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6491_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 0.32/0.62 ( ( finite_finite_real @ S4 )
% 0.32/0.62 => ( ( finite_finite_real @ T4 )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.62 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups8097168146408367636l_real @ G @ S2 )
% 0.32/0.62 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6492_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 0.32/0.62 ( ( finite_finite_real @ S4 )
% 0.32/0.62 => ( ( finite_finite_int @ T4 )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.62 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups8097168146408367636l_real @ G @ S2 )
% 0.32/0.62 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6493_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 0.32/0.62 ( ( finite_finite_real @ S4 )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ T4 )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.62 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: complex] :
% 0.32/0.62 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: complex] :
% 0.32/0.62 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.62 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [B3: complex] :
% 0.32/0.62 ( ( member_complex @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [A5: real] :
% 0.32/0.62 ( ( member_real @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups8097168146408367636l_real @ G @ S2 )
% 0.32/0.62 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6494_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 0.32/0.62 ( ( finite_finite_int @ S4 )
% 0.32/0.62 => ( ( finite_finite_real @ T4 )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.62 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [B3: real] :
% 0.32/0.62 ( ( member_real @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups8778361861064173332t_real @ G @ S2 )
% 0.32/0.62 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6495_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ S4 )
% 0.32/0.62 => ( ( finite_finite_int @ T4 )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.62 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [B3: int] :
% 0.32/0.62 ( ( member_int @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups8778361861064173332t_real @ G @ S2 )
% 0.32/0.62 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6496_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.62 ! [S4: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 0.32/0.62 ( ( finite_finite_int @ S4 )
% 0.32/0.62 => ( ( finite3207457112153483333omplex @ T4 )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( ( I @ ( J @ A5 ) )
% 0.32/0.62 = A5 ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.62 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.32/0.62 => ( ! [B3: complex] :
% 0.32/0.62 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.62 => ( ( J @ ( I @ B3 ) )
% 0.32/0.62 = B3 ) )
% 0.32/0.62 => ( ! [B3: complex] :
% 0.32/0.62 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.62 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S4 )
% 0.32/0.62 => ( ( G @ A5 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [B3: complex] :
% 0.32/0.62 ( ( member_complex @ B3 @ T4 )
% 0.32/0.62 => ( ( H2 @ B3 )
% 0.32/0.62 = zero_zero_real ) )
% 0.32/0.62 => ( ! [A5: int] :
% 0.32/0.62 ( ( member_int @ A5 @ S2 )
% 0.32/0.62 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.62 = ( G @ A5 ) ) )
% 0.32/0.62 => ( ( groups8778361861064173332t_real @ G @ S2 )
% 0.32/0.62 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.reindex_bij_witness_not_neutral
% 0.32/0.62 thf(fact_6497_take__bit__signed__take__bit,axiom,
% 0.32/0.62 ! [M: nat,N: nat,A: int] :
% 0.32/0.62 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 0.32/0.62 = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_signed_take_bit
% 0.32/0.62 thf(fact_6498_numeral__Bit1,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_Bit1
% 0.32/0.62 thf(fact_6499_numeral__Bit1,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_Bit1
% 0.32/0.62 thf(fact_6500_numeral__Bit1,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_Bit1
% 0.32/0.62 thf(fact_6501_numeral__Bit1,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_Bit1
% 0.32/0.62 thf(fact_6502_numeral__Bit1,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_Bit1
% 0.32/0.62 thf(fact_6503_take__bit__decr__eq,axiom,
% 0.32/0.62 ! [N: nat,K: int] :
% 0.32/0.62 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.32/0.62 != zero_zero_int )
% 0.32/0.62 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 0.32/0.62 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_decr_eq
% 0.32/0.62 thf(fact_6504_eval__nat__numeral_I3_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.32/0.62 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % eval_nat_numeral(3)
% 0.32/0.62 thf(fact_6505_cong__exp__iff__simps_I13_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.62 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.32/0.62 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(13)
% 0.32/0.62 thf(fact_6506_cong__exp__iff__simps_I13_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.62 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.32/0.62 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(13)
% 0.32/0.62 thf(fact_6507_cong__exp__iff__simps_I13_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.62 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.32/0.62 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(13)
% 0.32/0.62 thf(fact_6508_cong__exp__iff__simps_I12_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(12)
% 0.32/0.62 thf(fact_6509_cong__exp__iff__simps_I12_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(12)
% 0.32/0.62 thf(fact_6510_cong__exp__iff__simps_I12_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(12)
% 0.32/0.62 thf(fact_6511_cong__exp__iff__simps_I10_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(10)
% 0.32/0.62 thf(fact_6512_cong__exp__iff__simps_I10_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(10)
% 0.32/0.62 thf(fact_6513_cong__exp__iff__simps_I10_J,axiom,
% 0.32/0.62 ! [M: num,Q2: num,N: num] :
% 0.32/0.62 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.62 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % cong_exp_iff_simps(10)
% 0.32/0.62 thf(fact_6514_power__minus__Bit1,axiom,
% 0.32/0.62 ! [X: real,K: num] :
% 0.32/0.62 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.32/0.62 = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_Bit1
% 0.32/0.62 thf(fact_6515_power__minus__Bit1,axiom,
% 0.32/0.62 ! [X: int,K: num] :
% 0.32/0.62 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.32/0.62 = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_Bit1
% 0.32/0.62 thf(fact_6516_power__minus__Bit1,axiom,
% 0.32/0.62 ! [X: complex,K: num] :
% 0.32/0.62 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.32/0.62 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_Bit1
% 0.32/0.62 thf(fact_6517_power__minus__Bit1,axiom,
% 0.32/0.62 ! [X: code_integer,K: num] :
% 0.32/0.62 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.32/0.62 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_Bit1
% 0.32/0.62 thf(fact_6518_power__minus__Bit1,axiom,
% 0.32/0.62 ! [X: rat,K: num] :
% 0.32/0.62 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.32/0.62 = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_minus_Bit1
% 0.32/0.62 thf(fact_6519_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_real,F: real > real,I: real] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups8097168146408367636l_real @ F @ S )
% 0.32/0.62 = zero_zero_real )
% 0.32/0.62 => ( ( member_real @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_real ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6520_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_int,F: int > real,I: int] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups8778361861064173332t_real @ F @ S )
% 0.32/0.62 = zero_zero_real )
% 0.32/0.62 => ( ( member_int @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_real ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6521_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_complex,F: complex > real,I: complex] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups5808333547571424918x_real @ F @ S )
% 0.32/0.62 = zero_zero_real )
% 0.32/0.62 => ( ( member_complex @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_real ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6522_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_real,F: real > rat,I: real] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 => ( ( member_real @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6523_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_nat,F: nat > rat,I: nat] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ! [I2: nat] :
% 0.32/0.62 ( ( member_nat @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 => ( ( member_nat @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6524_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_int,F: int > rat,I: int] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 => ( ( member_int @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6525_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_complex,F: complex > rat,I: complex] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 0.32/0.62 = zero_zero_rat )
% 0.32/0.62 => ( ( member_complex @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_rat ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6526_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_real,F: real > nat,I: real] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 0.32/0.62 = zero_zero_nat )
% 0.32/0.62 => ( ( member_real @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6527_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_int,F: int > nat,I: int] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 0.32/0.62 = zero_zero_nat )
% 0.32/0.62 => ( ( member_int @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6528_sum__nonneg__0,axiom,
% 0.32/0.62 ! [S: set_complex,F: complex > nat,I: complex] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 0.32/0.62 = zero_zero_nat )
% 0.32/0.62 => ( ( member_complex @ I @ S )
% 0.32/0.62 => ( ( F @ I )
% 0.32/0.62 = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_0
% 0.32/0.62 thf(fact_6529_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_real,F: real > real,B5: real,I: real] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups8097168146408367636l_real @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_real @ I @ S )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6530_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_int,F: int > real,B5: real,I: int] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups8778361861064173332t_real @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_int @ I @ S )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6531_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_complex,F: complex > real,B5: real,I: complex] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups5808333547571424918x_real @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_complex @ I @ S )
% 0.32/0.62 => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6532_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_real,F: real > rat,B5: rat,I: real] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_real @ I @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6533_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_nat,F: nat > rat,B5: rat,I: nat] :
% 0.32/0.62 ( ( finite_finite_nat @ S )
% 0.32/0.62 => ( ! [I2: nat] :
% 0.32/0.62 ( ( member_nat @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_nat @ I @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6534_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_int,F: int > rat,B5: rat,I: int] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_int @ I @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6535_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_complex,F: complex > rat,B5: rat,I: complex] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_complex @ I @ S )
% 0.32/0.62 => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6536_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_real,F: real > nat,B5: nat,I: real] :
% 0.32/0.62 ( ( finite_finite_real @ S )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_real @ I @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6537_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_int,F: int > nat,B5: nat,I: int] :
% 0.32/0.62 ( ( finite_finite_int @ S )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_int @ I @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6538_sum__nonneg__leq__bound,axiom,
% 0.32/0.62 ! [S: set_complex,F: complex > nat,B5: nat,I: complex] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ S )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 0.32/0.62 = B5 )
% 0.32/0.62 => ( ( member_complex @ I @ S )
% 0.32/0.62 => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_nonneg_leq_bound
% 0.32/0.62 thf(fact_6539_dvd__imp__le__int,axiom,
% 0.32/0.62 ! [I: int,D: int] :
% 0.32/0.62 ( ( I != zero_zero_int )
% 0.32/0.62 => ( ( dvd_dvd_int @ D @ I )
% 0.32/0.62 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % dvd_imp_le_int
% 0.32/0.62 thf(fact_6540_take__bit__Suc__bit1,axiom,
% 0.32/0.62 ! [N: nat,K: num] :
% 0.32/0.62 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.32/0.62 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_Suc_bit1
% 0.32/0.62 thf(fact_6541_take__bit__Suc__bit1,axiom,
% 0.32/0.62 ! [N: nat,K: num] :
% 0.32/0.62 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.32/0.62 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % take_bit_Suc_bit1
% 0.32/0.62 thf(fact_6542_abs__mod__less,axiom,
% 0.32/0.62 ! [L2: int,K: int] :
% 0.32/0.62 ( ( L2 != zero_zero_int )
% 0.32/0.62 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % abs_mod_less
% 0.32/0.62 thf(fact_6543_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_real,G: real > complex] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( groups5754745047067104278omplex @ G
% 0.32/0.62 @ ( minus_minus_set_real @ A2
% 0.32/0.62 @ ( collect_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_complex ) ) ) )
% 0.32/0.62 = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6544_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_int,G: int > complex] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( groups3049146728041665814omplex @ G
% 0.32/0.62 @ ( minus_minus_set_int @ A2
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [X2: int] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_complex ) ) ) )
% 0.32/0.62 = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6545_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_real,G: real > real] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( groups8097168146408367636l_real @ G
% 0.32/0.62 @ ( minus_minus_set_real @ A2
% 0.32/0.62 @ ( collect_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_real ) ) ) )
% 0.32/0.62 = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6546_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_int,G: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( groups8778361861064173332t_real @ G
% 0.32/0.62 @ ( minus_minus_set_int @ A2
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [X2: int] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_real ) ) ) )
% 0.32/0.62 = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6547_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_complex,G: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ( groups5808333547571424918x_real @ G
% 0.32/0.62 @ ( minus_811609699411566653omplex @ A2
% 0.32/0.62 @ ( collect_complex
% 0.32/0.62 @ ^ [X2: complex] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_real ) ) ) )
% 0.32/0.62 = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6548_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_real,G: real > rat] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( groups1300246762558778688al_rat @ G
% 0.32/0.62 @ ( minus_minus_set_real @ A2
% 0.32/0.62 @ ( collect_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_rat ) ) ) )
% 0.32/0.62 = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6549_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_int,G: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( groups3906332499630173760nt_rat @ G
% 0.32/0.62 @ ( minus_minus_set_int @ A2
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [X2: int] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_rat ) ) ) )
% 0.32/0.62 = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6550_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_complex,G: complex > rat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.62 => ( ( groups5058264527183730370ex_rat @ G
% 0.32/0.62 @ ( minus_811609699411566653omplex @ A2
% 0.32/0.62 @ ( collect_complex
% 0.32/0.62 @ ^ [X2: complex] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_rat ) ) ) )
% 0.32/0.62 = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6551_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_real,G: real > nat] :
% 0.32/0.62 ( ( finite_finite_real @ A2 )
% 0.32/0.62 => ( ( groups1935376822645274424al_nat @ G
% 0.32/0.62 @ ( minus_minus_set_real @ A2
% 0.32/0.62 @ ( collect_real
% 0.32/0.62 @ ^ [X2: real] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_nat ) ) ) )
% 0.32/0.62 = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6552_sum_Osetdiff__irrelevant,axiom,
% 0.32/0.62 ! [A2: set_int,G: int > nat] :
% 0.32/0.62 ( ( finite_finite_int @ A2 )
% 0.32/0.62 => ( ( groups4541462559716669496nt_nat @ G
% 0.32/0.62 @ ( minus_minus_set_int @ A2
% 0.32/0.62 @ ( collect_int
% 0.32/0.62 @ ^ [X2: int] :
% 0.32/0.62 ( ( G @ X2 )
% 0.32/0.62 = zero_zero_nat ) ) ) )
% 0.32/0.62 = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum.setdiff_irrelevant
% 0.32/0.62 thf(fact_6553_numeral__code_I3_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_code(3)
% 0.32/0.62 thf(fact_6554_numeral__code_I3_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_code(3)
% 0.32/0.62 thf(fact_6555_numeral__code_I3_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_code(3)
% 0.32/0.62 thf(fact_6556_numeral__code_I3_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_code(3)
% 0.32/0.62 thf(fact_6557_numeral__code_I3_J,axiom,
% 0.32/0.62 ! [N: num] :
% 0.32/0.62 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 0.32/0.62 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 0.32/0.62
% 0.32/0.62 % numeral_code(3)
% 0.32/0.62 thf(fact_6558_power__numeral__odd,axiom,
% 0.32/0.62 ! [Z: complex,W: num] :
% 0.32/0.62 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.32/0.62 = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_numeral_odd
% 0.32/0.62 thf(fact_6559_power__numeral__odd,axiom,
% 0.32/0.62 ! [Z: real,W: num] :
% 0.32/0.62 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.32/0.62 = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_numeral_odd
% 0.32/0.62 thf(fact_6560_power__numeral__odd,axiom,
% 0.32/0.62 ! [Z: rat,W: num] :
% 0.32/0.62 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.32/0.62 = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_numeral_odd
% 0.32/0.62 thf(fact_6561_power__numeral__odd,axiom,
% 0.32/0.62 ! [Z: nat,W: num] :
% 0.32/0.62 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.32/0.62 = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_numeral_odd
% 0.32/0.62 thf(fact_6562_power__numeral__odd,axiom,
% 0.32/0.62 ! [Z: int,W: num] :
% 0.32/0.62 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 0.32/0.62 = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % power_numeral_odd
% 0.32/0.62 thf(fact_6563_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_real,I: real,F: real > real] :
% 0.32/0.62 ( ( finite_finite_real @ I5 )
% 0.32/0.62 => ( ( member_real @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6564_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_int,I: int,F: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ I5 )
% 0.32/0.62 => ( ( member_int @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6565_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_complex,I: complex,F: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.62 => ( ( member_complex @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6566_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_real,I: real,F: real > rat] :
% 0.32/0.62 ( ( finite_finite_real @ I5 )
% 0.32/0.62 => ( ( member_real @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6567_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_nat,I: nat,F: nat > rat] :
% 0.32/0.62 ( ( finite_finite_nat @ I5 )
% 0.32/0.62 => ( ( member_nat @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: nat] :
% 0.32/0.62 ( ( member_nat @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6568_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_int,I: int,F: int > rat] :
% 0.32/0.62 ( ( finite_finite_int @ I5 )
% 0.32/0.62 => ( ( member_int @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6569_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_complex,I: complex,F: complex > rat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.62 => ( ( member_complex @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6570_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_real,I: real,F: real > nat] :
% 0.32/0.62 ( ( finite_finite_real @ I5 )
% 0.32/0.62 => ( ( member_real @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: real] :
% 0.32/0.62 ( ( member_real @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6571_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_int,I: int,F: int > nat] :
% 0.32/0.62 ( ( finite_finite_int @ I5 )
% 0.32/0.62 => ( ( member_int @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6572_sum__pos2,axiom,
% 0.32/0.62 ! [I5: set_complex,I: complex,F: complex > nat] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.62 => ( ( member_complex @ I @ I5 )
% 0.32/0.62 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos2
% 0.32/0.62 thf(fact_6573_sum__pos,axiom,
% 0.32/0.62 ! [I5: set_complex,F: complex > real] :
% 0.32/0.62 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.62 => ( ( I5 != bot_bot_set_complex )
% 0.32/0.62 => ( ! [I2: complex] :
% 0.32/0.62 ( ( member_complex @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.62 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 0.32/0.62
% 0.32/0.62 % sum_pos
% 0.32/0.62 thf(fact_6574_sum__pos,axiom,
% 0.32/0.62 ! [I5: set_int,F: int > real] :
% 0.32/0.62 ( ( finite_finite_int @ I5 )
% 0.32/0.62 => ( ( I5 != bot_bot_set_int )
% 0.32/0.62 => ( ! [I2: int] :
% 0.32/0.62 ( ( member_int @ I2 @ I5 )
% 0.32/0.62 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6575_sum__pos,axiom,
% 0.32/0.63 ! [I5: set_real,F: real > real] :
% 0.32/0.63 ( ( finite_finite_real @ I5 )
% 0.32/0.63 => ( ( I5 != bot_bot_set_real )
% 0.32/0.63 => ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6576_sum__pos,axiom,
% 0.32/0.63 ! [I5: set_complex,F: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.63 => ( ( I5 != bot_bot_set_complex )
% 0.32/0.63 => ( ! [I2: complex] :
% 0.32/0.63 ( ( member_complex @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6577_sum__pos,axiom,
% 0.32/0.63 ! [I5: set_nat,F: nat > rat] :
% 0.32/0.63 ( ( finite_finite_nat @ I5 )
% 0.32/0.63 => ( ( I5 != bot_bot_set_nat )
% 0.32/0.63 => ( ! [I2: nat] :
% 0.32/0.63 ( ( member_nat @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6578_sum__pos,axiom,
% 0.32/0.63 ! [I5: set_int,F: int > rat] :
% 0.32/0.63 ( ( finite_finite_int @ I5 )
% 0.32/0.63 => ( ( I5 != bot_bot_set_int )
% 0.32/0.63 => ( ! [I2: int] :
% 0.32/0.63 ( ( member_int @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6579_sum__pos,axiom,
% 0.32/0.63 ! [I5: set_real,F: real > rat] :
% 0.32/0.63 ( ( finite_finite_real @ I5 )
% 0.32/0.63 => ( ( I5 != bot_bot_set_real )
% 0.32/0.63 => ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6580_sum__pos,axiom,
% 0.32/0.63 ! [I5: set_complex,F: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.63 => ( ( I5 != bot_bot_set_complex )
% 0.32/0.63 => ( ! [I2: complex] :
% 0.32/0.63 ( ( member_complex @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6581_sum__pos,axiom,
% 0.32/0.63 ! [I5: set_int,F: int > nat] :
% 0.32/0.63 ( ( finite_finite_int @ I5 )
% 0.32/0.63 => ( ( I5 != bot_bot_set_int )
% 0.32/0.63 => ( ! [I2: int] :
% 0.32/0.63 ( ( member_int @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6582_sum__pos,axiom,
% 0.32/0.63 ! [I5: set_real,F: real > nat] :
% 0.32/0.63 ( ( finite_finite_real @ I5 )
% 0.32/0.63 => ( ( I5 != bot_bot_set_real )
% 0.32/0.63 => ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 0.32/0.63 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_pos
% 0.32/0.63 thf(fact_6583_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5754745047067104278omplex @ G @ T3 )
% 0.32/0.63 = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6584_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups8097168146408367636l_real @ G @ T3 )
% 0.32/0.63 = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6585_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.32/0.63 = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6586_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 0.32/0.63 = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6587_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.32/0.63 = ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6588_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 0.32/0.63 = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6589_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.32/0.63 = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6590_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups1932886352136224148al_int @ G @ T3 )
% 0.32/0.63 = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6591_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.32/0.63 = ( groups5690904116761175830ex_int @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6592_sum_Omono__neutral__cong__right,axiom,
% 0.32/0.63 ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
% 0.32/0.63 ( ( finite_finite_nat @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.32/0.63 = ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_right
% 0.32/0.63 thf(fact_6593_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5754745047067104278omplex @ G @ S2 )
% 0.32/0.63 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6594_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups8097168146408367636l_real @ G @ S2 )
% 0.32/0.63 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6595_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5808333547571424918x_real @ G @ S2 )
% 0.32/0.63 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6596_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups1300246762558778688al_rat @ G @ S2 )
% 0.32/0.63 = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6597_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 0.32/0.63 = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6598_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups1935376822645274424al_nat @ G @ S2 )
% 0.32/0.63 = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6599_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 0.32/0.63 = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6600_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
% 0.32/0.63 ( ( finite_finite_real @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups1932886352136224148al_int @ G @ S2 )
% 0.32/0.63 = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6601_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 0.32/0.63 = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6602_sum_Omono__neutral__cong__left,axiom,
% 0.32/0.63 ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.32/0.63 ( ( finite_finite_nat @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.32/0.63 => ( ( H2 @ X3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ S2 )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = ( H2 @ X3 ) ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex @ G @ S2 )
% 0.32/0.63 = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_cong_left
% 0.32/0.63 thf(fact_6603_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.32/0.63 = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6604_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.32/0.63 = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6605_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.32/0.63 = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6606_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.32/0.63 = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6607_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 0.32/0.63 ( ( finite_finite_nat @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.32/0.63 = ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6608_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 0.32/0.63 ( ( finite_finite_nat @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 0.32/0.63 = ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6609_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 0.32/0.63 ( ( finite_finite_nat @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ G @ T3 )
% 0.32/0.63 = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6610_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_int,S2: set_int,G: int > complex] :
% 0.32/0.63 ( ( finite_finite_int @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: int] :
% 0.32/0.63 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ( groups3049146728041665814omplex @ G @ T3 )
% 0.32/0.63 = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6611_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_int,S2: set_int,G: int > real] :
% 0.32/0.63 ( ( finite_finite_int @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: int] :
% 0.32/0.63 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ( groups8778361861064173332t_real @ G @ T3 )
% 0.32/0.63 = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6612_sum_Omono__neutral__right,axiom,
% 0.32/0.63 ! [T3: set_int,S2: set_int,G: int > rat] :
% 0.32/0.63 ( ( finite_finite_int @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: int] :
% 0.32/0.63 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 0.32/0.63 = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_right
% 0.32/0.63 thf(fact_6613_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ( groups5808333547571424918x_real @ G @ S2 )
% 0.32/0.63 = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6614_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 0.32/0.63 = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6615_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 0.32/0.63 = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6616_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ T3 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 0.32/0.63 = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6617_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 0.32/0.63 ( ( finite_finite_nat @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex @ G @ S2 )
% 0.32/0.63 = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6618_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 0.32/0.63 ( ( finite_finite_nat @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ G @ S2 )
% 0.32/0.63 = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6619_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 0.32/0.63 ( ( finite_finite_nat @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ G @ S2 )
% 0.32/0.63 = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6620_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_int,S2: set_int,G: int > complex] :
% 0.32/0.63 ( ( finite_finite_int @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: int] :
% 0.32/0.63 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ( groups3049146728041665814omplex @ G @ S2 )
% 0.32/0.63 = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6621_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_int,S2: set_int,G: int > real] :
% 0.32/0.63 ( ( finite_finite_int @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: int] :
% 0.32/0.63 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ( groups8778361861064173332t_real @ G @ S2 )
% 0.32/0.63 = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6622_sum_Omono__neutral__left,axiom,
% 0.32/0.63 ! [T3: set_int,S2: set_int,G: int > rat] :
% 0.32/0.63 ( ( finite_finite_int @ T3 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 0.32/0.63 => ( ! [X3: int] :
% 0.32/0.63 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 0.32/0.63 => ( ( G @ X3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( groups3906332499630173760nt_rat @ G @ S2 )
% 0.32/0.63 = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.mono_neutral_left
% 0.32/0.63 thf(fact_6623_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ( ( groups5754745047067104278omplex @ G @ C4 )
% 0.32/0.63 = ( groups5754745047067104278omplex @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.32/0.63 = ( groups5754745047067104278omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6624_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ( ( groups8097168146408367636l_real @ G @ C4 )
% 0.32/0.63 = ( groups8097168146408367636l_real @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.32/0.63 = ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6625_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: complex] :
% 0.32/0.63 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ( ( groups5808333547571424918x_real @ G @ C4 )
% 0.32/0.63 = ( groups5808333547571424918x_real @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.32/0.63 = ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6626_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
% 0.32/0.63 = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.32/0.63 = ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6627_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: complex] :
% 0.32/0.63 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 0.32/0.63 = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.32/0.63 = ( groups5058264527183730370ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6628_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
% 0.32/0.63 = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.32/0.63 = ( groups1935376822645274424al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6629_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: complex] :
% 0.32/0.63 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 0.32/0.63 = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.32/0.63 = ( groups5693394587270226106ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6630_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > int,H2: real > int] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ( ( groups1932886352136224148al_int @ G @ C4 )
% 0.32/0.63 = ( groups1932886352136224148al_int @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.32/0.63 = ( groups1932886352136224148al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6631_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: complex] :
% 0.32/0.63 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
% 0.32/0.63 = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.32/0.63 = ( groups5690904116761175830ex_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6632_sum_Osame__carrierI,axiom,
% 0.32/0.63 ! [C4: set_nat,A2: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 0.32/0.63 ( ( finite_finite_nat @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: nat] :
% 0.32/0.63 ( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ! [B3: nat] :
% 0.32/0.63 ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ( ( groups2073611262835488442omplex @ G @ C4 )
% 0.32/0.63 = ( groups2073611262835488442omplex @ H2 @ C4 ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.32/0.63 = ( groups2073611262835488442omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrierI
% 0.32/0.63 thf(fact_6633_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.32/0.63 = ( groups5754745047067104278omplex @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups5754745047067104278omplex @ G @ C4 )
% 0.32/0.63 = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6634_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.32/0.63 = ( groups8097168146408367636l_real @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups8097168146408367636l_real @ G @ C4 )
% 0.32/0.63 = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6635_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: complex] :
% 0.32/0.63 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.32/0.63 = ( groups5808333547571424918x_real @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups5808333547571424918x_real @ G @ C4 )
% 0.32/0.63 = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6636_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.32/0.63 = ( groups1300246762558778688al_rat @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups1300246762558778688al_rat @ G @ C4 )
% 0.32/0.63 = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6637_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: complex] :
% 0.32/0.63 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.32/0.63 = ( groups5058264527183730370ex_rat @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 0.32/0.63 = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6638_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.32/0.63 = ( groups1935376822645274424al_nat @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups1935376822645274424al_nat @ G @ C4 )
% 0.32/0.63 = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6639_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: complex] :
% 0.32/0.63 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.32/0.63 = ( groups5693394587270226106ex_nat @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 0.32/0.63 = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6640_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_real,A2: set_real,B5: set_real,G: real > int,H2: real > int] :
% 0.32/0.63 ( ( finite_finite_real @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: real] :
% 0.32/0.63 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.32/0.63 = ( groups1932886352136224148al_int @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups1932886352136224148al_int @ G @ C4 )
% 0.32/0.63 = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6641_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: complex] :
% 0.32/0.63 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.32/0.63 = ( groups5690904116761175830ex_int @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups5690904116761175830ex_int @ G @ C4 )
% 0.32/0.63 = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6642_sum_Osame__carrier,axiom,
% 0.32/0.63 ! [C4: set_nat,A2: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 0.32/0.63 ( ( finite_finite_nat @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 0.32/0.63 => ( ! [A5: nat] :
% 0.32/0.63 ( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.32/0.63 => ( ( G @ A5 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ! [B3: nat] :
% 0.32/0.63 ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 0.32/0.63 => ( ( H2 @ B3 )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 => ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.32/0.63 = ( groups2073611262835488442omplex @ H2 @ B5 ) )
% 0.32/0.63 = ( ( groups2073611262835488442omplex @ G @ C4 )
% 0.32/0.63 = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.same_carrier
% 0.32/0.63 thf(fact_6643_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,G: complex > real] :
% 0.32/0.63 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.32/0.63 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6644_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,G: complex > rat] :
% 0.32/0.63 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.32/0.63 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6645_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,G: complex > nat] :
% 0.32/0.63 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.32/0.63 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6646_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,G: complex > int] :
% 0.32/0.63 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.32/0.63 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6647_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_nat,A2: set_nat,G: nat > rat] :
% 0.32/0.63 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.63 => ( ( finite_finite_nat @ A2 )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.32/0.63 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6648_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_nat,A2: set_nat,G: nat > int] :
% 0.32/0.63 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.63 => ( ( finite_finite_nat @ A2 )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ G @ A2 )
% 0.32/0.63 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6649_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_int,A2: set_int,G: int > real] :
% 0.32/0.63 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.63 => ( ( finite_finite_int @ A2 )
% 0.32/0.63 => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.32/0.63 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups8778361861064173332t_real @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6650_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_int,A2: set_int,G: int > rat] :
% 0.32/0.63 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.63 => ( ( finite_finite_int @ A2 )
% 0.32/0.63 => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.32/0.63 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6651_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_int,A2: set_int,G: int > nat] :
% 0.32/0.63 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.63 => ( ( finite_finite_int @ A2 )
% 0.32/0.63 => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.32/0.63 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6652_sum_Osubset__diff,axiom,
% 0.32/0.63 ! [B5: set_int,A2: set_int,G: int > int] :
% 0.32/0.63 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.63 => ( ( finite_finite_int @ A2 )
% 0.32/0.63 => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 0.32/0.63 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.subset_diff
% 0.32/0.63 thf(fact_6653_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_complex,B5: set_complex,F: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6654_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_complex,B5: set_complex,F: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6655_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_complex,B5: set_complex,F: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6656_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_nat,B5: set_nat,F: nat > rat] :
% 0.32/0.63 ( ( finite_finite_nat @ A2 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6657_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_nat,B5: set_nat,F: nat > int] :
% 0.32/0.63 ( ( finite_finite_nat @ A2 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6658_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_int,B5: set_int,F: int > real] :
% 0.32/0.63 ( ( finite_finite_int @ A2 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.63 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6659_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_int,B5: set_int,F: int > rat] :
% 0.32/0.63 ( ( finite_finite_int @ A2 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.63 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6660_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_int,B5: set_int,F: int > int] :
% 0.32/0.63 ( ( finite_finite_int @ A2 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.63 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6661_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_complex,B5: set_complex,F: complex > complex] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6662_sum__diff,axiom,
% 0.32/0.63 ! [A2: set_nat,B5: set_nat,F: nat > real] :
% 0.32/0.63 ( ( finite_finite_nat @ A2 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.63 => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff
% 0.32/0.63 thf(fact_6663_numeral__Bit1__div__2,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.63 = ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % numeral_Bit1_div_2
% 0.32/0.63 thf(fact_6664_numeral__Bit1__div__2,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.63 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % numeral_Bit1_div_2
% 0.32/0.63 thf(fact_6665_odd__numeral,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % odd_numeral
% 0.32/0.63 thf(fact_6666_odd__numeral,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % odd_numeral
% 0.32/0.63 thf(fact_6667_odd__numeral,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % odd_numeral
% 0.32/0.63 thf(fact_6668_cong__exp__iff__simps_I3_J,axiom,
% 0.32/0.63 ! [N: num,Q2: num] :
% 0.32/0.63 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 != zero_zero_nat ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(3)
% 0.32/0.63 thf(fact_6669_cong__exp__iff__simps_I3_J,axiom,
% 0.32/0.63 ! [N: num,Q2: num] :
% 0.32/0.63 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 != zero_zero_int ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(3)
% 0.32/0.63 thf(fact_6670_cong__exp__iff__simps_I3_J,axiom,
% 0.32/0.63 ! [N: num,Q2: num] :
% 0.32/0.63 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 != zero_z3403309356797280102nteger ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(3)
% 0.32/0.63 thf(fact_6671_power3__eq__cube,axiom,
% 0.32/0.63 ! [A: complex] :
% 0.32/0.63 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.32/0.63 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 0.32/0.63
% 0.32/0.63 % power3_eq_cube
% 0.32/0.63 thf(fact_6672_power3__eq__cube,axiom,
% 0.32/0.63 ! [A: real] :
% 0.32/0.63 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.32/0.63 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 0.32/0.63
% 0.32/0.63 % power3_eq_cube
% 0.32/0.63 thf(fact_6673_power3__eq__cube,axiom,
% 0.32/0.63 ! [A: rat] :
% 0.32/0.63 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.32/0.63 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 0.32/0.63
% 0.32/0.63 % power3_eq_cube
% 0.32/0.63 thf(fact_6674_power3__eq__cube,axiom,
% 0.32/0.63 ! [A: nat] :
% 0.32/0.63 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.32/0.63 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 0.32/0.63
% 0.32/0.63 % power3_eq_cube
% 0.32/0.63 thf(fact_6675_power3__eq__cube,axiom,
% 0.32/0.63 ! [A: int] :
% 0.32/0.63 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.32/0.63 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 0.32/0.63
% 0.32/0.63 % power3_eq_cube
% 0.32/0.63 thf(fact_6676_numeral__3__eq__3,axiom,
% 0.32/0.63 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.32/0.63 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % numeral_3_eq_3
% 0.32/0.63 thf(fact_6677_Suc3__eq__add__3,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % Suc3_eq_add_3
% 0.32/0.63 thf(fact_6678_zdvd__mult__cancel1,axiom,
% 0.32/0.63 ! [M: int,N: int] :
% 0.32/0.63 ( ( M != zero_zero_int )
% 0.32/0.63 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 0.32/0.63 = ( ( abs_abs_int @ N )
% 0.32/0.63 = one_one_int ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % zdvd_mult_cancel1
% 0.32/0.63 thf(fact_6679_take__bit__Suc__bit0,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.32/0.63 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc_bit0
% 0.32/0.63 thf(fact_6680_take__bit__Suc__bit0,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.32/0.63 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc_bit0
% 0.32/0.63 thf(fact_6681_take__bit__eq__mod,axiom,
% 0.32/0.63 ( bit_se1745604003318907178nteger
% 0.32/0.63 = ( ^ [N2: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_eq_mod
% 0.32/0.63 thf(fact_6682_take__bit__eq__mod,axiom,
% 0.32/0.63 ( bit_se2923211474154528505it_int
% 0.32/0.63 = ( ^ [N2: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_eq_mod
% 0.32/0.63 thf(fact_6683_take__bit__eq__mod,axiom,
% 0.32/0.63 ( bit_se2925701944663578781it_nat
% 0.32/0.63 = ( ^ [N2: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_eq_mod
% 0.32/0.63 thf(fact_6684_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_real,A2: set_real,F: real > real] :
% 0.32/0.63 ( ( finite_finite_real @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6685_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,F: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6686_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_real,A2: set_real,F: real > rat] :
% 0.32/0.63 ( ( finite_finite_real @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6687_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,F: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6688_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_nat,A2: set_nat,F: nat > rat] :
% 0.32/0.63 ( ( finite_finite_nat @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: nat] :
% 0.32/0.63 ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6689_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_real,A2: set_real,F: real > nat] :
% 0.32/0.63 ( ( finite_finite_real @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6690_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,F: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6691_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_real,A2: set_real,F: real > int] :
% 0.32/0.63 ( ( finite_finite_real @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: real] :
% 0.32/0.63 ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6692_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,F: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: complex] :
% 0.32/0.63 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6693_sum__mono2,axiom,
% 0.32/0.63 ! [B5: set_nat,A2: set_nat,F: nat > int] :
% 0.32/0.63 ( ( finite_finite_nat @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.63 => ( ! [B3: nat] :
% 0.32/0.63 ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A2 ) )
% 0.32/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 0.32/0.63 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_mono2
% 0.32/0.63 thf(fact_6694_take__bit__nat__eq__self,axiom,
% 0.32/0.63 ! [M: nat,N: nat] :
% 0.32/0.63 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.63 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.32/0.63 = M ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_nat_eq_self
% 0.32/0.63 thf(fact_6695_take__bit__nat__less__exp,axiom,
% 0.32/0.63 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_nat_less_exp
% 0.32/0.63 thf(fact_6696_take__bit__nat__eq__self__iff,axiom,
% 0.32/0.63 ! [N: nat,M: nat] :
% 0.32/0.63 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.32/0.63 = M )
% 0.32/0.63 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_nat_eq_self_iff
% 0.32/0.63 thf(fact_6697_num_Osize_I6_J,axiom,
% 0.32/0.63 ! [X32: num] :
% 0.32/0.63 ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.32/0.63 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % num.size(6)
% 0.32/0.63 thf(fact_6698_take__bit__nat__def,axiom,
% 0.32/0.63 ( bit_se2925701944663578781it_nat
% 0.32/0.63 = ( ^ [N2: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_nat_def
% 0.32/0.63 thf(fact_6699_take__bit__Suc__minus__bit0,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.32/0.63 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc_minus_bit0
% 0.32/0.63 thf(fact_6700_take__bit__int__less__exp,axiom,
% 0.32/0.63 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_int_less_exp
% 0.32/0.63 thf(fact_6701_take__bit__int__def,axiom,
% 0.32/0.63 ( bit_se2923211474154528505it_int
% 0.32/0.63 = ( ^ [N2: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_int_def
% 0.32/0.63 thf(fact_6702_cong__exp__iff__simps_I11_J,axiom,
% 0.32/0.63 ! [M: num,Q2: num] :
% 0.32/0.63 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.63 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.32/0.63 = zero_zero_nat ) ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(11)
% 0.32/0.63 thf(fact_6703_cong__exp__iff__simps_I11_J,axiom,
% 0.32/0.63 ! [M: num,Q2: num] :
% 0.32/0.63 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.63 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 0.32/0.63 = zero_zero_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(11)
% 0.32/0.63 thf(fact_6704_cong__exp__iff__simps_I11_J,axiom,
% 0.32/0.63 ! [M: num,Q2: num] :
% 0.32/0.63 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.63 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.32/0.63 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(11)
% 0.32/0.63 thf(fact_6705_cong__exp__iff__simps_I7_J,axiom,
% 0.32/0.63 ! [Q2: num,N: num] :
% 0.32/0.63 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.63 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 0.32/0.63 = zero_zero_nat ) ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(7)
% 0.32/0.63 thf(fact_6706_cong__exp__iff__simps_I7_J,axiom,
% 0.32/0.63 ! [Q2: num,N: num] :
% 0.32/0.63 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.63 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 0.32/0.63 = zero_zero_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(7)
% 0.32/0.63 thf(fact_6707_cong__exp__iff__simps_I7_J,axiom,
% 0.32/0.63 ! [Q2: num,N: num] :
% 0.32/0.63 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 0.32/0.63 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 0.32/0.63 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 0.32/0.63 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.63
% 0.32/0.63 % cong_exp_iff_simps(7)
% 0.32/0.63 thf(fact_6708_even__add__abs__iff,axiom,
% 0.32/0.63 ! [K: int,L2: int] :
% 0.32/0.63 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
% 0.32/0.63 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % even_add_abs_iff
% 0.32/0.63 thf(fact_6709_even__abs__add__iff,axiom,
% 0.32/0.63 ! [K: int,L2: int] :
% 0.32/0.63 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
% 0.32/0.63 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % even_abs_add_iff
% 0.32/0.63 thf(fact_6710_Suc__div__eq__add3__div,axiom,
% 0.32/0.63 ! [M: nat,N: nat] :
% 0.32/0.63 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.32/0.63 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % Suc_div_eq_add3_div
% 0.32/0.63 thf(fact_6711_Suc__mod__eq__add3__mod,axiom,
% 0.32/0.63 ! [M: nat,N: nat] :
% 0.32/0.63 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 0.32/0.63 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % Suc_mod_eq_add3_mod
% 0.32/0.63 thf(fact_6712_take__bit__eq__0__iff,axiom,
% 0.32/0.63 ! [N: nat,A: code_integer] :
% 0.32/0.63 ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.32/0.63 = zero_z3403309356797280102nteger )
% 0.32/0.63 = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_eq_0_iff
% 0.32/0.63 thf(fact_6713_take__bit__eq__0__iff,axiom,
% 0.32/0.63 ! [N: nat,A: int] :
% 0.32/0.63 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.32/0.63 = zero_zero_int )
% 0.32/0.63 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_eq_0_iff
% 0.32/0.63 thf(fact_6714_take__bit__eq__0__iff,axiom,
% 0.32/0.63 ! [N: nat,A: nat] :
% 0.32/0.63 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.32/0.63 = zero_zero_nat )
% 0.32/0.63 = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_eq_0_iff
% 0.32/0.63 thf(fact_6715_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_real,A2: set_real,B: real,F: real > real] :
% 0.32/0.63 ( ( finite_finite_real @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.63 => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6716_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.63 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6717_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_real,A2: set_real,B: real,F: real > rat] :
% 0.32/0.63 ( ( finite_finite_real @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.63 => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6718_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.63 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6719_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_nat,A2: set_nat,B: nat,F: nat > rat] :
% 0.32/0.63 ( ( finite_finite_nat @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.63 => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6720_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_real,A2: set_real,B: real,F: real > nat] :
% 0.32/0.63 ( ( finite_finite_real @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.63 => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6721_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.63 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6722_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_real,A2: set_real,B: real,F: real > int] :
% 0.32/0.63 ( ( finite_finite_real @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.63 => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6723_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,B: complex,F: complex > int] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.63 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6724_sum__strict__mono2,axiom,
% 0.32/0.63 ! [B5: set_nat,A2: set_nat,B: nat,F: nat > int] :
% 0.32/0.63 ( ( finite_finite_nat @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.63 => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A2 ) )
% 0.32/0.63 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.32/0.63 => ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ B5 )
% 0.32/0.63 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_strict_mono2
% 0.32/0.63 thf(fact_6725_take__bit__nat__less__self__iff,axiom,
% 0.32/0.63 ! [N: nat,M: nat] :
% 0.32/0.63 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.32/0.63 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_nat_less_self_iff
% 0.32/0.63 thf(fact_6726_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 0.32/0.63 ( ! [I2: nat] :
% 0.32/0.63 ( ( member_nat @ I2 @ I5 )
% 0.32/0.63 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups7501900531339628137nteger @ X @ I5 )
% 0.32/0.63 = one_one_Code_integer )
% 0.32/0.63 => ( ! [I2: nat] :
% 0.32/0.63 ( ( member_nat @ I2 @ I5 )
% 0.32/0.63 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_le3102999989581377725nteger
% 0.32/0.63 @ ( abs_abs_Code_integer
% 0.32/0.63 @ ( minus_8373710615458151222nteger
% 0.32/0.63 @ ( groups7501900531339628137nteger
% 0.32/0.63 @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6727_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 0.32/0.63 ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups7713935264441627589nteger @ X @ I5 )
% 0.32/0.63 = one_one_Code_integer )
% 0.32/0.63 => ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_le3102999989581377725nteger
% 0.32/0.63 @ ( abs_abs_Code_integer
% 0.32/0.63 @ ( minus_8373710615458151222nteger
% 0.32/0.63 @ ( groups7713935264441627589nteger
% 0.32/0.63 @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6728_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 0.32/0.63 ( ! [I2: int] :
% 0.32/0.63 ( ( member_int @ I2 @ I5 )
% 0.32/0.63 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups7873554091576472773nteger @ X @ I5 )
% 0.32/0.63 = one_one_Code_integer )
% 0.32/0.63 => ( ! [I2: int] :
% 0.32/0.63 ( ( member_int @ I2 @ I5 )
% 0.32/0.63 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_le3102999989581377725nteger
% 0.32/0.63 @ ( abs_abs_Code_integer
% 0.32/0.63 @ ( minus_8373710615458151222nteger
% 0.32/0.63 @ ( groups7873554091576472773nteger
% 0.32/0.63 @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6729_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
% 0.32/0.63 ( ! [I2: complex] :
% 0.32/0.63 ( ( member_complex @ I2 @ I5 )
% 0.32/0.63 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups6621422865394947399nteger @ X @ I5 )
% 0.32/0.63 = one_one_Code_integer )
% 0.32/0.63 => ( ! [I2: complex] :
% 0.32/0.63 ( ( member_complex @ I2 @ I5 )
% 0.32/0.63 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_le3102999989581377725nteger
% 0.32/0.63 @ ( abs_abs_Code_integer
% 0.32/0.63 @ ( minus_8373710615458151222nteger
% 0.32/0.63 @ ( groups6621422865394947399nteger
% 0.32/0.63 @ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6730_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_real,X: real > real,A: real > real,B: real,Delta: real] :
% 0.32/0.63 ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups8097168146408367636l_real @ X @ I5 )
% 0.32/0.63 = one_one_real )
% 0.32/0.63 => ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_less_eq_real
% 0.32/0.63 @ ( abs_abs_real
% 0.32/0.63 @ ( minus_minus_real
% 0.32/0.63 @ ( groups8097168146408367636l_real
% 0.32/0.63 @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6731_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_int,X: int > real,A: int > real,B: real,Delta: real] :
% 0.32/0.63 ( ! [I2: int] :
% 0.32/0.63 ( ( member_int @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups8778361861064173332t_real @ X @ I5 )
% 0.32/0.63 = one_one_real )
% 0.32/0.63 => ( ! [I2: int] :
% 0.32/0.63 ( ( member_int @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_less_eq_real
% 0.32/0.63 @ ( abs_abs_real
% 0.32/0.63 @ ( minus_minus_real
% 0.32/0.63 @ ( groups8778361861064173332t_real
% 0.32/0.63 @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6732_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
% 0.32/0.63 ( ! [I2: complex] :
% 0.32/0.63 ( ( member_complex @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups5808333547571424918x_real @ X @ I5 )
% 0.32/0.63 = one_one_real )
% 0.32/0.63 => ( ! [I2: complex] :
% 0.32/0.63 ( ( member_complex @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_less_eq_real
% 0.32/0.63 @ ( abs_abs_real
% 0.32/0.63 @ ( minus_minus_real
% 0.32/0.63 @ ( groups5808333547571424918x_real
% 0.32/0.63 @ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6733_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 0.32/0.63 ( ! [I2: nat] :
% 0.32/0.63 ( ( member_nat @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups2906978787729119204at_rat @ X @ I5 )
% 0.32/0.63 = one_one_rat )
% 0.32/0.63 => ( ! [I2: nat] :
% 0.32/0.63 ( ( member_nat @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_less_eq_rat
% 0.32/0.63 @ ( abs_abs_rat
% 0.32/0.63 @ ( minus_minus_rat
% 0.32/0.63 @ ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6734_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
% 0.32/0.63 ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups1300246762558778688al_rat @ X @ I5 )
% 0.32/0.63 = one_one_rat )
% 0.32/0.63 => ( ! [I2: real] :
% 0.32/0.63 ( ( member_real @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_less_eq_rat
% 0.32/0.63 @ ( abs_abs_rat
% 0.32/0.63 @ ( minus_minus_rat
% 0.32/0.63 @ ( groups1300246762558778688al_rat
% 0.32/0.63 @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6735_convex__sum__bound__le,axiom,
% 0.32/0.63 ! [I5: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
% 0.32/0.63 ( ! [I2: int] :
% 0.32/0.63 ( ( member_int @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 0.32/0.63 => ( ( ( groups3906332499630173760nt_rat @ X @ I5 )
% 0.32/0.63 = one_one_rat )
% 0.32/0.63 => ( ! [I2: int] :
% 0.32/0.63 ( ( member_int @ I2 @ I5 )
% 0.32/0.63 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 0.32/0.63 => ( ord_less_eq_rat
% 0.32/0.63 @ ( abs_abs_rat
% 0.32/0.63 @ ( minus_minus_rat
% 0.32/0.63 @ ( groups3906332499630173760nt_rat
% 0.32/0.63 @ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 @ B ) )
% 0.32/0.63 @ Delta ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % convex_sum_bound_le
% 0.32/0.63 thf(fact_6736_take__bit__int__less__self__iff,axiom,
% 0.32/0.63 ! [N: nat,K: int] :
% 0.32/0.63 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.32/0.63 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_int_less_self_iff
% 0.32/0.63 thf(fact_6737_take__bit__int__greater__eq__self__iff,axiom,
% 0.32/0.63 ! [K: int,N: nat] :
% 0.32/0.63 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.32/0.63 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_int_greater_eq_self_iff
% 0.32/0.63 thf(fact_6738_nat__intermed__int__val,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.32/0.63 ( ! [I2: nat] :
% 0.32/0.63 ( ( ( ord_less_eq_nat @ M @ I2 )
% 0.32/0.63 & ( ord_less_nat @ I2 @ N ) )
% 0.32/0.63 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.32/0.63 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.32/0.63 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.32/0.63 => ? [I2: nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ I2 )
% 0.32/0.63 & ( ord_less_eq_nat @ I2 @ N )
% 0.32/0.63 & ( ( F @ I2 )
% 0.32/0.63 = K ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % nat_intermed_int_val
% 0.32/0.63 thf(fact_6739_mod__exhaust__less__4,axiom,
% 0.32/0.63 ! [M: nat] :
% 0.32/0.63 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.63 = zero_zero_nat )
% 0.32/0.63 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.63 = one_one_nat )
% 0.32/0.63 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.63 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.63 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.63 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % mod_exhaust_less_4
% 0.32/0.63 thf(fact_6740_incr__lemma,axiom,
% 0.32/0.63 ! [D: int,Z: int,X: int] :
% 0.32/0.63 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.63 => ( 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 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % incr_lemma
% 0.32/0.63 thf(fact_6741_decr__lemma,axiom,
% 0.32/0.63 ! [D: int,X: int,Z: int] :
% 0.32/0.63 ( ( ord_less_int @ zero_zero_int @ D )
% 0.32/0.63 => ( 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 ) ) ).
% 0.32/0.63
% 0.32/0.63 % decr_lemma
% 0.32/0.63 thf(fact_6742_take__bit__int__eq__self,axiom,
% 0.32/0.63 ! [K: int,N: nat] :
% 0.32/0.63 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.63 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.63 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.32/0.63 = K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_int_eq_self
% 0.32/0.63 thf(fact_6743_take__bit__int__eq__self__iff,axiom,
% 0.32/0.63 ! [N: nat,K: int] :
% 0.32/0.63 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.32/0.63 = K )
% 0.32/0.63 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.63 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_int_eq_self_iff
% 0.32/0.63 thf(fact_6744_take__bit__incr__eq,axiom,
% 0.32/0.63 ! [N: nat,K: int] :
% 0.32/0.63 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.32/0.63 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 0.32/0.63 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.32/0.63 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_incr_eq
% 0.32/0.63 thf(fact_6745_nat__ivt__aux,axiom,
% 0.32/0.63 ! [N: nat,F: nat > int,K: int] :
% 0.32/0.63 ( ! [I2: nat] :
% 0.32/0.63 ( ( ord_less_nat @ I2 @ N )
% 0.32/0.63 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.32/0.63 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.32/0.63 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.32/0.63 => ? [I2: nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ I2 @ N )
% 0.32/0.63 & ( ( F @ I2 )
% 0.32/0.63 = K ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % nat_ivt_aux
% 0.32/0.63 thf(fact_6746_take__bit__Suc__minus__1__eq,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.63 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc_minus_1_eq
% 0.32/0.63 thf(fact_6747_take__bit__Suc__minus__1__eq,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.63 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc_minus_1_eq
% 0.32/0.63 thf(fact_6748_take__bit__numeral__minus__1__eq,axiom,
% 0.32/0.63 ! [K: num] :
% 0.32/0.63 ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.63 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_numeral_minus_1_eq
% 0.32/0.63 thf(fact_6749_take__bit__numeral__minus__1__eq,axiom,
% 0.32/0.63 ! [K: num] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.63 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_numeral_minus_1_eq
% 0.32/0.63 thf(fact_6750_take__bit__Suc,axiom,
% 0.32/0.63 ! [N: nat,A: code_integer] :
% 0.32/0.63 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 0.32/0.63 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc
% 0.32/0.63 thf(fact_6751_take__bit__Suc,axiom,
% 0.32/0.63 ! [N: nat,A: int] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 0.32/0.63 = ( 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 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc
% 0.32/0.63 thf(fact_6752_take__bit__Suc,axiom,
% 0.32/0.63 ! [N: nat,A: nat] :
% 0.32/0.63 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 0.32/0.63 = ( 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 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc
% 0.32/0.63 thf(fact_6753_take__bit__int__less__eq,axiom,
% 0.32/0.63 ! [N: nat,K: int] :
% 0.32/0.63 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.32/0.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.63 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_int_less_eq
% 0.32/0.63 thf(fact_6754_take__bit__int__greater__eq,axiom,
% 0.32/0.63 ! [K: int,N: nat] :
% 0.32/0.63 ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.63 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_int_greater_eq
% 0.32/0.63 thf(fact_6755_take__bit__minus__small__eq,axiom,
% 0.32/0.63 ! [K: int,N: nat] :
% 0.32/0.63 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.63 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.63 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.32/0.63 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_minus_small_eq
% 0.32/0.63 thf(fact_6756_signed__take__bit__eq__take__bit__shift,axiom,
% 0.32/0.63 ( bit_ri631733984087533419it_int
% 0.32/0.63 = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % signed_take_bit_eq_take_bit_shift
% 0.32/0.63 thf(fact_6757_nat0__intermed__int__val,axiom,
% 0.32/0.63 ! [N: nat,F: nat > int,K: int] :
% 0.32/0.63 ( ! [I2: nat] :
% 0.32/0.63 ( ( ord_less_nat @ I2 @ N )
% 0.32/0.63 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.32/0.63 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.32/0.63 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.32/0.63 => ? [I2: nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ I2 @ N )
% 0.32/0.63 & ( ( F @ I2 )
% 0.32/0.63 = K ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % nat0_intermed_int_val
% 0.32/0.63 thf(fact_6758_stable__imp__take__bit__eq,axiom,
% 0.32/0.63 ! [A: code_integer,N: nat] :
% 0.32/0.63 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.63 = A )
% 0.32/0.63 => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.63 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.32/0.63 = zero_z3403309356797280102nteger ) )
% 0.32/0.63 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.63 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 0.32/0.63 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % stable_imp_take_bit_eq
% 0.32/0.63 thf(fact_6759_stable__imp__take__bit__eq,axiom,
% 0.32/0.63 ! [A: int,N: nat] :
% 0.32/0.63 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.63 = A )
% 0.32/0.63 => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.63 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.63 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 0.32/0.63 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % stable_imp_take_bit_eq
% 0.32/0.63 thf(fact_6760_stable__imp__take__bit__eq,axiom,
% 0.32/0.63 ! [A: nat,N: nat] :
% 0.32/0.63 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.63 = A )
% 0.32/0.63 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.63 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.63 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 0.32/0.63 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % stable_imp_take_bit_eq
% 0.32/0.63 thf(fact_6761_divmod__step__nat__def,axiom,
% 0.32/0.63 ( unique5026877609467782581ep_nat
% 0.32/0.63 = ( ^ [L: num] :
% 0.32/0.63 ( produc2626176000494625587at_nat
% 0.32/0.63 @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_step_nat_def
% 0.32/0.63 thf(fact_6762_arctan__add,axiom,
% 0.32/0.63 ! [X: real,Y3: real] :
% 0.32/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.63 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.32/0.63 => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y3 ) )
% 0.32/0.63 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % arctan_add
% 0.32/0.63 thf(fact_6763_divmod__step__int__def,axiom,
% 0.32/0.63 ( unique5024387138958732305ep_int
% 0.32/0.63 = ( ^ [L: num] :
% 0.32/0.63 ( produc4245557441103728435nt_int
% 0.32/0.63 @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_step_int_def
% 0.32/0.63 thf(fact_6764_odd__mod__4__div__2,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.63 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.32/0.63 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % odd_mod_4_div_2
% 0.32/0.63 thf(fact_6765_divmod__algorithm__code_I6_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.63 = ( produc4245557441103728435nt_int
% 0.32/0.63 @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 0.32/0.63 @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(6)
% 0.32/0.63 thf(fact_6766_divmod__algorithm__code_I6_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.63 = ( produc2626176000494625587at_nat
% 0.32/0.63 @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 0.32/0.63 @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(6)
% 0.32/0.63 thf(fact_6767_divmod__algorithm__code_I6_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.63 = ( produc6916734918728496179nteger
% 0.32/0.63 @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 0.32/0.63 @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(6)
% 0.32/0.63 thf(fact_6768_signed__take__bit__numeral__minus__bit1,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % signed_take_bit_numeral_minus_bit1
% 0.32/0.63 thf(fact_6769_dbl__dec__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.63 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(4)
% 0.32/0.63 thf(fact_6770_dbl__dec__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.63 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(4)
% 0.32/0.63 thf(fact_6771_dbl__dec__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.63 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(4)
% 0.32/0.63 thf(fact_6772_dbl__dec__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.63 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(4)
% 0.32/0.63 thf(fact_6773_dbl__dec__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.63 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(4)
% 0.32/0.63 thf(fact_6774_divmod__algorithm__code_I7_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( ( ord_less_eq_num @ M @ N )
% 0.32/0.63 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.32/0.63 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(7)
% 0.32/0.63 thf(fact_6775_divmod__algorithm__code_I7_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( ( ord_less_eq_num @ M @ N )
% 0.32/0.63 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.32/0.63 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(7)
% 0.32/0.63 thf(fact_6776_divmod__algorithm__code_I7_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( ( ord_less_eq_num @ M @ N )
% 0.32/0.63 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.32/0.63 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(7)
% 0.32/0.63 thf(fact_6777_divmod__algorithm__code_I8_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( ( ord_less_num @ M @ N )
% 0.32/0.63 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_num @ M @ N )
% 0.32/0.63 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(8)
% 0.32/0.63 thf(fact_6778_divmod__algorithm__code_I8_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( ( ord_less_num @ M @ N )
% 0.32/0.63 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_num @ M @ N )
% 0.32/0.63 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(8)
% 0.32/0.63 thf(fact_6779_divmod__algorithm__code_I8_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( ( ord_less_num @ M @ N )
% 0.32/0.63 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_num @ M @ N )
% 0.32/0.63 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.63 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(8)
% 0.32/0.63 thf(fact_6780_signed__take__bit__numeral__bit1,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % signed_take_bit_numeral_bit1
% 0.32/0.63 thf(fact_6781_dbl__dec__simps_I3_J,axiom,
% 0.32/0.63 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 0.32/0.63 = one_one_complex ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(3)
% 0.32/0.63 thf(fact_6782_dbl__dec__simps_I3_J,axiom,
% 0.32/0.63 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 0.32/0.63 = one_one_real ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(3)
% 0.32/0.63 thf(fact_6783_dbl__dec__simps_I3_J,axiom,
% 0.32/0.63 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 0.32/0.63 = one_one_rat ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(3)
% 0.32/0.63 thf(fact_6784_dbl__dec__simps_I3_J,axiom,
% 0.32/0.63 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 0.32/0.63 = one_one_int ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(3)
% 0.32/0.63 thf(fact_6785_pred__numeral__simps_I1_J,axiom,
% 0.32/0.63 ( ( pred_numeral @ one )
% 0.32/0.63 = zero_zero_nat ) ).
% 0.32/0.63
% 0.32/0.63 % pred_numeral_simps(1)
% 0.32/0.63 thf(fact_6786_eq__numeral__Suc,axiom,
% 0.32/0.63 ! [K: num,N: nat] :
% 0.32/0.63 ( ( ( numeral_numeral_nat @ K )
% 0.32/0.63 = ( suc @ N ) )
% 0.32/0.63 = ( ( pred_numeral @ K )
% 0.32/0.63 = N ) ) ).
% 0.32/0.63
% 0.32/0.63 % eq_numeral_Suc
% 0.32/0.63 thf(fact_6787_Suc__eq__numeral,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( ( suc @ N )
% 0.32/0.63 = ( numeral_numeral_nat @ K ) )
% 0.32/0.63 = ( N
% 0.32/0.63 = ( pred_numeral @ K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % Suc_eq_numeral
% 0.32/0.63 thf(fact_6788_less__numeral__Suc,axiom,
% 0.32/0.63 ! [K: num,N: nat] :
% 0.32/0.63 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.32/0.63 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % less_numeral_Suc
% 0.32/0.63 thf(fact_6789_less__Suc__numeral,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.63 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % less_Suc_numeral
% 0.32/0.63 thf(fact_6790_pred__numeral__simps_I3_J,axiom,
% 0.32/0.63 ! [K: num] :
% 0.32/0.63 ( ( pred_numeral @ ( bit1 @ K ) )
% 0.32/0.63 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % pred_numeral_simps(3)
% 0.32/0.63 thf(fact_6791_le__Suc__numeral,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.63 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % le_Suc_numeral
% 0.32/0.63 thf(fact_6792_le__numeral__Suc,axiom,
% 0.32/0.63 ! [K: num,N: nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.32/0.63 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % le_numeral_Suc
% 0.32/0.63 thf(fact_6793_diff__numeral__Suc,axiom,
% 0.32/0.63 ! [K: num,N: nat] :
% 0.32/0.63 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.32/0.63 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.32/0.63
% 0.32/0.63 % diff_numeral_Suc
% 0.32/0.63 thf(fact_6794_diff__Suc__numeral,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.63 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % diff_Suc_numeral
% 0.32/0.63 thf(fact_6795_max__Suc__numeral,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.63 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % max_Suc_numeral
% 0.32/0.63 thf(fact_6796_max__numeral__Suc,axiom,
% 0.32/0.63 ! [K: num,N: nat] :
% 0.32/0.63 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.32/0.63 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % max_numeral_Suc
% 0.32/0.63 thf(fact_6797_dbl__dec__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.32/0.63 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(2)
% 0.32/0.63 thf(fact_6798_dbl__dec__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.32/0.63 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(2)
% 0.32/0.63 thf(fact_6799_dbl__dec__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.32/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(2)
% 0.32/0.63 thf(fact_6800_dbl__dec__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.32/0.63 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(2)
% 0.32/0.63 thf(fact_6801_dbl__dec__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.32/0.63 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_simps(2)
% 0.32/0.63 thf(fact_6802_numeral__div__minus__numeral,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.63 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % numeral_div_minus_numeral
% 0.32/0.63 thf(fact_6803_minus__numeral__div__numeral,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.63 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % minus_numeral_div_numeral
% 0.32/0.63 thf(fact_6804_sum_Ocl__ivl__Suc,axiom,
% 0.32/0.63 ! [N: nat,M: nat,G: nat > complex] :
% 0.32/0.63 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = zero_zero_complex ) )
% 0.32/0.63 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.cl_ivl_Suc
% 0.32/0.63 thf(fact_6805_sum_Ocl__ivl__Suc,axiom,
% 0.32/0.63 ! [N: nat,M: nat,G: nat > rat] :
% 0.32/0.63 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = zero_zero_rat ) )
% 0.32/0.63 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.cl_ivl_Suc
% 0.32/0.63 thf(fact_6806_sum_Ocl__ivl__Suc,axiom,
% 0.32/0.63 ! [N: nat,M: nat,G: nat > int] :
% 0.32/0.63 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = zero_zero_int ) )
% 0.32/0.63 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.cl_ivl_Suc
% 0.32/0.63 thf(fact_6807_sum_Ocl__ivl__Suc,axiom,
% 0.32/0.63 ! [N: nat,M: nat,G: nat > nat] :
% 0.32/0.63 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = zero_zero_nat ) )
% 0.32/0.63 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.cl_ivl_Suc
% 0.32/0.63 thf(fact_6808_sum_Ocl__ivl__Suc,axiom,
% 0.32/0.63 ! [N: nat,M: nat,G: nat > real] :
% 0.32/0.63 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = zero_zero_real ) )
% 0.32/0.63 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.63 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.cl_ivl_Suc
% 0.32/0.63 thf(fact_6809_dvd__numeral__simp,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.63 = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dvd_numeral_simp
% 0.32/0.63 thf(fact_6810_dvd__numeral__simp,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.63 = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dvd_numeral_simp
% 0.32/0.63 thf(fact_6811_dvd__numeral__simp,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 0.32/0.63 = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dvd_numeral_simp
% 0.32/0.63 thf(fact_6812_divmod__algorithm__code_I2_J,axiom,
% 0.32/0.63 ! [M: num] :
% 0.32/0.63 ( ( unique5052692396658037445od_int @ M @ one )
% 0.32/0.63 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(2)
% 0.32/0.63 thf(fact_6813_divmod__algorithm__code_I2_J,axiom,
% 0.32/0.63 ! [M: num] :
% 0.32/0.63 ( ( unique5055182867167087721od_nat @ M @ one )
% 0.32/0.63 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(2)
% 0.32/0.63 thf(fact_6814_divmod__algorithm__code_I2_J,axiom,
% 0.32/0.63 ! [M: num] :
% 0.32/0.63 ( ( unique3479559517661332726nteger @ M @ one )
% 0.32/0.63 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(2)
% 0.32/0.63 thf(fact_6815_sum__zero__power,axiom,
% 0.32/0.63 ! [A2: set_nat,C: nat > complex] :
% 0.32/0.63 ( ( ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex
% 0.32/0.63 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( C @ zero_zero_nat ) ) )
% 0.32/0.63 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex
% 0.32/0.63 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = zero_zero_complex ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_zero_power
% 0.32/0.63 thf(fact_6816_sum__zero__power,axiom,
% 0.32/0.63 ! [A2: set_nat,C: nat > rat] :
% 0.32/0.63 ( ( ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( C @ zero_zero_nat ) ) )
% 0.32/0.63 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = zero_zero_rat ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_zero_power
% 0.32/0.63 thf(fact_6817_sum__zero__power,axiom,
% 0.32/0.63 ! [A2: set_nat,C: nat > real] :
% 0.32/0.63 ( ( ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( C @ zero_zero_nat ) ) )
% 0.32/0.63 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = zero_zero_real ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_zero_power
% 0.32/0.63 thf(fact_6818_divmod__algorithm__code_I3_J,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 0.32/0.63 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(3)
% 0.32/0.63 thf(fact_6819_divmod__algorithm__code_I3_J,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 0.32/0.63 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(3)
% 0.32/0.63 thf(fact_6820_divmod__algorithm__code_I3_J,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 0.32/0.63 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(3)
% 0.32/0.63 thf(fact_6821_divmod__algorithm__code_I4_J,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 0.32/0.63 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(4)
% 0.32/0.63 thf(fact_6822_divmod__algorithm__code_I4_J,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 0.32/0.63 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(4)
% 0.32/0.63 thf(fact_6823_divmod__algorithm__code_I4_J,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 0.32/0.63 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(4)
% 0.32/0.63 thf(fact_6824_sum__zero__power_H,axiom,
% 0.32/0.63 ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 0.32/0.63 ( ( ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex
% 0.32/0.63 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.32/0.63 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups2073611262835488442omplex
% 0.32/0.63 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = zero_zero_complex ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_zero_power'
% 0.32/0.63 thf(fact_6825_sum__zero__power_H,axiom,
% 0.32/0.63 ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 0.32/0.63 ( ( ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.32/0.63 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = zero_zero_rat ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_zero_power'
% 0.32/0.63 thf(fact_6826_sum__zero__power_H,axiom,
% 0.32/0.63 ! [A2: set_nat,C: nat > real,D: nat > real] :
% 0.32/0.63 ( ( ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.32/0.63 & ( ~ ( ( finite_finite_nat @ A2 )
% 0.32/0.63 & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.32/0.63 => ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = zero_zero_real ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_zero_power'
% 0.32/0.63 thf(fact_6827_one__div__minus__numeral,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.63 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % one_div_minus_numeral
% 0.32/0.63 thf(fact_6828_minus__one__div__numeral,axiom,
% 0.32/0.63 ! [N: num] :
% 0.32/0.63 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.63 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % minus_one_div_numeral
% 0.32/0.63 thf(fact_6829_signed__take__bit__numeral__bit0,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.32/0.63 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % signed_take_bit_numeral_bit0
% 0.32/0.63 thf(fact_6830_divmod__algorithm__code_I5_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.63 = ( produc4245557441103728435nt_int
% 0.32/0.63 @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 0.32/0.63 @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(5)
% 0.32/0.63 thf(fact_6831_divmod__algorithm__code_I5_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.63 = ( produc2626176000494625587at_nat
% 0.32/0.63 @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 0.32/0.63 @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(5)
% 0.32/0.63 thf(fact_6832_divmod__algorithm__code_I5_J,axiom,
% 0.32/0.63 ! [M: num,N: num] :
% 0.32/0.63 ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.63 = ( produc6916734918728496179nteger
% 0.32/0.63 @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 0.32/0.63 @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_algorithm_code(5)
% 0.32/0.63 thf(fact_6833_signed__take__bit__numeral__minus__bit0,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.32/0.63 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % signed_take_bit_numeral_minus_bit0
% 0.32/0.63 thf(fact_6834_numeral__eq__Suc,axiom,
% 0.32/0.63 ( numeral_numeral_nat
% 0.32/0.63 = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % numeral_eq_Suc
% 0.32/0.63 thf(fact_6835_sum__subtractf__nat,axiom,
% 0.32/0.63 ! [A2: set_real,G: real > nat,F: real > nat] :
% 0.32/0.63 ( ! [X3: real] :
% 0.32/0.63 ( ( member_real @ X3 @ A2 )
% 0.32/0.63 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ( groups1935376822645274424al_nat
% 0.32/0.63 @ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_subtractf_nat
% 0.32/0.63 thf(fact_6836_sum__subtractf__nat,axiom,
% 0.32/0.63 ! [A2: set_int,G: int > nat,F: int > nat] :
% 0.32/0.63 ( ! [X3: int] :
% 0.32/0.63 ( ( member_int @ X3 @ A2 )
% 0.32/0.63 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ( groups4541462559716669496nt_nat
% 0.32/0.63 @ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_subtractf_nat
% 0.32/0.63 thf(fact_6837_sum__subtractf__nat,axiom,
% 0.32/0.63 ! [A2: set_complex,G: complex > nat,F: complex > nat] :
% 0.32/0.63 ( ! [X3: complex] :
% 0.32/0.63 ( ( member_complex @ X3 @ A2 )
% 0.32/0.63 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ( groups5693394587270226106ex_nat
% 0.32/0.63 @ ^ [X2: complex] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_subtractf_nat
% 0.32/0.63 thf(fact_6838_sum__subtractf__nat,axiom,
% 0.32/0.63 ! [A2: set_Pr8693737435421807431at_nat,G: produc859450856879609959at_nat > nat,F: produc859450856879609959at_nat > nat] :
% 0.32/0.63 ( ! [X3: produc859450856879609959at_nat] :
% 0.32/0.63 ( ( member8206827879206165904at_nat @ X3 @ A2 )
% 0.32/0.63 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ( groups1900718384385340925at_nat
% 0.32/0.63 @ ^ [X2: produc859450856879609959at_nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( minus_minus_nat @ ( groups1900718384385340925at_nat @ F @ A2 ) @ ( groups1900718384385340925at_nat @ G @ A2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_subtractf_nat
% 0.32/0.63 thf(fact_6839_sum__subtractf__nat,axiom,
% 0.32/0.63 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 0.32/0.63 ( ! [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ A2 )
% 0.32/0.63 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 0.32/0.63 => ( ( groups3542108847815614940at_nat
% 0.32/0.63 @ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.63 @ A2 )
% 0.32/0.63 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_subtractf_nat
% 0.32/0.63 thf(fact_6840_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.32/0.63 ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 0.32/0.63 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.32/0.63 = ( groups3542108847815614940at_nat
% 0.32/0.63 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.shift_bounds_cl_nat_ivl
% 0.32/0.63 thf(fact_6841_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.32/0.63 ! [G: nat > real,M: nat,K: nat,N: nat] :
% 0.32/0.63 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.32/0.63 = ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.shift_bounds_cl_nat_ivl
% 0.32/0.63 thf(fact_6842_sum__SucD,axiom,
% 0.32/0.63 ! [F: nat > nat,A2: set_nat,N: nat] :
% 0.32/0.63 ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.32/0.63 = ( suc @ N ) )
% 0.32/0.63 => ? [X3: nat] :
% 0.32/0.63 ( ( member_nat @ X3 @ A2 )
% 0.32/0.63 & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_SucD
% 0.32/0.63 thf(fact_6843_sum__eq__1__iff,axiom,
% 0.32/0.63 ! [A2: set_int,F: int > nat] :
% 0.32/0.63 ( ( finite_finite_int @ A2 )
% 0.32/0.63 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.32/0.63 = one_one_nat )
% 0.32/0.63 = ( ? [X2: int] :
% 0.32/0.63 ( ( member_int @ X2 @ A2 )
% 0.32/0.63 & ( ( F @ X2 )
% 0.32/0.63 = one_one_nat )
% 0.32/0.63 & ! [Y: int] :
% 0.32/0.63 ( ( member_int @ Y @ A2 )
% 0.32/0.63 => ( ( X2 != Y )
% 0.32/0.63 => ( ( F @ Y )
% 0.32/0.63 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_eq_1_iff
% 0.32/0.63 thf(fact_6844_sum__eq__1__iff,axiom,
% 0.32/0.63 ! [A2: set_complex,F: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.63 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.32/0.63 = one_one_nat )
% 0.32/0.63 = ( ? [X2: complex] :
% 0.32/0.63 ( ( member_complex @ X2 @ A2 )
% 0.32/0.63 & ( ( F @ X2 )
% 0.32/0.63 = one_one_nat )
% 0.32/0.63 & ! [Y: complex] :
% 0.32/0.63 ( ( member_complex @ Y @ A2 )
% 0.32/0.63 => ( ( X2 != Y )
% 0.32/0.63 => ( ( F @ Y )
% 0.32/0.63 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_eq_1_iff
% 0.32/0.63 thf(fact_6845_sum__eq__1__iff,axiom,
% 0.32/0.63 ! [A2: set_nat,F: nat > nat] :
% 0.32/0.63 ( ( finite_finite_nat @ A2 )
% 0.32/0.63 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.32/0.63 = one_one_nat )
% 0.32/0.63 = ( ? [X2: nat] :
% 0.32/0.63 ( ( member_nat @ X2 @ A2 )
% 0.32/0.63 & ( ( F @ X2 )
% 0.32/0.63 = one_one_nat )
% 0.32/0.63 & ! [Y: nat] :
% 0.32/0.63 ( ( member_nat @ Y @ A2 )
% 0.32/0.63 => ( ( X2 != Y )
% 0.32/0.63 => ( ( F @ Y )
% 0.32/0.63 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_eq_1_iff
% 0.32/0.63 thf(fact_6846_sum__power__add,axiom,
% 0.32/0.63 ! [X: complex,M: nat,I5: set_nat] :
% 0.32/0.63 ( ( groups2073611262835488442omplex
% 0.32/0.63 @ ^ [I4: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_power_add
% 0.32/0.63 thf(fact_6847_sum__power__add,axiom,
% 0.32/0.63 ! [X: rat,M: nat,I5: set_nat] :
% 0.32/0.63 ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_power_add
% 0.32/0.63 thf(fact_6848_sum__power__add,axiom,
% 0.32/0.63 ! [X: int,M: nat,I5: set_nat] :
% 0.32/0.63 ( ( groups3539618377306564664at_int
% 0.32/0.63 @ ^ [I4: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_power_add
% 0.32/0.63 thf(fact_6849_sum__power__add,axiom,
% 0.32/0.63 ! [X: real,M: nat,I5: set_nat] :
% 0.32/0.63 ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I4 ) )
% 0.32/0.63 @ I5 )
% 0.32/0.63 = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_power_add
% 0.32/0.63 thf(fact_6850_sum_OatLeastAtMost__rev,axiom,
% 0.32/0.63 ! [G: nat > nat,N: nat,M: nat] :
% 0.32/0.63 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.32/0.63 = ( groups3542108847815614940at_nat
% 0.32/0.63 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeastAtMost_rev
% 0.32/0.63 thf(fact_6851_sum_OatLeastAtMost__rev,axiom,
% 0.32/0.63 ! [G: nat > real,N: nat,M: nat] :
% 0.32/0.63 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.32/0.63 = ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeastAtMost_rev
% 0.32/0.63 thf(fact_6852_pred__numeral__def,axiom,
% 0.32/0.63 ( pred_numeral
% 0.32/0.63 = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % pred_numeral_def
% 0.32/0.63 thf(fact_6853_sum__nth__roots,axiom,
% 0.32/0.63 ! [N: nat,C: complex] :
% 0.32/0.63 ( ( ord_less_nat @ one_one_nat @ N )
% 0.32/0.63 => ( ( groups7754918857620584856omplex
% 0.32/0.63 @ ^ [X2: complex] : X2
% 0.32/0.63 @ ( collect_complex
% 0.32/0.63 @ ^ [Z2: complex] :
% 0.32/0.63 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.63 = C ) ) )
% 0.32/0.63 = zero_zero_complex ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_nth_roots
% 0.32/0.63 thf(fact_6854_sum__roots__unity,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( ord_less_nat @ one_one_nat @ N )
% 0.32/0.63 => ( ( groups7754918857620584856omplex
% 0.32/0.63 @ ^ [X2: complex] : X2
% 0.32/0.63 @ ( collect_complex
% 0.32/0.63 @ ^ [Z2: complex] :
% 0.32/0.63 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.63 = one_one_complex ) ) )
% 0.32/0.63 = zero_zero_complex ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_roots_unity
% 0.32/0.63 thf(fact_6855_sum__diff__nat,axiom,
% 0.32/0.63 ! [B5: set_complex,A2: set_complex,F: complex > nat] :
% 0.32/0.63 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.63 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.63 => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff_nat
% 0.32/0.63 thf(fact_6856_sum__diff__nat,axiom,
% 0.32/0.63 ! [B5: set_int,A2: set_int,F: int > nat] :
% 0.32/0.63 ( ( finite_finite_int @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.63 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff_nat
% 0.32/0.63 thf(fact_6857_sum__diff__nat,axiom,
% 0.32/0.63 ! [B5: set_nat,A2: set_nat,F: nat > nat] :
% 0.32/0.63 ( ( finite_finite_nat @ B5 )
% 0.32/0.63 => ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.63 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 0.32/0.63 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_diff_nat
% 0.32/0.63 thf(fact_6858_sum_OatLeast0__atMost__Suc,axiom,
% 0.32/0.63 ! [G: nat > rat,N: nat] :
% 0.32/0.63 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeast0_atMost_Suc
% 0.32/0.63 thf(fact_6859_sum_OatLeast0__atMost__Suc,axiom,
% 0.32/0.63 ! [G: nat > int,N: nat] :
% 0.32/0.63 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeast0_atMost_Suc
% 0.32/0.63 thf(fact_6860_sum_OatLeast0__atMost__Suc,axiom,
% 0.32/0.63 ! [G: nat > nat,N: nat] :
% 0.32/0.63 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeast0_atMost_Suc
% 0.32/0.63 thf(fact_6861_sum_OatLeast0__atMost__Suc,axiom,
% 0.32/0.63 ! [G: nat > real,N: nat] :
% 0.32/0.63 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeast0_atMost_Suc
% 0.32/0.63 thf(fact_6862_sum_OatLeast__Suc__atMost,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > rat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeast_Suc_atMost
% 0.32/0.63 thf(fact_6863_sum_OatLeast__Suc__atMost,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > int] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeast_Suc_atMost
% 0.32/0.63 thf(fact_6864_sum_OatLeast__Suc__atMost,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeast_Suc_atMost
% 0.32/0.63 thf(fact_6865_sum_OatLeast__Suc__atMost,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > real] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.atLeast_Suc_atMost
% 0.32/0.63 thf(fact_6866_sum_Onat__ivl__Suc_H,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > rat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.nat_ivl_Suc'
% 0.32/0.63 thf(fact_6867_sum_Onat__ivl__Suc_H,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > int] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.nat_ivl_Suc'
% 0.32/0.63 thf(fact_6868_sum_Onat__ivl__Suc_H,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.63 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.nat_ivl_Suc'
% 0.32/0.63 thf(fact_6869_sum_Onat__ivl__Suc_H,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > real] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.63 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.nat_ivl_Suc'
% 0.32/0.63 thf(fact_6870_sum_OSuc__reindex__ivl,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > rat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_rat @ ( G @ M )
% 0.32/0.63 @ ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.Suc_reindex_ivl
% 0.32/0.63 thf(fact_6871_sum_OSuc__reindex__ivl,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > int] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( G @ M )
% 0.32/0.63 @ ( groups3539618377306564664at_int
% 0.32/0.63 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.Suc_reindex_ivl
% 0.32/0.63 thf(fact_6872_sum_OSuc__reindex__ivl,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_nat @ ( G @ M )
% 0.32/0.63 @ ( groups3542108847815614940at_nat
% 0.32/0.63 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.Suc_reindex_ivl
% 0.32/0.63 thf(fact_6873_sum_OSuc__reindex__ivl,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > real] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.32/0.63 = ( plus_plus_real @ ( G @ M )
% 0.32/0.63 @ ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.Suc_reindex_ivl
% 0.32/0.63 thf(fact_6874_sum__Suc__diff,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > rat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_Suc_diff
% 0.32/0.63 thf(fact_6875_sum__Suc__diff,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > int] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.63 => ( ( groups3539618377306564664at_int
% 0.32/0.63 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_Suc_diff
% 0.32/0.63 thf(fact_6876_sum__Suc__diff,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > real] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.63 => ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_Suc_diff
% 0.32/0.63 thf(fact_6877_sum_Oub__add__nat,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > rat,P6: nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.32/0.63 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.ub_add_nat
% 0.32/0.63 thf(fact_6878_sum_Oub__add__nat,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > int,P6: nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.32/0.63 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.ub_add_nat
% 0.32/0.63 thf(fact_6879_sum_Oub__add__nat,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > nat,P6: nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.32/0.63 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.32/0.63 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.ub_add_nat
% 0.32/0.63 thf(fact_6880_sum_Oub__add__nat,axiom,
% 0.32/0.63 ! [M: nat,N: nat,G: nat > real,P6: nat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.32/0.63 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.32/0.63 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.ub_add_nat
% 0.32/0.63 thf(fact_6881_set__encode__def,axiom,
% 0.32/0.63 ( nat_set_encode
% 0.32/0.63 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % set_encode_def
% 0.32/0.63 thf(fact_6882_divmod__int__def,axiom,
% 0.32/0.63 ( unique5052692396658037445od_int
% 0.32/0.63 = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_int_def
% 0.32/0.63 thf(fact_6883_divmod__def,axiom,
% 0.32/0.63 ( unique5052692396658037445od_int
% 0.32/0.63 = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_def
% 0.32/0.63 thf(fact_6884_divmod__def,axiom,
% 0.32/0.63 ( unique5055182867167087721od_nat
% 0.32/0.63 = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_def
% 0.32/0.63 thf(fact_6885_divmod__def,axiom,
% 0.32/0.63 ( unique3479559517661332726nteger
% 0.32/0.63 = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_def
% 0.32/0.63 thf(fact_6886_divmod_H__nat__def,axiom,
% 0.32/0.63 ( unique5055182867167087721od_nat
% 0.32/0.63 = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod'_nat_def
% 0.32/0.63 thf(fact_6887_dbl__dec__def,axiom,
% 0.32/0.63 ( neg_nu6511756317524482435omplex
% 0.32/0.63 = ( ^ [X2: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_def
% 0.32/0.63 thf(fact_6888_dbl__dec__def,axiom,
% 0.32/0.63 ( neg_nu6075765906172075777c_real
% 0.32/0.63 = ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_def
% 0.32/0.63 thf(fact_6889_dbl__dec__def,axiom,
% 0.32/0.63 ( neg_nu3179335615603231917ec_rat
% 0.32/0.63 = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_def
% 0.32/0.63 thf(fact_6890_dbl__dec__def,axiom,
% 0.32/0.63 ( neg_nu3811975205180677377ec_int
% 0.32/0.63 = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_dec_def
% 0.32/0.63 thf(fact_6891_sum__natinterval__diff,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > complex] :
% 0.32/0.63 ( ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups2073611262835488442omplex
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups2073611262835488442omplex
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = zero_zero_complex ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_natinterval_diff
% 0.32/0.63 thf(fact_6892_sum__natinterval__diff,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > rat] :
% 0.32/0.63 ( ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = zero_zero_rat ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_natinterval_diff
% 0.32/0.63 thf(fact_6893_sum__natinterval__diff,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > int] :
% 0.32/0.63 ( ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups3539618377306564664at_int
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups3539618377306564664at_int
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = zero_zero_int ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_natinterval_diff
% 0.32/0.63 thf(fact_6894_sum__natinterval__diff,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > real] :
% 0.32/0.63 ( ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.32/0.63 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = zero_zero_real ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_natinterval_diff
% 0.32/0.63 thf(fact_6895_sum__telescope_H_H,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > rat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.32/0.63 = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_telescope''
% 0.32/0.63 thf(fact_6896_sum__telescope_H_H,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > int] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups3539618377306564664at_int
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.32/0.63 = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_telescope''
% 0.32/0.63 thf(fact_6897_sum__telescope_H_H,axiom,
% 0.32/0.63 ! [M: nat,N: nat,F: nat > real] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.32/0.63 = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_telescope''
% 0.32/0.63 thf(fact_6898_take__bit__numeral__bit0,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.32/0.63 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_numeral_bit0
% 0.32/0.63 thf(fact_6899_take__bit__numeral__bit0,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.32/0.63 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_numeral_bit0
% 0.32/0.63 thf(fact_6900_take__bit__numeral__minus__bit0,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.32/0.63 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_numeral_minus_bit0
% 0.32/0.63 thf(fact_6901_mask__eq__sum__exp,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 0.32/0.63 = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.63 @ ( collect_nat
% 0.32/0.63 @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % mask_eq_sum_exp
% 0.32/0.63 thf(fact_6902_mask__eq__sum__exp,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 0.32/0.63 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.63 @ ( collect_nat
% 0.32/0.63 @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % mask_eq_sum_exp
% 0.32/0.63 thf(fact_6903_sum__gp__multiplied,axiom,
% 0.32/0.63 ! [M: nat,N: nat,X: complex] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.32/0.63 = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_gp_multiplied
% 0.32/0.63 thf(fact_6904_sum__gp__multiplied,axiom,
% 0.32/0.63 ! [M: nat,N: nat,X: rat] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.32/0.63 = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_gp_multiplied
% 0.32/0.63 thf(fact_6905_sum__gp__multiplied,axiom,
% 0.32/0.63 ! [M: nat,N: nat,X: int] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.32/0.63 = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_gp_multiplied
% 0.32/0.63 thf(fact_6906_sum__gp__multiplied,axiom,
% 0.32/0.63 ! [M: nat,N: nat,X: real] :
% 0.32/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.63 => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.32/0.63 = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum_gp_multiplied
% 0.32/0.63 thf(fact_6907_sum_Oin__pairs,axiom,
% 0.32/0.63 ! [G: nat > rat,M: nat,N: nat] :
% 0.32/0.63 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.63 = ( groups2906978787729119204at_rat
% 0.32/0.63 @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.in_pairs
% 0.32/0.63 thf(fact_6908_sum_Oin__pairs,axiom,
% 0.32/0.63 ! [G: nat > int,M: nat,N: nat] :
% 0.32/0.63 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.63 = ( groups3539618377306564664at_int
% 0.32/0.63 @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.in_pairs
% 0.32/0.63 thf(fact_6909_sum_Oin__pairs,axiom,
% 0.32/0.63 ! [G: nat > nat,M: nat,N: nat] :
% 0.32/0.63 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.63 = ( groups3542108847815614940at_nat
% 0.32/0.63 @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.in_pairs
% 0.32/0.63 thf(fact_6910_sum_Oin__pairs,axiom,
% 0.32/0.63 ! [G: nat > real,M: nat,N: nat] :
% 0.32/0.63 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.32/0.63 = ( groups6591440286371151544t_real
% 0.32/0.63 @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % sum.in_pairs
% 0.32/0.63 thf(fact_6911_mask__eq__sum__exp__nat,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 0.32/0.63 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.63 @ ( collect_nat
% 0.32/0.63 @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % mask_eq_sum_exp_nat
% 0.32/0.63 thf(fact_6912_gauss__sum__nat,axiom,
% 0.32/0.63 ! [N: nat] :
% 0.32/0.63 ( ( groups3542108847815614940at_nat
% 0.32/0.63 @ ^ [X2: nat] : X2
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.63 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % gauss_sum_nat
% 0.32/0.63 thf(fact_6913_arith__series__nat,axiom,
% 0.32/0.63 ! [A: nat,D: nat,N: nat] :
% 0.32/0.63 ( ( groups3542108847815614940at_nat
% 0.32/0.63 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.63 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % arith_series_nat
% 0.32/0.63 thf(fact_6914_Sum__Icc__nat,axiom,
% 0.32/0.63 ! [M: nat,N: nat] :
% 0.32/0.63 ( ( groups3542108847815614940at_nat
% 0.32/0.63 @ ^ [X2: nat] : X2
% 0.32/0.63 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.63 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % Sum_Icc_nat
% 0.32/0.63 thf(fact_6915_divmod__divmod__step,axiom,
% 0.32/0.63 ( unique5055182867167087721od_nat
% 0.32/0.63 = ( ^ [M3: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_divmod_step
% 0.32/0.63 thf(fact_6916_divmod__divmod__step,axiom,
% 0.32/0.63 ( unique5052692396658037445od_int
% 0.32/0.63 = ( ^ [M3: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_divmod_step
% 0.32/0.63 thf(fact_6917_divmod__divmod__step,axiom,
% 0.32/0.63 ( unique3479559517661332726nteger
% 0.32/0.63 = ( ^ [M3: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_divmod_step
% 0.32/0.63 thf(fact_6918_take__bit__numeral__bit1,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_numeral_bit1
% 0.32/0.63 thf(fact_6919_take__bit__numeral__bit1,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.32/0.63 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_numeral_bit1
% 0.32/0.63 thf(fact_6920_take__bit__numeral__minus__bit1,axiom,
% 0.32/0.63 ! [L2: num,K: num] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.32/0.63 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_numeral_minus_bit1
% 0.32/0.63 thf(fact_6921_divmod__nat__if,axiom,
% 0.32/0.63 ( divmod_nat
% 0.32/0.63 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.63 ( if_Pro6206227464963214023at_nat
% 0.32/0.63 @ ( ( N2 = zero_zero_nat )
% 0.32/0.63 | ( ord_less_nat @ M3 @ N2 ) )
% 0.32/0.63 @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
% 0.32/0.63 @ ( produc2626176000494625587at_nat
% 0.32/0.63 @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
% 0.32/0.63 @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % divmod_nat_if
% 0.32/0.63 thf(fact_6922_take__bit__Suc__minus__bit1,axiom,
% 0.32/0.63 ! [N: nat,K: num] :
% 0.32/0.63 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.32/0.63 = ( 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 ) ) ).
% 0.32/0.63
% 0.32/0.63 % take_bit_Suc_minus_bit1
% 0.32/0.63 thf(fact_6923_infinite__int__iff__unbounded,axiom,
% 0.32/0.63 ! [S2: set_int] :
% 0.32/0.63 ( ( ~ ( finite_finite_int @ S2 ) )
% 0.32/0.63 = ( ! [M3: int] :
% 0.32/0.63 ? [N2: int] :
% 0.32/0.63 ( ( ord_less_int @ M3 @ ( abs_abs_int @ N2 ) )
% 0.32/0.63 & ( member_int @ N2 @ S2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % infinite_int_iff_unbounded
% 0.32/0.63 thf(fact_6924_infinite__int__iff__unbounded__le,axiom,
% 0.32/0.63 ! [S2: set_int] :
% 0.32/0.63 ( ( ~ ( finite_finite_int @ S2 ) )
% 0.32/0.63 = ( ! [M3: int] :
% 0.32/0.63 ? [N2: int] :
% 0.32/0.63 ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N2 ) )
% 0.32/0.63 & ( member_int @ N2 @ S2 ) ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % infinite_int_iff_unbounded_le
% 0.32/0.63 thf(fact_6925_dbl__inc__simps_I3_J,axiom,
% 0.32/0.63 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 0.32/0.63 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(3)
% 0.32/0.63 thf(fact_6926_dbl__inc__simps_I3_J,axiom,
% 0.32/0.63 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 0.32/0.63 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(3)
% 0.32/0.63 thf(fact_6927_dbl__inc__simps_I3_J,axiom,
% 0.32/0.63 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 0.32/0.63 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(3)
% 0.32/0.63 thf(fact_6928_dbl__inc__simps_I3_J,axiom,
% 0.32/0.63 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 0.32/0.63 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(3)
% 0.32/0.63 thf(fact_6929_pred__numeral__inc,axiom,
% 0.32/0.63 ! [K: num] :
% 0.32/0.63 ( ( pred_numeral @ ( inc @ K ) )
% 0.32/0.63 = ( numeral_numeral_nat @ K ) ) ).
% 0.32/0.63
% 0.32/0.63 % pred_numeral_inc
% 0.32/0.63 thf(fact_6930_dbl__inc__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.32/0.63 = one_one_complex ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(2)
% 0.32/0.63 thf(fact_6931_dbl__inc__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.32/0.63 = one_one_real ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(2)
% 0.32/0.63 thf(fact_6932_dbl__inc__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.32/0.63 = one_one_rat ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(2)
% 0.32/0.63 thf(fact_6933_dbl__inc__simps_I2_J,axiom,
% 0.32/0.63 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.32/0.63 = one_one_int ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(2)
% 0.32/0.63 thf(fact_6934_dbl__inc__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.63 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(4)
% 0.32/0.63 thf(fact_6935_dbl__inc__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.63 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(4)
% 0.32/0.63 thf(fact_6936_dbl__inc__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.63 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(4)
% 0.32/0.63 thf(fact_6937_dbl__inc__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.63 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(4)
% 0.32/0.63 thf(fact_6938_dbl__inc__simps_I4_J,axiom,
% 0.32/0.63 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.63 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(4)
% 0.32/0.63 thf(fact_6939_dbl__inc__simps_I5_J,axiom,
% 0.32/0.63 ! [K: num] :
% 0.32/0.63 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.32/0.63 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(5)
% 0.32/0.63 thf(fact_6940_dbl__inc__simps_I5_J,axiom,
% 0.32/0.63 ! [K: num] :
% 0.32/0.63 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 0.32/0.63 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 0.32/0.63
% 0.32/0.63 % dbl_inc_simps(5)
% 0.32/0.63 thf(fact_6941_dbl__inc__simps_I5_J,axiom,
% 0.32/0.63 ! [K: num] :
% 0.32/0.63 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 0.32/0.64 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_simps(5)
% 0.32/0.64 thf(fact_6942_dbl__inc__simps_I5_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 0.32/0.64 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_simps(5)
% 0.32/0.64 thf(fact_6943_dbl__dec__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(1)
% 0.32/0.64 thf(fact_6944_dbl__dec__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(1)
% 0.32/0.64 thf(fact_6945_dbl__dec__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.32/0.64 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(1)
% 0.32/0.64 thf(fact_6946_dbl__dec__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.32/0.64 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(1)
% 0.32/0.64 thf(fact_6947_dbl__dec__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(1)
% 0.32/0.64 thf(fact_6948_dbl__inc__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_simps(1)
% 0.32/0.64 thf(fact_6949_dbl__inc__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_simps(1)
% 0.32/0.64 thf(fact_6950_dbl__inc__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 0.32/0.64 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_simps(1)
% 0.32/0.64 thf(fact_6951_dbl__inc__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 0.32/0.64 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_simps(1)
% 0.32/0.64 thf(fact_6952_dbl__inc__simps_I1_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_simps(1)
% 0.32/0.64 thf(fact_6953_add__neg__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.64 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(6)
% 0.32/0.64 thf(fact_6954_add__neg__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(6)
% 0.32/0.64 thf(fact_6955_add__neg__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.64 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(6)
% 0.32/0.64 thf(fact_6956_add__neg__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.64 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(6)
% 0.32/0.64 thf(fact_6957_add__neg__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.64 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(6)
% 0.32/0.64 thf(fact_6958_add__neg__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.64 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(5)
% 0.32/0.64 thf(fact_6959_add__neg__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(5)
% 0.32/0.64 thf(fact_6960_add__neg__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 0.32/0.64 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(5)
% 0.32/0.64 thf(fact_6961_add__neg__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.32/0.64 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(5)
% 0.32/0.64 thf(fact_6962_add__neg__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.64 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_neg_numeral_special(5)
% 0.32/0.64 thf(fact_6963_diff__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.64 = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(6)
% 0.32/0.64 thf(fact_6964_diff__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.64 = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(6)
% 0.32/0.64 thf(fact_6965_diff__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.64 = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(6)
% 0.32/0.64 thf(fact_6966_diff__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.32/0.64 = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(6)
% 0.32/0.64 thf(fact_6967_diff__numeral__special_I6_J,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.32/0.64 = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(6)
% 0.32/0.64 thf(fact_6968_diff__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.64 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(5)
% 0.32/0.64 thf(fact_6969_diff__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(5)
% 0.32/0.64 thf(fact_6970_diff__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 0.32/0.64 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(5)
% 0.32/0.64 thf(fact_6971_diff__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 0.32/0.64 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(5)
% 0.32/0.64 thf(fact_6972_diff__numeral__special_I5_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.64 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % diff_numeral_special(5)
% 0.32/0.64 thf(fact_6973_num__induct,axiom,
% 0.32/0.64 ! [P: num > $o,X: num] :
% 0.32/0.64 ( ( P @ one )
% 0.32/0.64 => ( ! [X3: num] :
% 0.32/0.64 ( ( P @ X3 )
% 0.32/0.64 => ( P @ ( inc @ X3 ) ) )
% 0.32/0.64 => ( P @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % num_induct
% 0.32/0.64 thf(fact_6974_add__inc,axiom,
% 0.32/0.64 ! [X: num,Y3: num] :
% 0.32/0.64 ( ( plus_plus_num @ X @ ( inc @ Y3 ) )
% 0.32/0.64 = ( inc @ ( plus_plus_num @ X @ Y3 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_inc
% 0.32/0.64 thf(fact_6975_inc_Osimps_I1_J,axiom,
% 0.32/0.64 ( ( inc @ one )
% 0.32/0.64 = ( bit0 @ one ) ) ).
% 0.32/0.64
% 0.32/0.64 % inc.simps(1)
% 0.32/0.64 thf(fact_6976_inc_Osimps_I3_J,axiom,
% 0.32/0.64 ! [X: num] :
% 0.32/0.64 ( ( inc @ ( bit1 @ X ) )
% 0.32/0.64 = ( bit0 @ ( inc @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % inc.simps(3)
% 0.32/0.64 thf(fact_6977_inc_Osimps_I2_J,axiom,
% 0.32/0.64 ! [X: num] :
% 0.32/0.64 ( ( inc @ ( bit0 @ X ) )
% 0.32/0.64 = ( bit1 @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % inc.simps(2)
% 0.32/0.64 thf(fact_6978_add__One,axiom,
% 0.32/0.64 ! [X: num] :
% 0.32/0.64 ( ( plus_plus_num @ X @ one )
% 0.32/0.64 = ( inc @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % add_One
% 0.32/0.64 thf(fact_6979_mult__inc,axiom,
% 0.32/0.64 ! [X: num,Y3: num] :
% 0.32/0.64 ( ( times_times_num @ X @ ( inc @ Y3 ) )
% 0.32/0.64 = ( plus_plus_num @ ( times_times_num @ X @ Y3 ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_inc
% 0.32/0.64 thf(fact_6980_numeral__inc,axiom,
% 0.32/0.64 ! [X: num] :
% 0.32/0.64 ( ( numera6690914467698888265omplex @ ( inc @ X ) )
% 0.32/0.64 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_inc
% 0.32/0.64 thf(fact_6981_numeral__inc,axiom,
% 0.32/0.64 ! [X: num] :
% 0.32/0.64 ( ( numeral_numeral_real @ ( inc @ X ) )
% 0.32/0.64 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_inc
% 0.32/0.64 thf(fact_6982_numeral__inc,axiom,
% 0.32/0.64 ! [X: num] :
% 0.32/0.64 ( ( numeral_numeral_rat @ ( inc @ X ) )
% 0.32/0.64 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_inc
% 0.32/0.64 thf(fact_6983_numeral__inc,axiom,
% 0.32/0.64 ! [X: num] :
% 0.32/0.64 ( ( numeral_numeral_nat @ ( inc @ X ) )
% 0.32/0.64 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_inc
% 0.32/0.64 thf(fact_6984_numeral__inc,axiom,
% 0.32/0.64 ! [X: num] :
% 0.32/0.64 ( ( numeral_numeral_int @ ( inc @ X ) )
% 0.32/0.64 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_inc
% 0.32/0.64 thf(fact_6985_dbl__inc__def,axiom,
% 0.32/0.64 ( neg_nu8557863876264182079omplex
% 0.32/0.64 = ( ^ [X2: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_def
% 0.32/0.64 thf(fact_6986_dbl__inc__def,axiom,
% 0.32/0.64 ( neg_nu8295874005876285629c_real
% 0.32/0.64 = ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_def
% 0.32/0.64 thf(fact_6987_dbl__inc__def,axiom,
% 0.32/0.64 ( neg_nu5219082963157363817nc_rat
% 0.32/0.64 = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_def
% 0.32/0.64 thf(fact_6988_dbl__inc__def,axiom,
% 0.32/0.64 ( neg_nu5851722552734809277nc_int
% 0.32/0.64 = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_inc_def
% 0.32/0.64 thf(fact_6989_divmod__nat__def,axiom,
% 0.32/0.64 ( divmod_nat
% 0.32/0.64 = ( ^ [M3: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N2 ) @ ( modulo_modulo_nat @ M3 @ N2 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % divmod_nat_def
% 0.32/0.64 thf(fact_6990_infinite__nat__iff__unbounded,axiom,
% 0.32/0.64 ! [S2: set_nat] :
% 0.32/0.64 ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.32/0.64 = ( ! [M3: nat] :
% 0.32/0.64 ? [N2: nat] :
% 0.32/0.64 ( ( ord_less_nat @ M3 @ N2 )
% 0.32/0.64 & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % infinite_nat_iff_unbounded
% 0.32/0.64 thf(fact_6991_unbounded__k__infinite,axiom,
% 0.32/0.64 ! [K: nat,S2: set_nat] :
% 0.32/0.64 ( ! [M4: nat] :
% 0.32/0.64 ( ( ord_less_nat @ K @ M4 )
% 0.32/0.64 => ? [N7: nat] :
% 0.32/0.64 ( ( ord_less_nat @ M4 @ N7 )
% 0.32/0.64 & ( member_nat @ N7 @ S2 ) ) )
% 0.32/0.64 => ~ ( finite_finite_nat @ S2 ) ) ).
% 0.32/0.64
% 0.32/0.64 % unbounded_k_infinite
% 0.32/0.64 thf(fact_6992_infinite__nat__iff__unbounded__le,axiom,
% 0.32/0.64 ! [S2: set_nat] :
% 0.32/0.64 ( ( ~ ( finite_finite_nat @ S2 ) )
% 0.32/0.64 = ( ! [M3: nat] :
% 0.32/0.64 ? [N2: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.64 & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % infinite_nat_iff_unbounded_le
% 0.32/0.64 thf(fact_6993_sum__gp,axiom,
% 0.32/0.64 ! [N: nat,M: nat,X: complex] :
% 0.32/0.64 ( ( ( ord_less_nat @ N @ M )
% 0.32/0.64 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = zero_zero_complex ) )
% 0.32/0.64 & ( ~ ( ord_less_nat @ N @ M )
% 0.32/0.64 => ( ( ( X = one_one_complex )
% 0.32/0.64 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.32/0.64 & ( ( X != one_one_complex )
% 0.32/0.64 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % sum_gp
% 0.32/0.64 thf(fact_6994_sum__gp,axiom,
% 0.32/0.64 ! [N: nat,M: nat,X: rat] :
% 0.32/0.64 ( ( ( ord_less_nat @ N @ M )
% 0.32/0.64 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = zero_zero_rat ) )
% 0.32/0.64 & ( ~ ( ord_less_nat @ N @ M )
% 0.32/0.64 => ( ( ( X = one_one_rat )
% 0.32/0.64 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.32/0.64 & ( ( X != one_one_rat )
% 0.32/0.64 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % sum_gp
% 0.32/0.64 thf(fact_6995_sum__gp,axiom,
% 0.32/0.64 ! [N: nat,M: nat,X: real] :
% 0.32/0.64 ( ( ( ord_less_nat @ N @ M )
% 0.32/0.64 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = zero_zero_real ) )
% 0.32/0.64 & ( ~ ( ord_less_nat @ N @ M )
% 0.32/0.64 => ( ( ( X = one_one_real )
% 0.32/0.64 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.32/0.64 & ( ( X != one_one_real )
% 0.32/0.64 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.32/0.64 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % sum_gp
% 0.32/0.64 thf(fact_6996_divmod__BitM__2__eq,axiom,
% 0.32/0.64 ! [M: num] :
% 0.32/0.64 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.32/0.64 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % divmod_BitM_2_eq
% 0.32/0.64 thf(fact_6997_gauss__sum__from__Suc__0,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.32/0.64 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % gauss_sum_from_Suc_0
% 0.32/0.64 thf(fact_6998_gauss__sum__from__Suc__0,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.32/0.64 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % gauss_sum_from_Suc_0
% 0.32/0.64 thf(fact_6999_of__int__code__if,axiom,
% 0.32/0.64 ( ring_1_of_int_real
% 0.32/0.64 = ( ^ [K2: int] :
% 0.32/0.64 ( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
% 0.32/0.64 @ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
% 0.32/0.64 @ ( if_real
% 0.32/0.64 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.64 = zero_zero_int )
% 0.32/0.64 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.32/0.64 @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_code_if
% 0.32/0.64 thf(fact_7000_of__int__code__if,axiom,
% 0.32/0.64 ( ring_1_of_int_int
% 0.32/0.64 = ( ^ [K2: int] :
% 0.32/0.64 ( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
% 0.32/0.64 @ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
% 0.32/0.64 @ ( if_int
% 0.32/0.64 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.64 = zero_zero_int )
% 0.32/0.64 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.32/0.64 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_code_if
% 0.32/0.64 thf(fact_7001_of__int__code__if,axiom,
% 0.32/0.64 ( ring_17405671764205052669omplex
% 0.32/0.64 = ( ^ [K2: int] :
% 0.32/0.64 ( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
% 0.32/0.64 @ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
% 0.32/0.64 @ ( if_complex
% 0.32/0.64 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.64 = zero_zero_int )
% 0.32/0.64 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.32/0.64 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_code_if
% 0.32/0.64 thf(fact_7002_of__int__code__if,axiom,
% 0.32/0.64 ( ring_18347121197199848620nteger
% 0.32/0.64 = ( ^ [K2: int] :
% 0.32/0.64 ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.32/0.64 @ ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K2 ) ) )
% 0.32/0.64 @ ( if_Code_integer
% 0.32/0.64 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.64 = zero_zero_int )
% 0.32/0.64 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.32/0.64 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_code_if
% 0.32/0.64 thf(fact_7003_of__int__code__if,axiom,
% 0.32/0.64 ( ring_1_of_int_rat
% 0.32/0.64 = ( ^ [K2: int] :
% 0.32/0.64 ( if_rat @ ( K2 = zero_zero_int ) @ zero_zero_rat
% 0.32/0.64 @ ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K2 ) ) )
% 0.32/0.64 @ ( if_rat
% 0.32/0.64 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.64 = zero_zero_int )
% 0.32/0.64 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.32/0.64 @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_code_if
% 0.32/0.64 thf(fact_7004_signed__take__bit__eq__take__bit__minus,axiom,
% 0.32/0.64 ( bit_ri631733984087533419it_int
% 0.32/0.64 = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % signed_take_bit_eq_take_bit_minus
% 0.32/0.64 thf(fact_7005_of__nat__eq__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ( semiri1314217659103216013at_int @ M )
% 0.32/0.64 = ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( M = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_iff
% 0.32/0.64 thf(fact_7006_of__nat__eq__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ( semiri5074537144036343181t_real @ M )
% 0.32/0.64 = ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 = ( M = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_iff
% 0.32/0.64 thf(fact_7007_of__nat__eq__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.32/0.64 = ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 = ( M = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_iff
% 0.32/0.64 thf(fact_7008_of__nat__eq__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ( semiri8010041392384452111omplex @ M )
% 0.32/0.64 = ( semiri8010041392384452111omplex @ N ) )
% 0.32/0.64 = ( M = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_iff
% 0.32/0.64 thf(fact_7009_of__nat__eq__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ( semiri681578069525770553at_rat @ M )
% 0.32/0.64 = ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.64 = ( M = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_iff
% 0.32/0.64 thf(fact_7010_int__eq__iff__numeral,axiom,
% 0.32/0.64 ! [M: nat,V: num] :
% 0.32/0.64 ( ( ( semiri1314217659103216013at_int @ M )
% 0.32/0.64 = ( numeral_numeral_int @ V ) )
% 0.32/0.64 = ( M
% 0.32/0.64 = ( numeral_numeral_nat @ V ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_eq_iff_numeral
% 0.32/0.64 thf(fact_7011_bit__0__eq,axiom,
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ zero_zero_int )
% 0.32/0.64 = bot_bot_nat_o ) ).
% 0.32/0.64
% 0.32/0.64 % bit_0_eq
% 0.32/0.64 thf(fact_7012_bit__0__eq,axiom,
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
% 0.32/0.64 = bot_bot_nat_o ) ).
% 0.32/0.64
% 0.32/0.64 % bit_0_eq
% 0.32/0.64 thf(fact_7013_abs__of__nat,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.32/0.64 = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % abs_of_nat
% 0.32/0.64 thf(fact_7014_abs__of__nat,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % abs_of_nat
% 0.32/0.64 thf(fact_7015_abs__of__nat,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % abs_of_nat
% 0.32/0.64 thf(fact_7016_abs__of__nat,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.64 = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % abs_of_nat
% 0.32/0.64 thf(fact_7017_negative__zle,axiom,
% 0.32/0.64 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.32/0.64
% 0.32/0.64 % negative_zle
% 0.32/0.64 thf(fact_7018_of__nat__eq__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ( semiri1314217659103216013at_int @ M )
% 0.32/0.64 = zero_zero_int )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_0_iff
% 0.32/0.64 thf(fact_7019_of__nat__eq__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ( semiri5074537144036343181t_real @ M )
% 0.32/0.64 = zero_zero_real )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_0_iff
% 0.32/0.64 thf(fact_7020_of__nat__eq__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.32/0.64 = zero_zero_nat )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_0_iff
% 0.32/0.64 thf(fact_7021_of__nat__eq__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ( semiri8010041392384452111omplex @ M )
% 0.32/0.64 = zero_zero_complex )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_0_iff
% 0.32/0.64 thf(fact_7022_of__nat__eq__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ( semiri681578069525770553at_rat @ M )
% 0.32/0.64 = zero_zero_rat )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_0_iff
% 0.32/0.64 thf(fact_7023_of__nat__0__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( zero_zero_int
% 0.32/0.64 = ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( zero_zero_nat = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_eq_iff
% 0.32/0.64 thf(fact_7024_of__nat__0__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( zero_zero_real
% 0.32/0.64 = ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 = ( zero_zero_nat = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_eq_iff
% 0.32/0.64 thf(fact_7025_of__nat__0__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( zero_zero_nat
% 0.32/0.64 = ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 = ( zero_zero_nat = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_eq_iff
% 0.32/0.64 thf(fact_7026_of__nat__0__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( zero_zero_complex
% 0.32/0.64 = ( semiri8010041392384452111omplex @ N ) )
% 0.32/0.64 = ( zero_zero_nat = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_eq_iff
% 0.32/0.64 thf(fact_7027_of__nat__0__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( zero_zero_rat
% 0.32/0.64 = ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.64 = ( zero_zero_nat = N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_eq_iff
% 0.32/0.64 thf(fact_7028_of__nat__0,axiom,
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.32/0.64 = zero_zero_int ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0
% 0.32/0.64 thf(fact_7029_of__nat__0,axiom,
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.32/0.64 = zero_zero_real ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0
% 0.32/0.64 thf(fact_7030_of__nat__0,axiom,
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.32/0.64 = zero_zero_nat ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0
% 0.32/0.64 thf(fact_7031_of__nat__0,axiom,
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.32/0.64 = zero_zero_complex ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0
% 0.32/0.64 thf(fact_7032_of__nat__0,axiom,
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.32/0.64 = zero_zero_rat ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0
% 0.32/0.64 thf(fact_7033_of__nat__less__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_iff
% 0.32/0.64 thf(fact_7034_of__nat__less__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_iff
% 0.32/0.64 thf(fact_7035_of__nat__less__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_iff
% 0.32/0.64 thf(fact_7036_of__nat__less__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.64 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_iff
% 0.32/0.64 thf(fact_7037_of__nat__numeral,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_numeral
% 0.32/0.64 thf(fact_7038_of__nat__numeral,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( numeral_numeral_real @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_numeral
% 0.32/0.64 thf(fact_7039_of__nat__numeral,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_numeral
% 0.32/0.64 thf(fact_7040_of__nat__numeral,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( numera6690914467698888265omplex @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_numeral
% 0.32/0.64 thf(fact_7041_of__nat__numeral,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( numeral_numeral_rat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_numeral
% 0.32/0.64 thf(fact_7042_of__nat__le__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_iff
% 0.32/0.64 thf(fact_7043_of__nat__le__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_iff
% 0.32/0.64 thf(fact_7044_of__nat__le__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_iff
% 0.32/0.64 thf(fact_7045_of__nat__le__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_iff
% 0.32/0.64 thf(fact_7046_of__nat__add,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.64 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_add
% 0.32/0.64 thf(fact_7047_of__nat__add,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.64 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_add
% 0.32/0.64 thf(fact_7048_of__nat__add,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.64 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_add
% 0.32/0.64 thf(fact_7049_of__nat__add,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.64 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_add
% 0.32/0.64 thf(fact_7050_of__nat__add,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.64 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_add
% 0.32/0.64 thf(fact_7051_of__nat__mult,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.32/0.64 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mult
% 0.32/0.64 thf(fact_7052_of__nat__mult,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.32/0.64 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mult
% 0.32/0.64 thf(fact_7053_of__nat__mult,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.32/0.64 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mult
% 0.32/0.64 thf(fact_7054_of__nat__mult,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 0.32/0.64 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mult
% 0.32/0.64 thf(fact_7055_of__nat__mult,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.32/0.64 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mult
% 0.32/0.64 thf(fact_7056_of__nat__1,axiom,
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.32/0.64 = one_one_int ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1
% 0.32/0.64 thf(fact_7057_of__nat__1,axiom,
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.32/0.64 = one_one_real ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1
% 0.32/0.64 thf(fact_7058_of__nat__1,axiom,
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.32/0.64 = one_one_nat ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1
% 0.32/0.64 thf(fact_7059_of__nat__1,axiom,
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.32/0.64 = one_one_complex ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1
% 0.32/0.64 thf(fact_7060_of__nat__1,axiom,
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.32/0.64 = one_one_rat ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1
% 0.32/0.64 thf(fact_7061_of__nat__1__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( one_one_int
% 0.32/0.64 = ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1_eq_iff
% 0.32/0.64 thf(fact_7062_of__nat__1__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( one_one_real
% 0.32/0.64 = ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1_eq_iff
% 0.32/0.64 thf(fact_7063_of__nat__1__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( one_one_nat
% 0.32/0.64 = ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1_eq_iff
% 0.32/0.64 thf(fact_7064_of__nat__1__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( one_one_complex
% 0.32/0.64 = ( semiri8010041392384452111omplex @ N ) )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1_eq_iff
% 0.32/0.64 thf(fact_7065_of__nat__1__eq__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( one_one_rat
% 0.32/0.64 = ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_1_eq_iff
% 0.32/0.64 thf(fact_7066_of__nat__eq__1__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ( semiri1314217659103216013at_int @ N )
% 0.32/0.64 = one_one_int )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_1_iff
% 0.32/0.64 thf(fact_7067_of__nat__eq__1__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ( semiri5074537144036343181t_real @ N )
% 0.32/0.64 = one_one_real )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_1_iff
% 0.32/0.64 thf(fact_7068_of__nat__eq__1__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ( semiri1316708129612266289at_nat @ N )
% 0.32/0.64 = one_one_nat )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_1_iff
% 0.32/0.64 thf(fact_7069_of__nat__eq__1__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ( semiri8010041392384452111omplex @ N )
% 0.32/0.64 = one_one_complex )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_1_iff
% 0.32/0.64 thf(fact_7070_of__nat__eq__1__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ( semiri681578069525770553at_rat @ N )
% 0.32/0.64 = one_one_rat )
% 0.32/0.64 = ( N = one_one_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_1_iff
% 0.32/0.64 thf(fact_7071_of__int__le__iff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_iff
% 0.32/0.64 thf(fact_7072_of__int__le__iff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_iff
% 0.32/0.64 thf(fact_7073_of__int__le__iff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ W @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_iff
% 0.32/0.64 thf(fact_7074_of__int__eq__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.32/0.64 = ( numera6690914467698888265omplex @ N ) )
% 0.32/0.64 = ( Z
% 0.32/0.64 = ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_numeral_iff
% 0.32/0.64 thf(fact_7075_of__int__eq__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ( ring_1_of_int_real @ Z )
% 0.32/0.64 = ( numeral_numeral_real @ N ) )
% 0.32/0.64 = ( Z
% 0.32/0.64 = ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_numeral_iff
% 0.32/0.64 thf(fact_7076_of__int__eq__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ( ring_1_of_int_rat @ Z )
% 0.32/0.64 = ( numeral_numeral_rat @ N ) )
% 0.32/0.64 = ( Z
% 0.32/0.64 = ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_numeral_iff
% 0.32/0.64 thf(fact_7077_of__int__eq__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ( ring_1_of_int_int @ Z )
% 0.32/0.64 = ( numeral_numeral_int @ N ) )
% 0.32/0.64 = ( Z
% 0.32/0.64 = ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_numeral_iff
% 0.32/0.64 thf(fact_7078_of__int__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 0.32/0.64 = ( numera6690914467698888265omplex @ K ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral
% 0.32/0.64 thf(fact_7079_of__int__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.32/0.64 = ( numeral_numeral_real @ K ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral
% 0.32/0.64 thf(fact_7080_of__int__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.32/0.64 = ( numeral_numeral_rat @ K ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral
% 0.32/0.64 thf(fact_7081_of__int__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.32/0.64 = ( numeral_numeral_int @ K ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral
% 0.32/0.64 thf(fact_7082_of__int__less__iff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 0.32/0.64 = ( ord_less_int @ W @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_iff
% 0.32/0.64 thf(fact_7083_of__int__less__iff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 0.32/0.64 = ( ord_less_int @ W @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_iff
% 0.32/0.64 thf(fact_7084_of__int__less__iff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 0.32/0.64 = ( ord_less_int @ W @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_iff
% 0.32/0.64 thf(fact_7085_of__int__1,axiom,
% 0.32/0.64 ( ( ring_17405671764205052669omplex @ one_one_int )
% 0.32/0.64 = one_one_complex ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1
% 0.32/0.64 thf(fact_7086_of__int__1,axiom,
% 0.32/0.64 ( ( ring_1_of_int_int @ one_one_int )
% 0.32/0.64 = one_one_int ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1
% 0.32/0.64 thf(fact_7087_of__int__1,axiom,
% 0.32/0.64 ( ( ring_1_of_int_real @ one_one_int )
% 0.32/0.64 = one_one_real ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1
% 0.32/0.64 thf(fact_7088_of__int__1,axiom,
% 0.32/0.64 ( ( ring_1_of_int_rat @ one_one_int )
% 0.32/0.64 = one_one_rat ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1
% 0.32/0.64 thf(fact_7089_of__int__eq__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ( ring_17405671764205052669omplex @ Z )
% 0.32/0.64 = one_one_complex )
% 0.32/0.64 = ( Z = one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_1_iff
% 0.32/0.64 thf(fact_7090_of__int__eq__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ( ring_1_of_int_int @ Z )
% 0.32/0.64 = one_one_int )
% 0.32/0.64 = ( Z = one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_1_iff
% 0.32/0.64 thf(fact_7091_of__int__eq__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ( ring_1_of_int_real @ Z )
% 0.32/0.64 = one_one_real )
% 0.32/0.64 = ( Z = one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_1_iff
% 0.32/0.64 thf(fact_7092_of__int__eq__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ( ring_1_of_int_rat @ Z )
% 0.32/0.64 = one_one_rat )
% 0.32/0.64 = ( Z = one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_1_iff
% 0.32/0.64 thf(fact_7093_of__nat__power,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 0.32/0.64 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power
% 0.32/0.64 thf(fact_7094_of__nat__power,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 0.32/0.64 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power
% 0.32/0.64 thf(fact_7095_of__nat__power,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 0.32/0.64 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power
% 0.32/0.64 thf(fact_7096_of__nat__power,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 0.32/0.64 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power
% 0.32/0.64 thf(fact_7097_of__nat__power,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 0.32/0.64 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power
% 0.32/0.64 thf(fact_7098_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 0.32/0.64 = ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.64 = ( ( power_power_nat @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7099_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 0.32/0.64 = ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.64 = ( ( power_power_nat @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7100_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 0.32/0.64 = ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.64 = ( ( power_power_nat @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7101_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 0.32/0.64 = ( semiri8010041392384452111omplex @ X ) )
% 0.32/0.64 = ( ( power_power_nat @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7102_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 0.32/0.64 = ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.64 = ( ( power_power_nat @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_eq_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7103_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ( semiri1314217659103216013at_int @ X )
% 0.32/0.64 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7104_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ( semiri5074537144036343181t_real @ X )
% 0.32/0.64 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7105_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ( semiri1316708129612266289at_nat @ X )
% 0.32/0.64 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7106_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ( semiri8010041392384452111omplex @ X )
% 0.32/0.64 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7107_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ( semiri681578069525770553at_rat @ X )
% 0.32/0.64 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7108_of__int__mult,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 0.32/0.64 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_mult
% 0.32/0.64 thf(fact_7109_of__int__mult,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 0.32/0.64 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_mult
% 0.32/0.64 thf(fact_7110_of__int__mult,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 0.32/0.64 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_mult
% 0.32/0.64 thf(fact_7111_of__int__add,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 0.32/0.64 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_add
% 0.32/0.64 thf(fact_7112_of__int__add,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 0.32/0.64 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_add
% 0.32/0.64 thf(fact_7113_of__int__add,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 0.32/0.64 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_add
% 0.32/0.64 thf(fact_7114_negative__zless,axiom,
% 0.32/0.64 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.32/0.64
% 0.32/0.64 % negative_zless
% 0.32/0.64 thf(fact_7115_of__int__diff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 0.32/0.64 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_diff
% 0.32/0.64 thf(fact_7116_of__int__diff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 0.32/0.64 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_diff
% 0.32/0.64 thf(fact_7117_of__int__diff,axiom,
% 0.32/0.64 ! [W: int,Z: int] :
% 0.32/0.64 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 0.32/0.64 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_diff
% 0.32/0.64 thf(fact_7118_of__int__power,axiom,
% 0.32/0.64 ! [Z: int,N: nat] :
% 0.32/0.64 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 0.32/0.64 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power
% 0.32/0.64 thf(fact_7119_of__int__power,axiom,
% 0.32/0.64 ! [Z: int,N: nat] :
% 0.32/0.64 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 0.32/0.64 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power
% 0.32/0.64 thf(fact_7120_of__int__power,axiom,
% 0.32/0.64 ! [Z: int,N: nat] :
% 0.32/0.64 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 0.32/0.64 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power
% 0.32/0.64 thf(fact_7121_of__int__power,axiom,
% 0.32/0.64 ! [Z: int,N: nat] :
% 0.32/0.64 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 0.32/0.64 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power
% 0.32/0.64 thf(fact_7122_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 0.32/0.64 = ( ring_1_of_int_rat @ X ) )
% 0.32/0.64 = ( ( power_power_int @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7123_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 0.32/0.64 = ( ring_1_of_int_real @ X ) )
% 0.32/0.64 = ( ( power_power_int @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7124_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 0.32/0.64 = ( ring_1_of_int_int @ X ) )
% 0.32/0.64 = ( ( power_power_int @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7125_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 0.32/0.64 = ( ring_17405671764205052669omplex @ X ) )
% 0.32/0.64 = ( ( power_power_int @ B @ W )
% 0.32/0.64 = X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7126_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_rat @ X )
% 0.32/0.64 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7127_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_real @ X )
% 0.32/0.64 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7128_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_int @ X )
% 0.32/0.64 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7129_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ( ring_17405671764205052669omplex @ X )
% 0.32/0.64 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 0.32/0.64 = ( X
% 0.32/0.64 = ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7130_of__nat__of__bool,axiom,
% 0.32/0.64 ! [P: $o] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.64 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_of_bool
% 0.32/0.64 thf(fact_7131_of__nat__of__bool,axiom,
% 0.32/0.64 ! [P: $o] :
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.64 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_of_bool
% 0.32/0.64 thf(fact_7132_of__nat__of__bool,axiom,
% 0.32/0.64 ! [P: $o] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.64 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_of_bool
% 0.32/0.64 thf(fact_7133_of__nat__of__bool,axiom,
% 0.32/0.64 ! [P: $o] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.64 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_of_bool
% 0.32/0.64 thf(fact_7134_of__nat__of__bool,axiom,
% 0.32/0.64 ! [P: $o] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.64 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_of_bool
% 0.32/0.64 thf(fact_7135_of__nat__of__bool,axiom,
% 0.32/0.64 ! [P: $o] :
% 0.32/0.64 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.32/0.64 = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_of_bool
% 0.32/0.64 thf(fact_7136_dbl__dec__simps_I5_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 0.32/0.64 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(5)
% 0.32/0.64 thf(fact_7137_dbl__dec__simps_I5_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 0.32/0.64 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(5)
% 0.32/0.64 thf(fact_7138_dbl__dec__simps_I5_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 0.32/0.64 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(5)
% 0.32/0.64 thf(fact_7139_dbl__dec__simps_I5_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 0.32/0.64 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % dbl_dec_simps(5)
% 0.32/0.64 thf(fact_7140_of__nat__le__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_0_iff
% 0.32/0.64 thf(fact_7141_of__nat__le__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_0_iff
% 0.32/0.64 thf(fact_7142_of__nat__le__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_0_iff
% 0.32/0.64 thf(fact_7143_of__nat__le__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.32/0.64 = ( M = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_0_iff
% 0.32/0.64 thf(fact_7144_of__nat__Suc,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.32/0.64 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_Suc
% 0.32/0.64 thf(fact_7145_of__nat__Suc,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.32/0.64 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_Suc
% 0.32/0.64 thf(fact_7146_of__nat__Suc,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.32/0.64 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_Suc
% 0.32/0.64 thf(fact_7147_of__nat__Suc,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.32/0.64 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_Suc
% 0.32/0.64 thf(fact_7148_of__nat__Suc,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.32/0.64 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_Suc
% 0.32/0.64 thf(fact_7149_bit__numeral__Bit0__Suc__iff,axiom,
% 0.32/0.64 ! [M: num,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.32/0.64 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_Bit0_Suc_iff
% 0.32/0.64 thf(fact_7150_bit__numeral__Bit0__Suc__iff,axiom,
% 0.32/0.64 ! [M: num,N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 0.32/0.64 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_Bit0_Suc_iff
% 0.32/0.64 thf(fact_7151_real__of__nat__less__numeral__iff,axiom,
% 0.32/0.64 ! [N: nat,W: num] :
% 0.32/0.64 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 0.32/0.64 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_nat_less_numeral_iff
% 0.32/0.64 thf(fact_7152_numeral__less__real__of__nat__iff,axiom,
% 0.32/0.64 ! [W: num,N: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_less_real_of_nat_iff
% 0.32/0.64 thf(fact_7153_bit__numeral__Bit1__Suc__iff,axiom,
% 0.32/0.64 ! [M: num,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.32/0.64 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_Bit1_Suc_iff
% 0.32/0.64 thf(fact_7154_bit__numeral__Bit1__Suc__iff,axiom,
% 0.32/0.64 ! [M: num,N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 0.32/0.64 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_Bit1_Suc_iff
% 0.32/0.64 thf(fact_7155_numeral__le__real__of__nat__iff,axiom,
% 0.32/0.64 ! [N: num,M: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_le_real_of_nat_iff
% 0.32/0.64 thf(fact_7156_pred__numeral__simps_I2_J,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( pred_numeral @ ( bit0 @ K ) )
% 0.32/0.64 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % pred_numeral_simps(2)
% 0.32/0.64 thf(fact_7157_signed__take__bit__nonnegative__iff,axiom,
% 0.32/0.64 ! [N: nat,K: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.32/0.64 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % signed_take_bit_nonnegative_iff
% 0.32/0.64 thf(fact_7158_signed__take__bit__negative__iff,axiom,
% 0.32/0.64 ! [N: nat,K: int] :
% 0.32/0.64 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.64 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % signed_take_bit_negative_iff
% 0.32/0.64 thf(fact_7159_of__nat__0__less__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_less_iff
% 0.32/0.64 thf(fact_7160_of__nat__0__less__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_less_iff
% 0.32/0.64 thf(fact_7161_of__nat__0__less__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_less_iff
% 0.32/0.64 thf(fact_7162_of__nat__0__less__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.64 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_less_iff
% 0.32/0.64 thf(fact_7163_of__int__0__le__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_0_le_iff
% 0.32/0.64 thf(fact_7164_of__int__0__le__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_0_le_iff
% 0.32/0.64 thf(fact_7165_of__int__0__le__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_0_le_iff
% 0.32/0.64 thf(fact_7166_of__int__le__0__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_0_iff
% 0.32/0.64 thf(fact_7167_of__int__le__0__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_0_iff
% 0.32/0.64 thf(fact_7168_of__int__le__0__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_0_iff
% 0.32/0.64 thf(fact_7169_of__int__less__0__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.32/0.64 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_0_iff
% 0.32/0.64 thf(fact_7170_of__int__less__0__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.32/0.64 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_0_iff
% 0.32/0.64 thf(fact_7171_of__int__less__0__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.32/0.64 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_0_iff
% 0.32/0.64 thf(fact_7172_of__int__0__less__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.32/0.64 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_0_less_iff
% 0.32/0.64 thf(fact_7173_of__int__0__less__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.32/0.64 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_0_less_iff
% 0.32/0.64 thf(fact_7174_of__int__0__less__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.32/0.64 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_0_less_iff
% 0.32/0.64 thf(fact_7175_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.64 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7176_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.64 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7177_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.64 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7178_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.64 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7179_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7180_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7181_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7182_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_less_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7183_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: nat,X: num,N: nat] :
% 0.32/0.64 ( ( ( semiri1314217659103216013at_int @ Y3 )
% 0.32/0.64 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7184_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: nat,X: num,N: nat] :
% 0.32/0.64 ( ( ( semiri5074537144036343181t_real @ Y3 )
% 0.32/0.64 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7185_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: nat,X: num,N: nat] :
% 0.32/0.64 ( ( ( semiri1316708129612266289at_nat @ Y3 )
% 0.32/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7186_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: nat,X: num,N: nat] :
% 0.32/0.64 ( ( ( semiri8010041392384452111omplex @ Y3 )
% 0.32/0.64 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7187_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: nat,X: num,N: nat] :
% 0.32/0.64 ( ( ( semiri681578069525770553at_rat @ Y3 )
% 0.32/0.64 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_nat_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7188_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: nat] :
% 0.32/0.64 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/0.64 = ( semiri1314217659103216013at_int @ Y3 ) )
% 0.32/0.64 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7189_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: nat] :
% 0.32/0.64 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.32/0.64 = ( semiri5074537144036343181t_real @ Y3 ) )
% 0.32/0.64 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7190_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: nat] :
% 0.32/0.64 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.64 = ( semiri1316708129612266289at_nat @ Y3 ) )
% 0.32/0.64 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7191_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: nat] :
% 0.32/0.64 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 0.32/0.64 = ( semiri8010041392384452111omplex @ Y3 ) )
% 0.32/0.64 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7192_numeral__power__eq__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: nat] :
% 0.32/0.64 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 0.32/0.64 = ( semiri681578069525770553at_rat @ Y3 ) )
% 0.32/0.64 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7193_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7194_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7195_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7196_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: nat,W: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_of_nat_power_cancel_iff
% 0.32/0.64 thf(fact_7197_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7198_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7199_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7200_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,B: nat,W: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_power_le_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7201_of__int__le__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_numeral_iff
% 0.32/0.64 thf(fact_7202_of__int__le__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_numeral_iff
% 0.32/0.64 thf(fact_7203_of__int__le__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_numeral_iff
% 0.32/0.64 thf(fact_7204_of__int__numeral__le__iff,axiom,
% 0.32/0.64 ! [N: num,Z: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral_le_iff
% 0.32/0.64 thf(fact_7205_of__int__numeral__le__iff,axiom,
% 0.32/0.64 ! [N: num,Z: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral_le_iff
% 0.32/0.64 thf(fact_7206_of__int__numeral__le__iff,axiom,
% 0.32/0.64 ! [N: num,Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral_le_iff
% 0.32/0.64 thf(fact_7207_of__int__numeral__less__iff,axiom,
% 0.32/0.64 ! [N: num,Z: int] :
% 0.32/0.64 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.32/0.64 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral_less_iff
% 0.32/0.64 thf(fact_7208_of__int__numeral__less__iff,axiom,
% 0.32/0.64 ! [N: num,Z: int] :
% 0.32/0.64 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.32/0.64 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral_less_iff
% 0.32/0.64 thf(fact_7209_of__int__numeral__less__iff,axiom,
% 0.32/0.64 ! [N: num,Z: int] :
% 0.32/0.64 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.32/0.64 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_numeral_less_iff
% 0.32/0.64 thf(fact_7210_of__int__less__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.64 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_numeral_iff
% 0.32/0.64 thf(fact_7211_of__int__less__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.32/0.64 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_numeral_iff
% 0.32/0.64 thf(fact_7212_of__int__less__numeral__iff,axiom,
% 0.32/0.64 ! [Z: int,N: num] :
% 0.32/0.64 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.64 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_numeral_iff
% 0.32/0.64 thf(fact_7213_of__int__le__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_1_iff
% 0.32/0.64 thf(fact_7214_of__int__le__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_1_iff
% 0.32/0.64 thf(fact_7215_of__int__le__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.32/0.64 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_1_iff
% 0.32/0.64 thf(fact_7216_of__int__1__le__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1_le_iff
% 0.32/0.64 thf(fact_7217_of__int__1__le__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1_le_iff
% 0.32/0.64 thf(fact_7218_of__int__1__le__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.32/0.64 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1_le_iff
% 0.32/0.64 thf(fact_7219_of__int__1__less__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.32/0.64 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1_less_iff
% 0.32/0.64 thf(fact_7220_of__int__1__less__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.32/0.64 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1_less_iff
% 0.32/0.64 thf(fact_7221_of__int__1__less__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.32/0.64 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_1_less_iff
% 0.32/0.64 thf(fact_7222_of__int__less__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.32/0.64 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_1_iff
% 0.32/0.64 thf(fact_7223_of__int__less__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.32/0.64 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_1_iff
% 0.32/0.64 thf(fact_7224_of__int__less__1__iff,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.32/0.64 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_1_iff
% 0.32/0.64 thf(fact_7225_of__int__le__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7226_of__int__le__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7227_of__int__le__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7228_of__int__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7229_of__int__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7230_of__int__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7231_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 0.32/0.64 = ( ring_17405671764205052669omplex @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7232_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 0.32/0.64 = ( ring_1_of_int_real @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7233_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 0.32/0.64 = ( ring_1_of_int_rat @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7234_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/0.64 = ( ring_1_of_int_int @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7235_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 0.32/0.64 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7236_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_real @ Y3 )
% 0.32/0.64 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7237_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_rat @ Y3 )
% 0.32/0.64 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7238_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_int @ Y3 )
% 0.32/0.64 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7239_of__int__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7240_of__int__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7241_of__int__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: int,B: int,W: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 0.32/0.64 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7242_of__int__less__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7243_of__int__less__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7244_of__int__less__of__int__power__cancel__iff,axiom,
% 0.32/0.64 ! [B: int,W: nat,X: int] :
% 0.32/0.64 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_of_int_power_cancel_iff
% 0.32/0.64 thf(fact_7245_bit__numeral__simps_I2_J,axiom,
% 0.32/0.64 ! [W: num,N: num] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_simps(2)
% 0.32/0.64 thf(fact_7246_bit__numeral__simps_I2_J,axiom,
% 0.32/0.64 ! [W: num,N: num] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_simps(2)
% 0.32/0.64 thf(fact_7247_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.32/0.64 ! [W: num,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 0.32/0.64 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_minus_numeral_Bit0_Suc_iff
% 0.32/0.64 thf(fact_7248_bit__numeral__simps_I3_J,axiom,
% 0.32/0.64 ! [W: num,N: num] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_simps(3)
% 0.32/0.64 thf(fact_7249_bit__numeral__simps_I3_J,axiom,
% 0.32/0.64 ! [W: num,N: num] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_simps(3)
% 0.32/0.64 thf(fact_7250_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.32/0.64 ! [W: num,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 0.32/0.64 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_minus_numeral_Bit1_Suc_iff
% 0.32/0.64 thf(fact_7251_of__nat__zero__less__power__iff,axiom,
% 0.32/0.64 ! [X: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 0.32/0.64 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.64 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_zero_less_power_iff
% 0.32/0.64 thf(fact_7252_of__nat__zero__less__power__iff,axiom,
% 0.32/0.64 ! [X: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 0.32/0.64 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.64 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_zero_less_power_iff
% 0.32/0.64 thf(fact_7253_of__nat__zero__less__power__iff,axiom,
% 0.32/0.64 ! [X: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 0.32/0.64 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.64 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_zero_less_power_iff
% 0.32/0.64 thf(fact_7254_of__nat__zero__less__power__iff,axiom,
% 0.32/0.64 ! [X: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 0.32/0.64 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.64 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_zero_less_power_iff
% 0.32/0.64 thf(fact_7255_bit__0,axiom,
% 0.32/0.64 ! [A: code_integer] :
% 0.32/0.64 ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 0.32/0.64 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_0
% 0.32/0.64 thf(fact_7256_bit__0,axiom,
% 0.32/0.64 ! [A: int] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 0.32/0.64 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_0
% 0.32/0.64 thf(fact_7257_bit__0,axiom,
% 0.32/0.64 ! [A: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 0.32/0.64 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_0
% 0.32/0.64 thf(fact_7258_bit__minus__numeral__int_I1_J,axiom,
% 0.32/0.64 ! [W: num,N: num] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_minus_numeral_int(1)
% 0.32/0.64 thf(fact_7259_bit__minus__numeral__int_I2_J,axiom,
% 0.32/0.64 ! [W: num,N: num] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_minus_numeral_int(2)
% 0.32/0.64 thf(fact_7260_even__of__nat,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.32/0.64 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % even_of_nat
% 0.32/0.64 thf(fact_7261_even__of__nat,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % even_of_nat
% 0.32/0.64 thf(fact_7262_even__of__nat,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % even_of_nat
% 0.32/0.64 thf(fact_7263_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [I: num,N: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.64 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_less_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7264_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [I: num,N: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.64 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_less_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7265_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [I: num,N: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.64 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_less_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7266_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [I: num,N: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.64 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_less_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7267_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,I: num,N: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.32/0.64 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7268_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,I: num,N: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.32/0.64 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7269_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,I: num,N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.32/0.64 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7270_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,I: num,N: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.32/0.64 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7271_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [I: num,N: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_le_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7272_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [I: num,N: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_le_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7273_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [I: num,N: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_le_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7274_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.32/0.64 ! [I: num,N: nat,X: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 0.32/0.64 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_le_of_nat_cancel_iff
% 0.32/0.64 thf(fact_7275_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,I: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7276_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,I: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7277_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,I: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7278_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [X: nat,I: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_le_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7279_bit__mod__2__iff,axiom,
% 0.32/0.64 ! [A: code_integer,N: nat] :
% 0.32/0.64 ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 0.32/0.64 = ( ( N = zero_zero_nat )
% 0.32/0.64 & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_mod_2_iff
% 0.32/0.64 thf(fact_7280_bit__mod__2__iff,axiom,
% 0.32/0.64 ! [A: int,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 0.32/0.64 = ( ( N = zero_zero_nat )
% 0.32/0.64 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_mod_2_iff
% 0.32/0.64 thf(fact_7281_bit__mod__2__iff,axiom,
% 0.32/0.64 ! [A: nat,N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.32/0.64 = ( ( N = zero_zero_nat )
% 0.32/0.64 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_mod_2_iff
% 0.32/0.64 thf(fact_7282_numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7283_numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7284_numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7285_of__int__le__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7286_of__int__le__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7287_of__int__le__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7288_numeral__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7289_numeral__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7290_numeral__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % numeral_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7291_of__int__less__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 0.32/0.64 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7292_of__int__less__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 0.32/0.64 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7293_of__int__less__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 0.32/0.64 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7294_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
% 0.32/0.64 = ( ring_1_of_int_real @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7295_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.32/0.64 = ( ring_1_of_int_int @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7296_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
% 0.32/0.64 = ( ring_17405671764205052669omplex @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7297_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
% 0.32/0.64 = ( ring_18347121197199848620nteger @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7298_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,Y3: int] :
% 0.32/0.64 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
% 0.32/0.64 = ( ring_1_of_int_rat @ Y3 ) )
% 0.32/0.64 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 0.32/0.64 = Y3 ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_eq_of_int_cancel_iff
% 0.32/0.64 thf(fact_7299_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_real @ Y3 )
% 0.32/0.64 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7300_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_int @ Y3 )
% 0.32/0.64 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7301_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 0.32/0.64 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7302_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_18347121197199848620nteger @ Y3 )
% 0.32/0.64 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7303_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [Y3: int,X: num,N: nat] :
% 0.32/0.64 ( ( ( ring_1_of_int_rat @ Y3 )
% 0.32/0.64 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.32/0.64 = ( Y3
% 0.32/0.64 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_eq_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7304_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7305_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7306_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7307_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.32/0.64 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_le_of_int_cancel_iff
% 0.32/0.64 thf(fact_7308_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7309_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7310_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7311_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.32/0.64 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_le_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7312_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 0.32/0.64 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7313_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 0.32/0.64 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7314_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 0.32/0.64 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7315_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.32/0.64 ! [A: int,X: num,N: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 0.32/0.64 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_less_neg_numeral_power_cancel_iff
% 0.32/0.64 thf(fact_7316_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7317_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7318_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7319_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.32/0.64 ! [X: num,N: nat,A: int] :
% 0.32/0.64 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 0.32/0.64 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 0.32/0.64
% 0.32/0.64 % neg_numeral_power_less_of_int_cancel_iff
% 0.32/0.64 thf(fact_7320_bit__numeral__iff,axiom,
% 0.32/0.64 ! [M: num,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 0.32/0.64 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_iff
% 0.32/0.64 thf(fact_7321_bit__numeral__iff,axiom,
% 0.32/0.64 ! [M: num,N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 0.32/0.64 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_iff
% 0.32/0.64 thf(fact_7322_of__nat__less__of__int__iff,axiom,
% 0.32/0.64 ! [N: nat,X: int] :
% 0.32/0.64 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 0.32/0.64 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_of_int_iff
% 0.32/0.64 thf(fact_7323_of__nat__less__of__int__iff,axiom,
% 0.32/0.64 ! [N: nat,X: int] :
% 0.32/0.64 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 0.32/0.64 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_of_int_iff
% 0.32/0.64 thf(fact_7324_of__nat__less__of__int__iff,axiom,
% 0.32/0.64 ! [N: nat,X: int] :
% 0.32/0.64 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 0.32/0.64 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_of_int_iff
% 0.32/0.64 thf(fact_7325_bit__of__nat__iff__bit,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 0.32/0.64 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_of_nat_iff_bit
% 0.32/0.64 thf(fact_7326_bit__of__nat__iff__bit,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 0.32/0.64 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_of_nat_iff_bit
% 0.32/0.64 thf(fact_7327_bit__disjunctive__add__iff,axiom,
% 0.32/0.64 ! [A: int,B: int,N: nat] :
% 0.32/0.64 ( ! [N4: nat] :
% 0.32/0.64 ( ~ ( bit_se1146084159140164899it_int @ A @ N4 )
% 0.32/0.64 | ~ ( bit_se1146084159140164899it_int @ B @ N4 ) )
% 0.32/0.64 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
% 0.32/0.64 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.32/0.64 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_disjunctive_add_iff
% 0.32/0.64 thf(fact_7328_bit__disjunctive__add__iff,axiom,
% 0.32/0.64 ! [A: nat,B: nat,N: nat] :
% 0.32/0.64 ( ! [N4: nat] :
% 0.32/0.64 ( ~ ( bit_se1148574629649215175it_nat @ A @ N4 )
% 0.32/0.64 | ~ ( bit_se1148574629649215175it_nat @ B @ N4 ) )
% 0.32/0.64 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.32/0.64 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.32/0.64 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_disjunctive_add_iff
% 0.32/0.64 thf(fact_7329_mult__of__int__commute,axiom,
% 0.32/0.64 ! [X: int,Y3: real] :
% 0.32/0.64 ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y3 )
% 0.32/0.64 = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_of_int_commute
% 0.32/0.64 thf(fact_7330_mult__of__int__commute,axiom,
% 0.32/0.64 ! [X: int,Y3: rat] :
% 0.32/0.64 ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y3 )
% 0.32/0.64 = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_of_int_commute
% 0.32/0.64 thf(fact_7331_mult__of__int__commute,axiom,
% 0.32/0.64 ! [X: int,Y3: int] :
% 0.32/0.64 ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y3 )
% 0.32/0.64 = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_of_int_commute
% 0.32/0.64 thf(fact_7332_mult__of__nat__commute,axiom,
% 0.32/0.64 ! [X: nat,Y3: int] :
% 0.32/0.64 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y3 )
% 0.32/0.64 = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_of_nat_commute
% 0.32/0.64 thf(fact_7333_mult__of__nat__commute,axiom,
% 0.32/0.64 ! [X: nat,Y3: real] :
% 0.32/0.64 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y3 )
% 0.32/0.64 = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_of_nat_commute
% 0.32/0.64 thf(fact_7334_mult__of__nat__commute,axiom,
% 0.32/0.64 ! [X: nat,Y3: nat] :
% 0.32/0.64 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y3 )
% 0.32/0.64 = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_of_nat_commute
% 0.32/0.64 thf(fact_7335_mult__of__nat__commute,axiom,
% 0.32/0.64 ! [X: nat,Y3: complex] :
% 0.32/0.64 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y3 )
% 0.32/0.64 = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_of_nat_commute
% 0.32/0.64 thf(fact_7336_mult__of__nat__commute,axiom,
% 0.32/0.64 ! [X: nat,Y3: rat] :
% 0.32/0.64 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y3 )
% 0.32/0.64 = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mult_of_nat_commute
% 0.32/0.64 thf(fact_7337_int__diff__cases,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ~ ! [M4: nat,N4: nat] :
% 0.32/0.64 ( Z
% 0.32/0.64 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_diff_cases
% 0.32/0.64 thf(fact_7338_bit__unset__bit__iff,axiom,
% 0.32/0.64 ! [M: nat,A: int,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
% 0.32/0.64 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.32/0.64 & ( M != N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_unset_bit_iff
% 0.32/0.64 thf(fact_7339_bit__unset__bit__iff,axiom,
% 0.32/0.64 ! [M: nat,A: nat,N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
% 0.32/0.64 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.32/0.64 & ( M != N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_unset_bit_iff
% 0.32/0.64 thf(fact_7340_semiring__norm_I26_J,axiom,
% 0.32/0.64 ( ( bitM @ one )
% 0.32/0.64 = one ) ).
% 0.32/0.64
% 0.32/0.64 % semiring_norm(26)
% 0.32/0.64 thf(fact_7341_not__bit__1__Suc,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % not_bit_1_Suc
% 0.32/0.64 thf(fact_7342_not__bit__1__Suc,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % not_bit_1_Suc
% 0.32/0.64 thf(fact_7343_bit__1__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 0.32/0.64 = ( N = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_1_iff
% 0.32/0.64 thf(fact_7344_bit__1__iff,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 0.32/0.64 = ( N = zero_zero_nat ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_1_iff
% 0.32/0.64 thf(fact_7345_bit__numeral__simps_I1_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_simps(1)
% 0.32/0.64 thf(fact_7346_bit__numeral__simps_I1_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_numeral_simps(1)
% 0.32/0.64 thf(fact_7347_of__nat__0__le__iff,axiom,
% 0.32/0.64 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_le_iff
% 0.32/0.64 thf(fact_7348_of__nat__0__le__iff,axiom,
% 0.32/0.64 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_le_iff
% 0.32/0.64 thf(fact_7349_of__nat__0__le__iff,axiom,
% 0.32/0.64 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_le_iff
% 0.32/0.64 thf(fact_7350_of__nat__0__le__iff,axiom,
% 0.32/0.64 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_0_le_iff
% 0.32/0.64 thf(fact_7351_of__nat__less__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_0_iff
% 0.32/0.64 thf(fact_7352_of__nat__less__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_0_iff
% 0.32/0.64 thf(fact_7353_of__nat__less__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_0_iff
% 0.32/0.64 thf(fact_7354_of__nat__less__0__iff,axiom,
% 0.32/0.64 ! [M: nat] :
% 0.32/0.64 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_0_iff
% 0.32/0.64 thf(fact_7355_bit__take__bit__iff,axiom,
% 0.32/0.64 ! [M: nat,A: int,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 0.32/0.64 = ( ( ord_less_nat @ N @ M )
% 0.32/0.64 & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_take_bit_iff
% 0.32/0.64 thf(fact_7356_bit__take__bit__iff,axiom,
% 0.32/0.64 ! [M: nat,A: nat,N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 0.32/0.64 = ( ( ord_less_nat @ N @ M )
% 0.32/0.64 & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_take_bit_iff
% 0.32/0.64 thf(fact_7357_of__nat__neq__0,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.32/0.64 != zero_zero_int ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_neq_0
% 0.32/0.64 thf(fact_7358_of__nat__neq__0,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 0.32/0.64 != zero_zero_real ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_neq_0
% 0.32/0.64 thf(fact_7359_of__nat__neq__0,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 0.32/0.64 != zero_zero_nat ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_neq_0
% 0.32/0.64 thf(fact_7360_of__nat__neq__0,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 0.32/0.64 != zero_zero_complex ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_neq_0
% 0.32/0.64 thf(fact_7361_of__nat__neq__0,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 0.32/0.64 != zero_zero_rat ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_neq_0
% 0.32/0.64 thf(fact_7362_bit__of__bool__iff,axiom,
% 0.32/0.64 ! [B: $o,N: nat] :
% 0.32/0.64 ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
% 0.32/0.64 = ( B
% 0.32/0.64 & ( N = zero_zero_nat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_of_bool_iff
% 0.32/0.64 thf(fact_7363_bit__of__bool__iff,axiom,
% 0.32/0.64 ! [B: $o,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
% 0.32/0.64 = ( B
% 0.32/0.64 & ( N = zero_zero_nat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_of_bool_iff
% 0.32/0.64 thf(fact_7364_bit__of__bool__iff,axiom,
% 0.32/0.64 ! [B: $o,N: nat] :
% 0.32/0.64 ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
% 0.32/0.64 = ( B
% 0.32/0.64 & ( N = zero_zero_nat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_of_bool_iff
% 0.32/0.64 thf(fact_7365_div__mult2__eq_H,axiom,
% 0.32/0.64 ! [A: int,M: nat,N: nat] :
% 0.32/0.64 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.64 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % div_mult2_eq'
% 0.32/0.64 thf(fact_7366_div__mult2__eq_H,axiom,
% 0.32/0.64 ! [A: nat,M: nat,N: nat] :
% 0.32/0.64 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.32/0.64 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % div_mult2_eq'
% 0.32/0.64 thf(fact_7367_less__imp__of__nat__less,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ M @ N )
% 0.32/0.64 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % less_imp_of_nat_less
% 0.32/0.64 thf(fact_7368_less__imp__of__nat__less,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ M @ N )
% 0.32/0.64 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % less_imp_of_nat_less
% 0.32/0.64 thf(fact_7369_less__imp__of__nat__less,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ M @ N )
% 0.32/0.64 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % less_imp_of_nat_less
% 0.32/0.64 thf(fact_7370_less__imp__of__nat__less,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ M @ N )
% 0.32/0.64 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % less_imp_of_nat_less
% 0.32/0.64 thf(fact_7371_of__nat__less__imp__less,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_imp_less
% 0.32/0.64 thf(fact_7372_of__nat__less__imp__less,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.64 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_imp_less
% 0.32/0.64 thf(fact_7373_of__nat__less__imp__less,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_imp_less
% 0.32/0.64 thf(fact_7374_of__nat__less__imp__less,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.32/0.64 => ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_less_imp_less
% 0.32/0.64 thf(fact_7375_of__nat__mono,axiom,
% 0.32/0.64 ! [I: nat,J: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.64 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mono
% 0.32/0.64 thf(fact_7376_of__nat__mono,axiom,
% 0.32/0.64 ! [I: nat,J: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.64 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mono
% 0.32/0.64 thf(fact_7377_of__nat__mono,axiom,
% 0.32/0.64 ! [I: nat,J: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.64 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mono
% 0.32/0.64 thf(fact_7378_of__nat__mono,axiom,
% 0.32/0.64 ! [I: nat,J: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.64 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mono
% 0.32/0.64 thf(fact_7379_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.64 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.32/0.64 thf(fact_7380_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.64 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % unique_euclidean_semiring_with_nat_class.of_nat_div
% 0.32/0.64 thf(fact_7381_of__nat__dvd__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 0.32/0.64 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_dvd_iff
% 0.32/0.64 thf(fact_7382_of__nat__dvd__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_dvd_iff
% 0.32/0.64 thf(fact_7383_of__nat__dvd__iff,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.32/0.64 = ( dvd_dvd_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_dvd_iff
% 0.32/0.64 thf(fact_7384_int__ops_I1_J,axiom,
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.32/0.64 = zero_zero_int ) ).
% 0.32/0.64
% 0.32/0.64 % int_ops(1)
% 0.32/0.64 thf(fact_7385_signed__take__bit__eq__if__positive,axiom,
% 0.32/0.64 ! [A: int,N: nat] :
% 0.32/0.64 ( ~ ( bit_se1146084159140164899it_int @ A @ N )
% 0.32/0.64 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 0.32/0.64 = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % signed_take_bit_eq_if_positive
% 0.32/0.64 thf(fact_7386_int__ops_I3_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 0.32/0.64 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_ops(3)
% 0.32/0.64 thf(fact_7387_nat__int__comparison_I2_J,axiom,
% 0.32/0.64 ( ord_less_nat
% 0.32/0.64 = ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % nat_int_comparison(2)
% 0.32/0.64 thf(fact_7388_zle__int,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % zle_int
% 0.32/0.64 thf(fact_7389_nat__int__comparison_I3_J,axiom,
% 0.32/0.64 ( ord_less_eq_nat
% 0.32/0.64 = ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % nat_int_comparison(3)
% 0.32/0.64 thf(fact_7390_zero__le__imp__eq__int,axiom,
% 0.32/0.64 ! [K: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.64 => ? [N4: nat] :
% 0.32/0.64 ( K
% 0.32/0.64 = ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % zero_le_imp_eq_int
% 0.32/0.64 thf(fact_7391_nonneg__int__cases,axiom,
% 0.32/0.64 ! [K: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.64 => ~ ! [N4: nat] :
% 0.32/0.64 ( K
% 0.32/0.64 != ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % nonneg_int_cases
% 0.32/0.64 thf(fact_7392_of__nat__mod,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.64 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mod
% 0.32/0.64 thf(fact_7393_of__nat__mod,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.64 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mod
% 0.32/0.64 thf(fact_7394_of__nat__mod,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.32/0.64 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_mod
% 0.32/0.64 thf(fact_7395_zadd__int__left,axiom,
% 0.32/0.64 ! [M: nat,N: nat,Z: int] :
% 0.32/0.64 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 0.32/0.64 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 0.32/0.64
% 0.32/0.64 % zadd_int_left
% 0.32/0.64 thf(fact_7396_int__plus,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.64 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_plus
% 0.32/0.64 thf(fact_7397_int__ops_I5_J,axiom,
% 0.32/0.64 ! [A: nat,B: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 0.32/0.64 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_ops(5)
% 0.32/0.64 thf(fact_7398_int__ops_I2_J,axiom,
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.32/0.64 = one_one_int ) ).
% 0.32/0.64
% 0.32/0.64 % int_ops(2)
% 0.32/0.64 thf(fact_7399_int__ops_I7_J,axiom,
% 0.32/0.64 ! [A: nat,B: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 0.32/0.64 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_ops(7)
% 0.32/0.64 thf(fact_7400_zle__iff__zadd,axiom,
% 0.32/0.64 ( ord_less_eq_int
% 0.32/0.64 = ( ^ [W3: int,Z2: int] :
% 0.32/0.64 ? [N2: nat] :
% 0.32/0.64 ( Z2
% 0.32/0.64 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % zle_iff_zadd
% 0.32/0.64 thf(fact_7401_not__int__zless__negative,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % not_int_zless_negative
% 0.32/0.64 thf(fact_7402_take__bit__of__int,axiom,
% 0.32/0.64 ! [N: nat,K: int] :
% 0.32/0.64 ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 0.32/0.64 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % take_bit_of_int
% 0.32/0.64 thf(fact_7403_zdiv__int,axiom,
% 0.32/0.64 ! [A: nat,B: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 0.32/0.64 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % zdiv_int
% 0.32/0.64 thf(fact_7404_of__nat__max,axiom,
% 0.32/0.64 ! [X: nat,Y3: nat] :
% 0.32/0.64 ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y3 ) )
% 0.32/0.64 = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y3 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_max
% 0.32/0.64 thf(fact_7405_of__nat__max,axiom,
% 0.32/0.64 ! [X: nat,Y3: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y3 ) )
% 0.32/0.64 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_max
% 0.32/0.64 thf(fact_7406_of__nat__max,axiom,
% 0.32/0.64 ! [X: nat,Y3: nat] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y3 ) )
% 0.32/0.64 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_max
% 0.32/0.64 thf(fact_7407_of__nat__max,axiom,
% 0.32/0.64 ! [X: nat,Y3: nat] :
% 0.32/0.64 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y3 ) )
% 0.32/0.64 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_max
% 0.32/0.64 thf(fact_7408_of__nat__max,axiom,
% 0.32/0.64 ! [X: nat,Y3: nat] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y3 ) )
% 0.32/0.64 = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y3 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_max
% 0.32/0.64 thf(fact_7409_zmod__int,axiom,
% 0.32/0.64 ! [A: nat,B: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
% 0.32/0.64 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % zmod_int
% 0.32/0.64 thf(fact_7410_take__bit__of__nat,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 0.32/0.64 = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % take_bit_of_nat
% 0.32/0.64 thf(fact_7411_take__bit__of__nat,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 0.32/0.64 = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % take_bit_of_nat
% 0.32/0.64 thf(fact_7412_semiring__norm_I27_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( bitM @ ( bit0 @ N ) )
% 0.32/0.64 = ( bit1 @ ( bitM @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % semiring_norm(27)
% 0.32/0.64 thf(fact_7413_semiring__norm_I28_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( bitM @ ( bit1 @ N ) )
% 0.32/0.64 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % semiring_norm(28)
% 0.32/0.64 thf(fact_7414_nat__less__as__int,axiom,
% 0.32/0.64 ( ord_less_nat
% 0.32/0.64 = ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % nat_less_as_int
% 0.32/0.64 thf(fact_7415_nat__leq__as__int,axiom,
% 0.32/0.64 ( ord_less_eq_nat
% 0.32/0.64 = ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % nat_leq_as_int
% 0.32/0.64 thf(fact_7416_inc__BitM__eq,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( inc @ ( bitM @ N ) )
% 0.32/0.64 = ( bit0 @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % inc_BitM_eq
% 0.32/0.64 thf(fact_7417_BitM__inc__eq,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( bitM @ ( inc @ N ) )
% 0.32/0.64 = ( bit1 @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % BitM_inc_eq
% 0.32/0.64 thf(fact_7418_of__nat__diff,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.64 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.64 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_diff
% 0.32/0.64 thf(fact_7419_of__nat__diff,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.64 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.64 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_diff
% 0.32/0.64 thf(fact_7420_of__nat__diff,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.64 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.64 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_diff
% 0.32/0.64 thf(fact_7421_of__nat__diff,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.64 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.64 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_diff
% 0.32/0.64 thf(fact_7422_of__nat__diff,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.64 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.32/0.64 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_nat_diff
% 0.32/0.64 thf(fact_7423_bit__not__int__iff_H,axiom,
% 0.32/0.64 ! [K: int,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 0.32/0.64 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_not_int_iff'
% 0.32/0.64 thf(fact_7424_reals__Archimedean3,axiom,
% 0.32/0.64 ! [X: real] :
% 0.32/0.64 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.64 => ! [Y5: real] :
% 0.32/0.64 ? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % reals_Archimedean3
% 0.32/0.64 thf(fact_7425_real__of__int__div4,axiom,
% 0.32/0.64 ! [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 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_int_div4
% 0.32/0.64 thf(fact_7426_int__cases4,axiom,
% 0.32/0.64 ! [M: int] :
% 0.32/0.64 ( ! [N4: nat] :
% 0.32/0.64 ( M
% 0.32/0.64 != ( semiri1314217659103216013at_int @ N4 ) )
% 0.32/0.64 => ~ ! [N4: nat] :
% 0.32/0.64 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.64 => ( M
% 0.32/0.64 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_cases4
% 0.32/0.64 thf(fact_7427_real__of__nat__div4,axiom,
% 0.32/0.64 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_nat_div4
% 0.32/0.64 thf(fact_7428_int__zle__neg,axiom,
% 0.32/0.64 ! [N: nat,M: nat] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.32/0.64 = ( ( N = zero_zero_nat )
% 0.32/0.64 & ( M = zero_zero_nat ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_zle_neg
% 0.32/0.64 thf(fact_7429_int__ops_I4_J,axiom,
% 0.32/0.64 ! [A: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 0.32/0.64 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_ops(4)
% 0.32/0.64 thf(fact_7430_int__Suc,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.32/0.64 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_Suc
% 0.32/0.64 thf(fact_7431_zless__iff__Suc__zadd,axiom,
% 0.32/0.64 ( ord_less_int
% 0.32/0.64 = ( ^ [W3: int,Z2: int] :
% 0.32/0.64 ? [N2: nat] :
% 0.32/0.64 ( Z2
% 0.32/0.64 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % zless_iff_Suc_zadd
% 0.32/0.64 thf(fact_7432_nonpos__int__cases,axiom,
% 0.32/0.64 ! [K: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.32/0.64 => ~ ! [N4: nat] :
% 0.32/0.64 ( K
% 0.32/0.64 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % nonpos_int_cases
% 0.32/0.64 thf(fact_7433_negative__zle__0,axiom,
% 0.32/0.64 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 0.32/0.64
% 0.32/0.64 % negative_zle_0
% 0.32/0.64 thf(fact_7434_real__of__nat__div,axiom,
% 0.32/0.64 ! [D: nat,N: nat] :
% 0.32/0.64 ( ( dvd_dvd_nat @ D @ N )
% 0.32/0.64 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 0.32/0.64 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_nat_div
% 0.32/0.64 thf(fact_7435_real__of__int__div,axiom,
% 0.32/0.64 ! [D: int,N: int] :
% 0.32/0.64 ( ( dvd_dvd_int @ D @ N )
% 0.32/0.64 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 0.32/0.64 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % real_of_int_div
% 0.32/0.64 thf(fact_7436_flip__bit__eq__if,axiom,
% 0.32/0.64 ( bit_se2159334234014336723it_int
% 0.32/0.64 = ( ^ [N2: nat,A3: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A3 @ N2 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N2 @ A3 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % flip_bit_eq_if
% 0.32/0.64 thf(fact_7437_flip__bit__eq__if,axiom,
% 0.32/0.64 ( bit_se2161824704523386999it_nat
% 0.32/0.64 = ( ^ [N2: nat,A3: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A3 @ N2 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N2 @ A3 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % flip_bit_eq_if
% 0.32/0.64 thf(fact_7438_eval__nat__numeral_I2_J,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 0.32/0.64 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % eval_nat_numeral(2)
% 0.32/0.64 thf(fact_7439_one__plus__BitM,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 0.32/0.64 = ( bit0 @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % one_plus_BitM
% 0.32/0.64 thf(fact_7440_BitM__plus__one,axiom,
% 0.32/0.64 ! [N: num] :
% 0.32/0.64 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 0.32/0.64 = ( bit0 @ N ) ) ).
% 0.32/0.64
% 0.32/0.64 % BitM_plus_one
% 0.32/0.64 thf(fact_7441_of__int__nonneg,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.64 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_nonneg
% 0.32/0.64 thf(fact_7442_of__int__nonneg,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_nonneg
% 0.32/0.64 thf(fact_7443_of__int__nonneg,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.64 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_nonneg
% 0.32/0.64 thf(fact_7444_of__int__leD,axiom,
% 0.32/0.64 ! [N: int,X: code_integer] :
% 0.32/0.64 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.32/0.64 => ( ( N = zero_zero_int )
% 0.32/0.64 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_leD
% 0.32/0.64 thf(fact_7445_of__int__leD,axiom,
% 0.32/0.64 ! [N: int,X: real] :
% 0.32/0.64 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.32/0.64 => ( ( N = zero_zero_int )
% 0.32/0.64 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_leD
% 0.32/0.64 thf(fact_7446_of__int__leD,axiom,
% 0.32/0.64 ! [N: int,X: rat] :
% 0.32/0.64 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.32/0.64 => ( ( N = zero_zero_int )
% 0.32/0.64 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_leD
% 0.32/0.64 thf(fact_7447_of__int__leD,axiom,
% 0.32/0.64 ! [N: int,X: int] :
% 0.32/0.64 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.32/0.64 => ( ( N = zero_zero_int )
% 0.32/0.64 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_leD
% 0.32/0.64 thf(fact_7448_of__int__pos,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.64 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_pos
% 0.32/0.64 thf(fact_7449_of__int__pos,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.64 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_pos
% 0.32/0.64 thf(fact_7450_of__int__pos,axiom,
% 0.32/0.64 ! [Z: int] :
% 0.32/0.64 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.64 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_pos
% 0.32/0.64 thf(fact_7451_of__int__lessD,axiom,
% 0.32/0.64 ! [N: int,X: code_integer] :
% 0.32/0.64 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 0.32/0.64 => ( ( N = zero_zero_int )
% 0.32/0.64 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_lessD
% 0.32/0.64 thf(fact_7452_of__int__lessD,axiom,
% 0.32/0.64 ! [N: int,X: real] :
% 0.32/0.64 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 0.32/0.64 => ( ( N = zero_zero_int )
% 0.32/0.64 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_lessD
% 0.32/0.64 thf(fact_7453_of__int__lessD,axiom,
% 0.32/0.64 ! [N: int,X: rat] :
% 0.32/0.64 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 0.32/0.64 => ( ( N = zero_zero_int )
% 0.32/0.64 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_lessD
% 0.32/0.64 thf(fact_7454_of__int__lessD,axiom,
% 0.32/0.64 ! [N: int,X: int] :
% 0.32/0.64 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 0.32/0.64 => ( ( N = zero_zero_int )
% 0.32/0.64 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_lessD
% 0.32/0.64 thf(fact_7455_mod__mult2__eq_H,axiom,
% 0.32/0.64 ! [A: code_integer,M: nat,N: nat] :
% 0.32/0.64 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 0.32/0.64 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mod_mult2_eq'
% 0.32/0.64 thf(fact_7456_mod__mult2__eq_H,axiom,
% 0.32/0.64 ! [A: int,M: nat,N: nat] :
% 0.32/0.64 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.64 = ( 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 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mod_mult2_eq'
% 0.32/0.64 thf(fact_7457_mod__mult2__eq_H,axiom,
% 0.32/0.64 ! [A: nat,M: nat,N: nat] :
% 0.32/0.64 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 0.32/0.64 = ( 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 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % mod_mult2_eq'
% 0.32/0.64 thf(fact_7458_of__int__neg__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_neg_numeral
% 0.32/0.64 thf(fact_7459_of__int__neg__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_neg_numeral
% 0.32/0.64 thf(fact_7460_of__int__neg__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.64 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_neg_numeral
% 0.32/0.64 thf(fact_7461_of__int__neg__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.64 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_neg_numeral
% 0.32/0.64 thf(fact_7462_of__int__neg__numeral,axiom,
% 0.32/0.64 ! [K: num] :
% 0.32/0.64 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.64 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % of_int_neg_numeral
% 0.32/0.64 thf(fact_7463_bit__imp__take__bit__positive,axiom,
% 0.32/0.64 ! [N: nat,M: nat,K: int] :
% 0.32/0.64 ( ( ord_less_nat @ N @ M )
% 0.32/0.64 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.32/0.64 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_imp_take_bit_positive
% 0.32/0.64 thf(fact_7464_field__char__0__class_Oof__nat__div,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.64 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % field_char_0_class.of_nat_div
% 0.32/0.64 thf(fact_7465_field__char__0__class_Oof__nat__div,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.64 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % field_char_0_class.of_nat_div
% 0.32/0.64 thf(fact_7466_field__char__0__class_Oof__nat__div,axiom,
% 0.32/0.64 ! [M: nat,N: nat] :
% 0.32/0.64 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 0.32/0.64 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % field_char_0_class.of_nat_div
% 0.32/0.64 thf(fact_7467_int__le__real__less,axiom,
% 0.32/0.64 ( ord_less_eq_int
% 0.32/0.64 = ( ^ [N2: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_le_real_less
% 0.32/0.64 thf(fact_7468_int__less__real__le,axiom,
% 0.32/0.64 ( ord_less_int
% 0.32/0.64 = ( ^ [N2: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_less_real_le
% 0.32/0.64 thf(fact_7469_bit__concat__bit__iff,axiom,
% 0.32/0.64 ! [M: nat,K: int,L2: int,N: nat] :
% 0.32/0.64 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
% 0.32/0.64 = ( ( ( ord_less_nat @ N @ M )
% 0.32/0.64 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.32/0.64 | ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.64 & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % bit_concat_bit_iff
% 0.32/0.64 thf(fact_7470_zero__less__imp__eq__int,axiom,
% 0.32/0.64 ! [K: int] :
% 0.32/0.64 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.64 => ? [N4: nat] :
% 0.32/0.64 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.64 & ( K
% 0.32/0.64 = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % zero_less_imp_eq_int
% 0.32/0.64 thf(fact_7471_pos__int__cases,axiom,
% 0.32/0.64 ! [K: int] :
% 0.32/0.64 ( ( ord_less_int @ zero_zero_int @ K )
% 0.32/0.64 => ~ ! [N4: nat] :
% 0.32/0.64 ( ( K
% 0.32/0.64 = ( semiri1314217659103216013at_int @ N4 ) )
% 0.32/0.64 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % pos_int_cases
% 0.32/0.64 thf(fact_7472_int__cases3,axiom,
% 0.32/0.64 ! [K: int] :
% 0.32/0.64 ( ( K != zero_zero_int )
% 0.32/0.64 => ( ! [N4: nat] :
% 0.32/0.64 ( ( K
% 0.32/0.64 = ( semiri1314217659103216013at_int @ N4 ) )
% 0.32/0.64 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
% 0.32/0.64 => ~ ! [N4: nat] :
% 0.32/0.64 ( ( K
% 0.32/0.64 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 0.32/0.64 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % int_cases3
% 0.32/0.64 thf(fact_7473_nat__less__real__le,axiom,
% 0.32/0.64 ( ord_less_nat
% 0.32/0.64 = ( ^ [N2: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % nat_less_real_le
% 0.32/0.64 thf(fact_7474_nat__le__real__less,axiom,
% 0.32/0.64 ( ord_less_eq_nat
% 0.32/0.64 = ( ^ [N2: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % nat_le_real_less
% 0.32/0.64 thf(fact_7475_zmult__zless__mono2__lemma,axiom,
% 0.32/0.64 ! [I: int,J: int,K: nat] :
% 0.32/0.64 ( ( ord_less_int @ I @ J )
% 0.32/0.64 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.64 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % zmult_zless_mono2_lemma
% 0.32/0.64 thf(fact_7476_not__zle__0__negative,axiom,
% 0.32/0.64 ! [N: nat] :
% 0.32/0.64 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % not_zle_0_negative
% 0.32/0.64 thf(fact_7477_negative__zless__0,axiom,
% 0.32/0.64 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.32/0.64
% 0.32/0.64 % negative_zless_0
% 0.32/0.64 thf(fact_7478_negD,axiom,
% 0.32/0.64 ! [X: int] :
% 0.32/0.64 ( ( ord_less_int @ X @ zero_zero_int )
% 0.32/0.64 => ? [N4: nat] :
% 0.32/0.64 ( X
% 0.32/0.64 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 0.32/0.64
% 0.32/0.64 % negD
% 0.32/0.64 thf(fact_7479_int__ops_I6_J,axiom,
% 0.32/0.64 ! [A: nat,B: nat] :
% 0.32/0.64 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.32/0.64 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.64 = zero_zero_int ) )
% 0.32/0.65 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.32/0.65 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.32/0.65 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % int_ops(6)
% 0.32/0.65 thf(fact_7480_real__of__int__div__aux,axiom,
% 0.32/0.65 ! [X: int,D: int] :
% 0.32/0.65 ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 0.32/0.65 = ( 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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_of_int_div_aux
% 0.32/0.65 thf(fact_7481_real__of__nat__div__aux,axiom,
% 0.32/0.65 ! [X: nat,D: nat] :
% 0.32/0.65 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 0.32/0.65 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_of_nat_div_aux
% 0.32/0.65 thf(fact_7482_signed__take__bit__eq__concat__bit,axiom,
% 0.32/0.65 ( bit_ri631733984087533419it_int
% 0.32/0.65 = ( ^ [N2: nat,K2: int] : ( bit_concat_bit @ N2 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % signed_take_bit_eq_concat_bit
% 0.32/0.65 thf(fact_7483_exp__eq__0__imp__not__bit,axiom,
% 0.32/0.65 ! [N: nat,A: int] :
% 0.32/0.65 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.65 = zero_zero_int )
% 0.32/0.65 => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % exp_eq_0_imp_not_bit
% 0.32/0.65 thf(fact_7484_exp__eq__0__imp__not__bit,axiom,
% 0.32/0.65 ! [N: nat,A: nat] :
% 0.32/0.65 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.65 = zero_zero_nat )
% 0.32/0.65 => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % exp_eq_0_imp_not_bit
% 0.32/0.65 thf(fact_7485_bit__Suc,axiom,
% 0.32/0.65 ! [A: int,N: nat] :
% 0.32/0.65 ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 0.32/0.65 = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_Suc
% 0.32/0.65 thf(fact_7486_bit__Suc,axiom,
% 0.32/0.65 ! [A: nat,N: nat] :
% 0.32/0.65 ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 0.32/0.65 = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_Suc
% 0.32/0.65 thf(fact_7487_stable__imp__bit__iff__odd,axiom,
% 0.32/0.65 ! [A: code_integer,N: nat] :
% 0.32/0.65 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.65 = A )
% 0.32/0.65 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.32/0.65 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % stable_imp_bit_iff_odd
% 0.32/0.65 thf(fact_7488_stable__imp__bit__iff__odd,axiom,
% 0.32/0.65 ! [A: int,N: nat] :
% 0.32/0.65 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.65 = A )
% 0.32/0.65 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.32/0.65 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % stable_imp_bit_iff_odd
% 0.32/0.65 thf(fact_7489_stable__imp__bit__iff__odd,axiom,
% 0.32/0.65 ! [A: nat,N: nat] :
% 0.32/0.65 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.65 = A )
% 0.32/0.65 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.32/0.65 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % stable_imp_bit_iff_odd
% 0.32/0.65 thf(fact_7490_bit__iff__idd__imp__stable,axiom,
% 0.32/0.65 ! [A: code_integer] :
% 0.32/0.65 ( ! [N4: nat] :
% 0.32/0.65 ( ( bit_se9216721137139052372nteger @ A @ N4 )
% 0.32/0.65 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.32/0.65 => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.65 = A ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_iff_idd_imp_stable
% 0.32/0.65 thf(fact_7491_bit__iff__idd__imp__stable,axiom,
% 0.32/0.65 ! [A: int] :
% 0.32/0.65 ( ! [N4: nat] :
% 0.32/0.65 ( ( bit_se1146084159140164899it_int @ A @ N4 )
% 0.32/0.65 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.32/0.65 => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.65 = A ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_iff_idd_imp_stable
% 0.32/0.65 thf(fact_7492_bit__iff__idd__imp__stable,axiom,
% 0.32/0.65 ! [A: nat] :
% 0.32/0.65 ( ! [N4: nat] :
% 0.32/0.65 ( ( bit_se1148574629649215175it_nat @ A @ N4 )
% 0.32/0.65 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.32/0.65 => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.65 = A ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_iff_idd_imp_stable
% 0.32/0.65 thf(fact_7493_numeral__BitM,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
% 0.32/0.65 = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% 0.32/0.65
% 0.32/0.65 % numeral_BitM
% 0.32/0.65 thf(fact_7494_numeral__BitM,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ( ( numeral_numeral_real @ ( bitM @ N ) )
% 0.32/0.65 = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% 0.32/0.65
% 0.32/0.65 % numeral_BitM
% 0.32/0.65 thf(fact_7495_numeral__BitM,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ( ( numeral_numeral_rat @ ( bitM @ N ) )
% 0.32/0.65 = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% 0.32/0.65
% 0.32/0.65 % numeral_BitM
% 0.32/0.65 thf(fact_7496_numeral__BitM,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ( ( numeral_numeral_int @ ( bitM @ N ) )
% 0.32/0.65 = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% 0.32/0.65
% 0.32/0.65 % numeral_BitM
% 0.32/0.65 thf(fact_7497_odd__numeral__BitM,axiom,
% 0.32/0.65 ! [W: num] :
% 0.32/0.65 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % odd_numeral_BitM
% 0.32/0.65 thf(fact_7498_odd__numeral__BitM,axiom,
% 0.32/0.65 ! [W: num] :
% 0.32/0.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % odd_numeral_BitM
% 0.32/0.65 thf(fact_7499_odd__numeral__BitM,axiom,
% 0.32/0.65 ! [W: num] :
% 0.32/0.65 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % odd_numeral_BitM
% 0.32/0.65 thf(fact_7500_of__nat__less__two__power,axiom,
% 0.32/0.65 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_nat_less_two_power
% 0.32/0.65 thf(fact_7501_of__nat__less__two__power,axiom,
% 0.32/0.65 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_nat_less_two_power
% 0.32/0.65 thf(fact_7502_of__nat__less__two__power,axiom,
% 0.32/0.65 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_nat_less_two_power
% 0.32/0.65 thf(fact_7503_int__bit__bound,axiom,
% 0.32/0.65 ! [K: int] :
% 0.32/0.65 ~ ! [N4: nat] :
% 0.32/0.65 ( ! [M2: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N4 @ M2 )
% 0.32/0.65 => ( ( bit_se1146084159140164899it_int @ K @ M2 )
% 0.32/0.65 = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
% 0.32/0.65 => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 0.32/0.65 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
% 0.32/0.65 = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % int_bit_bound
% 0.32/0.65 thf(fact_7504_inverse__of__nat__le,axiom,
% 0.32/0.65 ! [N: nat,M: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.65 => ( ( N != zero_zero_nat )
% 0.32/0.65 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % inverse_of_nat_le
% 0.32/0.65 thf(fact_7505_inverse__of__nat__le,axiom,
% 0.32/0.65 ! [N: nat,M: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.65 => ( ( N != zero_zero_nat )
% 0.32/0.65 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % inverse_of_nat_le
% 0.32/0.65 thf(fact_7506_real__archimedian__rdiv__eq__0,axiom,
% 0.32/0.65 ! [X: real,C: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.32/0.65 => ( ! [M4: nat] :
% 0.32/0.65 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
% 0.32/0.65 => ( X = zero_zero_real ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_archimedian_rdiv_eq_0
% 0.32/0.65 thf(fact_7507_real__of__int__div2,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_of_int_div2
% 0.32/0.65 thf(fact_7508_real__of__int__div3,axiom,
% 0.32/0.65 ! [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 ) ).
% 0.32/0.65
% 0.32/0.65 % real_of_int_div3
% 0.32/0.65 thf(fact_7509_neg__int__cases,axiom,
% 0.32/0.65 ! [K: int] :
% 0.32/0.65 ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.65 => ~ ! [N4: nat] :
% 0.32/0.65 ( ( K
% 0.32/0.65 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 0.32/0.65 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % neg_int_cases
% 0.32/0.65 thf(fact_7510_zdiff__int__split,axiom,
% 0.32/0.65 ! [P: int > $o,X: nat,Y3: nat] :
% 0.32/0.65 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y3 ) ) )
% 0.32/0.65 = ( ( ( ord_less_eq_nat @ Y3 @ X )
% 0.32/0.65 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 0.32/0.65 & ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.65 => ( P @ zero_zero_int ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % zdiff_int_split
% 0.32/0.65 thf(fact_7511_real__of__nat__div2,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_of_nat_div2
% 0.32/0.65 thf(fact_7512_real__of__nat__div3,axiom,
% 0.32/0.65 ! [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 ) ).
% 0.32/0.65
% 0.32/0.65 % real_of_nat_div3
% 0.32/0.65 thf(fact_7513_ln__realpow,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.65 => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 0.32/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % ln_realpow
% 0.32/0.65 thf(fact_7514_bit__iff__odd,axiom,
% 0.32/0.65 ( bit_se9216721137139052372nteger
% 0.32/0.65 = ( ^ [A3: code_integer,N2: nat] :
% 0.32/0.65 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_iff_odd
% 0.32/0.65 thf(fact_7515_bit__iff__odd,axiom,
% 0.32/0.65 ( bit_se1146084159140164899it_int
% 0.32/0.65 = ( ^ [A3: int,N2: nat] :
% 0.32/0.65 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_iff_odd
% 0.32/0.65 thf(fact_7516_bit__iff__odd,axiom,
% 0.32/0.65 ( bit_se1148574629649215175it_nat
% 0.32/0.65 = ( ^ [A3: nat,N2: nat] :
% 0.32/0.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_iff_odd
% 0.32/0.65 thf(fact_7517_even__of__int__iff,axiom,
% 0.32/0.65 ! [K: int] :
% 0.32/0.65 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 0.32/0.65 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.32/0.65
% 0.32/0.65 % even_of_int_iff
% 0.32/0.65 thf(fact_7518_even__of__int__iff,axiom,
% 0.32/0.65 ! [K: int] :
% 0.32/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 0.32/0.65 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 0.32/0.65
% 0.32/0.65 % even_of_int_iff
% 0.32/0.65 thf(fact_7519_bit__int__def,axiom,
% 0.32/0.65 ( bit_se1146084159140164899it_int
% 0.32/0.65 = ( ^ [K2: int,N2: nat] :
% 0.32/0.65 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_int_def
% 0.32/0.65 thf(fact_7520_linear__plus__1__le__power,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.65 => ( 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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % linear_plus_1_le_power
% 0.32/0.65 thf(fact_7521_Bernoulli__inequality,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.65 => ( 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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % Bernoulli_inequality
% 0.32/0.65 thf(fact_7522_even__bit__succ__iff,axiom,
% 0.32/0.65 ! [A: code_integer,N: nat] :
% 0.32/0.65 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.65 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 0.32/0.65 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.32/0.65 | ( N = zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % even_bit_succ_iff
% 0.32/0.65 thf(fact_7523_even__bit__succ__iff,axiom,
% 0.32/0.65 ! [A: int,N: nat] :
% 0.32/0.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.65 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 0.32/0.65 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.32/0.65 | ( N = zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % even_bit_succ_iff
% 0.32/0.65 thf(fact_7524_even__bit__succ__iff,axiom,
% 0.32/0.65 ! [A: nat,N: nat] :
% 0.32/0.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.65 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 0.32/0.65 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.32/0.65 | ( N = zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % even_bit_succ_iff
% 0.32/0.65 thf(fact_7525_odd__bit__iff__bit__pred,axiom,
% 0.32/0.65 ! [A: code_integer,N: nat] :
% 0.32/0.65 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.32/0.65 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 0.32/0.65 = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 0.32/0.65 | ( N = zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % odd_bit_iff_bit_pred
% 0.32/0.65 thf(fact_7526_odd__bit__iff__bit__pred,axiom,
% 0.32/0.65 ! [A: int,N: nat] :
% 0.32/0.65 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.32/0.65 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 0.32/0.65 = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 0.32/0.65 | ( N = zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % odd_bit_iff_bit_pred
% 0.32/0.65 thf(fact_7527_odd__bit__iff__bit__pred,axiom,
% 0.32/0.65 ! [A: nat,N: nat] :
% 0.32/0.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.32/0.65 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 0.32/0.65 = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 0.32/0.65 | ( N = zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % odd_bit_iff_bit_pred
% 0.32/0.65 thf(fact_7528_bit__sum__mult__2__cases,axiom,
% 0.32/0.65 ! [A: code_integer,B: code_integer,N: nat] :
% 0.32/0.65 ( ! [J2: nat] :
% 0.32/0.65 ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 0.32/0.65 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.32/0.65 = ( ( ( N = zero_zero_nat )
% 0.32/0.65 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.65 & ( ( N != zero_zero_nat )
% 0.32/0.65 => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_sum_mult_2_cases
% 0.32/0.65 thf(fact_7529_bit__sum__mult__2__cases,axiom,
% 0.32/0.65 ! [A: int,B: int,N: nat] :
% 0.32/0.65 ( ! [J2: nat] :
% 0.32/0.65 ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 0.32/0.65 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.32/0.65 = ( ( ( N = zero_zero_nat )
% 0.32/0.65 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.65 & ( ( N != zero_zero_nat )
% 0.32/0.65 => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_sum_mult_2_cases
% 0.32/0.65 thf(fact_7530_bit__sum__mult__2__cases,axiom,
% 0.32/0.65 ! [A: nat,B: nat,N: nat] :
% 0.32/0.65 ( ! [J2: nat] :
% 0.32/0.65 ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 0.32/0.65 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
% 0.32/0.65 = ( ( ( N = zero_zero_nat )
% 0.32/0.65 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.32/0.65 & ( ( N != zero_zero_nat )
% 0.32/0.65 => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_sum_mult_2_cases
% 0.32/0.65 thf(fact_7531_bit__rec,axiom,
% 0.32/0.65 ( bit_se9216721137139052372nteger
% 0.32/0.65 = ( ^ [A3: code_integer,N2: nat] :
% 0.32/0.65 ( ( ( N2 = zero_zero_nat )
% 0.32/0.65 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
% 0.32/0.65 & ( ( N2 != zero_zero_nat )
% 0.32/0.65 => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_rec
% 0.32/0.65 thf(fact_7532_bit__rec,axiom,
% 0.32/0.65 ( bit_se1146084159140164899it_int
% 0.32/0.65 = ( ^ [A3: int,N2: nat] :
% 0.32/0.65 ( ( ( N2 = zero_zero_nat )
% 0.32/0.65 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
% 0.32/0.65 & ( ( N2 != zero_zero_nat )
% 0.32/0.65 => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_rec
% 0.32/0.65 thf(fact_7533_bit__rec,axiom,
% 0.32/0.65 ( bit_se1148574629649215175it_nat
% 0.32/0.65 = ( ^ [A3: nat,N2: nat] :
% 0.32/0.65 ( ( ( N2 = zero_zero_nat )
% 0.32/0.65 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
% 0.32/0.65 & ( ( N2 != zero_zero_nat )
% 0.32/0.65 => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_rec
% 0.32/0.65 thf(fact_7534_set__bit__eq,axiom,
% 0.32/0.65 ( bit_se7879613467334960850it_int
% 0.32/0.65 = ( ^ [N2: nat,K2: int] :
% 0.32/0.65 ( plus_plus_int @ K2
% 0.32/0.65 @ ( times_times_int
% 0.32/0.65 @ ( zero_n2684676970156552555ol_int
% 0.32/0.65 @ ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) )
% 0.32/0.65 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % set_bit_eq
% 0.32/0.65 thf(fact_7535_unset__bit__eq,axiom,
% 0.32/0.65 ( bit_se4203085406695923979it_int
% 0.32/0.65 = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % unset_bit_eq
% 0.32/0.65 thf(fact_7536_take__bit__Suc__from__most,axiom,
% 0.32/0.65 ! [N: nat,K: int] :
% 0.32/0.65 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 0.32/0.65 = ( 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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % take_bit_Suc_from_most
% 0.32/0.65 thf(fact_7537_Bernoulli__inequality__even,axiom,
% 0.32/0.65 ! [N: nat,X: real] :
% 0.32/0.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.65 => ( 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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % Bernoulli_inequality_even
% 0.32/0.65 thf(fact_7538_double__arith__series,axiom,
% 0.32/0.65 ! [A: int,D: int,N: nat] :
% 0.32/0.65 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.32/0.65 @ ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.32/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_arith_series
% 0.32/0.65 thf(fact_7539_double__arith__series,axiom,
% 0.32/0.65 ! [A: complex,D: complex,N: nat] :
% 0.32/0.65 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.32/0.65 @ ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 0.32/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_arith_series
% 0.32/0.65 thf(fact_7540_double__arith__series,axiom,
% 0.32/0.65 ! [A: rat,D: rat,N: nat] :
% 0.32/0.65 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.32/0.65 @ ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 0.32/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_arith_series
% 0.32/0.65 thf(fact_7541_double__arith__series,axiom,
% 0.32/0.65 ! [A: nat,D: nat,N: nat] :
% 0.32/0.65 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.32/0.65 @ ( groups3542108847815614940at_nat
% 0.32/0.65 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.32/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_arith_series
% 0.32/0.65 thf(fact_7542_double__arith__series,axiom,
% 0.32/0.65 ! [A: real,D: real,N: nat] :
% 0.32/0.65 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 0.32/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_arith_series
% 0.32/0.65 thf(fact_7543_double__gauss__sum,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum
% 0.32/0.65 thf(fact_7544_double__gauss__sum,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum
% 0.32/0.65 thf(fact_7545_double__gauss__sum,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum
% 0.32/0.65 thf(fact_7546_double__gauss__sum,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum
% 0.32/0.65 thf(fact_7547_double__gauss__sum,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum
% 0.32/0.65 thf(fact_7548_arith__series,axiom,
% 0.32/0.65 ! [A: int,D: int,N: nat] :
% 0.32/0.65 ( ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.32/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.65 = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % arith_series
% 0.32/0.65 thf(fact_7549_arith__series,axiom,
% 0.32/0.65 ! [A: nat,D: nat,N: nat] :
% 0.32/0.65 ( ( groups3542108847815614940at_nat
% 0.32/0.65 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.32/0.65 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.65 = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % arith_series
% 0.32/0.65 thf(fact_7550_gauss__sum,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.65 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % gauss_sum
% 0.32/0.65 thf(fact_7551_gauss__sum,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.32/0.65 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % gauss_sum
% 0.32/0.65 thf(fact_7552_double__gauss__sum__from__Suc__0,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.32/0.65 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum_from_Suc_0
% 0.32/0.65 thf(fact_7553_double__gauss__sum__from__Suc__0,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.32/0.65 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum_from_Suc_0
% 0.32/0.65 thf(fact_7554_double__gauss__sum__from__Suc__0,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.32/0.65 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum_from_Suc_0
% 0.32/0.65 thf(fact_7555_double__gauss__sum__from__Suc__0,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.32/0.65 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum_from_Suc_0
% 0.32/0.65 thf(fact_7556_double__gauss__sum__from__Suc__0,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.32/0.65 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % double_gauss_sum_from_Suc_0
% 0.32/0.65 thf(fact_7557_sum__gp__offset,axiom,
% 0.32/0.65 ! [X: complex,M: nat,N: nat] :
% 0.32/0.65 ( ( ( X = one_one_complex )
% 0.32/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.32/0.65 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
% 0.32/0.65 & ( ( X != one_one_complex )
% 0.32/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.32/0.65 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_gp_offset
% 0.32/0.65 thf(fact_7558_sum__gp__offset,axiom,
% 0.32/0.65 ! [X: rat,M: nat,N: nat] :
% 0.32/0.65 ( ( ( X = one_one_rat )
% 0.32/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.32/0.65 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
% 0.32/0.65 & ( ( X != one_one_rat )
% 0.32/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.32/0.65 = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_gp_offset
% 0.32/0.65 thf(fact_7559_sum__gp__offset,axiom,
% 0.32/0.65 ! [X: real,M: nat,N: nat] :
% 0.32/0.65 ( ( ( X = one_one_real )
% 0.32/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.32/0.65 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
% 0.32/0.65 & ( ( X != one_one_real )
% 0.32/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.32/0.65 = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_gp_offset
% 0.32/0.65 thf(fact_7560_of__nat__code__if,axiom,
% 0.32/0.65 ( semiri1314217659103216013at_int
% 0.32/0.65 = ( ^ [N2: nat] :
% 0.32/0.65 ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
% 0.32/0.65 @ ( produc6840382203811409530at_int
% 0.32/0.65 @ ^ [M3: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
% 0.32/0.65 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_nat_code_if
% 0.32/0.65 thf(fact_7561_of__nat__code__if,axiom,
% 0.32/0.65 ( semiri5074537144036343181t_real
% 0.32/0.65 = ( ^ [N2: nat] :
% 0.32/0.65 ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 0.32/0.65 @ ( produc1703576794950452218t_real
% 0.32/0.65 @ ^ [M3: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
% 0.32/0.65 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_nat_code_if
% 0.32/0.65 thf(fact_7562_of__nat__code__if,axiom,
% 0.32/0.65 ( semiri8010041392384452111omplex
% 0.32/0.65 = ( ^ [N2: nat] :
% 0.32/0.65 ( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
% 0.32/0.65 @ ( produc1917071388513777916omplex
% 0.32/0.65 @ ^ [M3: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
% 0.32/0.65 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_nat_code_if
% 0.32/0.65 thf(fact_7563_of__nat__code__if,axiom,
% 0.32/0.65 ( semiri681578069525770553at_rat
% 0.32/0.65 = ( ^ [N2: nat] :
% 0.32/0.65 ( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
% 0.32/0.65 @ ( produc6207742614233964070at_rat
% 0.32/0.65 @ ^ [M3: nat,Q5: nat] : ( if_rat @ ( Q5 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
% 0.32/0.65 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_nat_code_if
% 0.32/0.65 thf(fact_7564_of__nat__code__if,axiom,
% 0.32/0.65 ( semiri1316708129612266289at_nat
% 0.32/0.65 = ( ^ [N2: nat] :
% 0.32/0.65 ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 0.32/0.65 @ ( produc6842872674320459806at_nat
% 0.32/0.65 @ ^ [M3: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
% 0.32/0.65 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_nat_code_if
% 0.32/0.65 thf(fact_7565_nat__approx__posE,axiom,
% 0.32/0.65 ! [E: real] :
% 0.32/0.65 ( ( ord_less_real @ zero_zero_real @ E )
% 0.32/0.65 => ~ ! [N4: nat] :
% 0.32/0.65 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).
% 0.32/0.65
% 0.32/0.65 % nat_approx_posE
% 0.32/0.65 thf(fact_7566_nat__approx__posE,axiom,
% 0.32/0.65 ! [E: rat] :
% 0.32/0.65 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.32/0.65 => ~ ! [N4: nat] :
% 0.32/0.65 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).
% 0.32/0.65
% 0.32/0.65 % nat_approx_posE
% 0.32/0.65 thf(fact_7567_floor__exists1,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ? [X3: int] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
% 0.32/0.65 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 0.32/0.65 & ! [Y5: int] :
% 0.32/0.65 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
% 0.32/0.65 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.32/0.65 => ( Y5 = X3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % floor_exists1
% 0.32/0.65 thf(fact_7568_floor__exists1,axiom,
% 0.32/0.65 ! [X: rat] :
% 0.32/0.65 ? [X3: int] :
% 0.32/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
% 0.32/0.65 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 0.32/0.65 & ! [Y5: int] :
% 0.32/0.65 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
% 0.32/0.65 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.32/0.65 => ( Y5 = X3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % floor_exists1
% 0.32/0.65 thf(fact_7569_floor__exists,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ? [Z5: int] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
% 0.32/0.65 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % floor_exists
% 0.32/0.65 thf(fact_7570_floor__exists,axiom,
% 0.32/0.65 ! [X: rat] :
% 0.32/0.65 ? [Z5: int] :
% 0.32/0.65 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
% 0.32/0.65 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % floor_exists
% 0.32/0.65 thf(fact_7571_monoseq__arctan__series,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.65 => ( topolo6980174941875973593q_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % monoseq_arctan_series
% 0.32/0.65 thf(fact_7572_lemma__termdiff3,axiom,
% 0.32/0.65 ! [H2: real,Z: real,K5: real,N: nat] :
% 0.32/0.65 ( ( H2 != zero_zero_real )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff3
% 0.32/0.65 thf(fact_7573_lemma__termdiff3,axiom,
% 0.32/0.65 ! [H2: complex,Z: complex,K5: real,N: nat] :
% 0.32/0.65 ( ( H2 != zero_zero_complex )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff3
% 0.32/0.65 thf(fact_7574_int__if,axiom,
% 0.32/0.65 ! [P: $o,A: nat,B: nat] :
% 0.32/0.65 ( ( P
% 0.32/0.65 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 0.32/0.65 = ( semiri1314217659103216013at_int @ A ) ) )
% 0.32/0.65 & ( ~ P
% 0.32/0.65 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 0.32/0.65 = ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % int_if
% 0.32/0.65 thf(fact_7575_nat__int__comparison_I1_J,axiom,
% 0.32/0.65 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.65 = ( ^ [A3: nat,B4: nat] :
% 0.32/0.65 ( ( semiri1314217659103216013at_int @ A3 )
% 0.32/0.65 = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % nat_int_comparison(1)
% 0.32/0.65 thf(fact_7576_bit__Suc__0__iff,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.65 = ( N = zero_zero_nat ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_Suc_0_iff
% 0.32/0.65 thf(fact_7577_not__bit__Suc__0__Suc,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % not_bit_Suc_0_Suc
% 0.32/0.65 thf(fact_7578_complex__mod__minus__le__complex__mod,axiom,
% 0.32/0.65 ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.32/0.65
% 0.32/0.65 % complex_mod_minus_le_complex_mod
% 0.32/0.65 thf(fact_7579_complex__mod__triangle__ineq2,axiom,
% 0.32/0.65 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 0.32/0.65
% 0.32/0.65 % complex_mod_triangle_ineq2
% 0.32/0.65 thf(fact_7580_not__bit__Suc__0__numeral,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % not_bit_Suc_0_numeral
% 0.32/0.65 thf(fact_7581_bit__nat__def,axiom,
% 0.32/0.65 ( bit_se1148574629649215175it_nat
% 0.32/0.65 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % bit_nat_def
% 0.32/0.65 thf(fact_7582_monoseq__realpow,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.65 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.65 => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % monoseq_realpow
% 0.32/0.65 thf(fact_7583_real__arch__simple,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ? [N4: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_arch_simple
% 0.32/0.65 thf(fact_7584_real__arch__simple,axiom,
% 0.32/0.65 ! [X: rat] :
% 0.32/0.65 ? [N4: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_arch_simple
% 0.32/0.65 thf(fact_7585_reals__Archimedean2,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 0.32/0.65
% 0.32/0.65 % reals_Archimedean2
% 0.32/0.65 thf(fact_7586_reals__Archimedean2,axiom,
% 0.32/0.65 ! [X: rat] :
% 0.32/0.65 ? [N4: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 0.32/0.65
% 0.32/0.65 % reals_Archimedean2
% 0.32/0.65 thf(fact_7587_ex__le__of__int,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ? [Z5: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% 0.32/0.65
% 0.32/0.65 % ex_le_of_int
% 0.32/0.65 thf(fact_7588_ex__le__of__int,axiom,
% 0.32/0.65 ! [X: rat] :
% 0.32/0.65 ? [Z5: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).
% 0.32/0.65
% 0.32/0.65 % ex_le_of_int
% 0.32/0.65 thf(fact_7589_ex__of__int__less,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ? [Z5: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z5 ) @ X ) ).
% 0.32/0.65
% 0.32/0.65 % ex_of_int_less
% 0.32/0.65 thf(fact_7590_ex__of__int__less,axiom,
% 0.32/0.65 ! [X: rat] :
% 0.32/0.65 ? [Z5: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z5 ) @ X ) ).
% 0.32/0.65
% 0.32/0.65 % ex_of_int_less
% 0.32/0.65 thf(fact_7591_ex__less__of__int,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ? [Z5: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% 0.32/0.65
% 0.32/0.65 % ex_less_of_int
% 0.32/0.65 thf(fact_7592_ex__less__of__int,axiom,
% 0.32/0.65 ! [X: rat] :
% 0.32/0.65 ? [Z5: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).
% 0.32/0.65
% 0.32/0.65 % ex_less_of_int
% 0.32/0.65 thf(fact_7593_ex__less__of__nat__mult,axiom,
% 0.32/0.65 ! [X: real,Y3: real] :
% 0.32/0.65 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.65 => ? [N4: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % ex_less_of_nat_mult
% 0.32/0.65 thf(fact_7594_ex__less__of__nat__mult,axiom,
% 0.32/0.65 ! [X: rat,Y3: rat] :
% 0.32/0.65 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.65 => ? [N4: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % ex_less_of_nat_mult
% 0.32/0.65 thf(fact_7595_norm__divide__numeral,axiom,
% 0.32/0.65 ! [A: real,W: num] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.65 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_divide_numeral
% 0.32/0.65 thf(fact_7596_norm__divide__numeral,axiom,
% 0.32/0.65 ! [A: complex,W: num] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.65 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_divide_numeral
% 0.32/0.65 thf(fact_7597_norm__mult__numeral2,axiom,
% 0.32/0.65 ! [A: real,W: num] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.65 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult_numeral2
% 0.32/0.65 thf(fact_7598_norm__mult__numeral2,axiom,
% 0.32/0.65 ! [A: complex,W: num] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.65 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult_numeral2
% 0.32/0.65 thf(fact_7599_norm__mult__numeral1,axiom,
% 0.32/0.65 ! [W: num,A: real] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 0.32/0.65 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult_numeral1
% 0.32/0.65 thf(fact_7600_norm__mult__numeral1,axiom,
% 0.32/0.65 ! [W: num,A: complex] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 0.32/0.65 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult_numeral1
% 0.32/0.65 thf(fact_7601_norm__neg__numeral,axiom,
% 0.32/0.65 ! [W: num] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.65 = ( numeral_numeral_real @ W ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_neg_numeral
% 0.32/0.65 thf(fact_7602_norm__neg__numeral,axiom,
% 0.32/0.65 ! [W: num] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.65 = ( numeral_numeral_real @ W ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_neg_numeral
% 0.32/0.65 thf(fact_7603_norm__le__zero__iff,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 0.32/0.65 = ( X = zero_zero_real ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_le_zero_iff
% 0.32/0.65 thf(fact_7604_norm__le__zero__iff,axiom,
% 0.32/0.65 ! [X: complex] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 0.32/0.65 = ( X = zero_zero_complex ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_le_zero_iff
% 0.32/0.65 thf(fact_7605_norm__one,axiom,
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ one_one_real )
% 0.32/0.65 = one_one_real ) ).
% 0.32/0.65
% 0.32/0.65 % norm_one
% 0.32/0.65 thf(fact_7606_norm__one,axiom,
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 0.32/0.65 = one_one_real ) ).
% 0.32/0.65
% 0.32/0.65 % norm_one
% 0.32/0.65 thf(fact_7607_norm__numeral,axiom,
% 0.32/0.65 ! [W: num] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 0.32/0.65 = ( numeral_numeral_real @ W ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_numeral
% 0.32/0.65 thf(fact_7608_norm__numeral,axiom,
% 0.32/0.65 ! [W: num] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.65 = ( numeral_numeral_real @ W ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_numeral
% 0.32/0.65 thf(fact_7609_zero__less__norm__iff,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 0.32/0.65 = ( X != zero_zero_real ) ) ).
% 0.32/0.65
% 0.32/0.65 % zero_less_norm_iff
% 0.32/0.65 thf(fact_7610_zero__less__norm__iff,axiom,
% 0.32/0.65 ! [X: complex] :
% 0.32/0.65 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 0.32/0.65 = ( X != zero_zero_complex ) ) ).
% 0.32/0.65
% 0.32/0.65 % zero_less_norm_iff
% 0.32/0.65 thf(fact_7611_norm__minus__commute,axiom,
% 0.32/0.65 ! [A: real,B: real] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 0.32/0.65 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_minus_commute
% 0.32/0.65 thf(fact_7612_norm__minus__commute,axiom,
% 0.32/0.65 ! [A: complex,B: complex] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 0.32/0.65 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_minus_commute
% 0.32/0.65 thf(fact_7613_norm__not__less__zero,axiom,
% 0.32/0.65 ! [X: complex] :
% 0.32/0.65 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 0.32/0.65
% 0.32/0.65 % norm_not_less_zero
% 0.32/0.65 thf(fact_7614_norm__ge__zero,axiom,
% 0.32/0.65 ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_ge_zero
% 0.32/0.65 thf(fact_7615_norm__mult,axiom,
% 0.32/0.65 ! [X: real,Y3: real] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y3 ) )
% 0.32/0.65 = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult
% 0.32/0.65 thf(fact_7616_norm__mult,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y3 ) )
% 0.32/0.65 = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult
% 0.32/0.65 thf(fact_7617_sum__norm__le,axiom,
% 0.32/0.65 ! [S2: set_real,F: real > complex,G: real > real] :
% 0.32/0.65 ( ! [X3: real] :
% 0.32/0.65 ( ( member_real @ X3 @ S2 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_norm_le
% 0.32/0.65 thf(fact_7618_sum__norm__le,axiom,
% 0.32/0.65 ! [S2: set_int,F: int > complex,G: int > real] :
% 0.32/0.65 ( ! [X3: int] :
% 0.32/0.65 ( ( member_int @ X3 @ S2 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_norm_le
% 0.32/0.65 thf(fact_7619_sum__norm__le,axiom,
% 0.32/0.65 ! [S2: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > complex,G: produc859450856879609959at_nat > real] :
% 0.32/0.65 ( ! [X3: produc859450856879609959at_nat] :
% 0.32/0.65 ( ( member8206827879206165904at_nat @ X3 @ S2 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7963564606188603611omplex @ F @ S2 ) ) @ ( groups7817252763086508633t_real @ G @ S2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_norm_le
% 0.32/0.65 thf(fact_7620_sum__norm__le,axiom,
% 0.32/0.65 ! [S2: set_nat,F: nat > complex,G: nat > real] :
% 0.32/0.65 ( ! [X3: nat] :
% 0.32/0.65 ( ( member_nat @ X3 @ S2 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_norm_le
% 0.32/0.65 thf(fact_7621_sum__norm__le,axiom,
% 0.32/0.65 ! [S2: set_complex,F: complex > complex,G: complex > real] :
% 0.32/0.65 ( ! [X3: complex] :
% 0.32/0.65 ( ( member_complex @ X3 @ S2 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_norm_le
% 0.32/0.65 thf(fact_7622_sum__norm__le,axiom,
% 0.32/0.65 ! [S2: set_nat,F: nat > real,G: nat > real] :
% 0.32/0.65 ( ! [X3: nat] :
% 0.32/0.65 ( ( member_nat @ X3 @ S2 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_norm_le
% 0.32/0.65 thf(fact_7623_norm__divide,axiom,
% 0.32/0.65 ! [A: real,B: real] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.65 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_divide
% 0.32/0.65 thf(fact_7624_norm__divide,axiom,
% 0.32/0.65 ! [A: complex,B: complex] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.65 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_divide
% 0.32/0.65 thf(fact_7625_norm__power,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
% 0.32/0.65 = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_power
% 0.32/0.65 thf(fact_7626_norm__power,axiom,
% 0.32/0.65 ! [X: complex,N: nat] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
% 0.32/0.65 = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_power
% 0.32/0.65 thf(fact_7627_norm__sum,axiom,
% 0.32/0.65 ! [F: nat > complex,A2: set_nat] :
% 0.32/0.65 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 0.32/0.65 @ A2 ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_sum
% 0.32/0.65 thf(fact_7628_norm__sum,axiom,
% 0.32/0.65 ! [F: complex > complex,A2: set_complex] :
% 0.32/0.65 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 0.32/0.65 @ ( groups5808333547571424918x_real
% 0.32/0.65 @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 0.32/0.65 @ A2 ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_sum
% 0.32/0.65 thf(fact_7629_norm__sum,axiom,
% 0.32/0.65 ! [F: nat > real,A2: set_nat] :
% 0.32/0.65 ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
% 0.32/0.65 @ A2 ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_sum
% 0.32/0.65 thf(fact_7630_norm__uminus__minus,axiom,
% 0.32/0.65 ! [X: real,Y3: real] :
% 0.32/0.65 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y3 ) )
% 0.32/0.65 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_uminus_minus
% 0.32/0.65 thf(fact_7631_norm__uminus__minus,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex] :
% 0.32/0.65 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y3 ) )
% 0.32/0.65 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_uminus_minus
% 0.32/0.65 thf(fact_7632_nonzero__norm__divide,axiom,
% 0.32/0.65 ! [B: real,A: real] :
% 0.32/0.65 ( ( B != zero_zero_real )
% 0.32/0.65 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.32/0.65 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % nonzero_norm_divide
% 0.32/0.65 thf(fact_7633_nonzero__norm__divide,axiom,
% 0.32/0.65 ! [B: complex,A: complex] :
% 0.32/0.65 ( ( B != zero_zero_complex )
% 0.32/0.65 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.65 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % nonzero_norm_divide
% 0.32/0.65 thf(fact_7634_power__eq__imp__eq__norm,axiom,
% 0.32/0.65 ! [W: real,N: nat,Z: real] :
% 0.32/0.65 ( ( ( power_power_real @ W @ N )
% 0.32/0.65 = ( power_power_real @ Z @ N ) )
% 0.32/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.65 => ( ( real_V7735802525324610683m_real @ W )
% 0.32/0.65 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_eq_imp_eq_norm
% 0.32/0.65 thf(fact_7635_power__eq__imp__eq__norm,axiom,
% 0.32/0.65 ! [W: complex,N: nat,Z: complex] :
% 0.32/0.65 ( ( ( power_power_complex @ W @ N )
% 0.32/0.65 = ( power_power_complex @ Z @ N ) )
% 0.32/0.65 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.65 => ( ( real_V1022390504157884413omplex @ W )
% 0.32/0.65 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_eq_imp_eq_norm
% 0.32/0.65 thf(fact_7636_norm__mult__less,axiom,
% 0.32/0.65 ! [X: real,R: real,Y3: real,S: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 0.32/0.65 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y3 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult_less
% 0.32/0.65 thf(fact_7637_norm__mult__less,axiom,
% 0.32/0.65 ! [X: complex,R: real,Y3: complex,S: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 0.32/0.65 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y3 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult_less
% 0.32/0.65 thf(fact_7638_norm__mult__ineq,axiom,
% 0.32/0.65 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult_ineq
% 0.32/0.65 thf(fact_7639_norm__mult__ineq,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_mult_ineq
% 0.32/0.65 thf(fact_7640_norm__triangle__lt,axiom,
% 0.32/0.65 ! [X: real,Y3: real,E: real] :
% 0.32/0.65 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.32/0.65 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) @ E ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_lt
% 0.32/0.65 thf(fact_7641_norm__triangle__lt,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex,E: real] :
% 0.32/0.65 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.32/0.65 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) @ E ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_lt
% 0.32/0.65 thf(fact_7642_norm__add__less,axiom,
% 0.32/0.65 ! [X: real,R: real,Y3: real,S: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 0.32/0.65 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_add_less
% 0.32/0.65 thf(fact_7643_norm__add__less,axiom,
% 0.32/0.65 ! [X: complex,R: real,Y3: complex,S: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 0.32/0.65 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_add_less
% 0.32/0.65 thf(fact_7644_norm__power__ineq,axiom,
% 0.32/0.65 ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_power_ineq
% 0.32/0.65 thf(fact_7645_norm__power__ineq,axiom,
% 0.32/0.65 ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_power_ineq
% 0.32/0.65 thf(fact_7646_norm__triangle__mono,axiom,
% 0.32/0.65 ! [A: real,R: real,B: real,S: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_mono
% 0.32/0.65 thf(fact_7647_norm__triangle__mono,axiom,
% 0.32/0.65 ! [A: complex,R: real,B: complex,S: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_mono
% 0.32/0.65 thf(fact_7648_norm__triangle__ineq,axiom,
% 0.32/0.65 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_ineq
% 0.32/0.65 thf(fact_7649_norm__triangle__ineq,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_ineq
% 0.32/0.65 thf(fact_7650_norm__triangle__le,axiom,
% 0.32/0.65 ! [X: real,Y3: real,E: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) @ E ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_le
% 0.32/0.65 thf(fact_7651_norm__triangle__le,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex,E: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) @ E ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_le
% 0.32/0.65 thf(fact_7652_norm__add__leD,axiom,
% 0.32/0.65 ! [A: real,B: real,C: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_add_leD
% 0.32/0.65 thf(fact_7653_norm__add__leD,axiom,
% 0.32/0.65 ! [A: complex,B: complex,C: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_add_leD
% 0.32/0.65 thf(fact_7654_norm__diff__triangle__less,axiom,
% 0.32/0.65 ! [X: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y3 ) ) @ E1 )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.32/0.65 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_diff_triangle_less
% 0.32/0.65 thf(fact_7655_norm__diff__triangle__less,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y3 ) ) @ E1 )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.32/0.65 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_diff_triangle_less
% 0.32/0.65 thf(fact_7656_norm__triangle__sub,axiom,
% 0.32/0.65 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_sub
% 0.32/0.65 thf(fact_7657_norm__triangle__sub,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y3 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_sub
% 0.32/0.65 thf(fact_7658_norm__triangle__ineq4,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_ineq4
% 0.32/0.65 thf(fact_7659_norm__triangle__ineq4,axiom,
% 0.32/0.65 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_ineq4
% 0.32/0.65 thf(fact_7660_norm__diff__triangle__le,axiom,
% 0.32/0.65 ! [X: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y3 ) ) @ E1 )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_diff_triangle_le
% 0.32/0.65 thf(fact_7661_norm__diff__triangle__le,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y3 ) ) @ E1 )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_diff_triangle_le
% 0.32/0.65 thf(fact_7662_norm__triangle__le__diff,axiom,
% 0.32/0.65 ! [X: real,Y3: real,E: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y3 ) ) @ E ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_le_diff
% 0.32/0.65 thf(fact_7663_norm__triangle__le__diff,axiom,
% 0.32/0.65 ! [X: complex,Y3: complex,E: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y3 ) ) @ E ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_le_diff
% 0.32/0.65 thf(fact_7664_norm__diff__ineq,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_diff_ineq
% 0.32/0.65 thf(fact_7665_norm__diff__ineq,axiom,
% 0.32/0.65 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_diff_ineq
% 0.32/0.65 thf(fact_7666_norm__triangle__ineq2,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_ineq2
% 0.32/0.65 thf(fact_7667_norm__triangle__ineq2,axiom,
% 0.32/0.65 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_ineq2
% 0.32/0.65 thf(fact_7668_power__eq__1__iff,axiom,
% 0.32/0.65 ! [W: real,N: nat] :
% 0.32/0.65 ( ( ( power_power_real @ W @ N )
% 0.32/0.65 = one_one_real )
% 0.32/0.65 => ( ( ( real_V7735802525324610683m_real @ W )
% 0.32/0.65 = one_one_real )
% 0.32/0.65 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_eq_1_iff
% 0.32/0.65 thf(fact_7669_power__eq__1__iff,axiom,
% 0.32/0.65 ! [W: complex,N: nat] :
% 0.32/0.65 ( ( ( power_power_complex @ W @ N )
% 0.32/0.65 = one_one_complex )
% 0.32/0.65 => ( ( ( real_V1022390504157884413omplex @ W )
% 0.32/0.65 = one_one_real )
% 0.32/0.65 | ( N = zero_zero_nat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_eq_1_iff
% 0.32/0.65 thf(fact_7670_norm__diff__triangle__ineq,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_diff_triangle_ineq
% 0.32/0.65 thf(fact_7671_norm__diff__triangle__ineq,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_diff_triangle_ineq
% 0.32/0.65 thf(fact_7672_norm__triangle__ineq3,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_ineq3
% 0.32/0.65 thf(fact_7673_norm__triangle__ineq3,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_triangle_ineq3
% 0.32/0.65 thf(fact_7674_square__norm__one,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.65 = one_one_real )
% 0.32/0.65 => ( ( real_V7735802525324610683m_real @ X )
% 0.32/0.65 = one_one_real ) ) ).
% 0.32/0.65
% 0.32/0.65 % square_norm_one
% 0.32/0.65 thf(fact_7675_square__norm__one,axiom,
% 0.32/0.65 ! [X: complex] :
% 0.32/0.65 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.65 = one_one_complex )
% 0.32/0.65 => ( ( real_V1022390504157884413omplex @ X )
% 0.32/0.65 = one_one_real ) ) ).
% 0.32/0.65
% 0.32/0.65 % square_norm_one
% 0.32/0.65 thf(fact_7676_norm__power__diff,axiom,
% 0.32/0.65 ! [Z: real,W: real,M: nat] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 0.32/0.65 => ( 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 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_power_diff
% 0.32/0.65 thf(fact_7677_norm__power__diff,axiom,
% 0.32/0.65 ! [Z: complex,W: complex,M: nat] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 0.32/0.65 => ( 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 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % norm_power_diff
% 0.32/0.65 thf(fact_7678_ln__series,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.65 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.65 => ( ( ln_ln_real @ X )
% 0.32/0.65 = ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % ln_series
% 0.32/0.65 thf(fact_7679_arctan__series,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.65 => ( ( arctan @ X )
% 0.32/0.65 = ( suminf_real
% 0.32/0.65 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % arctan_series
% 0.32/0.65 thf(fact_7680_round__unique,axiom,
% 0.32/0.65 ! [X: real,Y3: int] :
% 0.32/0.65 ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 0.32/0.65 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.32/0.65 => ( ( archim8280529875227126926d_real @ X )
% 0.32/0.65 = Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_unique
% 0.32/0.65 thf(fact_7681_round__unique,axiom,
% 0.32/0.65 ! [X: rat,Y3: int] :
% 0.32/0.65 ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 0.32/0.65 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.65 => ( ( archim7778729529865785530nd_rat @ X )
% 0.32/0.65 = Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_unique
% 0.32/0.65 thf(fact_7682_lemma__termdiff2,axiom,
% 0.32/0.65 ! [H2: complex,Z: complex,N: nat] :
% 0.32/0.65 ( ( H2 != zero_zero_complex )
% 0.32/0.65 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.32/0.65 = ( times_times_complex @ H2
% 0.32/0.65 @ ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [P4: nat] :
% 0.32/0.65 ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q5 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff2
% 0.32/0.65 thf(fact_7683_lemma__termdiff2,axiom,
% 0.32/0.65 ! [H2: rat,Z: rat,N: nat] :
% 0.32/0.65 ( ( H2 != zero_zero_rat )
% 0.32/0.65 => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.32/0.65 = ( times_times_rat @ H2
% 0.32/0.65 @ ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [P4: nat] :
% 0.32/0.65 ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q5 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff2
% 0.32/0.65 thf(fact_7684_lemma__termdiff2,axiom,
% 0.32/0.65 ! [H2: real,Z: real,N: nat] :
% 0.32/0.65 ( ( H2 != zero_zero_real )
% 0.32/0.65 => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.32/0.65 = ( times_times_real @ H2
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [P4: nat] :
% 0.32/0.65 ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q5 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff2
% 0.32/0.65 thf(fact_7685_summable__arctan__series,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_arctan_series
% 0.32/0.65 thf(fact_7686_lessThan__iff,axiom,
% 0.32/0.65 ! [I: rat,K: rat] :
% 0.32/0.65 ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 0.32/0.65 = ( ord_less_rat @ I @ K ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_iff
% 0.32/0.65 thf(fact_7687_lessThan__iff,axiom,
% 0.32/0.65 ! [I: num,K: num] :
% 0.32/0.65 ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 0.32/0.65 = ( ord_less_num @ I @ K ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_iff
% 0.32/0.65 thf(fact_7688_lessThan__iff,axiom,
% 0.32/0.65 ! [I: nat,K: nat] :
% 0.32/0.65 ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 0.32/0.65 = ( ord_less_nat @ I @ K ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_iff
% 0.32/0.65 thf(fact_7689_lessThan__iff,axiom,
% 0.32/0.65 ! [I: int,K: int] :
% 0.32/0.65 ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 0.32/0.65 = ( ord_less_int @ I @ K ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_iff
% 0.32/0.65 thf(fact_7690_lessThan__iff,axiom,
% 0.32/0.65 ! [I: real,K: real] :
% 0.32/0.65 ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 0.32/0.65 = ( ord_less_real @ I @ K ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_iff
% 0.32/0.65 thf(fact_7691_lessThan__subset__iff,axiom,
% 0.32/0.65 ! [X: rat,Y3: rat] :
% 0.32/0.65 ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 0.32/0.65 = ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_subset_iff
% 0.32/0.65 thf(fact_7692_lessThan__subset__iff,axiom,
% 0.32/0.65 ! [X: num,Y3: num] :
% 0.32/0.65 ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y3 ) )
% 0.32/0.65 = ( ord_less_eq_num @ X @ Y3 ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_subset_iff
% 0.32/0.65 thf(fact_7693_lessThan__subset__iff,axiom,
% 0.32/0.65 ! [X: nat,Y3: nat] :
% 0.32/0.65 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 0.32/0.65 = ( ord_less_eq_nat @ X @ Y3 ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_subset_iff
% 0.32/0.65 thf(fact_7694_lessThan__subset__iff,axiom,
% 0.32/0.65 ! [X: int,Y3: int] :
% 0.32/0.65 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y3 ) )
% 0.32/0.65 = ( ord_less_eq_int @ X @ Y3 ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_subset_iff
% 0.32/0.65 thf(fact_7695_lessThan__subset__iff,axiom,
% 0.32/0.65 ! [X: real,Y3: real] :
% 0.32/0.65 ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 0.32/0.65 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_subset_iff
% 0.32/0.65 thf(fact_7696_round__numeral,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 0.32/0.65 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_numeral
% 0.32/0.65 thf(fact_7697_round__numeral,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 0.32/0.65 = ( numeral_numeral_int @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_numeral
% 0.32/0.65 thf(fact_7698_round__1,axiom,
% 0.32/0.65 ( ( archim8280529875227126926d_real @ one_one_real )
% 0.32/0.65 = one_one_int ) ).
% 0.32/0.65
% 0.32/0.65 % round_1
% 0.32/0.65 thf(fact_7699_round__1,axiom,
% 0.32/0.65 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 0.32/0.65 = one_one_int ) ).
% 0.32/0.65
% 0.32/0.65 % round_1
% 0.32/0.65 thf(fact_7700_sum_OlessThan__Suc,axiom,
% 0.32/0.65 ! [G: nat > rat,N: nat] :
% 0.32/0.65 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.lessThan_Suc
% 0.32/0.65 thf(fact_7701_sum_OlessThan__Suc,axiom,
% 0.32/0.65 ! [G: nat > int,N: nat] :
% 0.32/0.65 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.lessThan_Suc
% 0.32/0.65 thf(fact_7702_sum_OlessThan__Suc,axiom,
% 0.32/0.65 ! [G: nat > nat,N: nat] :
% 0.32/0.65 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.lessThan_Suc
% 0.32/0.65 thf(fact_7703_sum_OlessThan__Suc,axiom,
% 0.32/0.65 ! [G: nat > real,N: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.lessThan_Suc
% 0.32/0.65 thf(fact_7704_round__neg__numeral,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_neg_numeral
% 0.32/0.65 thf(fact_7705_round__neg__numeral,axiom,
% 0.32/0.65 ! [N: num] :
% 0.32/0.65 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.32/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_neg_numeral
% 0.32/0.65 thf(fact_7706_powser__zero,axiom,
% 0.32/0.65 ! [F: nat > complex] :
% 0.32/0.65 ( ( suminf_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 0.32/0.65 = ( F @ zero_zero_nat ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_zero
% 0.32/0.65 thf(fact_7707_powser__zero,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 0.32/0.65 = ( F @ zero_zero_nat ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_zero
% 0.32/0.65 thf(fact_7708_lessThan__def,axiom,
% 0.32/0.65 ( set_ord_lessThan_rat
% 0.32/0.65 = ( ^ [U3: rat] :
% 0.32/0.65 ( collect_rat
% 0.32/0.65 @ ^ [X2: rat] : ( ord_less_rat @ X2 @ U3 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_def
% 0.32/0.65 thf(fact_7709_lessThan__def,axiom,
% 0.32/0.65 ( set_ord_lessThan_num
% 0.32/0.65 = ( ^ [U3: num] :
% 0.32/0.65 ( collect_num
% 0.32/0.65 @ ^ [X2: num] : ( ord_less_num @ X2 @ U3 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_def
% 0.32/0.65 thf(fact_7710_lessThan__def,axiom,
% 0.32/0.65 ( set_ord_lessThan_nat
% 0.32/0.65 = ( ^ [U3: nat] :
% 0.32/0.65 ( collect_nat
% 0.32/0.65 @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U3 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_def
% 0.32/0.65 thf(fact_7711_lessThan__def,axiom,
% 0.32/0.65 ( set_ord_lessThan_int
% 0.32/0.65 = ( ^ [U3: int] :
% 0.32/0.65 ( collect_int
% 0.32/0.65 @ ^ [X2: int] : ( ord_less_int @ X2 @ U3 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_def
% 0.32/0.65 thf(fact_7712_lessThan__def,axiom,
% 0.32/0.65 ( set_or5984915006950818249n_real
% 0.32/0.65 = ( ^ [U3: real] :
% 0.32/0.65 ( collect_real
% 0.32/0.65 @ ^ [X2: real] : ( ord_less_real @ X2 @ U3 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_def
% 0.32/0.65 thf(fact_7713_powser__insidea,axiom,
% 0.32/0.65 ! [F: nat > real,X: real,Z: real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_insidea
% 0.32/0.65 thf(fact_7714_powser__insidea,axiom,
% 0.32/0.65 ! [F: nat > complex,X: complex,Z: complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_insidea
% 0.32/0.65 thf(fact_7715_lessThan__strict__subset__iff,axiom,
% 0.32/0.65 ! [M: rat,N: rat] :
% 0.32/0.65 ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 0.32/0.65 = ( ord_less_rat @ M @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_strict_subset_iff
% 0.32/0.65 thf(fact_7716_lessThan__strict__subset__iff,axiom,
% 0.32/0.65 ! [M: num,N: num] :
% 0.32/0.65 ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 0.32/0.65 = ( ord_less_num @ M @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_strict_subset_iff
% 0.32/0.65 thf(fact_7717_lessThan__strict__subset__iff,axiom,
% 0.32/0.65 ! [M: nat,N: nat] :
% 0.32/0.65 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( ord_less_nat @ M @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_strict_subset_iff
% 0.32/0.65 thf(fact_7718_lessThan__strict__subset__iff,axiom,
% 0.32/0.65 ! [M: int,N: int] :
% 0.32/0.65 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 0.32/0.65 = ( ord_less_int @ M @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_strict_subset_iff
% 0.32/0.65 thf(fact_7719_lessThan__strict__subset__iff,axiom,
% 0.32/0.65 ! [M: real,N: real] :
% 0.32/0.65 ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 0.32/0.65 = ( ord_less_real @ M @ N ) ) ).
% 0.32/0.65
% 0.32/0.65 % lessThan_strict_subset_iff
% 0.32/0.65 thf(fact_7720_sum__pos__lt__pair,axiom,
% 0.32/0.65 ! [F: nat > real,K: nat] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
% 0.32/0.65 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_pos_lt_pair
% 0.32/0.65 thf(fact_7721_finite__nat__iff__bounded,axiom,
% 0.32/0.65 ( finite_finite_nat
% 0.32/0.65 = ( ^ [S5: set_nat] :
% 0.32/0.65 ? [K2: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % finite_nat_iff_bounded
% 0.32/0.65 thf(fact_7722_finite__nat__bounded,axiom,
% 0.32/0.65 ! [S2: set_nat] :
% 0.32/0.65 ( ( finite_finite_nat @ S2 )
% 0.32/0.65 => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % finite_nat_bounded
% 0.32/0.65 thf(fact_7723_round__mono,axiom,
% 0.32/0.65 ! [X: rat,Y3: rat] :
% 0.32/0.65 ( ( ord_less_eq_rat @ X @ Y3 )
% 0.32/0.65 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_mono
% 0.32/0.65 thf(fact_7724_sum_Onat__diff__reindex,axiom,
% 0.32/0.65 ! [G: nat > nat,N: nat] :
% 0.32/0.65 ( ( groups3542108847815614940at_nat
% 0.32/0.65 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.nat_diff_reindex
% 0.32/0.65 thf(fact_7725_sum_Onat__diff__reindex,axiom,
% 0.32/0.65 ! [G: nat > real,N: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.nat_diff_reindex
% 0.32/0.65 thf(fact_7726_sum__diff__distrib,axiom,
% 0.32/0.65 ! [Q: int > nat,P: int > nat,N: int] :
% 0.32/0.65 ( ! [X3: int] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
% 0.32/0.65 => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 0.32/0.65 = ( groups4541462559716669496nt_nat
% 0.32/0.65 @ ^ [X2: int] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 0.32/0.65 @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_diff_distrib
% 0.32/0.65 thf(fact_7727_sum__diff__distrib,axiom,
% 0.32/0.65 ! [Q: real > nat,P: real > nat,N: real] :
% 0.32/0.65 ( ! [X3: real] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
% 0.32/0.65 => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 0.32/0.65 = ( groups1935376822645274424al_nat
% 0.32/0.65 @ ^ [X2: real] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 0.32/0.65 @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_diff_distrib
% 0.32/0.65 thf(fact_7728_sum__diff__distrib,axiom,
% 0.32/0.65 ! [Q: nat > nat,P: nat > nat,N: nat] :
% 0.32/0.65 ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
% 0.32/0.65 => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 0.32/0.65 = ( groups3542108847815614940at_nat
% 0.32/0.65 @ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_diff_distrib
% 0.32/0.65 thf(fact_7729_powser__inside,axiom,
% 0.32/0.65 ! [F: nat > real,X: real,Z: real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_inside
% 0.32/0.65 thf(fact_7730_powser__inside,axiom,
% 0.32/0.65 ! [F: nat > complex,X: complex,Z: complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 0.32/0.65 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
% 0.32/0.65 => ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_inside
% 0.32/0.65 thf(fact_7731_sum_OlessThan__Suc__shift,axiom,
% 0.32/0.65 ! [G: nat > rat,N: nat] :
% 0.32/0.65 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.32/0.65 @ ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.lessThan_Suc_shift
% 0.32/0.65 thf(fact_7732_sum_OlessThan__Suc__shift,axiom,
% 0.32/0.65 ! [G: nat > int,N: nat] :
% 0.32/0.65 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.32/0.65 @ ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.lessThan_Suc_shift
% 0.32/0.65 thf(fact_7733_sum_OlessThan__Suc__shift,axiom,
% 0.32/0.65 ! [G: nat > nat,N: nat] :
% 0.32/0.65 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.32/0.65 @ ( groups3542108847815614940at_nat
% 0.32/0.65 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.lessThan_Suc_shift
% 0.32/0.65 thf(fact_7734_sum_OlessThan__Suc__shift,axiom,
% 0.32/0.65 ! [G: nat > real,N: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum.lessThan_Suc_shift
% 0.32/0.65 thf(fact_7735_sumr__diff__mult__const2,axiom,
% 0.32/0.65 ! [F: nat > int,N: nat,R: int] :
% 0.32/0.65 ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R ) )
% 0.32/0.65 = ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sumr_diff_mult_const2
% 0.32/0.65 thf(fact_7736_sumr__diff__mult__const2,axiom,
% 0.32/0.65 ! [F: nat > complex,N: nat,R: complex] :
% 0.32/0.65 ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ R ) )
% 0.32/0.65 = ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ I4 ) @ R )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sumr_diff_mult_const2
% 0.32/0.65 thf(fact_7737_sumr__diff__mult__const2,axiom,
% 0.32/0.65 ! [F: nat > rat,N: nat,R: rat] :
% 0.32/0.65 ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R ) )
% 0.32/0.65 = ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ R )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sumr_diff_mult_const2
% 0.32/0.65 thf(fact_7738_sumr__diff__mult__const2,axiom,
% 0.32/0.65 ! [F: nat > real,N: nat,R: real] :
% 0.32/0.65 ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R ) )
% 0.32/0.65 = ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sumr_diff_mult_const2
% 0.32/0.65 thf(fact_7739_sum__lessThan__telescope_H,axiom,
% 0.32/0.65 ! [F: nat > rat,M: nat] :
% 0.32/0.65 ( ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_lessThan_telescope'
% 0.32/0.65 thf(fact_7740_sum__lessThan__telescope_H,axiom,
% 0.32/0.65 ! [F: nat > int,M: nat] :
% 0.32/0.65 ( ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_lessThan_telescope'
% 0.32/0.65 thf(fact_7741_sum__lessThan__telescope_H,axiom,
% 0.32/0.65 ! [F: nat > real,M: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_lessThan_telescope'
% 0.32/0.65 thf(fact_7742_sum__lessThan__telescope,axiom,
% 0.32/0.65 ! [F: nat > rat,M: nat] :
% 0.32/0.65 ( ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_lessThan_telescope
% 0.32/0.65 thf(fact_7743_sum__lessThan__telescope,axiom,
% 0.32/0.65 ! [F: nat > int,M: nat] :
% 0.32/0.65 ( ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_lessThan_telescope
% 0.32/0.65 thf(fact_7744_sum__lessThan__telescope,axiom,
% 0.32/0.65 ! [F: nat > real,M: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_lessThan_telescope
% 0.32/0.65 thf(fact_7745_one__diff__power__eq,axiom,
% 0.32/0.65 ! [X: complex,N: nat] :
% 0.32/0.65 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
% 0.32/0.65 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % one_diff_power_eq
% 0.32/0.65 thf(fact_7746_one__diff__power__eq,axiom,
% 0.32/0.65 ! [X: rat,N: nat] :
% 0.32/0.65 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
% 0.32/0.65 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % one_diff_power_eq
% 0.32/0.65 thf(fact_7747_one__diff__power__eq,axiom,
% 0.32/0.65 ! [X: int,N: nat] :
% 0.32/0.65 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
% 0.32/0.65 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % one_diff_power_eq
% 0.32/0.65 thf(fact_7748_one__diff__power__eq,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
% 0.32/0.65 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % one_diff_power_eq
% 0.32/0.65 thf(fact_7749_power__diff__1__eq,axiom,
% 0.32/0.65 ! [X: complex,N: nat] :
% 0.32/0.65 ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex )
% 0.32/0.65 = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_diff_1_eq
% 0.32/0.65 thf(fact_7750_power__diff__1__eq,axiom,
% 0.32/0.65 ! [X: rat,N: nat] :
% 0.32/0.65 ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
% 0.32/0.65 = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_diff_1_eq
% 0.32/0.65 thf(fact_7751_power__diff__1__eq,axiom,
% 0.32/0.65 ! [X: int,N: nat] :
% 0.32/0.65 ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
% 0.32/0.65 = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_diff_1_eq
% 0.32/0.65 thf(fact_7752_power__diff__1__eq,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real )
% 0.32/0.65 = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_diff_1_eq
% 0.32/0.65 thf(fact_7753_geometric__sum,axiom,
% 0.32/0.65 ! [X: complex,N: nat] :
% 0.32/0.65 ( ( X != one_one_complex )
% 0.32/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % geometric_sum
% 0.32/0.65 thf(fact_7754_geometric__sum,axiom,
% 0.32/0.65 ! [X: rat,N: nat] :
% 0.32/0.65 ( ( X != one_one_rat )
% 0.32/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % geometric_sum
% 0.32/0.65 thf(fact_7755_geometric__sum,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( X != one_one_real )
% 0.32/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % geometric_sum
% 0.32/0.65 thf(fact_7756_round__diff__minimal,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_diff_minimal
% 0.32/0.65 thf(fact_7757_round__diff__minimal,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_diff_minimal
% 0.32/0.65 thf(fact_7758_sum__gp__strict,axiom,
% 0.32/0.65 ! [X: complex,N: nat] :
% 0.32/0.65 ( ( ( X = one_one_complex )
% 0.32/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( semiri8010041392384452111omplex @ N ) ) )
% 0.32/0.65 & ( ( X != one_one_complex )
% 0.32/0.65 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_gp_strict
% 0.32/0.65 thf(fact_7759_sum__gp__strict,axiom,
% 0.32/0.65 ! [X: rat,N: nat] :
% 0.32/0.65 ( ( ( X = one_one_rat )
% 0.32/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( semiri681578069525770553at_rat @ N ) ) )
% 0.32/0.65 & ( ( X != one_one_rat )
% 0.32/0.65 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_gp_strict
% 0.32/0.65 thf(fact_7760_sum__gp__strict,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( ( X = one_one_real )
% 0.32/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( semiri5074537144036343181t_real @ N ) ) )
% 0.32/0.65 & ( ( X != one_one_real )
% 0.32/0.65 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_gp_strict
% 0.32/0.65 thf(fact_7761_lemma__termdiff1,axiom,
% 0.32/0.65 ! [Z: complex,H2: complex,M: nat] :
% 0.32/0.65 ( ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [P4: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ P4 ) ) @ ( power_power_complex @ Z @ M ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P4 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff1
% 0.32/0.65 thf(fact_7762_lemma__termdiff1,axiom,
% 0.32/0.65 ! [Z: rat,H2: rat,M: nat] :
% 0.32/0.65 ( ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [P4: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ P4 ) ) @ ( power_power_rat @ Z @ M ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P4 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff1
% 0.32/0.65 thf(fact_7763_lemma__termdiff1,axiom,
% 0.32/0.65 ! [Z: int,H2: int,M: nat] :
% 0.32/0.65 ( ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [P4: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ P4 ) ) @ ( power_power_int @ Z @ M ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ Z @ P4 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff1
% 0.32/0.65 thf(fact_7764_lemma__termdiff1,axiom,
% 0.32/0.65 ! [Z: real,H2: real,M: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [P4: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ P4 ) ) @ ( power_power_real @ Z @ M ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) )
% 0.32/0.65 = ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ Z @ P4 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % lemma_termdiff1
% 0.32/0.65 thf(fact_7765_diff__power__eq__sum,axiom,
% 0.32/0.65 ! [X: complex,N: nat,Y3: complex] :
% 0.32/0.65 ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
% 0.32/0.65 = ( times_times_complex @ ( minus_minus_complex @ X @ Y3 )
% 0.32/0.65 @ ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ X @ P4 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % diff_power_eq_sum
% 0.32/0.65 thf(fact_7766_diff__power__eq__sum,axiom,
% 0.32/0.65 ! [X: rat,N: nat,Y3: rat] :
% 0.32/0.65 ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
% 0.32/0.65 = ( times_times_rat @ ( minus_minus_rat @ X @ Y3 )
% 0.32/0.65 @ ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ X @ P4 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % diff_power_eq_sum
% 0.32/0.65 thf(fact_7767_diff__power__eq__sum,axiom,
% 0.32/0.65 ! [X: int,N: nat,Y3: int] :
% 0.32/0.65 ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
% 0.32/0.65 = ( times_times_int @ ( minus_minus_int @ X @ Y3 )
% 0.32/0.65 @ ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ X @ P4 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % diff_power_eq_sum
% 0.32/0.65 thf(fact_7768_diff__power__eq__sum,axiom,
% 0.32/0.65 ! [X: real,N: nat,Y3: real] :
% 0.32/0.65 ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
% 0.32/0.65 = ( times_times_real @ ( minus_minus_real @ X @ Y3 )
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ X @ P4 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % diff_power_eq_sum
% 0.32/0.65 thf(fact_7769_power__diff__sumr2,axiom,
% 0.32/0.65 ! [X: complex,N: nat,Y3: complex] :
% 0.32/0.65 ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 0.32/0.65 = ( times_times_complex @ ( minus_minus_complex @ X @ Y3 )
% 0.32/0.65 @ ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_diff_sumr2
% 0.32/0.65 thf(fact_7770_power__diff__sumr2,axiom,
% 0.32/0.65 ! [X: rat,N: nat,Y3: rat] :
% 0.32/0.65 ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 0.32/0.65 = ( times_times_rat @ ( minus_minus_rat @ X @ Y3 )
% 0.32/0.65 @ ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_diff_sumr2
% 0.32/0.65 thf(fact_7771_power__diff__sumr2,axiom,
% 0.32/0.65 ! [X: int,N: nat,Y3: int] :
% 0.32/0.65 ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) )
% 0.32/0.65 = ( times_times_int @ ( minus_minus_int @ X @ Y3 )
% 0.32/0.65 @ ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_diff_sumr2
% 0.32/0.65 thf(fact_7772_power__diff__sumr2,axiom,
% 0.32/0.65 ! [X: real,N: nat,Y3: real] :
% 0.32/0.65 ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y3 @ N ) )
% 0.32/0.65 = ( times_times_real @ ( minus_minus_real @ X @ Y3 )
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % power_diff_sumr2
% 0.32/0.65 thf(fact_7773_real__sum__nat__ivl__bounded2,axiom,
% 0.32/0.65 ! [N: nat,F: nat > rat,K5: rat,K: nat] :
% 0.32/0.65 ( ! [P7: nat] :
% 0.32/0.65 ( ( ord_less_nat @ P7 @ N )
% 0.32/0.65 => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 0.32/0.65 => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 0.32/0.65 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_sum_nat_ivl_bounded2
% 0.32/0.65 thf(fact_7774_real__sum__nat__ivl__bounded2,axiom,
% 0.32/0.65 ! [N: nat,F: nat > int,K5: int,K: nat] :
% 0.32/0.65 ( ! [P7: nat] :
% 0.32/0.65 ( ( ord_less_nat @ P7 @ N )
% 0.32/0.65 => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 0.32/0.65 => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 0.32/0.65 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_sum_nat_ivl_bounded2
% 0.32/0.65 thf(fact_7775_real__sum__nat__ivl__bounded2,axiom,
% 0.32/0.65 ! [N: nat,F: nat > nat,K5: nat,K: nat] :
% 0.32/0.65 ( ! [P7: nat] :
% 0.32/0.65 ( ( ord_less_nat @ P7 @ N )
% 0.32/0.65 => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 0.32/0.65 => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 0.32/0.65 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_sum_nat_ivl_bounded2
% 0.32/0.65 thf(fact_7776_real__sum__nat__ivl__bounded2,axiom,
% 0.32/0.65 ! [N: nat,F: nat > real,K5: real,K: nat] :
% 0.32/0.65 ( ! [P7: nat] :
% 0.32/0.65 ( ( ord_less_nat @ P7 @ N )
% 0.32/0.65 => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 0.32/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 0.32/0.65 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % real_sum_nat_ivl_bounded2
% 0.32/0.65 thf(fact_7777_one__diff__power__eq_H,axiom,
% 0.32/0.65 ! [X: complex,N: nat] :
% 0.32/0.65 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
% 0.32/0.65 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
% 0.32/0.65 @ ( groups2073611262835488442omplex
% 0.32/0.65 @ ^ [I4: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % one_diff_power_eq'
% 0.32/0.65 thf(fact_7778_one__diff__power__eq_H,axiom,
% 0.32/0.65 ! [X: rat,N: nat] :
% 0.32/0.65 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
% 0.32/0.65 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
% 0.32/0.65 @ ( groups2906978787729119204at_rat
% 0.32/0.65 @ ^ [I4: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % one_diff_power_eq'
% 0.32/0.65 thf(fact_7779_one__diff__power__eq_H,axiom,
% 0.32/0.65 ! [X: int,N: nat] :
% 0.32/0.65 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
% 0.32/0.65 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
% 0.32/0.65 @ ( groups3539618377306564664at_int
% 0.32/0.65 @ ^ [I4: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % one_diff_power_eq'
% 0.32/0.65 thf(fact_7780_one__diff__power__eq_H,axiom,
% 0.32/0.65 ! [X: real,N: nat] :
% 0.32/0.65 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
% 0.32/0.65 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % one_diff_power_eq'
% 0.32/0.65 thf(fact_7781_sum__split__even__odd,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real,N: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.65 = ( plus_plus_real
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_split_even_odd
% 0.32/0.65 thf(fact_7782_of__int__round__le,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_int_round_le
% 0.32/0.65 thf(fact_7783_of__int__round__le,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_int_round_le
% 0.32/0.65 thf(fact_7784_of__int__round__ge,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_int_round_ge
% 0.32/0.65 thf(fact_7785_of__int__round__ge,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_int_round_ge
% 0.32/0.65 thf(fact_7786_of__int__round__gt,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_int_round_gt
% 0.32/0.65 thf(fact_7787_of__int__round__gt,axiom,
% 0.32/0.65 ! [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 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_int_round_gt
% 0.32/0.65 thf(fact_7788_of__int__round__abs__le,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_int_round_abs_le
% 0.32/0.65 thf(fact_7789_of__int__round__abs__le,axiom,
% 0.32/0.65 ! [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 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % of_int_round_abs_le
% 0.32/0.65 thf(fact_7790_round__unique_H,axiom,
% 0.32/0.65 ! [X: real,N: int] :
% 0.32/0.65 ( ( 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 ) ) ) )
% 0.32/0.65 => ( ( archim8280529875227126926d_real @ X )
% 0.32/0.65 = N ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_unique'
% 0.32/0.65 thf(fact_7791_round__unique_H,axiom,
% 0.32/0.65 ! [X: rat,N: int] :
% 0.32/0.65 ( ( 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 ) ) ) )
% 0.32/0.65 => ( ( archim7778729529865785530nd_rat @ X )
% 0.32/0.65 = N ) ) ).
% 0.32/0.65
% 0.32/0.65 % round_unique'
% 0.32/0.65 thf(fact_7792_summable__geometric__iff,axiom,
% 0.32/0.65 ! [C: real] :
% 0.32/0.65 ( ( summable_real @ ( power_power_real @ C ) )
% 0.32/0.65 = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_geometric_iff
% 0.32/0.65 thf(fact_7793_summable__geometric__iff,axiom,
% 0.32/0.65 ! [C: complex] :
% 0.32/0.65 ( ( summable_complex @ ( power_power_complex @ C ) )
% 0.32/0.65 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_geometric_iff
% 0.32/0.65 thf(fact_7794_summable__divide__iff,axiom,
% 0.32/0.65 ! [F: nat > complex,C: complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 0.32/0.65 = ( ( C = zero_zero_complex )
% 0.32/0.65 | ( summable_complex @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_divide_iff
% 0.32/0.65 thf(fact_7795_summable__divide__iff,axiom,
% 0.32/0.65 ! [F: nat > real,C: real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 0.32/0.65 = ( ( C = zero_zero_real )
% 0.32/0.65 | ( summable_real @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_divide_iff
% 0.32/0.65 thf(fact_7796_suminf__geometric,axiom,
% 0.32/0.65 ! [C: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.32/0.65 => ( ( suminf_real @ ( power_power_real @ C ) )
% 0.32/0.65 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_geometric
% 0.32/0.65 thf(fact_7797_suminf__geometric,axiom,
% 0.32/0.65 ! [C: complex] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.32/0.65 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 0.32/0.65 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_geometric
% 0.32/0.65 thf(fact_7798_summable__cmult__iff,axiom,
% 0.32/0.65 ! [C: complex,F: nat > complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 0.32/0.65 = ( ( C = zero_zero_complex )
% 0.32/0.65 | ( summable_complex @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_cmult_iff
% 0.32/0.65 thf(fact_7799_summable__cmult__iff,axiom,
% 0.32/0.65 ! [C: real,F: nat > real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.32/0.65 = ( ( C = zero_zero_real )
% 0.32/0.65 | ( summable_real @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_cmult_iff
% 0.32/0.65 thf(fact_7800_sum__less__suminf2,axiom,
% 0.32/0.65 ! [F: nat > int,N: nat,I: nat] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ! [M4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N @ M4 )
% 0.32/0.65 => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
% 0.32/0.65 => ( ( ord_less_eq_nat @ N @ I )
% 0.32/0.65 => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 0.32/0.65 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_less_suminf2
% 0.32/0.65 thf(fact_7801_sum__less__suminf2,axiom,
% 0.32/0.65 ! [F: nat > nat,N: nat,I: nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ! [M4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N @ M4 )
% 0.32/0.65 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 0.32/0.65 => ( ( ord_less_eq_nat @ N @ I )
% 0.32/0.65 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.32/0.65 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_less_suminf2
% 0.32/0.65 thf(fact_7802_sum__less__suminf2,axiom,
% 0.32/0.65 ! [F: nat > real,N: nat,I: nat] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [M4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N @ M4 )
% 0.32/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
% 0.32/0.65 => ( ( ord_less_eq_nat @ N @ I )
% 0.32/0.65 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.32/0.65 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_less_suminf2
% 0.32/0.65 thf(fact_7803_summable__iff__shift,axiom,
% 0.32/0.65 ! [F: nat > real,K: nat] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.32/0.65 = ( summable_real @ F ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_iff_shift
% 0.32/0.65 thf(fact_7804_summable__mult2,axiom,
% 0.32/0.65 ! [F: nat > real,C: real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_mult2
% 0.32/0.65 thf(fact_7805_summable__mult,axiom,
% 0.32/0.65 ! [F: nat > real,C: real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_mult
% 0.32/0.65 thf(fact_7806_summable__add,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_add
% 0.32/0.65 thf(fact_7807_summable__add,axiom,
% 0.32/0.65 ! [F: nat > nat,G: nat > nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ( summable_nat @ G )
% 0.32/0.65 => ( summable_nat
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_add
% 0.32/0.65 thf(fact_7808_summable__add,axiom,
% 0.32/0.65 ! [F: nat > int,G: nat > int] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ( summable_int @ G )
% 0.32/0.65 => ( summable_int
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_add
% 0.32/0.65 thf(fact_7809_summable__diff,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_diff
% 0.32/0.65 thf(fact_7810_summable__divide,axiom,
% 0.32/0.65 ! [F: nat > complex,C: complex] :
% 0.32/0.65 ( ( summable_complex @ F )
% 0.32/0.65 => ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_divide
% 0.32/0.65 thf(fact_7811_summable__divide,axiom,
% 0.32/0.65 ! [F: nat > real,C: real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_divide
% 0.32/0.65 thf(fact_7812_summable__ignore__initial__segment,axiom,
% 0.32/0.65 ! [F: nat > real,K: nat] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_ignore_initial_segment
% 0.32/0.65 thf(fact_7813_suminf__le,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.32/0.65 => ( ( summable_real @ F )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_le
% 0.32/0.65 thf(fact_7814_suminf__le,axiom,
% 0.32/0.65 ! [F: nat > nat,G: nat > nat] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.32/0.65 => ( ( summable_nat @ F )
% 0.32/0.65 => ( ( summable_nat @ G )
% 0.32/0.65 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_le
% 0.32/0.65 thf(fact_7815_suminf__le,axiom,
% 0.32/0.65 ! [F: nat > int,G: nat > int] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.32/0.65 => ( ( summable_int @ F )
% 0.32/0.65 => ( ( summable_int @ G )
% 0.32/0.65 => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_le
% 0.32/0.65 thf(fact_7816_summable__mult__D,axiom,
% 0.32/0.65 ! [C: complex,F: nat > complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 0.32/0.65 => ( ( C != zero_zero_complex )
% 0.32/0.65 => ( summable_complex @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_mult_D
% 0.32/0.65 thf(fact_7817_summable__mult__D,axiom,
% 0.32/0.65 ! [C: real,F: nat > real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.32/0.65 => ( ( C != zero_zero_real )
% 0.32/0.65 => ( summable_real @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_mult_D
% 0.32/0.65 thf(fact_7818_summable__zero__power,axiom,
% 0.32/0.65 summable_real @ ( power_power_real @ zero_zero_real ) ).
% 0.32/0.65
% 0.32/0.65 % summable_zero_power
% 0.32/0.65 thf(fact_7819_summable__zero__power,axiom,
% 0.32/0.65 summable_int @ ( power_power_int @ zero_zero_int ) ).
% 0.32/0.65
% 0.32/0.65 % summable_zero_power
% 0.32/0.65 thf(fact_7820_summable__zero__power,axiom,
% 0.32/0.65 summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 0.32/0.65
% 0.32/0.65 % summable_zero_power
% 0.32/0.65 thf(fact_7821_suminf__mult2,axiom,
% 0.32/0.65 ! [F: nat > real,C: real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( times_times_real @ ( suminf_real @ F ) @ C )
% 0.32/0.65 = ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_mult2
% 0.32/0.65 thf(fact_7822_suminf__mult,axiom,
% 0.32/0.65 ! [F: nat > real,C: real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.32/0.65 = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_mult
% 0.32/0.65 thf(fact_7823_suminf__add,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 0.32/0.65 = ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_add
% 0.32/0.65 thf(fact_7824_suminf__add,axiom,
% 0.32/0.65 ! [F: nat > nat,G: nat > nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ( summable_nat @ G )
% 0.32/0.65 => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 0.32/0.65 = ( suminf_nat
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_add
% 0.32/0.65 thf(fact_7825_suminf__add,axiom,
% 0.32/0.65 ! [F: nat > int,G: nat > int] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ( summable_int @ G )
% 0.32/0.65 => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 0.32/0.65 = ( suminf_int
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_add
% 0.32/0.65 thf(fact_7826_suminf__diff,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 0.32/0.65 = ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_diff
% 0.32/0.65 thf(fact_7827_suminf__divide,axiom,
% 0.32/0.65 ! [F: nat > complex,C: complex] :
% 0.32/0.65 ( ( summable_complex @ F )
% 0.32/0.65 => ( ( suminf_complex
% 0.32/0.65 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 0.32/0.65 = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_divide
% 0.32/0.65 thf(fact_7828_suminf__divide,axiom,
% 0.32/0.65 ! [F: nat > real,C: real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 0.32/0.65 = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_divide
% 0.32/0.65 thf(fact_7829_suminf__eq__zero__iff,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ( suminf_real @ F )
% 0.32/0.65 = zero_zero_real )
% 0.32/0.65 = ( ! [N2: nat] :
% 0.32/0.65 ( ( F @ N2 )
% 0.32/0.65 = zero_zero_real ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_eq_zero_iff
% 0.32/0.65 thf(fact_7830_suminf__eq__zero__iff,axiom,
% 0.32/0.65 ! [F: nat > nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ( suminf_nat @ F )
% 0.32/0.65 = zero_zero_nat )
% 0.32/0.65 = ( ! [N2: nat] :
% 0.32/0.65 ( ( F @ N2 )
% 0.32/0.65 = zero_zero_nat ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_eq_zero_iff
% 0.32/0.65 thf(fact_7831_suminf__eq__zero__iff,axiom,
% 0.32/0.65 ! [F: nat > int] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ( suminf_int @ F )
% 0.32/0.65 = zero_zero_int )
% 0.32/0.65 = ( ! [N2: nat] :
% 0.32/0.65 ( ( F @ N2 )
% 0.32/0.65 = zero_zero_int ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_eq_zero_iff
% 0.32/0.65 thf(fact_7832_suminf__nonneg,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.32/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_nonneg
% 0.32/0.65 thf(fact_7833_suminf__nonneg,axiom,
% 0.32/0.65 ! [F: nat > nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.32/0.65 => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_nonneg
% 0.32/0.65 thf(fact_7834_suminf__nonneg,axiom,
% 0.32/0.65 ! [F: nat > int] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.32/0.65 => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_nonneg
% 0.32/0.65 thf(fact_7835_suminf__pos,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N4 ) )
% 0.32/0.65 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos
% 0.32/0.65 thf(fact_7836_suminf__pos,axiom,
% 0.32/0.65 ! [F: nat > nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.32/0.65 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos
% 0.32/0.65 thf(fact_7837_suminf__pos,axiom,
% 0.32/0.65 ! [F: nat > int] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N4 ) )
% 0.32/0.65 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos
% 0.32/0.65 thf(fact_7838_summable__zero__power_H,axiom,
% 0.32/0.65 ! [F: nat > complex] :
% 0.32/0.65 ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_zero_power'
% 0.32/0.65 thf(fact_7839_summable__zero__power_H,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_zero_power'
% 0.32/0.65 thf(fact_7840_summable__zero__power_H,axiom,
% 0.32/0.65 ! [F: nat > int] :
% 0.32/0.65 ( summable_int
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_zero_power'
% 0.32/0.65 thf(fact_7841_summable__0__powser,axiom,
% 0.32/0.65 ! [F: nat > complex] :
% 0.32/0.65 ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_0_powser
% 0.32/0.65 thf(fact_7842_summable__0__powser,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_0_powser
% 0.32/0.65 thf(fact_7843_powser__split__head_I3_J,axiom,
% 0.32/0.65 ! [F: nat > complex,Z: complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 => ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_split_head(3)
% 0.32/0.65 thf(fact_7844_powser__split__head_I3_J,axiom,
% 0.32/0.65 ! [F: nat > real,Z: real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_split_head(3)
% 0.32/0.65 thf(fact_7845_summable__powser__split__head,axiom,
% 0.32/0.65 ! [F: nat > complex,Z: complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 = ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_powser_split_head
% 0.32/0.65 thf(fact_7846_summable__powser__split__head,axiom,
% 0.32/0.65 ! [F: nat > real,Z: real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 = ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_powser_split_head
% 0.32/0.65 thf(fact_7847_summable__powser__ignore__initial__segment,axiom,
% 0.32/0.65 ! [F: nat > complex,M: nat,Z: complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 = ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_powser_ignore_initial_segment
% 0.32/0.65 thf(fact_7848_summable__powser__ignore__initial__segment,axiom,
% 0.32/0.65 ! [F: nat > real,M: nat,Z: real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 = ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_powser_ignore_initial_segment
% 0.32/0.65 thf(fact_7849_summable__norm__comparison__test,axiom,
% 0.32/0.65 ! [F: nat > complex,G: nat > real] :
% 0.32/0.65 ( ? [N8: nat] :
% 0.32/0.65 ! [N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N8 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_norm_comparison_test
% 0.32/0.65 thf(fact_7850_summable__rabs__comparison__test,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real] :
% 0.32/0.65 ( ? [N8: nat] :
% 0.32/0.65 ! [N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N8 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_rabs_comparison_test
% 0.32/0.65 thf(fact_7851_summable__rabs,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 0.32/0.65 @ ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_rabs
% 0.32/0.65 thf(fact_7852_suminf__pos__iff,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 0.32/0.65 = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos_iff
% 0.32/0.65 thf(fact_7853_suminf__pos__iff,axiom,
% 0.32/0.65 ! [F: nat > nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 0.32/0.65 = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos_iff
% 0.32/0.65 thf(fact_7854_suminf__pos__iff,axiom,
% 0.32/0.65 ! [F: nat > int] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 0.32/0.65 = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos_iff
% 0.32/0.65 thf(fact_7855_suminf__pos2,axiom,
% 0.32/0.65 ! [F: nat > real,I: nat] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.32/0.65 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos2
% 0.32/0.65 thf(fact_7856_suminf__pos2,axiom,
% 0.32/0.65 ! [F: nat > nat,I: nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.32/0.65 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos2
% 0.32/0.65 thf(fact_7857_suminf__pos2,axiom,
% 0.32/0.65 ! [F: nat > int,I: nat] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.32/0.65 => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 0.32/0.65 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_pos2
% 0.32/0.65 thf(fact_7858_suminf__le__const,axiom,
% 0.32/0.65 ! [F: nat > int,X: int] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.32/0.65 => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_le_const
% 0.32/0.65 thf(fact_7859_suminf__le__const,axiom,
% 0.32/0.65 ! [F: nat > nat,X: nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.32/0.65 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_le_const
% 0.32/0.65 thf(fact_7860_suminf__le__const,axiom,
% 0.32/0.65 ! [F: nat > real,X: real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.32/0.65 => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_le_const
% 0.32/0.65 thf(fact_7861_summable__comparison__test,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real] :
% 0.32/0.65 ( ? [N8: nat] :
% 0.32/0.65 ! [N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N8 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( summable_real @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_comparison_test
% 0.32/0.65 thf(fact_7862_summable__comparison__test,axiom,
% 0.32/0.65 ! [F: nat > complex,G: nat > real] :
% 0.32/0.65 ( ? [N8: nat] :
% 0.32/0.65 ! [N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N8 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.32/0.65 => ( ( summable_real @ G )
% 0.32/0.65 => ( summable_complex @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_comparison_test
% 0.32/0.65 thf(fact_7863_summable__comparison__test_H,axiom,
% 0.32/0.65 ! [G: nat > real,N3: nat,F: nat > real] :
% 0.32/0.65 ( ( summable_real @ G )
% 0.32/0.65 => ( ! [N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N3 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.32/0.65 => ( summable_real @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_comparison_test'
% 0.32/0.65 thf(fact_7864_summable__comparison__test_H,axiom,
% 0.32/0.65 ! [G: nat > real,N3: nat,F: nat > complex] :
% 0.32/0.65 ( ( summable_real @ G )
% 0.32/0.65 => ( ! [N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N3 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 0.32/0.65 => ( summable_complex @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_comparison_test'
% 0.32/0.65 thf(fact_7865_summableI__nonneg__bounded,axiom,
% 0.32/0.65 ! [F: nat > int,X: int] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.32/0.65 => ( summable_int @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summableI_nonneg_bounded
% 0.32/0.65 thf(fact_7866_summableI__nonneg__bounded,axiom,
% 0.32/0.65 ! [F: nat > nat,X: nat] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.32/0.65 => ( summable_nat @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summableI_nonneg_bounded
% 0.32/0.65 thf(fact_7867_summableI__nonneg__bounded,axiom,
% 0.32/0.65 ! [F: nat > real,X: real] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 0.32/0.65 => ( summable_real @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summableI_nonneg_bounded
% 0.32/0.65 thf(fact_7868_complete__algebra__summable__geometric,axiom,
% 0.32/0.65 ! [X: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
% 0.32/0.65 => ( summable_real @ ( power_power_real @ X ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % complete_algebra_summable_geometric
% 0.32/0.65 thf(fact_7869_complete__algebra__summable__geometric,axiom,
% 0.32/0.65 ! [X: complex] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
% 0.32/0.65 => ( summable_complex @ ( power_power_complex @ X ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % complete_algebra_summable_geometric
% 0.32/0.65 thf(fact_7870_summable__geometric,axiom,
% 0.32/0.65 ! [C: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.32/0.65 => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_geometric
% 0.32/0.65 thf(fact_7871_summable__geometric,axiom,
% 0.32/0.65 ! [C: complex] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.32/0.65 => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_geometric
% 0.32/0.65 thf(fact_7872_suminf__split__head,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 0.32/0.65 = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_split_head
% 0.32/0.65 thf(fact_7873_summable__norm,axiom,
% 0.32/0.65 ! [F: nat > real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 0.32/0.65 @ ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_norm
% 0.32/0.65 thf(fact_7874_summable__norm,axiom,
% 0.32/0.65 ! [F: nat > complex] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 0.32/0.65 @ ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_norm
% 0.32/0.65 thf(fact_7875_sum__le__suminf,axiom,
% 0.32/0.65 ! [F: nat > int,I5: set_nat] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ( finite_finite_nat @ I5 )
% 0.32/0.65 => ( ! [N4: nat] :
% 0.32/0.65 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.32/0.65 => ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) ) )
% 0.32/0.65 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_le_suminf
% 0.32/0.65 thf(fact_7876_sum__le__suminf,axiom,
% 0.32/0.65 ! [F: nat > nat,I5: set_nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ( finite_finite_nat @ I5 )
% 0.32/0.65 => ( ! [N4: nat] :
% 0.32/0.65 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.32/0.65 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) ) )
% 0.32/0.65 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_le_suminf
% 0.32/0.65 thf(fact_7877_sum__le__suminf,axiom,
% 0.32/0.65 ! [F: nat > real,I5: set_nat] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( finite_finite_nat @ I5 )
% 0.32/0.65 => ( ! [N4: nat] :
% 0.32/0.65 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.32/0.65 => ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) ) )
% 0.32/0.65 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_le_suminf
% 0.32/0.65 thf(fact_7878_suminf__split__initial__segment,axiom,
% 0.32/0.65 ! [F: nat > real,K: nat] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( suminf_real @ F )
% 0.32/0.65 = ( plus_plus_real
% 0.32/0.65 @ ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.32/0.65 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_split_initial_segment
% 0.32/0.65 thf(fact_7879_suminf__minus__initial__segment,axiom,
% 0.32/0.65 ! [F: nat > real,K: nat] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.32/0.65 = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_minus_initial_segment
% 0.32/0.65 thf(fact_7880_sum__less__suminf,axiom,
% 0.32/0.65 ! [F: nat > int,N: nat] :
% 0.32/0.65 ( ( summable_int @ F )
% 0.32/0.65 => ( ! [M4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N @ M4 )
% 0.32/0.65 => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
% 0.32/0.65 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_less_suminf
% 0.32/0.65 thf(fact_7881_sum__less__suminf,axiom,
% 0.32/0.65 ! [F: nat > nat,N: nat] :
% 0.32/0.65 ( ( summable_nat @ F )
% 0.32/0.65 => ( ! [M4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N @ M4 )
% 0.32/0.65 => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 0.32/0.65 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_less_suminf
% 0.32/0.65 thf(fact_7882_sum__less__suminf,axiom,
% 0.32/0.65 ! [F: nat > real,N: nat] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ! [M4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N @ M4 )
% 0.32/0.65 => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
% 0.32/0.65 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_less_suminf
% 0.32/0.65 thf(fact_7883_powser__split__head_I1_J,axiom,
% 0.32/0.65 ! [F: nat > complex,Z: complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 => ( ( suminf_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 0.32/0.65 @ ( times_times_complex
% 0.32/0.65 @ ( suminf_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 @ Z ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_split_head(1)
% 0.32/0.65 thf(fact_7884_powser__split__head_I1_J,axiom,
% 0.32/0.65 ! [F: nat > real,Z: real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 => ( ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 = ( plus_plus_real @ ( F @ zero_zero_nat )
% 0.32/0.65 @ ( times_times_real
% 0.32/0.65 @ ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 @ Z ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_split_head(1)
% 0.32/0.65 thf(fact_7885_powser__split__head_I2_J,axiom,
% 0.32/0.65 ! [F: nat > complex,Z: complex] :
% 0.32/0.65 ( ( summable_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 => ( ( times_times_complex
% 0.32/0.65 @ ( suminf_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 @ Z )
% 0.32/0.65 = ( minus_minus_complex
% 0.32/0.65 @ ( suminf_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.32/0.65 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_split_head(2)
% 0.32/0.65 thf(fact_7886_powser__split__head_I2_J,axiom,
% 0.32/0.65 ! [F: nat > real,Z: real] :
% 0.32/0.65 ( ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 => ( ( times_times_real
% 0.32/0.65 @ ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 @ Z )
% 0.32/0.65 = ( minus_minus_real
% 0.32/0.65 @ ( suminf_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.32/0.65 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_split_head(2)
% 0.32/0.65 thf(fact_7887_summable__partial__sum__bound,axiom,
% 0.32/0.65 ! [F: nat > complex,E: real] :
% 0.32/0.65 ( ( summable_complex @ F )
% 0.32/0.65 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.32/0.65 => ~ ! [N9: nat] :
% 0.32/0.65 ~ ! [M2: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N9 @ M2 )
% 0.32/0.65 => ! [N7: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_partial_sum_bound
% 0.32/0.65 thf(fact_7888_summable__partial__sum__bound,axiom,
% 0.32/0.65 ! [F: nat > real,E: real] :
% 0.32/0.65 ( ( summable_real @ F )
% 0.32/0.65 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.32/0.65 => ~ ! [N9: nat] :
% 0.32/0.65 ~ ! [M2: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N9 @ M2 )
% 0.32/0.65 => ! [N7: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_partial_sum_bound
% 0.32/0.65 thf(fact_7889_suminf__exist__split,axiom,
% 0.32/0.65 ! [R: real,F: nat > real] :
% 0.32/0.65 ( ( ord_less_real @ zero_zero_real @ R )
% 0.32/0.65 => ( ( summable_real @ F )
% 0.32/0.65 => ? [N9: nat] :
% 0.32/0.65 ! [N7: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N9 @ N7 )
% 0.32/0.65 => ( ord_less_real
% 0.32/0.65 @ ( real_V7735802525324610683m_real
% 0.32/0.65 @ ( suminf_real
% 0.32/0.65 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N7 ) ) ) )
% 0.32/0.65 @ R ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_exist_split
% 0.32/0.65 thf(fact_7890_suminf__exist__split,axiom,
% 0.32/0.65 ! [R: real,F: nat > complex] :
% 0.32/0.65 ( ( ord_less_real @ zero_zero_real @ R )
% 0.32/0.65 => ( ( summable_complex @ F )
% 0.32/0.65 => ? [N9: nat] :
% 0.32/0.65 ! [N7: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N9 @ N7 )
% 0.32/0.65 => ( ord_less_real
% 0.32/0.65 @ ( real_V1022390504157884413omplex
% 0.32/0.65 @ ( suminf_complex
% 0.32/0.65 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N7 ) ) ) )
% 0.32/0.65 @ R ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % suminf_exist_split
% 0.32/0.65 thf(fact_7891_summable__power__series,axiom,
% 0.32/0.65 ! [F: nat > real,Z: real] :
% 0.32/0.65 ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 0.32/0.65 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.32/0.65 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 0.32/0.65 => ( ( ord_less_real @ Z @ one_one_real )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_power_series
% 0.32/0.65 thf(fact_7892_Abel__lemma,axiom,
% 0.32/0.65 ! [R: real,R0: real,A: nat > complex,M5: real] :
% 0.32/0.65 ( ( ord_less_eq_real @ zero_zero_real @ R )
% 0.32/0.65 => ( ( ord_less_real @ R @ R0 )
% 0.32/0.65 => ( ! [N4: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R0 @ N4 ) ) @ M5 )
% 0.32/0.65 => ( summable_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R @ N2 ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % Abel_lemma
% 0.32/0.65 thf(fact_7893_summable__ratio__test,axiom,
% 0.32/0.65 ! [C: real,N3: nat,F: nat > real] :
% 0.32/0.65 ( ( ord_less_real @ C @ one_one_real )
% 0.32/0.65 => ( ! [N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N3 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) )
% 0.32/0.65 => ( summable_real @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_ratio_test
% 0.32/0.65 thf(fact_7894_summable__ratio__test,axiom,
% 0.32/0.65 ! [C: real,N3: nat,F: nat > complex] :
% 0.32/0.65 ( ( ord_less_real @ C @ one_one_real )
% 0.32/0.65 => ( ! [N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ N3 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) )
% 0.32/0.65 => ( summable_complex @ F ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % summable_ratio_test
% 0.32/0.65 thf(fact_7895_sum__bounds__lt__plus1,axiom,
% 0.32/0.65 ! [F: nat > nat,Mm: nat] :
% 0.32/0.65 ( ( groups3542108847815614940at_nat
% 0.32/0.65 @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ Mm ) )
% 0.32/0.65 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_bounds_lt_plus1
% 0.32/0.65 thf(fact_7896_sum__bounds__lt__plus1,axiom,
% 0.32/0.65 ! [F: nat > real,Mm: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ Mm ) )
% 0.32/0.65 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sum_bounds_lt_plus1
% 0.32/0.65 thf(fact_7897_sumr__cos__zero__one,axiom,
% 0.32/0.65 ! [N: nat] :
% 0.32/0.65 ( ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
% 0.32/0.65 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.65 = one_one_real ) ).
% 0.32/0.65
% 0.32/0.65 % sumr_cos_zero_one
% 0.32/0.65 thf(fact_7898_pi__series,axiom,
% 0.32/0.65 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.65 = ( suminf_real
% 0.32/0.65 @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % pi_series
% 0.32/0.65 thf(fact_7899_geometric__deriv__sums,axiom,
% 0.32/0.65 ! [Z: real] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.32/0.65 => ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
% 0.32/0.65 @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % geometric_deriv_sums
% 0.32/0.65 thf(fact_7900_geometric__deriv__sums,axiom,
% 0.32/0.65 ! [Z: complex] :
% 0.32/0.65 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.32/0.65 => ( sums_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
% 0.32/0.65 @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % geometric_deriv_sums
% 0.32/0.65 thf(fact_7901_monoI1,axiom,
% 0.32/0.65 ! [X7: nat > real] :
% 0.32/0.65 ( ! [M4: nat,N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.65 => ( ord_less_eq_real @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 0.32/0.65 => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 0.32/0.65
% 0.32/0.65 % monoI1
% 0.32/0.65 thf(fact_7902_monoI1,axiom,
% 0.32/0.65 ! [X7: nat > set_int] :
% 0.32/0.65 ( ! [M4: nat,N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.65 => ( ord_less_eq_set_int @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 0.32/0.65 => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 0.32/0.65
% 0.32/0.65 % monoI1
% 0.32/0.65 thf(fact_7903_monoI1,axiom,
% 0.32/0.65 ! [X7: nat > rat] :
% 0.32/0.65 ( ! [M4: nat,N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.65 => ( ord_less_eq_rat @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 0.32/0.65 => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 0.32/0.65
% 0.32/0.65 % monoI1
% 0.32/0.65 thf(fact_7904_monoI1,axiom,
% 0.32/0.65 ! [X7: nat > num] :
% 0.32/0.65 ( ! [M4: nat,N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.65 => ( ord_less_eq_num @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 0.32/0.65 => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 0.32/0.65
% 0.32/0.65 % monoI1
% 0.32/0.65 thf(fact_7905_monoI1,axiom,
% 0.32/0.65 ! [X7: nat > nat] :
% 0.32/0.65 ( ! [M4: nat,N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.65 => ( ord_less_eq_nat @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 0.32/0.65 => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 0.32/0.65
% 0.32/0.65 % monoI1
% 0.32/0.65 thf(fact_7906_monoI1,axiom,
% 0.32/0.65 ! [X7: nat > int] :
% 0.32/0.65 ( ! [M4: nat,N4: nat] :
% 0.32/0.65 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.65 => ( ord_less_eq_int @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 0.32/0.65 => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 0.32/0.65
% 0.32/0.65 % monoI1
% 0.32/0.65 thf(fact_7907_cos__coeff__0,axiom,
% 0.32/0.65 ( ( cos_coeff @ zero_zero_nat )
% 0.32/0.65 = one_one_real ) ).
% 0.32/0.65
% 0.32/0.65 % cos_coeff_0
% 0.32/0.65 thf(fact_7908_powser__sums__zero__iff,axiom,
% 0.32/0.65 ! [A: nat > complex,X: complex] :
% 0.32/0.65 ( ( sums_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 0.32/0.65 @ X )
% 0.32/0.65 = ( ( A @ zero_zero_nat )
% 0.32/0.65 = X ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_sums_zero_iff
% 0.32/0.65 thf(fact_7909_powser__sums__zero__iff,axiom,
% 0.32/0.65 ! [A: nat > real,X: real] :
% 0.32/0.65 ( ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 0.32/0.65 @ X )
% 0.32/0.65 = ( ( A @ zero_zero_nat )
% 0.32/0.65 = X ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_sums_zero_iff
% 0.32/0.65 thf(fact_7910_sums__le,axiom,
% 0.32/0.65 ! [F: nat > real,G: nat > real,S: real,T: real] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.32/0.65 => ( ( sums_real @ F @ S )
% 0.32/0.65 => ( ( sums_real @ G @ T )
% 0.32/0.65 => ( ord_less_eq_real @ S @ T ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_le
% 0.32/0.65 thf(fact_7911_sums__le,axiom,
% 0.32/0.65 ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.32/0.65 => ( ( sums_nat @ F @ S )
% 0.32/0.65 => ( ( sums_nat @ G @ T )
% 0.32/0.65 => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_le
% 0.32/0.65 thf(fact_7912_sums__le,axiom,
% 0.32/0.65 ! [F: nat > int,G: nat > int,S: int,T: int] :
% 0.32/0.65 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.32/0.65 => ( ( sums_int @ F @ S )
% 0.32/0.65 => ( ( sums_int @ G @ T )
% 0.32/0.65 => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_le
% 0.32/0.65 thf(fact_7913_sums__mult,axiom,
% 0.32/0.65 ! [F: nat > real,A: real,C: real] :
% 0.32/0.65 ( ( sums_real @ F @ A )
% 0.32/0.65 => ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 0.32/0.65 @ ( times_times_real @ C @ A ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_mult
% 0.32/0.65 thf(fact_7914_sums__mult2,axiom,
% 0.32/0.65 ! [F: nat > real,A: real,C: real] :
% 0.32/0.65 ( ( sums_real @ F @ A )
% 0.32/0.65 => ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 0.32/0.65 @ ( times_times_real @ A @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_mult2
% 0.32/0.65 thf(fact_7915_sums__add,axiom,
% 0.32/0.65 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 0.32/0.65 ( ( sums_real @ F @ A )
% 0.32/0.65 => ( ( sums_real @ G @ B )
% 0.32/0.65 => ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.32/0.65 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_add
% 0.32/0.65 thf(fact_7916_sums__add,axiom,
% 0.32/0.65 ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
% 0.32/0.65 ( ( sums_nat @ F @ A )
% 0.32/0.65 => ( ( sums_nat @ G @ B )
% 0.32/0.65 => ( sums_nat
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.32/0.65 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_add
% 0.32/0.65 thf(fact_7917_sums__add,axiom,
% 0.32/0.65 ! [F: nat > int,A: int,G: nat > int,B: int] :
% 0.32/0.65 ( ( sums_int @ F @ A )
% 0.32/0.65 => ( ( sums_int @ G @ B )
% 0.32/0.65 => ( sums_int
% 0.32/0.65 @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.32/0.65 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_add
% 0.32/0.65 thf(fact_7918_sums__diff,axiom,
% 0.32/0.65 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 0.32/0.65 ( ( sums_real @ F @ A )
% 0.32/0.65 => ( ( sums_real @ G @ B )
% 0.32/0.65 => ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.32/0.65 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_diff
% 0.32/0.65 thf(fact_7919_sums__divide,axiom,
% 0.32/0.65 ! [F: nat > complex,A: complex,C: complex] :
% 0.32/0.65 ( ( sums_complex @ F @ A )
% 0.32/0.65 => ( sums_complex
% 0.32/0.65 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C )
% 0.32/0.65 @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_divide
% 0.32/0.65 thf(fact_7920_sums__divide,axiom,
% 0.32/0.65 ! [F: nat > real,A: real,C: real] :
% 0.32/0.65 ( ( sums_real @ F @ A )
% 0.32/0.65 => ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C )
% 0.32/0.65 @ ( divide_divide_real @ A @ C ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_divide
% 0.32/0.65 thf(fact_7921_sums__mult__iff,axiom,
% 0.32/0.65 ! [C: complex,F: nat > complex,D: complex] :
% 0.32/0.65 ( ( C != zero_zero_complex )
% 0.32/0.65 => ( ( sums_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 0.32/0.65 @ ( times_times_complex @ C @ D ) )
% 0.32/0.65 = ( sums_complex @ F @ D ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_mult_iff
% 0.32/0.65 thf(fact_7922_sums__mult__iff,axiom,
% 0.32/0.65 ! [C: real,F: nat > real,D: real] :
% 0.32/0.65 ( ( C != zero_zero_real )
% 0.32/0.65 => ( ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 0.32/0.65 @ ( times_times_real @ C @ D ) )
% 0.32/0.65 = ( sums_real @ F @ D ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_mult_iff
% 0.32/0.65 thf(fact_7923_sums__mult2__iff,axiom,
% 0.32/0.65 ! [C: complex,F: nat > complex,D: complex] :
% 0.32/0.65 ( ( C != zero_zero_complex )
% 0.32/0.65 => ( ( sums_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
% 0.32/0.65 @ ( times_times_complex @ D @ C ) )
% 0.32/0.65 = ( sums_complex @ F @ D ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_mult2_iff
% 0.32/0.65 thf(fact_7924_sums__mult2__iff,axiom,
% 0.32/0.65 ! [C: real,F: nat > real,D: real] :
% 0.32/0.65 ( ( C != zero_zero_real )
% 0.32/0.65 => ( ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 0.32/0.65 @ ( times_times_real @ D @ C ) )
% 0.32/0.65 = ( sums_real @ F @ D ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_mult2_iff
% 0.32/0.65 thf(fact_7925_pi__gt__zero,axiom,
% 0.32/0.65 ord_less_real @ zero_zero_real @ pi ).
% 0.32/0.65
% 0.32/0.65 % pi_gt_zero
% 0.32/0.65 thf(fact_7926_pi__not__less__zero,axiom,
% 0.32/0.65 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 0.32/0.65
% 0.32/0.65 % pi_not_less_zero
% 0.32/0.65 thf(fact_7927_pi__ge__zero,axiom,
% 0.32/0.65 ord_less_eq_real @ zero_zero_real @ pi ).
% 0.32/0.65
% 0.32/0.65 % pi_ge_zero
% 0.32/0.65 thf(fact_7928_sums__mult__D,axiom,
% 0.32/0.65 ! [C: complex,F: nat > complex,A: complex] :
% 0.32/0.65 ( ( sums_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 0.32/0.65 @ A )
% 0.32/0.65 => ( ( C != zero_zero_complex )
% 0.32/0.65 => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_mult_D
% 0.32/0.65 thf(fact_7929_sums__mult__D,axiom,
% 0.32/0.65 ! [C: real,F: nat > real,A: real] :
% 0.32/0.65 ( ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 0.32/0.65 @ A )
% 0.32/0.65 => ( ( C != zero_zero_real )
% 0.32/0.65 => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_mult_D
% 0.32/0.65 thf(fact_7930_sums__Suc,axiom,
% 0.32/0.65 ! [F: nat > real,L2: real] :
% 0.32/0.65 ( ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.32/0.65 @ L2 )
% 0.32/0.65 => ( sums_real @ F @ ( plus_plus_real @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_Suc
% 0.32/0.65 thf(fact_7931_sums__Suc,axiom,
% 0.32/0.65 ! [F: nat > nat,L2: nat] :
% 0.32/0.65 ( ( sums_nat
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.32/0.65 @ L2 )
% 0.32/0.65 => ( sums_nat @ F @ ( plus_plus_nat @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_Suc
% 0.32/0.65 thf(fact_7932_sums__Suc,axiom,
% 0.32/0.65 ! [F: nat > int,L2: int] :
% 0.32/0.65 ( ( sums_int
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.32/0.65 @ L2 )
% 0.32/0.65 => ( sums_int @ F @ ( plus_plus_int @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_Suc
% 0.32/0.65 thf(fact_7933_sums__Suc__iff,axiom,
% 0.32/0.65 ! [F: nat > real,S: real] :
% 0.32/0.65 ( ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.32/0.65 @ S )
% 0.32/0.65 = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_Suc_iff
% 0.32/0.65 thf(fact_7934_sums__zero__iff__shift,axiom,
% 0.32/0.65 ! [N: nat,F: nat > complex,S: complex] :
% 0.32/0.65 ( ! [I2: nat] :
% 0.32/0.65 ( ( ord_less_nat @ I2 @ N )
% 0.32/0.65 => ( ( F @ I2 )
% 0.32/0.65 = zero_zero_complex ) )
% 0.32/0.65 => ( ( sums_complex
% 0.32/0.65 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.32/0.65 @ S )
% 0.32/0.65 = ( sums_complex @ F @ S ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_zero_iff_shift
% 0.32/0.65 thf(fact_7935_sums__zero__iff__shift,axiom,
% 0.32/0.65 ! [N: nat,F: nat > real,S: real] :
% 0.32/0.65 ( ! [I2: nat] :
% 0.32/0.65 ( ( ord_less_nat @ I2 @ N )
% 0.32/0.65 => ( ( F @ I2 )
% 0.32/0.65 = zero_zero_real ) )
% 0.32/0.65 => ( ( sums_real
% 0.32/0.65 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.32/0.65 @ S )
% 0.32/0.65 = ( sums_real @ F @ S ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_zero_iff_shift
% 0.32/0.65 thf(fact_7936_powser__sums__if,axiom,
% 0.32/0.65 ! [M: nat,Z: complex] :
% 0.32/0.65 ( sums_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
% 0.32/0.65 @ ( power_power_complex @ Z @ M ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_sums_if
% 0.32/0.65 thf(fact_7937_powser__sums__if,axiom,
% 0.32/0.65 ! [M: nat,Z: real] :
% 0.32/0.65 ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
% 0.32/0.65 @ ( power_power_real @ Z @ M ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_sums_if
% 0.32/0.65 thf(fact_7938_powser__sums__if,axiom,
% 0.32/0.65 ! [M: nat,Z: int] :
% 0.32/0.65 ( sums_int
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
% 0.32/0.65 @ ( power_power_int @ Z @ M ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_sums_if
% 0.32/0.65 thf(fact_7939_powser__sums__zero,axiom,
% 0.32/0.65 ! [A: nat > complex] :
% 0.32/0.65 ( sums_complex
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 0.32/0.65 @ ( A @ zero_zero_nat ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_sums_zero
% 0.32/0.65 thf(fact_7940_powser__sums__zero,axiom,
% 0.32/0.65 ! [A: nat > real] :
% 0.32/0.65 ( sums_real
% 0.32/0.65 @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 0.32/0.65 @ ( A @ zero_zero_nat ) ) ).
% 0.32/0.65
% 0.32/0.65 % powser_sums_zero
% 0.32/0.65 thf(fact_7941_sums__iff__shift,axiom,
% 0.32/0.65 ! [F: nat > real,N: nat,S: real] :
% 0.32/0.65 ( ( sums_real
% 0.32/0.65 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.32/0.65 @ S )
% 0.32/0.65 = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_iff_shift
% 0.32/0.65 thf(fact_7942_sums__iff__shift_H,axiom,
% 0.32/0.65 ! [F: nat > real,N: nat,S: real] :
% 0.32/0.65 ( ( sums_real
% 0.32/0.65 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.32/0.65 @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.32/0.65 = ( sums_real @ F @ S ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_iff_shift'
% 0.32/0.65 thf(fact_7943_sums__split__initial__segment,axiom,
% 0.32/0.65 ! [F: nat > real,S: real,N: nat] :
% 0.32/0.65 ( ( sums_real @ F @ S )
% 0.32/0.65 => ( sums_real
% 0.32/0.65 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.32/0.65 @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 0.32/0.65
% 0.32/0.65 % sums_split_initial_segment
% 0.32/0.65 thf(fact_7944_sums__If__finite__set_H,axiom,
% 0.32/0.65 ! [G: nat > real,S2: real,A2: set_nat,S4: real,F: nat > real] :
% 0.32/0.65 ( ( sums_real @ G @ S2 )
% 0.32/0.65 => ( ( finite_finite_nat @ A2 )
% 0.32/0.65 => ( ( S4
% 0.32/0.65 = ( plus_plus_real @ S2
% 0.32/0.65 @ ( groups6591440286371151544t_real
% 0.32/0.65 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.32/0.66 @ A2 ) ) )
% 0.32/0.66 => ( sums_real
% 0.32/0.66 @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A2 ) @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.32/0.66 @ S4 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sums_If_finite_set'
% 0.32/0.66 thf(fact_7945_pi__less__4,axiom,
% 0.32/0.66 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pi_less_4
% 0.32/0.66 thf(fact_7946_pi__ge__two,axiom,
% 0.32/0.66 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.32/0.66
% 0.32/0.66 % pi_ge_two
% 0.32/0.66 thf(fact_7947_pi__half__neq__two,axiom,
% 0.32/0.66 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.66 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pi_half_neq_two
% 0.32/0.66 thf(fact_7948_pi__half__neq__zero,axiom,
% 0.32/0.66 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.66 != zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % pi_half_neq_zero
% 0.32/0.66 thf(fact_7949_pi__half__less__two,axiom,
% 0.32/0.66 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.32/0.66
% 0.32/0.66 % pi_half_less_two
% 0.32/0.66 thf(fact_7950_pi__half__le__two,axiom,
% 0.32/0.66 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.32/0.66
% 0.32/0.66 % pi_half_le_two
% 0.32/0.66 thf(fact_7951_pi__half__gt__zero,axiom,
% 0.32/0.66 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pi_half_gt_zero
% 0.32/0.66 thf(fact_7952_pi__half__ge__zero,axiom,
% 0.32/0.66 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pi_half_ge_zero
% 0.32/0.66 thf(fact_7953_m2pi__less__pi,axiom,
% 0.32/0.66 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.32/0.66
% 0.32/0.66 % m2pi_less_pi
% 0.32/0.66 thf(fact_7954_arctan__ubound,axiom,
% 0.32/0.66 ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % arctan_ubound
% 0.32/0.66 thf(fact_7955_arctan__one,axiom,
% 0.32/0.66 ( ( arctan @ one_one_real )
% 0.32/0.66 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % arctan_one
% 0.32/0.66 thf(fact_7956_geometric__sums,axiom,
% 0.32/0.66 ! [C: real] :
% 0.32/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.32/0.66 => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % geometric_sums
% 0.32/0.66 thf(fact_7957_geometric__sums,axiom,
% 0.32/0.66 ! [C: complex] :
% 0.32/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.32/0.66 => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % geometric_sums
% 0.32/0.66 thf(fact_7958_power__half__series,axiom,
% 0.32/0.66 ( sums_real
% 0.32/0.66 @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
% 0.32/0.66 @ one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % power_half_series
% 0.32/0.66 thf(fact_7959_minus__pi__half__less__zero,axiom,
% 0.32/0.66 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.32/0.66
% 0.32/0.66 % minus_pi_half_less_zero
% 0.32/0.66 thf(fact_7960_arctan__bounded,axiom,
% 0.32/0.66 ! [Y3: real] :
% 0.32/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.32/0.66 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % arctan_bounded
% 0.32/0.66 thf(fact_7961_arctan__lbound,axiom,
% 0.32/0.66 ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 0.32/0.66
% 0.32/0.66 % arctan_lbound
% 0.32/0.66 thf(fact_7962_sums__if_H,axiom,
% 0.32/0.66 ! [G: nat > real,X: real] :
% 0.32/0.66 ( ( sums_real @ G @ X )
% 0.32/0.66 => ( sums_real
% 0.32/0.66 @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.66 @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % sums_if'
% 0.32/0.66 thf(fact_7963_sums__if,axiom,
% 0.32/0.66 ! [G: nat > real,X: real,F: nat > real,Y3: real] :
% 0.32/0.66 ( ( sums_real @ G @ X )
% 0.32/0.66 => ( ( sums_real @ F @ Y3 )
% 0.32/0.66 => ( sums_real
% 0.32/0.66 @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.66 @ ( plus_plus_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sums_if
% 0.32/0.66 thf(fact_7964_machin__Euler,axiom,
% 0.32/0.66 ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 0.32/0.66 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % machin_Euler
% 0.32/0.66 thf(fact_7965_machin,axiom,
% 0.32/0.66 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % machin
% 0.32/0.66 thf(fact_7966_monoseq__Suc,axiom,
% 0.32/0.66 ( topolo6980174941875973593q_real
% 0.32/0.66 = ( ^ [X6: nat > real] :
% 0.32/0.66 ( ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 0.32/0.66 | ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_Suc
% 0.32/0.66 thf(fact_7967_monoseq__Suc,axiom,
% 0.32/0.66 ( topolo3100542954746470799et_int
% 0.32/0.66 = ( ^ [X6: nat > set_int] :
% 0.32/0.66 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 0.32/0.66 | ! [N2: nat] : ( ord_less_eq_set_int @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_Suc
% 0.32/0.66 thf(fact_7968_monoseq__Suc,axiom,
% 0.32/0.66 ( topolo4267028734544971653eq_rat
% 0.32/0.66 = ( ^ [X6: nat > rat] :
% 0.32/0.66 ( ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 0.32/0.66 | ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_Suc
% 0.32/0.66 thf(fact_7969_monoseq__Suc,axiom,
% 0.32/0.66 ( topolo1459490580787246023eq_num
% 0.32/0.66 = ( ^ [X6: nat > num] :
% 0.32/0.66 ( ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 0.32/0.66 | ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_Suc
% 0.32/0.66 thf(fact_7970_monoseq__Suc,axiom,
% 0.32/0.66 ( topolo4902158794631467389eq_nat
% 0.32/0.66 = ( ^ [X6: nat > nat] :
% 0.32/0.66 ( ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 0.32/0.66 | ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_Suc
% 0.32/0.66 thf(fact_7971_monoseq__Suc,axiom,
% 0.32/0.66 ( topolo4899668324122417113eq_int
% 0.32/0.66 = ( ^ [X6: nat > int] :
% 0.32/0.66 ( ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 0.32/0.66 | ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_Suc
% 0.32/0.66 thf(fact_7972_mono__SucI2,axiom,
% 0.32/0.66 ! [X7: nat > real] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 0.32/0.66 => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI2
% 0.32/0.66 thf(fact_7973_mono__SucI2,axiom,
% 0.32/0.66 ! [X7: nat > set_int] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 0.32/0.66 => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI2
% 0.32/0.66 thf(fact_7974_mono__SucI2,axiom,
% 0.32/0.66 ! [X7: nat > rat] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 0.32/0.66 => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI2
% 0.32/0.66 thf(fact_7975_mono__SucI2,axiom,
% 0.32/0.66 ! [X7: nat > num] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 0.32/0.66 => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI2
% 0.32/0.66 thf(fact_7976_mono__SucI2,axiom,
% 0.32/0.66 ! [X7: nat > nat] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 0.32/0.66 => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI2
% 0.32/0.66 thf(fact_7977_mono__SucI2,axiom,
% 0.32/0.66 ! [X7: nat > int] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 0.32/0.66 => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI2
% 0.32/0.66 thf(fact_7978_mono__SucI1,axiom,
% 0.32/0.66 ! [X7: nat > real] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_real @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 0.32/0.66 => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI1
% 0.32/0.66 thf(fact_7979_mono__SucI1,axiom,
% 0.32/0.66 ! [X7: nat > set_int] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_set_int @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 0.32/0.66 => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI1
% 0.32/0.66 thf(fact_7980_mono__SucI1,axiom,
% 0.32/0.66 ! [X7: nat > rat] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_rat @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 0.32/0.66 => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI1
% 0.32/0.66 thf(fact_7981_mono__SucI1,axiom,
% 0.32/0.66 ! [X7: nat > num] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_num @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 0.32/0.66 => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI1
% 0.32/0.66 thf(fact_7982_mono__SucI1,axiom,
% 0.32/0.66 ! [X7: nat > nat] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_nat @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 0.32/0.66 => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI1
% 0.32/0.66 thf(fact_7983_mono__SucI1,axiom,
% 0.32/0.66 ! [X7: nat > int] :
% 0.32/0.66 ( ! [N4: nat] : ( ord_less_eq_int @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 0.32/0.66 => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % mono_SucI1
% 0.32/0.66 thf(fact_7984_monoseq__def,axiom,
% 0.32/0.66 ( topolo6980174941875973593q_real
% 0.32/0.66 = ( ^ [X6: nat > real] :
% 0.32/0.66 ( ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_real @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 0.32/0.66 | ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_def
% 0.32/0.66 thf(fact_7985_monoseq__def,axiom,
% 0.32/0.66 ( topolo3100542954746470799et_int
% 0.32/0.66 = ( ^ [X6: nat > set_int] :
% 0.32/0.66 ( ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_set_int @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 0.32/0.66 | ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_set_int @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_def
% 0.32/0.66 thf(fact_7986_monoseq__def,axiom,
% 0.32/0.66 ( topolo4267028734544971653eq_rat
% 0.32/0.66 = ( ^ [X6: nat > rat] :
% 0.32/0.66 ( ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_rat @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 0.32/0.66 | ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_def
% 0.32/0.66 thf(fact_7987_monoseq__def,axiom,
% 0.32/0.66 ( topolo1459490580787246023eq_num
% 0.32/0.66 = ( ^ [X6: nat > num] :
% 0.32/0.66 ( ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_num @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 0.32/0.66 | ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_def
% 0.32/0.66 thf(fact_7988_monoseq__def,axiom,
% 0.32/0.66 ( topolo4902158794631467389eq_nat
% 0.32/0.66 = ( ^ [X6: nat > nat] :
% 0.32/0.66 ( ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_nat @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 0.32/0.66 | ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_def
% 0.32/0.66 thf(fact_7989_monoseq__def,axiom,
% 0.32/0.66 ( topolo4899668324122417113eq_int
% 0.32/0.66 = ( ^ [X6: nat > int] :
% 0.32/0.66 ( ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_int @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 0.32/0.66 | ! [M3: nat,N2: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M3 @ N2 )
% 0.32/0.66 => ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoseq_def
% 0.32/0.66 thf(fact_7990_monoI2,axiom,
% 0.32/0.66 ! [X7: nat > real] :
% 0.32/0.66 ( ! [M4: nat,N4: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.66 => ( ord_less_eq_real @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 0.32/0.66 => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoI2
% 0.32/0.66 thf(fact_7991_monoI2,axiom,
% 0.32/0.66 ! [X7: nat > set_int] :
% 0.32/0.66 ( ! [M4: nat,N4: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.66 => ( ord_less_eq_set_int @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 0.32/0.66 => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoI2
% 0.32/0.66 thf(fact_7992_monoI2,axiom,
% 0.32/0.66 ! [X7: nat > rat] :
% 0.32/0.66 ( ! [M4: nat,N4: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.66 => ( ord_less_eq_rat @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 0.32/0.66 => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoI2
% 0.32/0.66 thf(fact_7993_monoI2,axiom,
% 0.32/0.66 ! [X7: nat > num] :
% 0.32/0.66 ( ! [M4: nat,N4: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.66 => ( ord_less_eq_num @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 0.32/0.66 => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoI2
% 0.32/0.66 thf(fact_7994_monoI2,axiom,
% 0.32/0.66 ! [X7: nat > nat] :
% 0.32/0.66 ( ! [M4: nat,N4: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.66 => ( ord_less_eq_nat @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 0.32/0.66 => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoI2
% 0.32/0.66 thf(fact_7995_monoI2,axiom,
% 0.32/0.66 ! [X7: nat > int] :
% 0.32/0.66 ( ! [M4: nat,N4: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.66 => ( ord_less_eq_int @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 0.32/0.66 => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 0.32/0.66
% 0.32/0.66 % monoI2
% 0.32/0.66 thf(fact_7996_sin__cos__npi,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( 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 ) ) ) )
% 0.32/0.66 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_cos_npi
% 0.32/0.66 thf(fact_7997_diffs__equiv,axiom,
% 0.32/0.66 ! [C: nat > real,X: real] :
% 0.32/0.66 ( ( summable_real
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 0.32/0.66 => ( sums_real
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.32/0.66 @ ( suminf_real
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % diffs_equiv
% 0.32/0.66 thf(fact_7998_diffs__equiv,axiom,
% 0.32/0.66 ! [C: nat > complex,X: complex] :
% 0.32/0.66 ( ( summable_complex
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 0.32/0.66 => ( sums_complex
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.32/0.66 @ ( suminf_complex
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % diffs_equiv
% 0.32/0.66 thf(fact_7999_cos__pi__eq__zero,axiom,
% 0.32/0.66 ! [M: nat] :
% 0.32/0.66 ( ( 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 ) ) ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_pi_eq_zero
% 0.32/0.66 thf(fact_8000_pochhammer__double,axiom,
% 0.32/0.66 ! [Z: real,N: nat] :
% 0.32/0.66 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.66 = ( 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 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_double
% 0.32/0.66 thf(fact_8001_pochhammer__double,axiom,
% 0.32/0.66 ! [Z: complex,N: nat] :
% 0.32/0.66 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.66 = ( 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 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_double
% 0.32/0.66 thf(fact_8002_pochhammer__double,axiom,
% 0.32/0.66 ! [Z: rat,N: nat] :
% 0.32/0.66 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.66 = ( 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 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_double
% 0.32/0.66 thf(fact_8003_of__nat__code,axiom,
% 0.32/0.66 ( semiri1314217659103216013at_int
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri8420488043553186161ux_int
% 0.32/0.66 @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 0.32/0.66 @ N2
% 0.32/0.66 @ zero_zero_int ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % of_nat_code
% 0.32/0.66 thf(fact_8004_of__nat__code,axiom,
% 0.32/0.66 ( semiri5074537144036343181t_real
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri7260567687927622513x_real
% 0.32/0.66 @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 0.32/0.66 @ N2
% 0.32/0.66 @ zero_zero_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % of_nat_code
% 0.32/0.66 thf(fact_8005_of__nat__code,axiom,
% 0.32/0.66 ( semiri1316708129612266289at_nat
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri8422978514062236437ux_nat
% 0.32/0.66 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 0.32/0.66 @ N2
% 0.32/0.66 @ zero_zero_nat ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % of_nat_code
% 0.32/0.66 thf(fact_8006_of__nat__code,axiom,
% 0.32/0.66 ( semiri8010041392384452111omplex
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri2816024913162550771omplex
% 0.32/0.66 @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
% 0.32/0.66 @ N2
% 0.32/0.66 @ zero_zero_complex ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % of_nat_code
% 0.32/0.66 thf(fact_8007_of__nat__code,axiom,
% 0.32/0.66 ( semiri681578069525770553at_rat
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri7787848453975740701ux_rat
% 0.32/0.66 @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 0.32/0.66 @ N2
% 0.32/0.66 @ zero_zero_rat ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % of_nat_code
% 0.32/0.66 thf(fact_8008_cos__zero,axiom,
% 0.32/0.66 ( ( cos_complex @ zero_zero_complex )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % cos_zero
% 0.32/0.66 thf(fact_8009_cos__zero,axiom,
% 0.32/0.66 ( ( cos_real @ zero_zero_real )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_zero
% 0.32/0.66 thf(fact_8010_pochhammer__0,axiom,
% 0.32/0.66 ! [A: complex] :
% 0.32/0.66 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0
% 0.32/0.66 thf(fact_8011_pochhammer__0,axiom,
% 0.32/0.66 ! [A: real] :
% 0.32/0.66 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0
% 0.32/0.66 thf(fact_8012_pochhammer__0,axiom,
% 0.32/0.66 ! [A: rat] :
% 0.32/0.66 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.32/0.66 = one_one_rat ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0
% 0.32/0.66 thf(fact_8013_pochhammer__0,axiom,
% 0.32/0.66 ! [A: nat] :
% 0.32/0.66 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.32/0.66 = one_one_nat ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0
% 0.32/0.66 thf(fact_8014_pochhammer__0,axiom,
% 0.32/0.66 ! [A: int] :
% 0.32/0.66 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.32/0.66 = one_one_int ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0
% 0.32/0.66 thf(fact_8015_sin__pi__minus,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 0.32/0.66 = ( sin_real @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_pi_minus
% 0.32/0.66 thf(fact_8016_cos__pi,axiom,
% 0.32/0.66 ( ( cos_real @ pi )
% 0.32/0.66 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_pi
% 0.32/0.66 thf(fact_8017_cos__periodic__pi2,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_periodic_pi2
% 0.32/0.66 thf(fact_8018_cos__periodic__pi,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_periodic_pi
% 0.32/0.66 thf(fact_8019_sin__periodic__pi2,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_periodic_pi2
% 0.32/0.66 thf(fact_8020_sin__periodic__pi,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_periodic_pi
% 0.32/0.66 thf(fact_8021_cos__minus__pi,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_minus_pi
% 0.32/0.66 thf(fact_8022_cos__pi__minus,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_pi_minus
% 0.32/0.66 thf(fact_8023_sin__minus__pi,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_minus_pi
% 0.32/0.66 thf(fact_8024_sin__cos__squared__add3,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % sin_cos_squared_add3
% 0.32/0.66 thf(fact_8025_sin__cos__squared__add3,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_cos_squared_add3
% 0.32/0.66 thf(fact_8026_sin__npi2,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_npi2
% 0.32/0.66 thf(fact_8027_sin__npi,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_npi
% 0.32/0.66 thf(fact_8028_sin__npi__int,axiom,
% 0.32/0.66 ! [N: int] :
% 0.32/0.66 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_npi_int
% 0.32/0.66 thf(fact_8029_cos__pi__half,axiom,
% 0.32/0.66 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_pi_half
% 0.32/0.66 thf(fact_8030_sin__two__pi,axiom,
% 0.32/0.66 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_two_pi
% 0.32/0.66 thf(fact_8031_sin__pi__half,axiom,
% 0.32/0.66 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_pi_half
% 0.32/0.66 thf(fact_8032_cos__two__pi,axiom,
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_two_pi
% 0.32/0.66 thf(fact_8033_cos__periodic,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.32/0.66 = ( cos_real @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_periodic
% 0.32/0.66 thf(fact_8034_sin__periodic,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.32/0.66 = ( sin_real @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_periodic
% 0.32/0.66 thf(fact_8035_cos__2pi__minus,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.32/0.66 = ( cos_real @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_2pi_minus
% 0.32/0.66 thf(fact_8036_cos__npi,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.32/0.66 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_npi
% 0.32/0.66 thf(fact_8037_cos__npi2,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.32/0.66 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_npi2
% 0.32/0.66 thf(fact_8038_sin__cos__squared__add,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( 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 ) ) ) )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_cos_squared_add
% 0.32/0.66 thf(fact_8039_sin__cos__squared__add,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( 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 ) ) ) )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % sin_cos_squared_add
% 0.32/0.66 thf(fact_8040_sin__cos__squared__add2,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( 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 ) ) ) )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_cos_squared_add2
% 0.32/0.66 thf(fact_8041_sin__cos__squared__add2,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( 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 ) ) ) )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % sin_cos_squared_add2
% 0.32/0.66 thf(fact_8042_sin__2npi,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_2npi
% 0.32/0.66 thf(fact_8043_cos__2npi,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_2npi
% 0.32/0.66 thf(fact_8044_sin__2pi__minus,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_2pi_minus
% 0.32/0.66 thf(fact_8045_sin__int__2pin,axiom,
% 0.32/0.66 ! [N: int] :
% 0.32/0.66 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_int_2pin
% 0.32/0.66 thf(fact_8046_cos__int__2pin,axiom,
% 0.32/0.66 ! [N: int] :
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_int_2pin
% 0.32/0.66 thf(fact_8047_cos__3over2__pi,axiom,
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.32/0.66 = zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_3over2_pi
% 0.32/0.66 thf(fact_8048_sin__3over2__pi,axiom,
% 0.32/0.66 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.32/0.66 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_3over2_pi
% 0.32/0.66 thf(fact_8049_cos__npi__int,axiom,
% 0.32/0.66 ! [N: int] :
% 0.32/0.66 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.66 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.32/0.66 = one_one_real ) )
% 0.32/0.66 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.32/0.66 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 0.32/0.66 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_npi_int
% 0.32/0.66 thf(fact_8050_cos__one__sin__zero,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( ( cos_complex @ X )
% 0.32/0.66 = one_one_complex )
% 0.32/0.66 => ( ( sin_complex @ X )
% 0.32/0.66 = zero_zero_complex ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_one_sin_zero
% 0.32/0.66 thf(fact_8051_cos__one__sin__zero,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( cos_real @ X )
% 0.32/0.66 = one_one_real )
% 0.32/0.66 => ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_one_sin_zero
% 0.32/0.66 thf(fact_8052_sin__diff,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( sin_real @ ( minus_minus_real @ X @ Y3 ) )
% 0.32/0.66 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_diff
% 0.32/0.66 thf(fact_8053_polar__Ex,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ? [R2: real,A5: real] :
% 0.32/0.66 ( ( X
% 0.32/0.66 = ( times_times_real @ R2 @ ( cos_real @ A5 ) ) )
% 0.32/0.66 & ( Y3
% 0.32/0.66 = ( times_times_real @ R2 @ ( sin_real @ A5 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % polar_Ex
% 0.32/0.66 thf(fact_8054_sin__add,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( sin_real @ ( plus_plus_real @ X @ Y3 ) )
% 0.32/0.66 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_add
% 0.32/0.66 thf(fact_8055_cos__diff,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( cos_real @ ( minus_minus_real @ X @ Y3 ) )
% 0.32/0.66 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_diff
% 0.32/0.66 thf(fact_8056_cos__add,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( cos_real @ ( plus_plus_real @ X @ Y3 ) )
% 0.32/0.66 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_add
% 0.32/0.66 thf(fact_8057_sin__zero__norm__cos__one,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 0.32/0.66 = one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_zero_norm_cos_one
% 0.32/0.66 thf(fact_8058_sin__zero__norm__cos__one,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( ( sin_complex @ X )
% 0.32/0.66 = zero_zero_complex )
% 0.32/0.66 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 0.32/0.66 = one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_zero_norm_cos_one
% 0.32/0.66 thf(fact_8059_sin__zero__abs__cos__one,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 => ( ( abs_abs_real @ ( cos_real @ X ) )
% 0.32/0.66 = one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_zero_abs_cos_one
% 0.32/0.66 thf(fact_8060_sin__double,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.32/0.66 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_double
% 0.32/0.66 thf(fact_8061_sin__double,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.32/0.66 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_double
% 0.32/0.66 thf(fact_8062_sincos__principal__value,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ? [Y4: real] :
% 0.32/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.32/0.66 & ( ord_less_eq_real @ Y4 @ pi )
% 0.32/0.66 & ( ( sin_real @ Y4 )
% 0.32/0.66 = ( sin_real @ X ) )
% 0.32/0.66 & ( ( cos_real @ Y4 )
% 0.32/0.66 = ( cos_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sincos_principal_value
% 0.32/0.66 thf(fact_8063_sin__x__le__x,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_x_le_x
% 0.32/0.66 thf(fact_8064_sin__le__one,axiom,
% 0.32/0.66 ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_le_one
% 0.32/0.66 thf(fact_8065_cos__le__one,axiom,
% 0.32/0.66 ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_le_one
% 0.32/0.66 thf(fact_8066_abs__sin__x__le__abs__x,axiom,
% 0.32/0.66 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % abs_sin_x_le_abs_x
% 0.32/0.66 thf(fact_8067_pochhammer__pos,axiom,
% 0.32/0.66 ! [X: real,N: nat] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_pos
% 0.32/0.66 thf(fact_8068_pochhammer__pos,axiom,
% 0.32/0.66 ! [X: rat,N: nat] :
% 0.32/0.66 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.66 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_pos
% 0.32/0.66 thf(fact_8069_pochhammer__pos,axiom,
% 0.32/0.66 ! [X: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.66 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_pos
% 0.32/0.66 thf(fact_8070_pochhammer__pos,axiom,
% 0.32/0.66 ! [X: int,N: nat] :
% 0.32/0.66 ( ( ord_less_int @ zero_zero_int @ X )
% 0.32/0.66 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_pos
% 0.32/0.66 thf(fact_8071_pochhammer__eq__0__mono,axiom,
% 0.32/0.66 ! [A: complex,N: nat,M: nat] :
% 0.32/0.66 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.32/0.66 = zero_zero_complex )
% 0.32/0.66 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.32/0.66 = zero_zero_complex ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_eq_0_mono
% 0.32/0.66 thf(fact_8072_pochhammer__eq__0__mono,axiom,
% 0.32/0.66 ! [A: real,N: nat,M: nat] :
% 0.32/0.66 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.32/0.66 = zero_zero_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_eq_0_mono
% 0.32/0.66 thf(fact_8073_pochhammer__eq__0__mono,axiom,
% 0.32/0.66 ! [A: rat,N: nat,M: nat] :
% 0.32/0.66 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.32/0.66 = zero_zero_rat )
% 0.32/0.66 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.32/0.66 = zero_zero_rat ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_eq_0_mono
% 0.32/0.66 thf(fact_8074_pochhammer__neq__0__mono,axiom,
% 0.32/0.66 ! [A: complex,M: nat,N: nat] :
% 0.32/0.66 ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.32/0.66 != zero_zero_complex )
% 0.32/0.66 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.32/0.66 != zero_zero_complex ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_neq_0_mono
% 0.32/0.66 thf(fact_8075_pochhammer__neq__0__mono,axiom,
% 0.32/0.66 ! [A: real,M: nat,N: nat] :
% 0.32/0.66 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.32/0.66 != zero_zero_real )
% 0.32/0.66 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.32/0.66 != zero_zero_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_neq_0_mono
% 0.32/0.66 thf(fact_8076_pochhammer__neq__0__mono,axiom,
% 0.32/0.66 ! [A: rat,M: nat,N: nat] :
% 0.32/0.66 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.32/0.66 != zero_zero_rat )
% 0.32/0.66 => ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.32/0.66 != zero_zero_rat ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_neq_0_mono
% 0.32/0.66 thf(fact_8077_sin__cos__le1,axiom,
% 0.32/0.66 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % sin_cos_le1
% 0.32/0.66 thf(fact_8078_cos__squared__eq,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_squared_eq
% 0.32/0.66 thf(fact_8079_cos__squared__eq,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_squared_eq
% 0.32/0.66 thf(fact_8080_sin__squared__eq,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_squared_eq
% 0.32/0.66 thf(fact_8081_sin__squared__eq,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_squared_eq
% 0.32/0.66 thf(fact_8082_sin__gt__zero,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ pi )
% 0.32/0.66 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_gt_zero
% 0.32/0.66 thf(fact_8083_sin__x__ge__neg__x,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_x_ge_neg_x
% 0.32/0.66 thf(fact_8084_sin__ge__zero,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.32/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_ge_zero
% 0.32/0.66 thf(fact_8085_sin__ge__minus__one,axiom,
% 0.32/0.66 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_ge_minus_one
% 0.32/0.66 thf(fact_8086_cos__monotone__0__pi__le,axiom,
% 0.32/0.66 ! [Y3: real,X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.32/0.66 => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_monotone_0_pi_le
% 0.32/0.66 thf(fact_8087_cos__mono__le__eq,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.32/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.32/0.66 => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) )
% 0.32/0.66 = ( ord_less_eq_real @ Y3 @ X ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_mono_le_eq
% 0.32/0.66 thf(fact_8088_cos__inj__pi,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.32/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.32/0.66 => ( ( ( cos_real @ X )
% 0.32/0.66 = ( cos_real @ Y3 ) )
% 0.32/0.66 => ( X = Y3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_inj_pi
% 0.32/0.66 thf(fact_8089_cos__ge__minus__one,axiom,
% 0.32/0.66 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_ge_minus_one
% 0.32/0.66 thf(fact_8090_abs__sin__le__one,axiom,
% 0.32/0.66 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % abs_sin_le_one
% 0.32/0.66 thf(fact_8091_abs__cos__le__one,axiom,
% 0.32/0.66 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % abs_cos_le_one
% 0.32/0.66 thf(fact_8092_sin__times__sin,axiom,
% 0.32/0.66 ! [W: complex,Z: complex] :
% 0.32/0.66 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.32/0.66 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_times_sin
% 0.32/0.66 thf(fact_8093_sin__times__sin,axiom,
% 0.32/0.66 ! [W: real,Z: real] :
% 0.32/0.66 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_times_sin
% 0.32/0.66 thf(fact_8094_sin__times__cos,axiom,
% 0.32/0.66 ! [W: complex,Z: complex] :
% 0.32/0.66 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 0.32/0.66 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_times_cos
% 0.32/0.66 thf(fact_8095_sin__times__cos,axiom,
% 0.32/0.66 ! [W: real,Z: real] :
% 0.32/0.66 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_times_cos
% 0.32/0.66 thf(fact_8096_cos__times__sin,axiom,
% 0.32/0.66 ! [W: complex,Z: complex] :
% 0.32/0.66 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 0.32/0.66 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_times_sin
% 0.32/0.66 thf(fact_8097_cos__times__sin,axiom,
% 0.32/0.66 ! [W: real,Z: real] :
% 0.32/0.66 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_times_sin
% 0.32/0.66 thf(fact_8098_sin__plus__sin,axiom,
% 0.32/0.66 ! [W: complex,Z: complex] :
% 0.32/0.66 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_plus_sin
% 0.32/0.66 thf(fact_8099_sin__plus__sin,axiom,
% 0.32/0.66 ! [W: real,Z: real] :
% 0.32/0.66 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_plus_sin
% 0.32/0.66 thf(fact_8100_sin__diff__sin,axiom,
% 0.32/0.66 ! [W: complex,Z: complex] :
% 0.32/0.66 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_diff_sin
% 0.32/0.66 thf(fact_8101_sin__diff__sin,axiom,
% 0.32/0.66 ! [W: real,Z: real] :
% 0.32/0.66 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_diff_sin
% 0.32/0.66 thf(fact_8102_cos__diff__cos,axiom,
% 0.32/0.66 ! [W: complex,Z: complex] :
% 0.32/0.66 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_diff_cos
% 0.32/0.66 thf(fact_8103_cos__diff__cos,axiom,
% 0.32/0.66 ! [W: real,Z: real] :
% 0.32/0.66 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_diff_cos
% 0.32/0.66 thf(fact_8104_pochhammer__nonneg,axiom,
% 0.32/0.66 ! [X: real,N: nat] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_nonneg
% 0.32/0.66 thf(fact_8105_pochhammer__nonneg,axiom,
% 0.32/0.66 ! [X: rat,N: nat] :
% 0.32/0.66 ( ( ord_less_rat @ zero_zero_rat @ X )
% 0.32/0.66 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_nonneg
% 0.32/0.66 thf(fact_8106_pochhammer__nonneg,axiom,
% 0.32/0.66 ! [X: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_nat @ zero_zero_nat @ X )
% 0.32/0.66 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_nonneg
% 0.32/0.66 thf(fact_8107_pochhammer__nonneg,axiom,
% 0.32/0.66 ! [X: int,N: nat] :
% 0.32/0.66 ( ( ord_less_int @ zero_zero_int @ X )
% 0.32/0.66 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_nonneg
% 0.32/0.66 thf(fact_8108_cos__double,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_double
% 0.32/0.66 thf(fact_8109_cos__double,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_double
% 0.32/0.66 thf(fact_8110_pochhammer__0__left,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( ( N = zero_zero_nat )
% 0.32/0.66 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.32/0.66 = one_one_complex ) )
% 0.32/0.66 & ( ( N != zero_zero_nat )
% 0.32/0.66 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 0.32/0.66 = zero_zero_complex ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0_left
% 0.32/0.66 thf(fact_8111_pochhammer__0__left,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( ( N = zero_zero_nat )
% 0.32/0.66 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.32/0.66 = one_one_real ) )
% 0.32/0.66 & ( ( N != zero_zero_nat )
% 0.32/0.66 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 0.32/0.66 = zero_zero_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0_left
% 0.32/0.66 thf(fact_8112_pochhammer__0__left,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( ( N = zero_zero_nat )
% 0.32/0.66 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.32/0.66 = one_one_rat ) )
% 0.32/0.66 & ( ( N != zero_zero_nat )
% 0.32/0.66 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 0.32/0.66 = zero_zero_rat ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0_left
% 0.32/0.66 thf(fact_8113_pochhammer__0__left,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( ( N = zero_zero_nat )
% 0.32/0.66 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.32/0.66 = one_one_nat ) )
% 0.32/0.66 & ( ( N != zero_zero_nat )
% 0.32/0.66 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 0.32/0.66 = zero_zero_nat ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0_left
% 0.32/0.66 thf(fact_8114_pochhammer__0__left,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( ( N = zero_zero_nat )
% 0.32/0.66 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.32/0.66 = one_one_int ) )
% 0.32/0.66 & ( ( N != zero_zero_nat )
% 0.32/0.66 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 0.32/0.66 = zero_zero_int ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_0_left
% 0.32/0.66 thf(fact_8115_cos__double__sin,axiom,
% 0.32/0.66 ! [W: complex] :
% 0.32/0.66 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.32/0.66 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_double_sin
% 0.32/0.66 thf(fact_8116_cos__double__sin,axiom,
% 0.32/0.66 ! [W: real] :
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_double_sin
% 0.32/0.66 thf(fact_8117_cos__two__neq__zero,axiom,
% 0.32/0.66 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.66 != zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % cos_two_neq_zero
% 0.32/0.66 thf(fact_8118_cos__monotone__0__pi,axiom,
% 0.32/0.66 ! [Y3: real,X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.66 => ( ( ord_less_real @ Y3 @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.32/0.66 => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_monotone_0_pi
% 0.32/0.66 thf(fact_8119_cos__mono__less__eq,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ pi )
% 0.32/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.32/0.66 => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) )
% 0.32/0.66 = ( ord_less_real @ Y3 @ X ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_mono_less_eq
% 0.32/0.66 thf(fact_8120_sin__eq__0__pi,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ pi )
% 0.32/0.66 => ( ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 => ( X = zero_zero_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_eq_0_pi
% 0.32/0.66 thf(fact_8121_sin__zero__pi__iff,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 0.32/0.66 => ( ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 = ( X = zero_zero_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_zero_pi_iff
% 0.32/0.66 thf(fact_8122_cos__monotone__minus__pi__0_H,axiom,
% 0.32/0.66 ! [Y3: real,X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.66 => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_monotone_minus_pi_0'
% 0.32/0.66 thf(fact_8123_sin__zero__iff__int2,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 = ( ? [I4: int] :
% 0.32/0.66 ( X
% 0.32/0.66 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_zero_iff_int2
% 0.32/0.66 thf(fact_8124_pochhammer__rec,axiom,
% 0.32/0.66 ! [A: complex,N: nat] :
% 0.32/0.66 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec
% 0.32/0.66 thf(fact_8125_pochhammer__rec,axiom,
% 0.32/0.66 ! [A: real,N: nat] :
% 0.32/0.66 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec
% 0.32/0.66 thf(fact_8126_pochhammer__rec,axiom,
% 0.32/0.66 ! [A: rat,N: nat] :
% 0.32/0.66 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec
% 0.32/0.66 thf(fact_8127_pochhammer__rec,axiom,
% 0.32/0.66 ! [A: nat,N: nat] :
% 0.32/0.66 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec
% 0.32/0.66 thf(fact_8128_pochhammer__rec,axiom,
% 0.32/0.66 ! [A: int,N: nat] :
% 0.32/0.66 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec
% 0.32/0.66 thf(fact_8129_diffs__def,axiom,
% 0.32/0.66 ( diffs_int
% 0.32/0.66 = ( ^ [C3: nat > int,N2: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % diffs_def
% 0.32/0.66 thf(fact_8130_diffs__def,axiom,
% 0.32/0.66 ( diffs_real
% 0.32/0.66 = ( ^ [C3: nat > real,N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % diffs_def
% 0.32/0.66 thf(fact_8131_diffs__def,axiom,
% 0.32/0.66 ( diffs_complex
% 0.32/0.66 = ( ^ [C3: nat > complex,N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % diffs_def
% 0.32/0.66 thf(fact_8132_diffs__def,axiom,
% 0.32/0.66 ( diffs_rat
% 0.32/0.66 = ( ^ [C3: nat > rat,N2: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % diffs_def
% 0.32/0.66 thf(fact_8133_pochhammer__Suc,axiom,
% 0.32/0.66 ! [A: int,N: nat] :
% 0.32/0.66 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_Suc
% 0.32/0.66 thf(fact_8134_pochhammer__Suc,axiom,
% 0.32/0.66 ! [A: real,N: nat] :
% 0.32/0.66 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_Suc
% 0.32/0.66 thf(fact_8135_pochhammer__Suc,axiom,
% 0.32/0.66 ! [A: nat,N: nat] :
% 0.32/0.66 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_Suc
% 0.32/0.66 thf(fact_8136_pochhammer__Suc,axiom,
% 0.32/0.66 ! [A: complex,N: nat] :
% 0.32/0.66 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_Suc
% 0.32/0.66 thf(fact_8137_pochhammer__Suc,axiom,
% 0.32/0.66 ! [A: rat,N: nat] :
% 0.32/0.66 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_Suc
% 0.32/0.66 thf(fact_8138_pochhammer__rec_H,axiom,
% 0.32/0.66 ! [Z: int,N: nat] :
% 0.32/0.66 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec'
% 0.32/0.66 thf(fact_8139_pochhammer__rec_H,axiom,
% 0.32/0.66 ! [Z: real,N: nat] :
% 0.32/0.66 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec'
% 0.32/0.66 thf(fact_8140_pochhammer__rec_H,axiom,
% 0.32/0.66 ! [Z: nat,N: nat] :
% 0.32/0.66 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec'
% 0.32/0.66 thf(fact_8141_pochhammer__rec_H,axiom,
% 0.32/0.66 ! [Z: complex,N: nat] :
% 0.32/0.66 ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec'
% 0.32/0.66 thf(fact_8142_pochhammer__rec_H,axiom,
% 0.32/0.66 ! [Z: rat,N: nat] :
% 0.32/0.66 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_rec'
% 0.32/0.66 thf(fact_8143_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ord_less_nat @ N @ K )
% 0.32/0.66 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.32/0.66 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma
% 0.32/0.66 thf(fact_8144_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ord_less_nat @ N @ K )
% 0.32/0.66 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.32/0.66 = zero_zero_int ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma
% 0.32/0.66 thf(fact_8145_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ord_less_nat @ N @ K )
% 0.32/0.66 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.32/0.66 = zero_zero_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma
% 0.32/0.66 thf(fact_8146_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ord_less_nat @ N @ K )
% 0.32/0.66 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.32/0.66 = zero_zero_complex ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma
% 0.32/0.66 thf(fact_8147_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ord_less_nat @ N @ K )
% 0.32/0.66 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.32/0.66 = zero_zero_rat ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma
% 0.32/0.66 thf(fact_8148_pochhammer__of__nat__eq__0__iff,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.32/0.66 = zero_z3403309356797280102nteger )
% 0.32/0.66 = ( ord_less_nat @ N @ K ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_iff
% 0.32/0.66 thf(fact_8149_pochhammer__of__nat__eq__0__iff,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.32/0.66 = zero_zero_int )
% 0.32/0.66 = ( ord_less_nat @ N @ K ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_iff
% 0.32/0.66 thf(fact_8150_pochhammer__of__nat__eq__0__iff,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 = ( ord_less_nat @ N @ K ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_iff
% 0.32/0.66 thf(fact_8151_pochhammer__of__nat__eq__0__iff,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.32/0.66 = zero_zero_complex )
% 0.32/0.66 = ( ord_less_nat @ N @ K ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_iff
% 0.32/0.66 thf(fact_8152_pochhammer__of__nat__eq__0__iff,axiom,
% 0.32/0.66 ! [N: nat,K: nat] :
% 0.32/0.66 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.32/0.66 = zero_zero_rat )
% 0.32/0.66 = ( ord_less_nat @ N @ K ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_iff
% 0.32/0.66 thf(fact_8153_pochhammer__eq__0__iff,axiom,
% 0.32/0.66 ! [A: real,N: nat] :
% 0.32/0.66 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 = ( ? [K2: nat] :
% 0.32/0.66 ( ( ord_less_nat @ K2 @ N )
% 0.32/0.66 & ( A
% 0.32/0.66 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_eq_0_iff
% 0.32/0.66 thf(fact_8154_pochhammer__eq__0__iff,axiom,
% 0.32/0.66 ! [A: complex,N: nat] :
% 0.32/0.66 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 0.32/0.66 = zero_zero_complex )
% 0.32/0.66 = ( ? [K2: nat] :
% 0.32/0.66 ( ( ord_less_nat @ K2 @ N )
% 0.32/0.66 & ( A
% 0.32/0.66 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_eq_0_iff
% 0.32/0.66 thf(fact_8155_pochhammer__eq__0__iff,axiom,
% 0.32/0.66 ! [A: rat,N: nat] :
% 0.32/0.66 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 0.32/0.66 = zero_zero_rat )
% 0.32/0.66 = ( ? [K2: nat] :
% 0.32/0.66 ( ( ord_less_nat @ K2 @ N )
% 0.32/0.66 & ( A
% 0.32/0.66 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_eq_0_iff
% 0.32/0.66 thf(fact_8156_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.32/0.66 ! [K: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.66 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 0.32/0.66 != zero_z3403309356797280102nteger ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma'
% 0.32/0.66 thf(fact_8157_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.32/0.66 ! [K: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.66 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 0.32/0.66 != zero_zero_int ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma'
% 0.32/0.66 thf(fact_8158_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.32/0.66 ! [K: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.66 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 0.32/0.66 != zero_zero_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma'
% 0.32/0.66 thf(fact_8159_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.32/0.66 ! [K: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.66 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 0.32/0.66 != zero_zero_complex ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma'
% 0.32/0.66 thf(fact_8160_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.32/0.66 ! [K: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.66 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 0.32/0.66 != zero_zero_rat ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_of_nat_eq_0_lemma'
% 0.32/0.66 thf(fact_8161_sincos__total__pi,axiom,
% 0.32/0.66 ! [Y3: real,X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.66 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.66 = one_one_real )
% 0.32/0.66 => ? [T5: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.32/0.66 & ( ord_less_eq_real @ T5 @ pi )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( cos_real @ T5 ) )
% 0.32/0.66 & ( Y3
% 0.32/0.66 = ( sin_real @ T5 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sincos_total_pi
% 0.32/0.66 thf(fact_8162_pochhammer__product_H,axiom,
% 0.32/0.66 ! [Z: int,N: nat,M: nat] :
% 0.32/0.66 ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.66 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product'
% 0.32/0.66 thf(fact_8163_pochhammer__product_H,axiom,
% 0.32/0.66 ! [Z: real,N: nat,M: nat] :
% 0.32/0.66 ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.66 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product'
% 0.32/0.66 thf(fact_8164_pochhammer__product_H,axiom,
% 0.32/0.66 ! [Z: nat,N: nat,M: nat] :
% 0.32/0.66 ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.66 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product'
% 0.32/0.66 thf(fact_8165_pochhammer__product_H,axiom,
% 0.32/0.66 ! [Z: complex,N: nat,M: nat] :
% 0.32/0.66 ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.66 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product'
% 0.32/0.66 thf(fact_8166_pochhammer__product_H,axiom,
% 0.32/0.66 ! [Z: rat,N: nat,M: nat] :
% 0.32/0.66 ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 0.32/0.66 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product'
% 0.32/0.66 thf(fact_8167_sin__expansion__lemma,axiom,
% 0.32/0.66 ! [X: real,M: nat] :
% 0.32/0.66 ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.32/0.66 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_expansion_lemma
% 0.32/0.66 thf(fact_8168_cos__expansion__lemma,axiom,
% 0.32/0.66 ! [X: real,M: nat] :
% 0.32/0.66 ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.32/0.66 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_expansion_lemma
% 0.32/0.66 thf(fact_8169_termdiff__converges__all,axiom,
% 0.32/0.66 ! [C: nat > complex,X: complex] :
% 0.32/0.66 ( ! [X3: complex] :
% 0.32/0.66 ( summable_complex
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
% 0.32/0.66 => ( summable_complex
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % termdiff_converges_all
% 0.32/0.66 thf(fact_8170_termdiff__converges__all,axiom,
% 0.32/0.66 ! [C: nat > real,X: real] :
% 0.32/0.66 ( ! [X3: real] :
% 0.32/0.66 ( summable_real
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
% 0.32/0.66 => ( summable_real
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % termdiff_converges_all
% 0.32/0.66 thf(fact_8171_sin__gt__zero__02,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.66 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_gt_zero_02
% 0.32/0.66 thf(fact_8172_cos__two__less__zero,axiom,
% 0.32/0.66 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.32/0.66
% 0.32/0.66 % cos_two_less_zero
% 0.32/0.66 thf(fact_8173_cos__is__zero,axiom,
% 0.32/0.66 ? [X3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.32/0.66 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.66 & ( ( cos_real @ X3 )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 & ! [Y5: real] :
% 0.32/0.66 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.32/0.66 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.66 & ( ( cos_real @ Y5 )
% 0.32/0.66 = zero_zero_real ) )
% 0.32/0.66 => ( Y5 = X3 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_is_zero
% 0.32/0.66 thf(fact_8174_cos__two__le__zero,axiom,
% 0.32/0.66 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.32/0.66
% 0.32/0.66 % cos_two_le_zero
% 0.32/0.66 thf(fact_8175_cos__monotone__minus__pi__0,axiom,
% 0.32/0.66 ! [Y3: real,X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_real @ Y3 @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.66 => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_monotone_minus_pi_0
% 0.32/0.66 thf(fact_8176_cos__total,axiom,
% 0.32/0.66 ! [Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.66 => ? [X3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.32/0.66 & ( ord_less_eq_real @ X3 @ pi )
% 0.32/0.66 & ( ( cos_real @ X3 )
% 0.32/0.66 = Y3 )
% 0.32/0.66 & ! [Y5: real] :
% 0.32/0.66 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.32/0.66 & ( ord_less_eq_real @ Y5 @ pi )
% 0.32/0.66 & ( ( cos_real @ Y5 )
% 0.32/0.66 = Y3 ) )
% 0.32/0.66 => ( Y5 = X3 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_total
% 0.32/0.66 thf(fact_8177_sincos__total__pi__half,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.66 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.66 = one_one_real )
% 0.32/0.66 => ? [T5: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.32/0.66 & ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( cos_real @ T5 ) )
% 0.32/0.66 & ( Y3
% 0.32/0.66 = ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sincos_total_pi_half
% 0.32/0.66 thf(fact_8178_sincos__total__2pi__le,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.66 = one_one_real )
% 0.32/0.66 => ? [T5: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.32/0.66 & ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( cos_real @ T5 ) )
% 0.32/0.66 & ( Y3
% 0.32/0.66 = ( sin_real @ T5 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sincos_total_2pi_le
% 0.32/0.66 thf(fact_8179_pochhammer__product,axiom,
% 0.32/0.66 ! [M: nat,N: nat,Z: int] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 0.32/0.66 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product
% 0.32/0.66 thf(fact_8180_pochhammer__product,axiom,
% 0.32/0.66 ! [M: nat,N: nat,Z: real] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 0.32/0.66 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product
% 0.32/0.66 thf(fact_8181_pochhammer__product,axiom,
% 0.32/0.66 ! [M: nat,N: nat,Z: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 0.32/0.66 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product
% 0.32/0.66 thf(fact_8182_pochhammer__product,axiom,
% 0.32/0.66 ! [M: nat,N: nat,Z: complex] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ( comm_s2602460028002588243omplex @ Z @ N )
% 0.32/0.66 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product
% 0.32/0.66 thf(fact_8183_pochhammer__product,axiom,
% 0.32/0.66 ! [M: nat,N: nat,Z: rat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 0.32/0.66 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_product
% 0.32/0.66 thf(fact_8184_sincos__total__2pi,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.66 = one_one_real )
% 0.32/0.66 => ~ ! [T5: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.32/0.66 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.32/0.66 => ( ( X
% 0.32/0.66 = ( cos_real @ T5 ) )
% 0.32/0.66 => ( Y3
% 0.32/0.66 != ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sincos_total_2pi
% 0.32/0.66 thf(fact_8185_sin__pi__divide__n__ge__0,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( N != zero_zero_nat )
% 0.32/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_pi_divide_n_ge_0
% 0.32/0.66 thf(fact_8186_pochhammer__absorb__comp,axiom,
% 0.32/0.66 ! [R: code_integer,K: nat] :
% 0.32/0.66 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
% 0.32/0.66 = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_absorb_comp
% 0.32/0.66 thf(fact_8187_pochhammer__absorb__comp,axiom,
% 0.32/0.66 ! [R: int,K: nat] :
% 0.32/0.66 ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
% 0.32/0.66 = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_absorb_comp
% 0.32/0.66 thf(fact_8188_pochhammer__absorb__comp,axiom,
% 0.32/0.66 ! [R: real,K: nat] :
% 0.32/0.66 ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
% 0.32/0.66 = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_absorb_comp
% 0.32/0.66 thf(fact_8189_pochhammer__absorb__comp,axiom,
% 0.32/0.66 ! [R: complex,K: nat] :
% 0.32/0.66 ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
% 0.32/0.66 = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_absorb_comp
% 0.32/0.66 thf(fact_8190_pochhammer__absorb__comp,axiom,
% 0.32/0.66 ! [R: rat,K: nat] :
% 0.32/0.66 ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
% 0.32/0.66 = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_absorb_comp
% 0.32/0.66 thf(fact_8191_cos__times__cos,axiom,
% 0.32/0.66 ! [W: complex,Z: complex] :
% 0.32/0.66 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.32/0.66 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_times_cos
% 0.32/0.66 thf(fact_8192_cos__times__cos,axiom,
% 0.32/0.66 ! [W: real,Z: real] :
% 0.32/0.66 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_times_cos
% 0.32/0.66 thf(fact_8193_cos__plus__cos,axiom,
% 0.32/0.66 ! [W: complex,Z: complex] :
% 0.32/0.66 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_plus_cos
% 0.32/0.66 thf(fact_8194_cos__plus__cos,axiom,
% 0.32/0.66 ! [W: real,Z: real] :
% 0.32/0.66 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 0.32/0.66 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_plus_cos
% 0.32/0.66 thf(fact_8195_sin__gt__zero2,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_gt_zero2
% 0.32/0.66 thf(fact_8196_sin__lt__zero,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ pi @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.32/0.66 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_lt_zero
% 0.32/0.66 thf(fact_8197_cos__double__less__one,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.66 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_double_less_one
% 0.32/0.66 thf(fact_8198_sin__30,axiom,
% 0.32/0.66 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.32/0.66 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_30
% 0.32/0.66 thf(fact_8199_cos__gt__zero,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_gt_zero
% 0.32/0.66 thf(fact_8200_sin__inj__pi,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ( ( sin_real @ X )
% 0.32/0.66 = ( sin_real @ Y3 ) )
% 0.32/0.66 => ( X = Y3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_inj_pi
% 0.32/0.66 thf(fact_8201_sin__mono__le__eq,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) )
% 0.32/0.66 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_mono_le_eq
% 0.32/0.66 thf(fact_8202_sin__monotone__2pi__le,axiom,
% 0.32/0.66 ! [Y3: real,X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_monotone_2pi_le
% 0.32/0.66 thf(fact_8203_cos__60,axiom,
% 0.32/0.66 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.32/0.66 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_60
% 0.32/0.66 thf(fact_8204_cos__one__2pi__int,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( cos_real @ X )
% 0.32/0.66 = one_one_real )
% 0.32/0.66 = ( ? [X2: int] :
% 0.32/0.66 ( X
% 0.32/0.66 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_one_2pi_int
% 0.32/0.66 thf(fact_8205_cos__double__cos,axiom,
% 0.32/0.66 ! [W: complex] :
% 0.32/0.66 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 0.32/0.66 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_double_cos
% 0.32/0.66 thf(fact_8206_cos__double__cos,axiom,
% 0.32/0.66 ! [W: real] :
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 0.32/0.66 = ( 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 ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_double_cos
% 0.32/0.66 thf(fact_8207_cos__treble__cos,axiom,
% 0.32/0.66 ! [X: complex] :
% 0.32/0.66 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 0.32/0.66 = ( 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 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_treble_cos
% 0.32/0.66 thf(fact_8208_cos__treble__cos,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 0.32/0.66 = ( 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 ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_treble_cos
% 0.32/0.66 thf(fact_8209_pochhammer__minus_H,axiom,
% 0.32/0.66 ! [B: code_integer,K: nat] :
% 0.32/0.66 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 0.32/0.66 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus'
% 0.32/0.66 thf(fact_8210_pochhammer__minus_H,axiom,
% 0.32/0.66 ! [B: int,K: nat] :
% 0.32/0.66 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 0.32/0.66 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus'
% 0.32/0.66 thf(fact_8211_pochhammer__minus_H,axiom,
% 0.32/0.66 ! [B: real,K: nat] :
% 0.32/0.66 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 0.32/0.66 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus'
% 0.32/0.66 thf(fact_8212_pochhammer__minus_H,axiom,
% 0.32/0.66 ! [B: complex,K: nat] :
% 0.32/0.66 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 0.32/0.66 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus'
% 0.32/0.66 thf(fact_8213_pochhammer__minus_H,axiom,
% 0.32/0.66 ! [B: rat,K: nat] :
% 0.32/0.66 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 0.32/0.66 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus'
% 0.32/0.66 thf(fact_8214_pochhammer__minus,axiom,
% 0.32/0.66 ! [B: code_integer,K: nat] :
% 0.32/0.66 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 0.32/0.66 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus
% 0.32/0.66 thf(fact_8215_pochhammer__minus,axiom,
% 0.32/0.66 ! [B: int,K: nat] :
% 0.32/0.66 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 0.32/0.66 = ( 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 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus
% 0.32/0.66 thf(fact_8216_pochhammer__minus,axiom,
% 0.32/0.66 ! [B: real,K: nat] :
% 0.32/0.66 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 0.32/0.66 = ( 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 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus
% 0.32/0.66 thf(fact_8217_pochhammer__minus,axiom,
% 0.32/0.66 ! [B: complex,K: nat] :
% 0.32/0.66 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 0.32/0.66 = ( 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 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus
% 0.32/0.66 thf(fact_8218_pochhammer__minus,axiom,
% 0.32/0.66 ! [B: rat,K: nat] :
% 0.32/0.66 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 0.32/0.66 = ( 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 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_minus
% 0.32/0.66 thf(fact_8219_termdiff__converges,axiom,
% 0.32/0.66 ! [X: real,K5: real,C: nat > real] :
% 0.32/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
% 0.32/0.66 => ( ! [X3: real] :
% 0.32/0.66 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
% 0.32/0.66 => ( summable_real
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) )
% 0.32/0.66 => ( summable_real
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % termdiff_converges
% 0.32/0.66 thf(fact_8220_termdiff__converges,axiom,
% 0.32/0.66 ! [X: complex,K5: real,C: nat > complex] :
% 0.32/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
% 0.32/0.66 => ( ! [X3: complex] :
% 0.32/0.66 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
% 0.32/0.66 => ( summable_complex
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) )
% 0.32/0.66 => ( summable_complex
% 0.32/0.66 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % termdiff_converges
% 0.32/0.66 thf(fact_8221_sin__le__zero,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ pi @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.32/0.66 => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_le_zero
% 0.32/0.66 thf(fact_8222_sin__less__zero,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.66 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_less_zero
% 0.32/0.66 thf(fact_8223_sin__monotone__2pi,axiom,
% 0.32/0.66 ! [Y3: real,X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_real @ Y3 @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_monotone_2pi
% 0.32/0.66 thf(fact_8224_sin__mono__less__eq,axiom,
% 0.32/0.66 ! [X: real,Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) )
% 0.32/0.66 = ( ord_less_real @ X @ Y3 ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_mono_less_eq
% 0.32/0.66 thf(fact_8225_sin__total,axiom,
% 0.32/0.66 ! [Y3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.66 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.66 => ? [X3: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.32/0.66 & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 & ( ( sin_real @ X3 )
% 0.32/0.66 = Y3 )
% 0.32/0.66 & ! [Y5: real] :
% 0.32/0.66 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.32/0.66 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 & ( ( sin_real @ Y5 )
% 0.32/0.66 = Y3 ) )
% 0.32/0.66 => ( Y5 = X3 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_total
% 0.32/0.66 thf(fact_8226_cos__gt__zero__pi,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.66 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_gt_zero_pi
% 0.32/0.66 thf(fact_8227_cos__ge__zero,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.66 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_ge_zero
% 0.32/0.66 thf(fact_8228_cos__one__2pi,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( cos_real @ X )
% 0.32/0.66 = one_one_real )
% 0.32/0.66 = ( ? [X2: nat] :
% 0.32/0.66 ( X
% 0.32/0.66 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 0.32/0.66 | ? [X2: nat] :
% 0.32/0.66 ( X
% 0.32/0.66 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_one_2pi
% 0.32/0.66 thf(fact_8229_sin__pi__divide__n__gt__0,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.66 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_pi_divide_n_gt_0
% 0.32/0.66 thf(fact_8230_sin__zero__iff__int,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 = ( ? [I4: int] :
% 0.32/0.66 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_zero_iff_int
% 0.32/0.66 thf(fact_8231_cos__zero__iff__int,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( cos_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 = ( ? [I4: int] :
% 0.32/0.66 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_zero_iff_int
% 0.32/0.66 thf(fact_8232_sin__zero__lemma,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 => ? [N4: nat] :
% 0.32/0.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_zero_lemma
% 0.32/0.66 thf(fact_8233_sin__zero__iff,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( sin_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 = ( ? [N2: nat] :
% 0.32/0.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.66 | ? [N2: nat] :
% 0.32/0.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % sin_zero_iff
% 0.32/0.66 thf(fact_8234_cos__zero__lemma,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ( cos_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 => ? [N4: nat] :
% 0.32/0.66 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_zero_lemma
% 0.32/0.66 thf(fact_8235_cos__zero__iff,axiom,
% 0.32/0.66 ! [X: real] :
% 0.32/0.66 ( ( ( cos_real @ X )
% 0.32/0.66 = zero_zero_real )
% 0.32/0.66 = ( ? [N2: nat] :
% 0.32/0.66 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.66 | ? [N2: nat] :
% 0.32/0.66 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.32/0.66 & ( X
% 0.32/0.66 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % cos_zero_iff
% 0.32/0.66 thf(fact_8236_Maclaurin__cos__expansion2,axiom,
% 0.32/0.66 ! [X: real,N: nat] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.66 => ? [T5: real] :
% 0.32/0.66 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.32/0.66 & ( ord_less_real @ T5 @ X )
% 0.32/0.66 & ( ( cos_real @ X )
% 0.32/0.66 = ( plus_plus_real
% 0.32/0.66 @ ( groups6591440286371151544t_real
% 0.32/0.66 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % Maclaurin_cos_expansion2
% 0.32/0.66 thf(fact_8237_Maclaurin__minus__cos__expansion,axiom,
% 0.32/0.66 ! [N: nat,X: real] :
% 0.32/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.66 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.66 => ? [T5: real] :
% 0.32/0.66 ( ( ord_less_real @ X @ T5 )
% 0.32/0.66 & ( ord_less_real @ T5 @ zero_zero_real )
% 0.32/0.66 & ( ( cos_real @ X )
% 0.32/0.66 = ( plus_plus_real
% 0.32/0.66 @ ( groups6591440286371151544t_real
% 0.32/0.66 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % Maclaurin_minus_cos_expansion
% 0.32/0.66 thf(fact_8238_Maclaurin__cos__expansion,axiom,
% 0.32/0.66 ! [X: real,N: nat] :
% 0.32/0.66 ? [T5: real] :
% 0.32/0.66 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.32/0.66 & ( ( cos_real @ X )
% 0.32/0.66 = ( plus_plus_real
% 0.32/0.66 @ ( groups6591440286371151544t_real
% 0.32/0.66 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.66 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.66 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % Maclaurin_cos_expansion
% 0.32/0.66 thf(fact_8239_pochhammer__times__pochhammer__half,axiom,
% 0.32/0.66 ! [Z: real,N: nat] :
% 0.32/0.66 ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.32/0.66 = ( groups129246275422532515t_real
% 0.32/0.66 @ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.66 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_times_pochhammer_half
% 0.32/0.66 thf(fact_8240_pochhammer__times__pochhammer__half,axiom,
% 0.32/0.66 ! [Z: complex,N: nat] :
% 0.32/0.66 ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.32/0.66 = ( groups6464643781859351333omplex
% 0.32/0.66 @ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.32/0.66 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_times_pochhammer_half
% 0.32/0.66 thf(fact_8241_pochhammer__times__pochhammer__half,axiom,
% 0.32/0.66 ! [Z: rat,N: nat] :
% 0.32/0.66 ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 0.32/0.66 = ( groups73079841787564623at_rat
% 0.32/0.66 @ ^ [K2: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.32/0.66 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_times_pochhammer_half
% 0.32/0.66 thf(fact_8242_pochhammer__code,axiom,
% 0.32/0.66 ( comm_s4660882817536571857er_int
% 0.32/0.66 = ( ^ [A3: int,N2: nat] :
% 0.32/0.66 ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
% 0.32/0.66 @ ( set_fo2581907887559384638at_int
% 0.32/0.66 @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
% 0.32/0.66 @ zero_zero_nat
% 0.32/0.66 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.32/0.66 @ one_one_int ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_code
% 0.32/0.66 thf(fact_8243_pochhammer__code,axiom,
% 0.32/0.66 ( comm_s7457072308508201937r_real
% 0.32/0.66 = ( ^ [A3: real,N2: nat] :
% 0.32/0.66 ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
% 0.32/0.66 @ ( set_fo3111899725591712190t_real
% 0.32/0.66 @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ O ) ) )
% 0.32/0.66 @ zero_zero_nat
% 0.32/0.66 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.32/0.66 @ one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_code
% 0.32/0.66 thf(fact_8244_pochhammer__code,axiom,
% 0.32/0.66 ( comm_s2602460028002588243omplex
% 0.32/0.66 = ( ^ [A3: complex,N2: nat] :
% 0.32/0.66 ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
% 0.32/0.66 @ ( set_fo1517530859248394432omplex
% 0.32/0.66 @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ O ) ) )
% 0.32/0.66 @ zero_zero_nat
% 0.32/0.66 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.32/0.66 @ one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_code
% 0.32/0.66 thf(fact_8245_pochhammer__code,axiom,
% 0.32/0.66 ( comm_s4028243227959126397er_rat
% 0.32/0.66 = ( ^ [A3: rat,N2: nat] :
% 0.32/0.66 ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
% 0.32/0.66 @ ( set_fo1949268297981939178at_rat
% 0.32/0.66 @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ O ) ) )
% 0.32/0.66 @ zero_zero_nat
% 0.32/0.66 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.32/0.66 @ one_one_rat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_code
% 0.32/0.66 thf(fact_8246_pochhammer__code,axiom,
% 0.32/0.66 ( comm_s4663373288045622133er_nat
% 0.32/0.66 = ( ^ [A3: nat,N2: nat] :
% 0.32/0.66 ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
% 0.32/0.66 @ ( set_fo2584398358068434914at_nat
% 0.32/0.66 @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 0.32/0.66 @ zero_zero_nat
% 0.32/0.66 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 0.32/0.66 @ one_one_nat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % pochhammer_code
% 0.32/0.66 thf(fact_8247_prod_Oneutral__const,axiom,
% 0.32/0.66 ! [A2: set_nat] :
% 0.32/0.66 ( ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [Uu3: nat] : one_one_nat
% 0.32/0.66 @ A2 )
% 0.32/0.66 = one_one_nat ) ).
% 0.32/0.66
% 0.32/0.66 % prod.neutral_const
% 0.32/0.66 thf(fact_8248_prod_Oneutral__const,axiom,
% 0.32/0.66 ! [A2: set_nat] :
% 0.32/0.66 ( ( groups705719431365010083at_int
% 0.32/0.66 @ ^ [Uu3: nat] : one_one_int
% 0.32/0.66 @ A2 )
% 0.32/0.66 = one_one_int ) ).
% 0.32/0.66
% 0.32/0.66 % prod.neutral_const
% 0.32/0.66 thf(fact_8249_prod_Oneutral__const,axiom,
% 0.32/0.66 ! [A2: set_int] :
% 0.32/0.66 ( ( groups1705073143266064639nt_int
% 0.32/0.66 @ ^ [Uu3: int] : one_one_int
% 0.32/0.66 @ A2 )
% 0.32/0.66 = one_one_int ) ).
% 0.32/0.66
% 0.32/0.66 % prod.neutral_const
% 0.32/0.66 thf(fact_8250_prod_Oempty,axiom,
% 0.32/0.66 ! [G: nat > complex] :
% 0.32/0.66 ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8251_prod_Oempty,axiom,
% 0.32/0.66 ! [G: nat > real] :
% 0.32/0.66 ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8252_prod_Oempty,axiom,
% 0.32/0.66 ! [G: nat > rat] :
% 0.32/0.66 ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
% 0.32/0.66 = one_one_rat ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8253_prod_Oempty,axiom,
% 0.32/0.66 ! [G: int > complex] :
% 0.32/0.66 ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8254_prod_Oempty,axiom,
% 0.32/0.66 ! [G: int > real] :
% 0.32/0.66 ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8255_prod_Oempty,axiom,
% 0.32/0.66 ! [G: int > rat] :
% 0.32/0.66 ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
% 0.32/0.66 = one_one_rat ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8256_prod_Oempty,axiom,
% 0.32/0.66 ! [G: int > nat] :
% 0.32/0.66 ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
% 0.32/0.66 = one_one_nat ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8257_prod_Oempty,axiom,
% 0.32/0.66 ! [G: real > complex] :
% 0.32/0.66 ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8258_prod_Oempty,axiom,
% 0.32/0.66 ! [G: real > real] :
% 0.32/0.66 ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8259_prod_Oempty,axiom,
% 0.32/0.66 ! [G: real > rat] :
% 0.32/0.66 ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 0.32/0.66 = one_one_rat ) ).
% 0.32/0.66
% 0.32/0.66 % prod.empty
% 0.32/0.66 thf(fact_8260_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_nat,G: nat > complex] :
% 0.32/0.66 ( ~ ( finite_finite_nat @ A2 )
% 0.32/0.66 => ( ( groups6464643781859351333omplex @ G @ A2 )
% 0.32/0.66 = one_one_complex ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8261_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_int,G: int > complex] :
% 0.32/0.66 ( ~ ( finite_finite_int @ A2 )
% 0.32/0.66 => ( ( groups7440179247065528705omplex @ G @ A2 )
% 0.32/0.66 = one_one_complex ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8262_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_complex,G: complex > complex] :
% 0.32/0.66 ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.32/0.66 => ( ( groups3708469109370488835omplex @ G @ A2 )
% 0.32/0.66 = one_one_complex ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8263_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_nat,G: nat > real] :
% 0.32/0.66 ( ~ ( finite_finite_nat @ A2 )
% 0.32/0.66 => ( ( groups129246275422532515t_real @ G @ A2 )
% 0.32/0.66 = one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8264_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_int,G: int > real] :
% 0.32/0.66 ( ~ ( finite_finite_int @ A2 )
% 0.32/0.66 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.32/0.66 = one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8265_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_complex,G: complex > real] :
% 0.32/0.66 ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.32/0.66 => ( ( groups766887009212190081x_real @ G @ A2 )
% 0.32/0.66 = one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8266_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_nat,G: nat > rat] :
% 0.32/0.66 ( ~ ( finite_finite_nat @ A2 )
% 0.32/0.66 => ( ( groups73079841787564623at_rat @ G @ A2 )
% 0.32/0.66 = one_one_rat ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8267_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_int,G: int > rat] :
% 0.32/0.66 ( ~ ( finite_finite_int @ A2 )
% 0.32/0.66 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 0.32/0.66 = one_one_rat ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8268_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_complex,G: complex > rat] :
% 0.32/0.66 ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.32/0.66 => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 0.32/0.66 = one_one_rat ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8269_prod_Oinfinite,axiom,
% 0.32/0.66 ! [A2: set_int,G: int > nat] :
% 0.32/0.66 ( ~ ( finite_finite_int @ A2 )
% 0.32/0.66 => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 0.32/0.66 = one_one_nat ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.infinite
% 0.32/0.66 thf(fact_8270_fact__0,axiom,
% 0.32/0.66 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % fact_0
% 0.32/0.66 thf(fact_8271_fact__0,axiom,
% 0.32/0.66 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.32/0.66 = one_one_rat ) ).
% 0.32/0.66
% 0.32/0.66 % fact_0
% 0.32/0.66 thf(fact_8272_fact__0,axiom,
% 0.32/0.66 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.32/0.66 = one_one_int ) ).
% 0.32/0.66
% 0.32/0.66 % fact_0
% 0.32/0.66 thf(fact_8273_fact__0,axiom,
% 0.32/0.66 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % fact_0
% 0.32/0.66 thf(fact_8274_fact__0,axiom,
% 0.32/0.66 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.32/0.66 = one_one_nat ) ).
% 0.32/0.66
% 0.32/0.66 % fact_0
% 0.32/0.66 thf(fact_8275_fact__1,axiom,
% 0.32/0.66 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % fact_1
% 0.32/0.66 thf(fact_8276_fact__1,axiom,
% 0.32/0.66 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 0.32/0.66 = one_one_rat ) ).
% 0.32/0.66
% 0.32/0.66 % fact_1
% 0.32/0.66 thf(fact_8277_fact__1,axiom,
% 0.32/0.66 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 0.32/0.66 = one_one_int ) ).
% 0.32/0.66
% 0.32/0.66 % fact_1
% 0.32/0.66 thf(fact_8278_fact__1,axiom,
% 0.32/0.66 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % fact_1
% 0.32/0.66 thf(fact_8279_fact__1,axiom,
% 0.32/0.66 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 0.32/0.66 = one_one_nat ) ).
% 0.32/0.66
% 0.32/0.66 % fact_1
% 0.32/0.66 thf(fact_8280_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_real,A: real,B: real > complex] :
% 0.32/0.66 ( ( finite_finite_real @ S2 )
% 0.32/0.66 => ( ( ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups713298508707869441omplex
% 0.32/0.66 @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups713298508707869441omplex
% 0.32/0.66 @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8281_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_nat,A: nat,B: nat > complex] :
% 0.32/0.66 ( ( finite_finite_nat @ S2 )
% 0.32/0.66 => ( ( ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups6464643781859351333omplex
% 0.32/0.66 @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups6464643781859351333omplex
% 0.32/0.66 @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8282_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_int,A: int,B: int > complex] :
% 0.32/0.66 ( ( finite_finite_int @ S2 )
% 0.32/0.66 => ( ( ( member_int @ A @ S2 )
% 0.32/0.66 => ( ( groups7440179247065528705omplex
% 0.32/0.66 @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_int @ A @ S2 )
% 0.32/0.66 => ( ( groups7440179247065528705omplex
% 0.32/0.66 @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8283_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_complex,A: complex,B: complex > complex] :
% 0.32/0.66 ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.66 => ( ( ( member_complex @ A @ S2 )
% 0.32/0.66 => ( ( groups3708469109370488835omplex
% 0.32/0.66 @ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_complex @ A @ S2 )
% 0.32/0.66 => ( ( groups3708469109370488835omplex
% 0.32/0.66 @ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8284_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_real,A: real,B: real > real] :
% 0.32/0.66 ( ( finite_finite_real @ S2 )
% 0.32/0.66 => ( ( ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups1681761925125756287l_real
% 0.32/0.66 @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups1681761925125756287l_real
% 0.32/0.66 @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8285_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_nat,A: nat,B: nat > real] :
% 0.32/0.66 ( ( finite_finite_nat @ S2 )
% 0.32/0.66 => ( ( ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups129246275422532515t_real
% 0.32/0.66 @ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups129246275422532515t_real
% 0.32/0.66 @ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8286_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_int,A: int,B: int > real] :
% 0.32/0.66 ( ( finite_finite_int @ S2 )
% 0.32/0.66 => ( ( ( member_int @ A @ S2 )
% 0.32/0.66 => ( ( groups2316167850115554303t_real
% 0.32/0.66 @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_int @ A @ S2 )
% 0.32/0.66 => ( ( groups2316167850115554303t_real
% 0.32/0.66 @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8287_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_complex,A: complex,B: complex > real] :
% 0.32/0.66 ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.66 => ( ( ( member_complex @ A @ S2 )
% 0.32/0.66 => ( ( groups766887009212190081x_real
% 0.32/0.66 @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_complex @ A @ S2 )
% 0.32/0.66 => ( ( groups766887009212190081x_real
% 0.32/0.66 @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8288_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_real,A: real,B: real > rat] :
% 0.32/0.66 ( ( finite_finite_real @ S2 )
% 0.32/0.66 => ( ( ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups4061424788464935467al_rat
% 0.32/0.66 @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups4061424788464935467al_rat
% 0.32/0.66 @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_rat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8289_prod_Odelta_H,axiom,
% 0.32/0.66 ! [S2: set_nat,A: nat,B: nat > rat] :
% 0.32/0.66 ( ( finite_finite_nat @ S2 )
% 0.32/0.66 => ( ( ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups73079841787564623at_rat
% 0.32/0.66 @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups73079841787564623at_rat
% 0.32/0.66 @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_rat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta'
% 0.32/0.66 thf(fact_8290_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_real,A: real,B: real > complex] :
% 0.32/0.66 ( ( finite_finite_real @ S2 )
% 0.32/0.66 => ( ( ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups713298508707869441omplex
% 0.32/0.66 @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups713298508707869441omplex
% 0.32/0.66 @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8291_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_nat,A: nat,B: nat > complex] :
% 0.32/0.66 ( ( finite_finite_nat @ S2 )
% 0.32/0.66 => ( ( ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups6464643781859351333omplex
% 0.32/0.66 @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups6464643781859351333omplex
% 0.32/0.66 @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8292_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_int,A: int,B: int > complex] :
% 0.32/0.66 ( ( finite_finite_int @ S2 )
% 0.32/0.66 => ( ( ( member_int @ A @ S2 )
% 0.32/0.66 => ( ( groups7440179247065528705omplex
% 0.32/0.66 @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_int @ A @ S2 )
% 0.32/0.66 => ( ( groups7440179247065528705omplex
% 0.32/0.66 @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8293_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_complex,A: complex,B: complex > complex] :
% 0.32/0.66 ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.66 => ( ( ( member_complex @ A @ S2 )
% 0.32/0.66 => ( ( groups3708469109370488835omplex
% 0.32/0.66 @ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_complex @ A @ S2 )
% 0.32/0.66 => ( ( groups3708469109370488835omplex
% 0.32/0.66 @ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_complex ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8294_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_real,A: real,B: real > real] :
% 0.32/0.66 ( ( finite_finite_real @ S2 )
% 0.32/0.66 => ( ( ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups1681761925125756287l_real
% 0.32/0.66 @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups1681761925125756287l_real
% 0.32/0.66 @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8295_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_nat,A: nat,B: nat > real] :
% 0.32/0.66 ( ( finite_finite_nat @ S2 )
% 0.32/0.66 => ( ( ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups129246275422532515t_real
% 0.32/0.66 @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups129246275422532515t_real
% 0.32/0.66 @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8296_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_int,A: int,B: int > real] :
% 0.32/0.66 ( ( finite_finite_int @ S2 )
% 0.32/0.66 => ( ( ( member_int @ A @ S2 )
% 0.32/0.66 => ( ( groups2316167850115554303t_real
% 0.32/0.66 @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_int @ A @ S2 )
% 0.32/0.66 => ( ( groups2316167850115554303t_real
% 0.32/0.66 @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8297_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_complex,A: complex,B: complex > real] :
% 0.32/0.66 ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.66 => ( ( ( member_complex @ A @ S2 )
% 0.32/0.66 => ( ( groups766887009212190081x_real
% 0.32/0.66 @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_complex @ A @ S2 )
% 0.32/0.66 => ( ( groups766887009212190081x_real
% 0.32/0.66 @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_real ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8298_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_real,A: real,B: real > rat] :
% 0.32/0.66 ( ( finite_finite_real @ S2 )
% 0.32/0.66 => ( ( ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups4061424788464935467al_rat
% 0.32/0.66 @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_real @ A @ S2 )
% 0.32/0.66 => ( ( groups4061424788464935467al_rat
% 0.32/0.66 @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_rat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8299_prod_Odelta,axiom,
% 0.32/0.66 ! [S2: set_nat,A: nat,B: nat > rat] :
% 0.32/0.66 ( ( finite_finite_nat @ S2 )
% 0.32/0.66 => ( ( ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups73079841787564623at_rat
% 0.32/0.66 @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = ( B @ A ) ) )
% 0.32/0.66 & ( ~ ( member_nat @ A @ S2 )
% 0.32/0.66 => ( ( groups73079841787564623at_rat
% 0.32/0.66 @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
% 0.32/0.66 @ S2 )
% 0.32/0.66 = one_one_rat ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.delta
% 0.32/0.66 thf(fact_8300_fact__Suc__0,axiom,
% 0.32/0.66 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.32/0.66 = one_one_complex ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc_0
% 0.32/0.66 thf(fact_8301_fact__Suc__0,axiom,
% 0.32/0.66 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.32/0.66 = one_one_rat ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc_0
% 0.32/0.66 thf(fact_8302_fact__Suc__0,axiom,
% 0.32/0.66 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.32/0.66 = one_one_int ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc_0
% 0.32/0.66 thf(fact_8303_fact__Suc__0,axiom,
% 0.32/0.66 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.32/0.66 = one_one_real ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc_0
% 0.32/0.66 thf(fact_8304_fact__Suc__0,axiom,
% 0.32/0.66 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.32/0.66 = one_one_nat ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc_0
% 0.32/0.66 thf(fact_8305_fact__Suc,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc
% 0.32/0.66 thf(fact_8306_fact__Suc,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc
% 0.32/0.66 thf(fact_8307_fact__Suc,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc
% 0.32/0.66 thf(fact_8308_fact__Suc,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc
% 0.32/0.66 thf(fact_8309_fact__Suc,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 0.32/0.66 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_Suc
% 0.32/0.66 thf(fact_8310_prod_OlessThan__Suc,axiom,
% 0.32/0.66 ! [G: nat > real,N: nat] :
% 0.32/0.66 ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.lessThan_Suc
% 0.32/0.66 thf(fact_8311_prod_OlessThan__Suc,axiom,
% 0.32/0.66 ! [G: nat > rat,N: nat] :
% 0.32/0.66 ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.lessThan_Suc
% 0.32/0.66 thf(fact_8312_prod_OlessThan__Suc,axiom,
% 0.32/0.66 ! [G: nat > nat,N: nat] :
% 0.32/0.66 ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.lessThan_Suc
% 0.32/0.66 thf(fact_8313_prod_OlessThan__Suc,axiom,
% 0.32/0.66 ! [G: nat > int,N: nat] :
% 0.32/0.66 ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.lessThan_Suc
% 0.32/0.66 thf(fact_8314_fact__2,axiom,
% 0.32/0.66 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_2
% 0.32/0.66 thf(fact_8315_fact__2,axiom,
% 0.32/0.66 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_2
% 0.32/0.66 thf(fact_8316_fact__2,axiom,
% 0.32/0.66 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_2
% 0.32/0.66 thf(fact_8317_fact__2,axiom,
% 0.32/0.66 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_2
% 0.32/0.66 thf(fact_8318_fact__2,axiom,
% 0.32/0.66 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.66 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_2
% 0.32/0.66 thf(fact_8319_prod_Ocl__ivl__Suc,axiom,
% 0.32/0.66 ! [N: nat,M: nat,G: nat > complex] :
% 0.32/0.66 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = one_one_complex ) )
% 0.32/0.66 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.cl_ivl_Suc
% 0.32/0.66 thf(fact_8320_prod_Ocl__ivl__Suc,axiom,
% 0.32/0.66 ! [N: nat,M: nat,G: nat > real] :
% 0.32/0.66 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = one_one_real ) )
% 0.32/0.66 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.cl_ivl_Suc
% 0.32/0.66 thf(fact_8321_prod_Ocl__ivl__Suc,axiom,
% 0.32/0.66 ! [N: nat,M: nat,G: nat > rat] :
% 0.32/0.66 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = one_one_rat ) )
% 0.32/0.66 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.cl_ivl_Suc
% 0.32/0.66 thf(fact_8322_prod_Ocl__ivl__Suc,axiom,
% 0.32/0.66 ! [N: nat,M: nat,G: nat > nat] :
% 0.32/0.66 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = one_one_nat ) )
% 0.32/0.66 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.cl_ivl_Suc
% 0.32/0.66 thf(fact_8323_prod_Ocl__ivl__Suc,axiom,
% 0.32/0.66 ! [N: nat,M: nat,G: nat > int] :
% 0.32/0.66 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = one_one_int ) )
% 0.32/0.66 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.32/0.66 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.32/0.66 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.cl_ivl_Suc
% 0.32/0.66 thf(fact_8324_prod_Oneutral,axiom,
% 0.32/0.66 ! [A2: set_nat,G: nat > nat] :
% 0.32/0.66 ( ! [X3: nat] :
% 0.32/0.66 ( ( member_nat @ X3 @ A2 )
% 0.32/0.66 => ( ( G @ X3 )
% 0.32/0.66 = one_one_nat ) )
% 0.32/0.66 => ( ( groups708209901874060359at_nat @ G @ A2 )
% 0.32/0.66 = one_one_nat ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.neutral
% 0.32/0.66 thf(fact_8325_prod_Oneutral,axiom,
% 0.32/0.66 ! [A2: set_nat,G: nat > int] :
% 0.32/0.66 ( ! [X3: nat] :
% 0.32/0.66 ( ( member_nat @ X3 @ A2 )
% 0.32/0.66 => ( ( G @ X3 )
% 0.32/0.66 = one_one_int ) )
% 0.32/0.66 => ( ( groups705719431365010083at_int @ G @ A2 )
% 0.32/0.66 = one_one_int ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.neutral
% 0.32/0.66 thf(fact_8326_prod_Oneutral,axiom,
% 0.32/0.66 ! [A2: set_int,G: int > int] :
% 0.32/0.66 ( ! [X3: int] :
% 0.32/0.66 ( ( member_int @ X3 @ A2 )
% 0.32/0.66 => ( ( G @ X3 )
% 0.32/0.66 = one_one_int ) )
% 0.32/0.66 => ( ( groups1705073143266064639nt_int @ G @ A2 )
% 0.32/0.66 = one_one_int ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.neutral
% 0.32/0.66 thf(fact_8327_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: nat > complex,A2: set_nat] :
% 0.32/0.66 ( ( ( groups6464643781859351333omplex @ G @ A2 )
% 0.32/0.66 != one_one_complex )
% 0.32/0.66 => ~ ! [A5: nat] :
% 0.32/0.66 ( ( member_nat @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_complex ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8328_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: real > complex,A2: set_real] :
% 0.32/0.66 ( ( ( groups713298508707869441omplex @ G @ A2 )
% 0.32/0.66 != one_one_complex )
% 0.32/0.66 => ~ ! [A5: real] :
% 0.32/0.66 ( ( member_real @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_complex ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8329_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: int > complex,A2: set_int] :
% 0.32/0.66 ( ( ( groups7440179247065528705omplex @ G @ A2 )
% 0.32/0.66 != one_one_complex )
% 0.32/0.66 => ~ ! [A5: int] :
% 0.32/0.66 ( ( member_int @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_complex ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8330_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: complex > complex,A2: set_complex] :
% 0.32/0.66 ( ( ( groups3708469109370488835omplex @ G @ A2 )
% 0.32/0.66 != one_one_complex )
% 0.32/0.66 => ~ ! [A5: complex] :
% 0.32/0.66 ( ( member_complex @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_complex ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8331_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: nat > real,A2: set_nat] :
% 0.32/0.66 ( ( ( groups129246275422532515t_real @ G @ A2 )
% 0.32/0.66 != one_one_real )
% 0.32/0.66 => ~ ! [A5: nat] :
% 0.32/0.66 ( ( member_nat @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8332_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: real > real,A2: set_real] :
% 0.32/0.66 ( ( ( groups1681761925125756287l_real @ G @ A2 )
% 0.32/0.66 != one_one_real )
% 0.32/0.66 => ~ ! [A5: real] :
% 0.32/0.66 ( ( member_real @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8333_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: int > real,A2: set_int] :
% 0.32/0.66 ( ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.32/0.66 != one_one_real )
% 0.32/0.66 => ~ ! [A5: int] :
% 0.32/0.66 ( ( member_int @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8334_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: complex > real,A2: set_complex] :
% 0.32/0.66 ( ( ( groups766887009212190081x_real @ G @ A2 )
% 0.32/0.66 != one_one_real )
% 0.32/0.66 => ~ ! [A5: complex] :
% 0.32/0.66 ( ( member_complex @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_real ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8335_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: nat > rat,A2: set_nat] :
% 0.32/0.66 ( ( ( groups73079841787564623at_rat @ G @ A2 )
% 0.32/0.66 != one_one_rat )
% 0.32/0.66 => ~ ! [A5: nat] :
% 0.32/0.66 ( ( member_nat @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_rat ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8336_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.32/0.66 ! [G: real > rat,A2: set_real] :
% 0.32/0.66 ( ( ( groups4061424788464935467al_rat @ G @ A2 )
% 0.32/0.66 != one_one_rat )
% 0.32/0.66 => ~ ! [A5: real] :
% 0.32/0.66 ( ( member_real @ A5 @ A2 )
% 0.32/0.66 => ( ( G @ A5 )
% 0.32/0.66 = one_one_rat ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.not_neutral_contains_not_neutral
% 0.32/0.66 thf(fact_8337_prod_Odistrib,axiom,
% 0.32/0.66 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 0.32/0.66 ( ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [X2: nat] : ( times_times_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.66 @ A2 )
% 0.32/0.66 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.distrib
% 0.32/0.66 thf(fact_8338_prod_Odistrib,axiom,
% 0.32/0.66 ! [G: nat > int,H2: nat > int,A2: set_nat] :
% 0.32/0.66 ( ( groups705719431365010083at_int
% 0.32/0.66 @ ^ [X2: nat] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.66 @ A2 )
% 0.32/0.66 = ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.distrib
% 0.32/0.66 thf(fact_8339_prod_Odistrib,axiom,
% 0.32/0.66 ! [G: int > int,H2: int > int,A2: set_int] :
% 0.32/0.66 ( ( groups1705073143266064639nt_int
% 0.32/0.66 @ ^ [X2: int] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 0.32/0.66 @ A2 )
% 0.32/0.66 = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.distrib
% 0.32/0.66 thf(fact_8340_prod__power__distrib,axiom,
% 0.32/0.66 ! [F: nat > nat,A2: set_nat,N: nat] :
% 0.32/0.66 ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N )
% 0.32/0.66 = ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [X2: nat] : ( power_power_nat @ ( F @ X2 ) @ N )
% 0.32/0.66 @ A2 ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_power_distrib
% 0.32/0.66 thf(fact_8341_prod__power__distrib,axiom,
% 0.32/0.66 ! [F: nat > int,A2: set_nat,N: nat] :
% 0.32/0.66 ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N )
% 0.32/0.66 = ( groups705719431365010083at_int
% 0.32/0.66 @ ^ [X2: nat] : ( power_power_int @ ( F @ X2 ) @ N )
% 0.32/0.66 @ A2 ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_power_distrib
% 0.32/0.66 thf(fact_8342_prod__power__distrib,axiom,
% 0.32/0.66 ! [F: int > int,A2: set_int,N: nat] :
% 0.32/0.66 ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N )
% 0.32/0.66 = ( groups1705073143266064639nt_int
% 0.32/0.66 @ ^ [X2: int] : ( power_power_int @ ( F @ X2 ) @ N )
% 0.32/0.66 @ A2 ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_power_distrib
% 0.32/0.66 thf(fact_8343_mod__prod__eq,axiom,
% 0.32/0.66 ! [F: nat > nat,A: nat,A2: set_nat] :
% 0.32/0.66 ( ( modulo_modulo_nat
% 0.32/0.66 @ ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 0.32/0.66 @ A2 )
% 0.32/0.66 @ A )
% 0.32/0.66 = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).
% 0.32/0.66
% 0.32/0.66 % mod_prod_eq
% 0.32/0.66 thf(fact_8344_mod__prod__eq,axiom,
% 0.32/0.66 ! [F: nat > int,A: int,A2: set_nat] :
% 0.32/0.66 ( ( modulo_modulo_int
% 0.32/0.66 @ ( groups705719431365010083at_int
% 0.32/0.66 @ ^ [I4: nat] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 0.32/0.66 @ A2 )
% 0.32/0.66 @ A )
% 0.32/0.66 = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).
% 0.32/0.66
% 0.32/0.66 % mod_prod_eq
% 0.32/0.66 thf(fact_8345_mod__prod__eq,axiom,
% 0.32/0.66 ! [F: int > int,A: int,A2: set_int] :
% 0.32/0.66 ( ( modulo_modulo_int
% 0.32/0.66 @ ( groups1705073143266064639nt_int
% 0.32/0.66 @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 0.32/0.66 @ A2 )
% 0.32/0.66 @ A )
% 0.32/0.66 = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).
% 0.32/0.66
% 0.32/0.66 % mod_prod_eq
% 0.32/0.66 thf(fact_8346_fact__prod,axiom,
% 0.32/0.66 ( semiri1406184849735516958ct_int
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri1314217659103216013at_int
% 0.32/0.66 @ ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [X2: nat] : X2
% 0.32/0.66 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_prod
% 0.32/0.66 thf(fact_8347_fact__prod,axiom,
% 0.32/0.66 ( semiri5044797733671781792omplex
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri8010041392384452111omplex
% 0.32/0.66 @ ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [X2: nat] : X2
% 0.32/0.66 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_prod
% 0.32/0.66 thf(fact_8348_fact__prod,axiom,
% 0.32/0.66 ( semiri773545260158071498ct_rat
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri681578069525770553at_rat
% 0.32/0.66 @ ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [X2: nat] : X2
% 0.32/0.66 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_prod
% 0.32/0.66 thf(fact_8349_fact__prod,axiom,
% 0.32/0.66 ( semiri2265585572941072030t_real
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri5074537144036343181t_real
% 0.32/0.66 @ ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [X2: nat] : X2
% 0.32/0.66 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_prod
% 0.32/0.66 thf(fact_8350_fact__prod,axiom,
% 0.32/0.66 ( semiri1408675320244567234ct_nat
% 0.32/0.66 = ( ^ [N2: nat] :
% 0.32/0.66 ( semiri1316708129612266289at_nat
% 0.32/0.66 @ ( groups708209901874060359at_nat
% 0.32/0.66 @ ^ [X2: nat] : X2
% 0.32/0.66 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_prod
% 0.32/0.66 thf(fact_8351_prod__nonneg,axiom,
% 0.32/0.66 ! [A2: set_nat,F: nat > nat] :
% 0.32/0.66 ( ! [X3: nat] :
% 0.32/0.66 ( ( member_nat @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_nonneg
% 0.32/0.66 thf(fact_8352_prod__nonneg,axiom,
% 0.32/0.66 ! [A2: set_nat,F: nat > int] :
% 0.32/0.66 ( ! [X3: nat] :
% 0.32/0.66 ( ( member_nat @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_nonneg
% 0.32/0.66 thf(fact_8353_prod__nonneg,axiom,
% 0.32/0.66 ! [A2: set_int,F: int > int] :
% 0.32/0.66 ( ! [X3: int] :
% 0.32/0.66 ( ( member_int @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_nonneg
% 0.32/0.66 thf(fact_8354_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_nat,F: nat > real,G: nat > real] :
% 0.32/0.66 ( ! [I2: nat] :
% 0.32/0.66 ( ( member_nat @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8355_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_real,F: real > real,G: real > real] :
% 0.32/0.66 ( ! [I2: real] :
% 0.32/0.66 ( ( member_real @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8356_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_int,F: int > real,G: int > real] :
% 0.32/0.66 ( ! [I2: int] :
% 0.32/0.66 ( ( member_int @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8357_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.32/0.66 ( ! [I2: complex] :
% 0.32/0.66 ( ( member_complex @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8358_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.32/0.66 ( ! [I2: nat] :
% 0.32/0.66 ( ( member_nat @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8359_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_real,F: real > rat,G: real > rat] :
% 0.32/0.66 ( ! [I2: real] :
% 0.32/0.66 ( ( member_real @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8360_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.32/0.66 ( ! [I2: int] :
% 0.32/0.66 ( ( member_int @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8361_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.32/0.66 ( ! [I2: complex] :
% 0.32/0.66 ( ( member_complex @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8362_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.32/0.66 ( ! [I2: real] :
% 0.32/0.66 ( ( member_real @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8363_prod__mono,axiom,
% 0.32/0.66 ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.32/0.66 ( ! [I2: int] :
% 0.32/0.66 ( ( member_int @ I2 @ A2 )
% 0.32/0.66 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
% 0.32/0.66 & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 0.32/0.66 => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_mono
% 0.32/0.66 thf(fact_8364_prod__pos,axiom,
% 0.32/0.66 ! [A2: set_nat,F: nat > nat] :
% 0.32/0.66 ( ! [X3: nat] :
% 0.32/0.66 ( ( member_nat @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_pos
% 0.32/0.66 thf(fact_8365_prod__pos,axiom,
% 0.32/0.66 ! [A2: set_nat,F: nat > int] :
% 0.32/0.66 ( ! [X3: nat] :
% 0.32/0.66 ( ( member_nat @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_pos
% 0.32/0.66 thf(fact_8366_prod__pos,axiom,
% 0.32/0.66 ! [A2: set_int,F: int > int] :
% 0.32/0.66 ( ! [X3: int] :
% 0.32/0.66 ( ( member_int @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_pos
% 0.32/0.66 thf(fact_8367_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_nat,F: nat > real] :
% 0.32/0.66 ( ! [X3: nat] :
% 0.32/0.66 ( ( member_nat @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8368_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_real,F: real > real] :
% 0.32/0.66 ( ! [X3: real] :
% 0.32/0.66 ( ( member_real @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8369_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_int,F: int > real] :
% 0.32/0.66 ( ! [X3: int] :
% 0.32/0.66 ( ( member_int @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8370_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_complex,F: complex > real] :
% 0.32/0.66 ( ! [X3: complex] :
% 0.32/0.66 ( ( member_complex @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8371_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_nat,F: nat > rat] :
% 0.32/0.66 ( ! [X3: nat] :
% 0.32/0.66 ( ( member_nat @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8372_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_real,F: real > rat] :
% 0.32/0.66 ( ! [X3: real] :
% 0.32/0.66 ( ( member_real @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8373_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_int,F: int > rat] :
% 0.32/0.66 ( ! [X3: int] :
% 0.32/0.66 ( ( member_int @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8374_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_complex,F: complex > rat] :
% 0.32/0.66 ( ! [X3: complex] :
% 0.32/0.66 ( ( member_complex @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8375_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_real,F: real > nat] :
% 0.32/0.66 ( ! [X3: real] :
% 0.32/0.66 ( ( member_real @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8376_prod__ge__1,axiom,
% 0.32/0.66 ! [A2: set_int,F: int > nat] :
% 0.32/0.66 ( ! [X3: int] :
% 0.32/0.66 ( ( member_int @ X3 @ A2 )
% 0.32/0.66 => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
% 0.32/0.66 => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_ge_1
% 0.32/0.66 thf(fact_8377_prod__atLeastAtMost__code,axiom,
% 0.32/0.66 ! [F: nat > complex,A: nat,B: nat] :
% 0.32/0.66 ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.32/0.66 = ( set_fo1517530859248394432omplex
% 0.32/0.66 @ ^ [A3: nat] : ( times_times_complex @ ( F @ A3 ) )
% 0.32/0.66 @ A
% 0.32/0.66 @ B
% 0.32/0.66 @ one_one_complex ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_atLeastAtMost_code
% 0.32/0.66 thf(fact_8378_prod__atLeastAtMost__code,axiom,
% 0.32/0.66 ! [F: nat > real,A: nat,B: nat] :
% 0.32/0.66 ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.32/0.66 = ( set_fo3111899725591712190t_real
% 0.32/0.66 @ ^ [A3: nat] : ( times_times_real @ ( F @ A3 ) )
% 0.32/0.66 @ A
% 0.32/0.66 @ B
% 0.32/0.66 @ one_one_real ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_atLeastAtMost_code
% 0.32/0.66 thf(fact_8379_prod__atLeastAtMost__code,axiom,
% 0.32/0.66 ! [F: nat > rat,A: nat,B: nat] :
% 0.32/0.66 ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.32/0.66 = ( set_fo1949268297981939178at_rat
% 0.32/0.66 @ ^ [A3: nat] : ( times_times_rat @ ( F @ A3 ) )
% 0.32/0.66 @ A
% 0.32/0.66 @ B
% 0.32/0.66 @ one_one_rat ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_atLeastAtMost_code
% 0.32/0.66 thf(fact_8380_prod__atLeastAtMost__code,axiom,
% 0.32/0.66 ! [F: nat > nat,A: nat,B: nat] :
% 0.32/0.66 ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.32/0.66 = ( set_fo2584398358068434914at_nat
% 0.32/0.66 @ ^ [A3: nat] : ( times_times_nat @ ( F @ A3 ) )
% 0.32/0.66 @ A
% 0.32/0.66 @ B
% 0.32/0.66 @ one_one_nat ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_atLeastAtMost_code
% 0.32/0.66 thf(fact_8381_prod__atLeastAtMost__code,axiom,
% 0.32/0.66 ! [F: nat > int,A: nat,B: nat] :
% 0.32/0.66 ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.32/0.66 = ( set_fo2581907887559384638at_int
% 0.32/0.66 @ ^ [A3: nat] : ( times_times_int @ ( F @ A3 ) )
% 0.32/0.66 @ A
% 0.32/0.66 @ B
% 0.32/0.66 @ one_one_int ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod_atLeastAtMost_code
% 0.32/0.66 thf(fact_8382_fact__ge__zero,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_ge_zero
% 0.32/0.66 thf(fact_8383_fact__ge__zero,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_ge_zero
% 0.32/0.66 thf(fact_8384_fact__ge__zero,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_ge_zero
% 0.32/0.66 thf(fact_8385_fact__ge__zero,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_ge_zero
% 0.32/0.66 thf(fact_8386_fact__gt__zero,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_gt_zero
% 0.32/0.66 thf(fact_8387_fact__gt__zero,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_gt_zero
% 0.32/0.66 thf(fact_8388_fact__gt__zero,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_gt_zero
% 0.32/0.66 thf(fact_8389_fact__gt__zero,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_gt_zero
% 0.32/0.66 thf(fact_8390_fact__not__neg,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 0.32/0.66
% 0.32/0.66 % fact_not_neg
% 0.32/0.66 thf(fact_8391_fact__not__neg,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 0.32/0.66
% 0.32/0.66 % fact_not_neg
% 0.32/0.66 thf(fact_8392_fact__not__neg,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 0.32/0.66
% 0.32/0.66 % fact_not_neg
% 0.32/0.66 thf(fact_8393_fact__not__neg,axiom,
% 0.32/0.66 ! [N: nat] :
% 0.32/0.66 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 0.32/0.66
% 0.32/0.66 % fact_not_neg
% 0.32/0.66 thf(fact_8394_fact__ge__1,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_ge_1
% 0.32/0.66 thf(fact_8395_fact__ge__1,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_ge_1
% 0.32/0.66 thf(fact_8396_fact__ge__1,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_ge_1
% 0.32/0.66 thf(fact_8397_fact__ge__1,axiom,
% 0.32/0.66 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_ge_1
% 0.32/0.66 thf(fact_8398_fact__mono,axiom,
% 0.32/0.66 ! [M: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_mono
% 0.32/0.66 thf(fact_8399_fact__mono,axiom,
% 0.32/0.66 ! [M: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_mono
% 0.32/0.66 thf(fact_8400_fact__mono,axiom,
% 0.32/0.66 ! [M: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_mono
% 0.32/0.66 thf(fact_8401_fact__mono,axiom,
% 0.32/0.66 ! [M: nat,N: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.66 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_mono
% 0.32/0.66 thf(fact_8402_fact__dvd,axiom,
% 0.32/0.66 ! [N: nat,M: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_dvd
% 0.32/0.66 thf(fact_8403_fact__dvd,axiom,
% 0.32/0.66 ! [N: nat,M: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_dvd
% 0.32/0.66 thf(fact_8404_fact__dvd,axiom,
% 0.32/0.66 ! [N: nat,M: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_dvd
% 0.32/0.66 thf(fact_8405_fact__dvd,axiom,
% 0.32/0.66 ! [N: nat,M: nat] :
% 0.32/0.66 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.66 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % fact_dvd
% 0.32/0.66 thf(fact_8406_prod_Ointer__filter,axiom,
% 0.32/0.66 ! [A2: set_real,G: real > complex,P: real > $o] :
% 0.32/0.66 ( ( finite_finite_real @ A2 )
% 0.32/0.66 => ( ( groups713298508707869441omplex @ G
% 0.32/0.66 @ ( collect_real
% 0.32/0.66 @ ^ [X2: real] :
% 0.32/0.66 ( ( member_real @ X2 @ A2 )
% 0.32/0.66 & ( P @ X2 ) ) ) )
% 0.32/0.66 = ( groups713298508707869441omplex
% 0.32/0.66 @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 0.32/0.66 @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.inter_filter
% 0.32/0.66 thf(fact_8407_prod_Ointer__filter,axiom,
% 0.32/0.66 ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 0.32/0.66 ( ( finite_finite_nat @ A2 )
% 0.32/0.66 => ( ( groups6464643781859351333omplex @ G
% 0.32/0.66 @ ( collect_nat
% 0.32/0.66 @ ^ [X2: nat] :
% 0.32/0.66 ( ( member_nat @ X2 @ A2 )
% 0.32/0.66 & ( P @ X2 ) ) ) )
% 0.32/0.66 = ( groups6464643781859351333omplex
% 0.32/0.66 @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 0.32/0.66 @ A2 ) ) ) ).
% 0.32/0.66
% 0.32/0.66 % prod.inter_filter
% 0.32/0.66 thf(fact_8408_prod_Ointer__filter,axiom,
% 0.32/0.66 ! [A2: set_int,G: int > complex,P: int > $o] :
% 0.32/0.66 ( ( finite_finite_int @ A2 )
% 0.32/0.66 => ( ( groups7440179247065528705omplex @ G
% 0.32/0.66 @ ( collect_int
% 0.32/0.66 @ ^ [X2: int] :
% 0.32/0.67 ( ( member_int @ X2 @ A2 )
% 0.32/0.67 & ( P @ X2 ) ) ) )
% 0.32/0.67 = ( groups7440179247065528705omplex
% 0.32/0.67 @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 0.32/0.67 @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.inter_filter
% 0.32/0.67 thf(fact_8409_prod_Ointer__filter,axiom,
% 0.32/0.67 ! [A2: set_complex,G: complex > complex,P: complex > $o] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.67 => ( ( groups3708469109370488835omplex @ G
% 0.32/0.67 @ ( collect_complex
% 0.32/0.67 @ ^ [X2: complex] :
% 0.32/0.67 ( ( member_complex @ X2 @ A2 )
% 0.32/0.67 & ( P @ X2 ) ) ) )
% 0.32/0.67 = ( groups3708469109370488835omplex
% 0.32/0.67 @ ^ [X2: complex] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 0.32/0.67 @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.inter_filter
% 0.32/0.67 thf(fact_8410_prod_Ointer__filter,axiom,
% 0.32/0.67 ! [A2: set_real,G: real > real,P: real > $o] :
% 0.32/0.67 ( ( finite_finite_real @ A2 )
% 0.32/0.67 => ( ( groups1681761925125756287l_real @ G
% 0.32/0.67 @ ( collect_real
% 0.32/0.67 @ ^ [X2: real] :
% 0.32/0.67 ( ( member_real @ X2 @ A2 )
% 0.32/0.67 & ( P @ X2 ) ) ) )
% 0.32/0.67 = ( groups1681761925125756287l_real
% 0.32/0.67 @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 0.32/0.67 @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.inter_filter
% 0.32/0.67 thf(fact_8411_prod_Ointer__filter,axiom,
% 0.32/0.67 ! [A2: set_nat,G: nat > real,P: nat > $o] :
% 0.32/0.67 ( ( finite_finite_nat @ A2 )
% 0.32/0.67 => ( ( groups129246275422532515t_real @ G
% 0.32/0.67 @ ( collect_nat
% 0.32/0.67 @ ^ [X2: nat] :
% 0.32/0.67 ( ( member_nat @ X2 @ A2 )
% 0.32/0.67 & ( P @ X2 ) ) ) )
% 0.32/0.67 = ( groups129246275422532515t_real
% 0.32/0.67 @ ^ [X2: nat] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 0.32/0.67 @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.inter_filter
% 0.32/0.67 thf(fact_8412_prod_Ointer__filter,axiom,
% 0.32/0.67 ! [A2: set_int,G: int > real,P: int > $o] :
% 0.32/0.67 ( ( finite_finite_int @ A2 )
% 0.32/0.67 => ( ( groups2316167850115554303t_real @ G
% 0.32/0.67 @ ( collect_int
% 0.32/0.67 @ ^ [X2: int] :
% 0.32/0.67 ( ( member_int @ X2 @ A2 )
% 0.32/0.67 & ( P @ X2 ) ) ) )
% 0.32/0.67 = ( groups2316167850115554303t_real
% 0.32/0.67 @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 0.32/0.67 @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.inter_filter
% 0.32/0.67 thf(fact_8413_prod_Ointer__filter,axiom,
% 0.32/0.67 ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.67 => ( ( groups766887009212190081x_real @ G
% 0.32/0.67 @ ( collect_complex
% 0.32/0.67 @ ^ [X2: complex] :
% 0.32/0.67 ( ( member_complex @ X2 @ A2 )
% 0.32/0.67 & ( P @ X2 ) ) ) )
% 0.32/0.67 = ( groups766887009212190081x_real
% 0.32/0.67 @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 0.32/0.67 @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.inter_filter
% 0.32/0.67 thf(fact_8414_prod_Ointer__filter,axiom,
% 0.32/0.67 ! [A2: set_real,G: real > rat,P: real > $o] :
% 0.32/0.67 ( ( finite_finite_real @ A2 )
% 0.32/0.67 => ( ( groups4061424788464935467al_rat @ G
% 0.32/0.67 @ ( collect_real
% 0.32/0.67 @ ^ [X2: real] :
% 0.32/0.67 ( ( member_real @ X2 @ A2 )
% 0.32/0.67 & ( P @ X2 ) ) ) )
% 0.32/0.67 = ( groups4061424788464935467al_rat
% 0.32/0.67 @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
% 0.32/0.67 @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.inter_filter
% 0.32/0.67 thf(fact_8415_prod_Ointer__filter,axiom,
% 0.32/0.67 ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 0.32/0.67 ( ( finite_finite_nat @ A2 )
% 0.32/0.67 => ( ( groups73079841787564623at_rat @ G
% 0.32/0.67 @ ( collect_nat
% 0.32/0.67 @ ^ [X2: nat] :
% 0.32/0.67 ( ( member_nat @ X2 @ A2 )
% 0.32/0.67 & ( P @ X2 ) ) ) )
% 0.32/0.67 = ( groups73079841787564623at_rat
% 0.32/0.67 @ ^ [X2: nat] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
% 0.32/0.67 @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.inter_filter
% 0.32/0.67 thf(fact_8416_pochhammer__fact,axiom,
% 0.32/0.67 ( semiri5044797733671781792omplex
% 0.32/0.67 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 0.32/0.67
% 0.32/0.67 % pochhammer_fact
% 0.32/0.67 thf(fact_8417_pochhammer__fact,axiom,
% 0.32/0.67 ( semiri773545260158071498ct_rat
% 0.32/0.67 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 0.32/0.67
% 0.32/0.67 % pochhammer_fact
% 0.32/0.67 thf(fact_8418_pochhammer__fact,axiom,
% 0.32/0.67 ( semiri1406184849735516958ct_int
% 0.32/0.67 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 0.32/0.67
% 0.32/0.67 % pochhammer_fact
% 0.32/0.67 thf(fact_8419_pochhammer__fact,axiom,
% 0.32/0.67 ( semiri2265585572941072030t_real
% 0.32/0.67 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % pochhammer_fact
% 0.32/0.67 thf(fact_8420_pochhammer__fact,axiom,
% 0.32/0.67 ( semiri1408675320244567234ct_nat
% 0.32/0.67 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 0.32/0.67
% 0.32/0.67 % pochhammer_fact
% 0.32/0.67 thf(fact_8421_power__sum,axiom,
% 0.32/0.67 ! [C: real,F: nat > nat,A2: set_nat] :
% 0.32/0.67 ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.32/0.67 = ( groups129246275422532515t_real
% 0.32/0.67 @ ^ [A3: nat] : ( power_power_real @ C @ ( F @ A3 ) )
% 0.32/0.67 @ A2 ) ) ).
% 0.32/0.67
% 0.32/0.67 % power_sum
% 0.32/0.67 thf(fact_8422_power__sum,axiom,
% 0.32/0.67 ! [C: complex,F: nat > nat,A2: set_nat] :
% 0.32/0.67 ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.32/0.67 = ( groups6464643781859351333omplex
% 0.32/0.67 @ ^ [A3: nat] : ( power_power_complex @ C @ ( F @ A3 ) )
% 0.32/0.67 @ A2 ) ) ).
% 0.32/0.67
% 0.32/0.67 % power_sum
% 0.32/0.67 thf(fact_8423_power__sum,axiom,
% 0.32/0.67 ! [C: nat,F: nat > nat,A2: set_nat] :
% 0.32/0.67 ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.32/0.67 = ( groups708209901874060359at_nat
% 0.32/0.67 @ ^ [A3: nat] : ( power_power_nat @ C @ ( F @ A3 ) )
% 0.32/0.67 @ A2 ) ) ).
% 0.32/0.67
% 0.32/0.67 % power_sum
% 0.32/0.67 thf(fact_8424_power__sum,axiom,
% 0.32/0.67 ! [C: int,F: nat > nat,A2: set_nat] :
% 0.32/0.67 ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.32/0.67 = ( groups705719431365010083at_int
% 0.32/0.67 @ ^ [A3: nat] : ( power_power_int @ C @ ( F @ A3 ) )
% 0.32/0.67 @ A2 ) ) ).
% 0.32/0.67
% 0.32/0.67 % power_sum
% 0.32/0.67 thf(fact_8425_power__sum,axiom,
% 0.32/0.67 ! [C: int,F: int > nat,A2: set_int] :
% 0.32/0.67 ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
% 0.32/0.67 = ( groups1705073143266064639nt_int
% 0.32/0.67 @ ^ [A3: int] : ( power_power_int @ C @ ( F @ A3 ) )
% 0.32/0.67 @ A2 ) ) ).
% 0.32/0.67
% 0.32/0.67 % power_sum
% 0.32/0.67 thf(fact_8426_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.32/0.67 ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 0.32/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.32/0.67 = ( groups708209901874060359at_nat
% 0.32/0.67 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.32/0.67 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.shift_bounds_cl_nat_ivl
% 0.32/0.67 thf(fact_8427_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.32/0.67 ! [G: nat > int,M: nat,K: nat,N: nat] :
% 0.32/0.67 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.32/0.67 = ( groups705719431365010083at_int
% 0.32/0.67 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 0.32/0.67 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.shift_bounds_cl_nat_ivl
% 0.32/0.67 thf(fact_8428_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_nat,F: nat > real] :
% 0.32/0.67 ( ! [X3: nat] :
% 0.32/0.67 ( ( member_nat @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.32/0.67 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8429_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_real,F: real > real] :
% 0.32/0.67 ( ! [X3: real] :
% 0.32/0.67 ( ( member_real @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.32/0.67 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8430_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_int,F: int > real] :
% 0.32/0.67 ( ! [X3: int] :
% 0.32/0.67 ( ( member_int @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.32/0.67 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8431_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_complex,F: complex > real] :
% 0.32/0.67 ( ! [X3: complex] :
% 0.32/0.67 ( ( member_complex @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 0.32/0.67 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8432_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_nat,F: nat > rat] :
% 0.32/0.67 ( ! [X3: nat] :
% 0.32/0.67 ( ( member_nat @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.32/0.67 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8433_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_real,F: real > rat] :
% 0.32/0.67 ( ! [X3: real] :
% 0.32/0.67 ( ( member_real @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.32/0.67 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8434_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_int,F: int > rat] :
% 0.32/0.67 ( ! [X3: int] :
% 0.32/0.67 ( ( member_int @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.32/0.67 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8435_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_complex,F: complex > rat] :
% 0.32/0.67 ( ! [X3: complex] :
% 0.32/0.67 ( ( member_complex @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 0.32/0.67 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8436_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_real,F: real > nat] :
% 0.32/0.67 ( ! [X3: real] :
% 0.32/0.67 ( ( member_real @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
% 0.32/0.67 => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8437_prod__le__1,axiom,
% 0.32/0.67 ! [A2: set_int,F: int > nat] :
% 0.32/0.67 ( ! [X3: int] :
% 0.32/0.67 ( ( member_int @ X3 @ A2 )
% 0.32/0.67 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
% 0.32/0.67 & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
% 0.32/0.67 => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_le_1
% 0.32/0.67 thf(fact_8438_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 0.32/0.67 ( ( R4 @ one_one_complex @ one_one_complex )
% 0.32/0.67 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite_finite_nat @ S2 )
% 0.32/0.67 => ( ! [X3: nat] :
% 0.32/0.67 ( ( member_nat @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8439_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 0.32/0.67 ( ( R4 @ one_one_complex @ one_one_complex )
% 0.32/0.67 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite_finite_int @ S2 )
% 0.32/0.67 => ( ! [X3: int] :
% 0.32/0.67 ( ( member_int @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8440_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
% 0.32/0.67 ( ( R4 @ one_one_complex @ one_one_complex )
% 0.32/0.67 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.67 => ( ! [X3: complex] :
% 0.32/0.67 ( ( member_complex @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8441_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
% 0.32/0.67 ( ( R4 @ one_one_real @ one_one_real )
% 0.32/0.67 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite_finite_nat @ S2 )
% 0.32/0.67 => ( ! [X3: nat] :
% 0.32/0.67 ( ( member_nat @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8442_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 0.32/0.67 ( ( R4 @ one_one_real @ one_one_real )
% 0.32/0.67 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite_finite_int @ S2 )
% 0.32/0.67 => ( ! [X3: int] :
% 0.32/0.67 ( ( member_int @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8443_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 0.32/0.67 ( ( R4 @ one_one_real @ one_one_real )
% 0.32/0.67 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.67 => ( ! [X3: complex] :
% 0.32/0.67 ( ( member_complex @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8444_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 0.32/0.67 ( ( R4 @ one_one_rat @ one_one_rat )
% 0.32/0.67 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite_finite_nat @ S2 )
% 0.32/0.67 => ( ! [X3: nat] :
% 0.32/0.67 ( ( member_nat @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8445_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 0.32/0.67 ( ( R4 @ one_one_rat @ one_one_rat )
% 0.32/0.67 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite_finite_int @ S2 )
% 0.32/0.67 => ( ! [X3: int] :
% 0.32/0.67 ( ( member_int @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8446_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 0.32/0.67 ( ( R4 @ one_one_rat @ one_one_rat )
% 0.32/0.67 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite3207457112153483333omplex @ S2 )
% 0.32/0.67 => ( ! [X3: complex] :
% 0.32/0.67 ( ( member_complex @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8447_prod_Orelated,axiom,
% 0.32/0.67 ! [R4: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 0.32/0.67 ( ( R4 @ one_one_nat @ one_one_nat )
% 0.32/0.67 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 0.32/0.67 ( ( ( R4 @ X1 @ X23 )
% 0.32/0.67 & ( R4 @ Y1 @ Y22 ) )
% 0.32/0.67 => ( R4 @ ( times_times_nat @ X1 @ Y1 ) @ ( times_times_nat @ X23 @ Y22 ) ) )
% 0.32/0.67 => ( ( finite_finite_int @ S2 )
% 0.32/0.67 => ( ! [X3: int] :
% 0.32/0.67 ( ( member_int @ X3 @ S2 )
% 0.32/0.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 0.32/0.67 => ( R4 @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.related
% 0.32/0.67 thf(fact_8448_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_complex,A2: set_complex,F: complex > nat] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8449_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_complex,A2: set_complex,F: complex > int] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8450_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_nat,A2: set_nat,F: nat > code_integer] :
% 0.32/0.67 ( ( finite_finite_nat @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8451_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_complex,A2: set_complex,F: complex > code_integer] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8452_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,F: int > nat] :
% 0.32/0.67 ( ( finite_finite_int @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8453_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,F: int > code_integer] :
% 0.32/0.67 ( ( finite_finite_int @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8454_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_nat,A2: set_nat,F: nat > nat] :
% 0.32/0.67 ( ( finite_finite_nat @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8455_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_nat,A2: set_nat,F: nat > int] :
% 0.32/0.67 ( ( finite_finite_nat @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8456_prod__dvd__prod__subset,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,F: int > int] :
% 0.32/0.67 ( ( finite_finite_int @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.67 => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset
% 0.32/0.67 thf(fact_8457_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_real,A2: set_real,F: real > nat,G: real > nat] :
% 0.32/0.67 ( ( finite_finite_real @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8458_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8459_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_real,A2: set_real,F: real > int,G: real > int] :
% 0.32/0.67 ( ( finite_finite_real @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8460_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8461_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_real,A2: set_real,F: real > code_integer,G: real > code_integer] :
% 0.32/0.67 ( ( finite_finite_real @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A2 ) @ ( groups6225526099057966256nteger @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8462_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_nat,A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 0.32/0.67 ( ( finite_finite_nat @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: nat] :
% 0.32/0.67 ( ( member_nat @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8463_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_complex,A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ B5 )
% 0.32/0.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8464_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,F: int > nat,G: int > nat] :
% 0.32/0.67 ( ( finite_finite_int @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8465_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,F: int > code_integer,G: int > code_integer] :
% 0.32/0.67 ( ( finite_finite_int @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8466_prod__dvd__prod__subset2,axiom,
% 0.32/0.67 ! [B5: set_nat,A2: set_nat,F: nat > nat,G: nat > nat] :
% 0.32/0.67 ( ( finite_finite_nat @ B5 )
% 0.32/0.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 0.32/0.67 => ( ! [A5: nat] :
% 0.32/0.67 ( ( member_nat @ A5 @ A2 )
% 0.32/0.67 => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 0.32/0.67 => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_dvd_prod_subset2
% 0.32/0.67 thf(fact_8467_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 0.32/0.67 ( ( finite_finite_real @ S4 )
% 0.32/0.67 => ( ( finite_finite_real @ T4 )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.67 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups713298508707869441omplex @ G @ S2 )
% 0.32/0.67 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8468_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 0.32/0.67 ( ( finite_finite_real @ S4 )
% 0.32/0.67 => ( ( finite_finite_int @ T4 )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.67 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups713298508707869441omplex @ G @ S2 )
% 0.32/0.67 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8469_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > complex,H2: complex > complex] :
% 0.32/0.67 ( ( finite_finite_real @ S4 )
% 0.32/0.67 => ( ( finite3207457112153483333omplex @ T4 )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.67 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups713298508707869441omplex @ G @ S2 )
% 0.32/0.67 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8470_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 0.32/0.67 ( ( finite_finite_int @ S4 )
% 0.32/0.67 => ( ( finite_finite_real @ T4 )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.67 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.67 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups7440179247065528705omplex @ G @ S2 )
% 0.32/0.67 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8471_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 0.32/0.67 ( ( finite_finite_int @ S4 )
% 0.32/0.67 => ( ( finite_finite_int @ T4 )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.67 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.67 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups7440179247065528705omplex @ G @ S2 )
% 0.32/0.67 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8472_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > complex,H2: complex > complex] :
% 0.32/0.67 ( ( finite_finite_int @ S4 )
% 0.32/0.67 => ( ( finite3207457112153483333omplex @ T4 )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 0.32/0.67 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.67 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: int] :
% 0.32/0.67 ( ( member_int @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups7440179247065528705omplex @ G @ S2 )
% 0.32/0.67 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8473_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_complex,T4: set_real,S2: set_complex,I: real > complex,J: complex > real,T3: set_real,G: complex > complex,H2: real > complex] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ S4 )
% 0.32/0.67 => ( ( finite_finite_real @ T4 )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 0.32/0.67 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.67 => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups3708469109370488835omplex @ G @ S2 )
% 0.32/0.67 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8474_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_complex,T4: set_int,S2: set_complex,I: int > complex,J: complex > int,T3: set_int,G: complex > complex,H2: int > complex] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ S4 )
% 0.32/0.67 => ( ( finite_finite_int @ T4 )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 0.32/0.67 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 0.32/0.67 => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: int] :
% 0.32/0.67 ( ( member_int @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups3708469109370488835omplex @ G @ S2 )
% 0.32/0.67 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8475_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_complex,T4: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T3: set_complex,G: complex > complex,H2: complex > complex] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ S4 )
% 0.32/0.67 => ( ( finite3207457112153483333omplex @ T4 )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 0.32/0.67 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 0.32/0.67 => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [B3: complex] :
% 0.32/0.67 ( ( member_complex @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_complex ) )
% 0.32/0.67 => ( ! [A5: complex] :
% 0.32/0.67 ( ( member_complex @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups3708469109370488835omplex @ G @ S2 )
% 0.32/0.67 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8476_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.32/0.67 ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 0.32/0.67 ( ( finite_finite_real @ S4 )
% 0.32/0.67 => ( ( finite_finite_real @ T4 )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.67 => ( ( I @ ( J @ A5 ) )
% 0.32/0.67 = A5 ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 0.32/0.67 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.67 => ( ( J @ ( I @ B3 ) )
% 0.32/0.67 = B3 ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 0.32/0.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ S4 )
% 0.32/0.67 => ( ( G @ A5 )
% 0.32/0.67 = one_one_real ) )
% 0.32/0.67 => ( ! [B3: real] :
% 0.32/0.67 ( ( member_real @ B3 @ T4 )
% 0.32/0.67 => ( ( H2 @ B3 )
% 0.32/0.67 = one_one_real ) )
% 0.32/0.67 => ( ! [A5: real] :
% 0.32/0.67 ( ( member_real @ A5 @ S2 )
% 0.32/0.67 => ( ( H2 @ ( J @ A5 ) )
% 0.32/0.67 = ( G @ A5 ) ) )
% 0.32/0.67 => ( ( groups1681761925125756287l_real @ G @ S2 )
% 0.32/0.67 = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.reindex_bij_witness_not_neutral
% 0.32/0.67 thf(fact_8477_fact__less__mono,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.67 => ( ( ord_less_nat @ M @ N )
% 0.32/0.67 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_less_mono
% 0.32/0.67 thf(fact_8478_fact__less__mono,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.67 => ( ( ord_less_nat @ M @ N )
% 0.32/0.67 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_less_mono
% 0.32/0.67 thf(fact_8479_fact__less__mono,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.67 => ( ( ord_less_nat @ M @ N )
% 0.32/0.67 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_less_mono
% 0.32/0.67 thf(fact_8480_fact__less__mono,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.67 => ( ( ord_less_nat @ M @ N )
% 0.32/0.67 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_less_mono
% 0.32/0.67 thf(fact_8481_fact__mod,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.67 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.32/0.67 = zero_zero_int ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_mod
% 0.32/0.67 thf(fact_8482_fact__mod,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.67 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.32/0.67 = zero_z3403309356797280102nteger ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_mod
% 0.32/0.67 thf(fact_8483_fact__mod,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.67 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.32/0.67 = zero_zero_nat ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_mod
% 0.32/0.67 thf(fact_8484_fact__fact__dvd__fact,axiom,
% 0.32/0.67 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_fact_dvd_fact
% 0.32/0.67 thf(fact_8485_fact__fact__dvd__fact,axiom,
% 0.32/0.67 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_fact_dvd_fact
% 0.32/0.67 thf(fact_8486_fact__fact__dvd__fact,axiom,
% 0.32/0.67 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_fact_dvd_fact
% 0.32/0.67 thf(fact_8487_fact__fact__dvd__fact,axiom,
% 0.32/0.67 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_fact_dvd_fact
% 0.32/0.67 thf(fact_8488_fact__fact__dvd__fact,axiom,
% 0.32/0.67 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_fact_dvd_fact
% 0.32/0.67 thf(fact_8489_fact__le__power,axiom,
% 0.32/0.67 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_le_power
% 0.32/0.67 thf(fact_8490_fact__le__power,axiom,
% 0.32/0.67 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_le_power
% 0.32/0.67 thf(fact_8491_fact__le__power,axiom,
% 0.32/0.67 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_le_power
% 0.32/0.67 thf(fact_8492_fact__le__power,axiom,
% 0.32/0.67 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_le_power
% 0.32/0.67 thf(fact_8493_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_real,G: real > complex] :
% 0.32/0.67 ( ( finite_finite_real @ A2 )
% 0.32/0.67 => ( ( groups713298508707869441omplex @ G
% 0.32/0.67 @ ( minus_minus_set_real @ A2
% 0.32/0.67 @ ( collect_real
% 0.32/0.67 @ ^ [X2: real] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_complex ) ) ) )
% 0.32/0.67 = ( groups713298508707869441omplex @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8494_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_int,G: int > complex] :
% 0.32/0.67 ( ( finite_finite_int @ A2 )
% 0.32/0.67 => ( ( groups7440179247065528705omplex @ G
% 0.32/0.67 @ ( minus_minus_set_int @ A2
% 0.32/0.67 @ ( collect_int
% 0.32/0.67 @ ^ [X2: int] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_complex ) ) ) )
% 0.32/0.67 = ( groups7440179247065528705omplex @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8495_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_complex,G: complex > complex] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.67 => ( ( groups3708469109370488835omplex @ G
% 0.32/0.67 @ ( minus_811609699411566653omplex @ A2
% 0.32/0.67 @ ( collect_complex
% 0.32/0.67 @ ^ [X2: complex] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_complex ) ) ) )
% 0.32/0.67 = ( groups3708469109370488835omplex @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8496_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_real,G: real > real] :
% 0.32/0.67 ( ( finite_finite_real @ A2 )
% 0.32/0.67 => ( ( groups1681761925125756287l_real @ G
% 0.32/0.67 @ ( minus_minus_set_real @ A2
% 0.32/0.67 @ ( collect_real
% 0.32/0.67 @ ^ [X2: real] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_real ) ) ) )
% 0.32/0.67 = ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8497_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_int,G: int > real] :
% 0.32/0.67 ( ( finite_finite_int @ A2 )
% 0.32/0.67 => ( ( groups2316167850115554303t_real @ G
% 0.32/0.67 @ ( minus_minus_set_int @ A2
% 0.32/0.67 @ ( collect_int
% 0.32/0.67 @ ^ [X2: int] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_real ) ) ) )
% 0.32/0.67 = ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8498_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_complex,G: complex > real] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.67 => ( ( groups766887009212190081x_real @ G
% 0.32/0.67 @ ( minus_811609699411566653omplex @ A2
% 0.32/0.67 @ ( collect_complex
% 0.32/0.67 @ ^ [X2: complex] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_real ) ) ) )
% 0.32/0.67 = ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8499_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_real,G: real > rat] :
% 0.32/0.67 ( ( finite_finite_real @ A2 )
% 0.32/0.67 => ( ( groups4061424788464935467al_rat @ G
% 0.32/0.67 @ ( minus_minus_set_real @ A2
% 0.32/0.67 @ ( collect_real
% 0.32/0.67 @ ^ [X2: real] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_rat ) ) ) )
% 0.32/0.67 = ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8500_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_int,G: int > rat] :
% 0.32/0.67 ( ( finite_finite_int @ A2 )
% 0.32/0.67 => ( ( groups1072433553688619179nt_rat @ G
% 0.32/0.67 @ ( minus_minus_set_int @ A2
% 0.32/0.67 @ ( collect_int
% 0.32/0.67 @ ^ [X2: int] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_rat ) ) ) )
% 0.32/0.67 = ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8501_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_complex,G: complex > rat] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.67 => ( ( groups225925009352817453ex_rat @ G
% 0.32/0.67 @ ( minus_811609699411566653omplex @ A2
% 0.32/0.67 @ ( collect_complex
% 0.32/0.67 @ ^ [X2: complex] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_rat ) ) ) )
% 0.32/0.67 = ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8502_prod_Osetdiff__irrelevant,axiom,
% 0.32/0.67 ! [A2: set_real,G: real > nat] :
% 0.32/0.67 ( ( finite_finite_real @ A2 )
% 0.32/0.67 => ( ( groups4696554848551431203al_nat @ G
% 0.32/0.67 @ ( minus_minus_set_real @ A2
% 0.32/0.67 @ ( collect_real
% 0.32/0.67 @ ^ [X2: real] :
% 0.32/0.67 ( ( G @ X2 )
% 0.32/0.67 = one_one_nat ) ) ) )
% 0.32/0.67 = ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.setdiff_irrelevant
% 0.32/0.67 thf(fact_8503_prod_Onat__diff__reindex,axiom,
% 0.32/0.67 ! [G: nat > nat,N: nat] :
% 0.32/0.67 ( ( groups708209901874060359at_nat
% 0.32/0.67 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.nat_diff_reindex
% 0.32/0.67 thf(fact_8504_prod_Onat__diff__reindex,axiom,
% 0.32/0.67 ! [G: nat > int,N: nat] :
% 0.32/0.67 ( ( groups705719431365010083at_int
% 0.32/0.67 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.nat_diff_reindex
% 0.32/0.67 thf(fact_8505_prod_OatLeastAtMost__rev,axiom,
% 0.32/0.67 ! [G: nat > nat,N: nat,M: nat] :
% 0.32/0.67 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.32/0.67 = ( groups708209901874060359at_nat
% 0.32/0.67 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.32/0.67 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.atLeastAtMost_rev
% 0.32/0.67 thf(fact_8506_prod_OatLeastAtMost__rev,axiom,
% 0.32/0.67 ! [G: nat > int,N: nat,M: nat] :
% 0.32/0.67 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.32/0.67 = ( groups705719431365010083at_int
% 0.32/0.67 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.32/0.67 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.atLeastAtMost_rev
% 0.32/0.67 thf(fact_8507_fact__code,axiom,
% 0.32/0.67 ( semiri1406184849735516958ct_int
% 0.32/0.67 = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_code
% 0.32/0.67 thf(fact_8508_fact__code,axiom,
% 0.32/0.67 ( semiri5044797733671781792omplex
% 0.32/0.67 = ( ^ [N2: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_code
% 0.32/0.67 thf(fact_8509_fact__code,axiom,
% 0.32/0.67 ( semiri773545260158071498ct_rat
% 0.32/0.67 = ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_code
% 0.32/0.67 thf(fact_8510_fact__code,axiom,
% 0.32/0.67 ( semiri2265585572941072030t_real
% 0.32/0.67 = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_code
% 0.32/0.67 thf(fact_8511_fact__code,axiom,
% 0.32/0.67 ( semiri1408675320244567234ct_nat
% 0.32/0.67 = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_code
% 0.32/0.67 thf(fact_8512_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_real,I: real,F: real > real] :
% 0.32/0.67 ( ( finite_finite_real @ I5 )
% 0.32/0.67 => ( ( member_real @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: real] :
% 0.32/0.67 ( ( member_real @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8513_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_nat,I: nat,F: nat > real] :
% 0.32/0.67 ( ( finite_finite_nat @ I5 )
% 0.32/0.67 => ( ( member_nat @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: nat] :
% 0.32/0.67 ( ( member_nat @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8514_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_int,I: int,F: int > real] :
% 0.32/0.67 ( ( finite_finite_int @ I5 )
% 0.32/0.67 => ( ( member_int @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: int] :
% 0.32/0.67 ( ( member_int @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8515_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_complex,I: complex,F: complex > real] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.67 => ( ( member_complex @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: complex] :
% 0.32/0.67 ( ( member_complex @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8516_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_real,I: real,F: real > rat] :
% 0.32/0.67 ( ( finite_finite_real @ I5 )
% 0.32/0.67 => ( ( member_real @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: real] :
% 0.32/0.67 ( ( member_real @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8517_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_nat,I: nat,F: nat > rat] :
% 0.32/0.67 ( ( finite_finite_nat @ I5 )
% 0.32/0.67 => ( ( member_nat @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: nat] :
% 0.32/0.67 ( ( member_nat @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8518_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_int,I: int,F: int > rat] :
% 0.32/0.67 ( ( finite_finite_int @ I5 )
% 0.32/0.67 => ( ( member_int @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: int] :
% 0.32/0.67 ( ( member_int @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8519_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_complex,I: complex,F: complex > rat] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.67 => ( ( member_complex @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: complex] :
% 0.32/0.67 ( ( member_complex @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8520_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_real,I: real,F: real > int] :
% 0.32/0.67 ( ( finite_finite_real @ I5 )
% 0.32/0.67 => ( ( member_real @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: real] :
% 0.32/0.67 ( ( member_real @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8521_less__1__prod2,axiom,
% 0.32/0.67 ! [I5: set_complex,I: complex,F: complex > int] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.67 => ( ( member_complex @ I @ I5 )
% 0.32/0.67 => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
% 0.32/0.67 => ( ! [I2: complex] :
% 0.32/0.67 ( ( member_complex @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod2
% 0.32/0.67 thf(fact_8522_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_complex,F: complex > real] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_complex )
% 0.32/0.67 => ( ! [I2: complex] :
% 0.32/0.67 ( ( member_complex @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8523_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_nat,F: nat > real] :
% 0.32/0.67 ( ( finite_finite_nat @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_nat )
% 0.32/0.67 => ( ! [I2: nat] :
% 0.32/0.67 ( ( member_nat @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8524_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_int,F: int > real] :
% 0.32/0.67 ( ( finite_finite_int @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_int )
% 0.32/0.67 => ( ! [I2: int] :
% 0.32/0.67 ( ( member_int @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8525_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_real,F: real > real] :
% 0.32/0.67 ( ( finite_finite_real @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_real )
% 0.32/0.67 => ( ! [I2: real] :
% 0.32/0.67 ( ( member_real @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8526_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_complex,F: complex > rat] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_complex )
% 0.32/0.67 => ( ! [I2: complex] :
% 0.32/0.67 ( ( member_complex @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8527_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_nat,F: nat > rat] :
% 0.32/0.67 ( ( finite_finite_nat @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_nat )
% 0.32/0.67 => ( ! [I2: nat] :
% 0.32/0.67 ( ( member_nat @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8528_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_int,F: int > rat] :
% 0.32/0.67 ( ( finite_finite_int @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_int )
% 0.32/0.67 => ( ! [I2: int] :
% 0.32/0.67 ( ( member_int @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8529_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_real,F: real > rat] :
% 0.32/0.67 ( ( finite_finite_real @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_real )
% 0.32/0.67 => ( ! [I2: real] :
% 0.32/0.67 ( ( member_real @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8530_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_complex,F: complex > int] :
% 0.32/0.67 ( ( finite3207457112153483333omplex @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_complex )
% 0.32/0.67 => ( ! [I2: complex] :
% 0.32/0.67 ( ( member_complex @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8531_less__1__prod,axiom,
% 0.32/0.67 ! [I5: set_real,F: real > int] :
% 0.32/0.67 ( ( finite_finite_real @ I5 )
% 0.32/0.67 => ( ( I5 != bot_bot_set_real )
% 0.32/0.67 => ( ! [I2: real] :
% 0.32/0.67 ( ( member_real @ I2 @ I5 )
% 0.32/0.67 => ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
% 0.32/0.67 => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % less_1_prod
% 0.32/0.67 thf(fact_8532_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_nat,A2: set_nat,G: nat > rat] :
% 0.32/0.67 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.67 => ( ( finite_finite_nat @ A2 )
% 0.32/0.67 => ( ( groups73079841787564623at_rat @ G @ A2 )
% 0.32/0.67 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8533_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_complex,A2: set_complex,G: complex > nat] :
% 0.32/0.67 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.67 => ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.67 => ( ( groups861055069439313189ex_nat @ G @ A2 )
% 0.32/0.67 = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8534_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_complex,A2: set_complex,G: complex > int] :
% 0.32/0.67 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 0.32/0.67 => ( ( finite3207457112153483333omplex @ A2 )
% 0.32/0.67 => ( ( groups858564598930262913ex_int @ G @ A2 )
% 0.32/0.67 = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8535_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,G: int > real] :
% 0.32/0.67 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.67 => ( ( finite_finite_int @ A2 )
% 0.32/0.67 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.32/0.67 = ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups2316167850115554303t_real @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8536_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,G: int > rat] :
% 0.32/0.67 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.67 => ( ( finite_finite_int @ A2 )
% 0.32/0.67 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 0.32/0.67 = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8537_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,G: int > nat] :
% 0.32/0.67 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.67 => ( ( finite_finite_int @ A2 )
% 0.32/0.67 => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 0.32/0.67 = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8538_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_nat,A2: set_nat,G: nat > nat] :
% 0.32/0.67 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.67 => ( ( finite_finite_nat @ A2 )
% 0.32/0.67 => ( ( groups708209901874060359at_nat @ G @ A2 )
% 0.32/0.67 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8539_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_nat,A2: set_nat,G: nat > int] :
% 0.32/0.67 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 0.32/0.67 => ( ( finite_finite_nat @ A2 )
% 0.32/0.67 => ( ( groups705719431365010083at_int @ G @ A2 )
% 0.32/0.67 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups705719431365010083at_int @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8540_prod_Osubset__diff,axiom,
% 0.32/0.67 ! [B5: set_int,A2: set_int,G: int > int] :
% 0.32/0.67 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 0.32/0.67 => ( ( finite_finite_int @ A2 )
% 0.32/0.67 => ( ( groups1705073143266064639nt_int @ G @ A2 )
% 0.32/0.67 = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups1705073143266064639nt_int @ G @ B5 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod.subset_diff
% 0.32/0.67 thf(fact_8541_square__fact__le__2__fact,axiom,
% 0.32/0.67 ! [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 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % square_fact_le_2_fact
% 0.32/0.67 thf(fact_8542_Maclaurin__lemma,axiom,
% 0.32/0.67 ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.32/0.67 => ? [B9: real] :
% 0.32/0.67 ( ( F @ H2 )
% 0.32/0.67 = ( plus_plus_real
% 0.32/0.67 @ ( groups6591440286371151544t_real
% 0.32/0.67 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Maclaurin_lemma
% 0.32/0.67 thf(fact_8543_cos__coeff__def,axiom,
% 0.32/0.67 ( cos_coeff
% 0.32/0.67 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_coeff_def
% 0.32/0.67 thf(fact_8544_cos__paired,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( sums_real
% 0.32/0.67 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.67 @ ( cos_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_paired
% 0.32/0.67 thf(fact_8545_sin__paired,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( sums_real
% 0.32/0.67 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.32/0.67 @ ( sin_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_paired
% 0.32/0.67 thf(fact_8546_Maclaurin__sin__expansion3,axiom,
% 0.32/0.67 ! [N: nat,X: real] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.67 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ? [T5: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.32/0.67 & ( ord_less_real @ T5 @ X )
% 0.32/0.67 & ( ( sin_real @ X )
% 0.32/0.67 = ( plus_plus_real
% 0.32/0.67 @ ( groups6591440286371151544t_real
% 0.32/0.67 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Maclaurin_sin_expansion3
% 0.32/0.67 thf(fact_8547_Maclaurin__sin__expansion4,axiom,
% 0.32/0.67 ! [X: real,N: nat] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ? [T5: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.32/0.67 & ( ord_less_eq_real @ T5 @ X )
% 0.32/0.67 & ( ( sin_real @ X )
% 0.32/0.67 = ( plus_plus_real
% 0.32/0.67 @ ( groups6591440286371151544t_real
% 0.32/0.67 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Maclaurin_sin_expansion4
% 0.32/0.67 thf(fact_8548_Maclaurin__sin__expansion2,axiom,
% 0.32/0.67 ! [X: real,N: nat] :
% 0.32/0.67 ? [T5: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.32/0.67 & ( ( sin_real @ X )
% 0.32/0.67 = ( plus_plus_real
% 0.32/0.67 @ ( groups6591440286371151544t_real
% 0.32/0.67 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Maclaurin_sin_expansion2
% 0.32/0.67 thf(fact_8549_Maclaurin__sin__expansion,axiom,
% 0.32/0.67 ! [X: real,N: nat] :
% 0.32/0.67 ? [T5: real] :
% 0.32/0.67 ( ( sin_real @ X )
% 0.32/0.67 = ( plus_plus_real
% 0.32/0.67 @ ( groups6591440286371151544t_real
% 0.32/0.67 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Maclaurin_sin_expansion
% 0.32/0.67 thf(fact_8550_sin__coeff__def,axiom,
% 0.32/0.67 ( sin_coeff
% 0.32/0.67 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_coeff_def
% 0.32/0.67 thf(fact_8551_fact__mono__nat,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.67 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_mono_nat
% 0.32/0.67 thf(fact_8552_fact__ge__self,axiom,
% 0.32/0.67 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_ge_self
% 0.32/0.67 thf(fact_8553_fact__less__mono__nat,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.67 => ( ( ord_less_nat @ M @ N )
% 0.32/0.67 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_less_mono_nat
% 0.32/0.67 thf(fact_8554_fact__ge__Suc__0__nat,axiom,
% 0.32/0.67 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_ge_Suc_0_nat
% 0.32/0.67 thf(fact_8555_dvd__fact,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.32/0.67 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.67 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % dvd_fact
% 0.32/0.67 thf(fact_8556_fact__diff__Suc,axiom,
% 0.32/0.67 ! [N: nat,M: nat] :
% 0.32/0.67 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.32/0.67 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.32/0.67 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_diff_Suc
% 0.32/0.67 thf(fact_8557_fact__div__fact__le__pow,axiom,
% 0.32/0.67 ! [R: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ R @ N )
% 0.32/0.67 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_div_fact_le_pow
% 0.32/0.67 thf(fact_8558_fact__eq__fact__times,axiom,
% 0.32/0.67 ! [N: nat,M: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.67 => ( ( semiri1408675320244567234ct_nat @ M )
% 0.32/0.67 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.32/0.67 @ ( groups708209901874060359at_nat
% 0.32/0.67 @ ^ [X2: nat] : X2
% 0.32/0.67 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_eq_fact_times
% 0.32/0.67 thf(fact_8559_prod__int__plus__eq,axiom,
% 0.32/0.67 ! [I: nat,J: nat] :
% 0.32/0.67 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 0.32/0.67 = ( groups1705073143266064639nt_int
% 0.32/0.67 @ ^ [X2: int] : X2
% 0.32/0.67 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % prod_int_plus_eq
% 0.32/0.67 thf(fact_8560_fact__div__fact,axiom,
% 0.32/0.67 ! [N: nat,M: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.67 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.32/0.67 = ( groups708209901874060359at_nat
% 0.32/0.67 @ ^ [X2: nat] : X2
% 0.32/0.67 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % fact_div_fact
% 0.32/0.67 thf(fact_8561_sin__coeff__Suc,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( sin_coeff @ ( suc @ N ) )
% 0.32/0.67 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_coeff_Suc
% 0.32/0.67 thf(fact_8562_cos__coeff__Suc,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( cos_coeff @ ( suc @ N ) )
% 0.32/0.67 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_coeff_Suc
% 0.32/0.67 thf(fact_8563_complex__unimodular__polar,axiom,
% 0.32/0.67 ! [Z: complex] :
% 0.32/0.67 ( ( ( real_V1022390504157884413omplex @ Z )
% 0.32/0.67 = one_one_real )
% 0.32/0.67 => ~ ! [T5: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.32/0.67 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.32/0.67 => ( Z
% 0.32/0.67 != ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % complex_unimodular_polar
% 0.32/0.67 thf(fact_8564_Maclaurin__exp__lt,axiom,
% 0.32/0.67 ! [X: real,N: nat] :
% 0.32/0.67 ( ( X != zero_zero_real )
% 0.32/0.67 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.67 => ? [T5: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 0.32/0.67 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.32/0.67 & ( ( exp_real @ X )
% 0.32/0.67 = ( plus_plus_real
% 0.32/0.67 @ ( groups6591440286371151544t_real
% 0.32/0.67 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Maclaurin_exp_lt
% 0.32/0.67 thf(fact_8565_binomial__code,axiom,
% 0.32/0.67 ( binomial
% 0.32/0.67 = ( ^ [N2: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K2 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_code
% 0.32/0.67 thf(fact_8566_tan__periodic__pi,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
% 0.32/0.67 = ( tan_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_periodic_pi
% 0.32/0.67 thf(fact_8567_exp__less__cancel__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y3 ) )
% 0.32/0.67 = ( ord_less_real @ X @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_less_cancel_iff
% 0.32/0.67 thf(fact_8568_exp__less__mono,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.67 => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_less_mono
% 0.32/0.67 thf(fact_8569_exp__le__cancel__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y3 ) )
% 0.32/0.67 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_le_cancel_iff
% 0.32/0.67 thf(fact_8570_binomial__n__n,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( binomial @ N @ N )
% 0.32/0.67 = one_one_nat ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_n_n
% 0.32/0.67 thf(fact_8571_binomial__eq__0__iff,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( ( binomial @ N @ K )
% 0.32/0.67 = zero_zero_nat )
% 0.32/0.67 = ( ord_less_nat @ N @ K ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_eq_0_iff
% 0.32/0.67 thf(fact_8572_binomial__Suc__Suc,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.32/0.67 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_Suc_Suc
% 0.32/0.67 thf(fact_8573_binomial__n__0,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( binomial @ N @ zero_zero_nat )
% 0.32/0.67 = one_one_nat ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_n_0
% 0.32/0.67 thf(fact_8574_exp__eq__one__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ( exp_real @ X )
% 0.32/0.67 = one_one_real )
% 0.32/0.67 = ( X = zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_eq_one_iff
% 0.32/0.67 thf(fact_8575_zero__less__binomial__iff,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.32/0.67 = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.32/0.67
% 0.32/0.67 % zero_less_binomial_iff
% 0.32/0.67 thf(fact_8576_tan__npi,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.32/0.67 = zero_zero_real ) ).
% 0.32/0.67
% 0.32/0.67 % tan_npi
% 0.32/0.67 thf(fact_8577_exp__less__one__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 0.32/0.67 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_less_one_iff
% 0.32/0.67 thf(fact_8578_one__less__exp__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 0.32/0.67 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % one_less_exp_iff
% 0.32/0.67 thf(fact_8579_exp__le__one__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 0.32/0.67 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_le_one_iff
% 0.32/0.67 thf(fact_8580_one__le__exp__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 0.32/0.67 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % one_le_exp_iff
% 0.32/0.67 thf(fact_8581_tan__periodic__n,axiom,
% 0.32/0.67 ! [X: real,N: num] :
% 0.32/0.67 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 0.32/0.67 = ( tan_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_periodic_n
% 0.32/0.67 thf(fact_8582_tan__periodic__nat,axiom,
% 0.32/0.67 ! [X: real,N: nat] :
% 0.32/0.67 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 0.32/0.67 = ( tan_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_periodic_nat
% 0.32/0.67 thf(fact_8583_exp__ln__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.32/0.67 = X )
% 0.32/0.67 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_ln_iff
% 0.32/0.67 thf(fact_8584_exp__ln,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( exp_real @ ( ln_ln_real @ X ) )
% 0.32/0.67 = X ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_ln
% 0.32/0.67 thf(fact_8585_tan__periodic__int,axiom,
% 0.32/0.67 ! [X: real,I: int] :
% 0.32/0.67 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 0.32/0.67 = ( tan_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_periodic_int
% 0.32/0.67 thf(fact_8586_norm__cos__sin,axiom,
% 0.32/0.67 ! [T: real] :
% 0.32/0.67 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 0.32/0.67 = one_one_real ) ).
% 0.32/0.67
% 0.32/0.67 % norm_cos_sin
% 0.32/0.67 thf(fact_8587_tan__periodic,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.32/0.67 = ( tan_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_periodic
% 0.32/0.67 thf(fact_8588_complex__diff,axiom,
% 0.32/0.67 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.67 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.32/0.67 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % complex_diff
% 0.32/0.67 thf(fact_8589_exp__less__cancel,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y3 ) )
% 0.32/0.67 => ( ord_less_real @ X @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_less_cancel
% 0.32/0.67 thf(fact_8590_choose__one,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( binomial @ N @ one_one_nat )
% 0.32/0.67 = N ) ).
% 0.32/0.67
% 0.32/0.67 % choose_one
% 0.32/0.67 thf(fact_8591_binomial__eq__0,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( ord_less_nat @ N @ K )
% 0.32/0.67 => ( ( binomial @ N @ K )
% 0.32/0.67 = zero_zero_nat ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_eq_0
% 0.32/0.67 thf(fact_8592_Suc__times__binomial__eq,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 0.32/0.67 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Suc_times_binomial_eq
% 0.32/0.67 thf(fact_8593_Suc__times__binomial,axiom,
% 0.32/0.67 ! [K: nat,N: nat] :
% 0.32/0.67 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 0.32/0.67 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Suc_times_binomial
% 0.32/0.67 thf(fact_8594_binomial__symmetric,axiom,
% 0.32/0.67 ! [K: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.67 => ( ( binomial @ N @ K )
% 0.32/0.67 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_symmetric
% 0.32/0.67 thf(fact_8595_exp__total,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ? [X3: real] :
% 0.32/0.67 ( ( exp_real @ X3 )
% 0.32/0.67 = Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_total
% 0.32/0.67 thf(fact_8596_exp__gt__zero,axiom,
% 0.32/0.67 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_gt_zero
% 0.32/0.67 thf(fact_8597_not__exp__less__zero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.32/0.67
% 0.32/0.67 % not_exp_less_zero
% 0.32/0.67 thf(fact_8598_not__exp__le__zero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 0.32/0.67
% 0.32/0.67 % not_exp_le_zero
% 0.32/0.67 thf(fact_8599_exp__ge__zero,axiom,
% 0.32/0.67 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_ge_zero
% 0.32/0.67 thf(fact_8600_choose__mult__lemma,axiom,
% 0.32/0.67 ! [M: nat,R: nat,K: nat] :
% 0.32/0.67 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 0.32/0.67 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_mult_lemma
% 0.32/0.67 thf(fact_8601_binomial__le__pow,axiom,
% 0.32/0.67 ! [R: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ R @ N )
% 0.32/0.67 => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_le_pow
% 0.32/0.67 thf(fact_8602_Complex__eq__numeral,axiom,
% 0.32/0.67 ! [A: real,B: real,W: num] :
% 0.32/0.67 ( ( ( complex2 @ A @ B )
% 0.32/0.67 = ( numera6690914467698888265omplex @ W ) )
% 0.32/0.67 = ( ( A
% 0.32/0.67 = ( numeral_numeral_real @ W ) )
% 0.32/0.67 & ( B = zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Complex_eq_numeral
% 0.32/0.67 thf(fact_8603_complex__add,axiom,
% 0.32/0.67 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.67 ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.32/0.67 = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % complex_add
% 0.32/0.67 thf(fact_8604_zero__less__binomial,axiom,
% 0.32/0.67 ! [K: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.67 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % zero_less_binomial
% 0.32/0.67 thf(fact_8605_Suc__times__binomial__add,axiom,
% 0.32/0.67 ! [A: nat,B: nat] :
% 0.32/0.67 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 0.32/0.67 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Suc_times_binomial_add
% 0.32/0.67 thf(fact_8606_binomial__Suc__Suc__eq__times,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.32/0.67 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_Suc_Suc_eq_times
% 0.32/0.67 thf(fact_8607_choose__mult,axiom,
% 0.32/0.67 ! [K: nat,M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ K @ M )
% 0.32/0.67 => ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.67 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.32/0.67 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_mult
% 0.32/0.67 thf(fact_8608_binomial__fact__lemma,axiom,
% 0.32/0.67 ! [K: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.67 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.32/0.67 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_fact_lemma
% 0.32/0.67 thf(fact_8609_exp__gt__one,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_gt_one
% 0.32/0.67 thf(fact_8610_binomial__absorb__comp,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.32/0.67 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_absorb_comp
% 0.32/0.67 thf(fact_8611_exp__ge__add__one__self,axiom,
% 0.32/0.67 ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_ge_add_one_self
% 0.32/0.67 thf(fact_8612_Complex__eq__neg__numeral,axiom,
% 0.32/0.67 ! [A: real,B: real,W: num] :
% 0.32/0.67 ( ( ( complex2 @ A @ B )
% 0.32/0.67 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.67 = ( ( A
% 0.32/0.67 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 0.32/0.67 & ( B = zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Complex_eq_neg_numeral
% 0.32/0.67 thf(fact_8613_complex__mult,axiom,
% 0.32/0.67 ! [A: real,B: real,C: real,D: real] :
% 0.32/0.67 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 0.32/0.67 = ( 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 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % complex_mult
% 0.32/0.67 thf(fact_8614_one__complex_Ocode,axiom,
% 0.32/0.67 ( one_one_complex
% 0.32/0.67 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % one_complex.code
% 0.32/0.67 thf(fact_8615_Complex__eq__1,axiom,
% 0.32/0.67 ! [A: real,B: real] :
% 0.32/0.67 ( ( ( complex2 @ A @ B )
% 0.32/0.67 = one_one_complex )
% 0.32/0.67 = ( ( A = one_one_real )
% 0.32/0.67 & ( B = zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Complex_eq_1
% 0.32/0.67 thf(fact_8616_binomial__absorption,axiom,
% 0.32/0.67 ! [K: nat,N: nat] :
% 0.32/0.67 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.32/0.67 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_absorption
% 0.32/0.67 thf(fact_8617_binomial__altdef__nat,axiom,
% 0.32/0.67 ! [K: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ K @ N )
% 0.32/0.67 => ( ( binomial @ N @ K )
% 0.32/0.67 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_altdef_nat
% 0.32/0.67 thf(fact_8618_exp__ge__add__one__self__aux,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_ge_add_one_self_aux
% 0.32/0.67 thf(fact_8619_lemma__exp__total,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 0.32/0.67 => ? [X3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.32/0.67 & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 0.32/0.67 & ( ( exp_real @ X3 )
% 0.32/0.67 = Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % lemma_exp_total
% 0.32/0.67 thf(fact_8620_ln__ge__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X ) )
% 0.32/0.67 = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % ln_ge_iff
% 0.32/0.67 thf(fact_8621_ln__x__over__x__mono,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.67 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % ln_x_over_x_mono
% 0.32/0.67 thf(fact_8622_Complex__eq__neg__1,axiom,
% 0.32/0.67 ! [A: real,B: real] :
% 0.32/0.67 ( ( ( complex2 @ A @ B )
% 0.32/0.67 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.32/0.67 = ( ( A
% 0.32/0.67 = ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.67 & ( B = zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Complex_eq_neg_1
% 0.32/0.67 thf(fact_8623_binomial__mono,axiom,
% 0.32/0.67 ! [K: nat,K6: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ K @ K6 )
% 0.32/0.67 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.32/0.67 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_mono
% 0.32/0.67 thf(fact_8624_binomial__maximum_H,axiom,
% 0.32/0.67 ! [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 ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_maximum'
% 0.32/0.67 thf(fact_8625_binomial__maximum,axiom,
% 0.32/0.67 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_maximum
% 0.32/0.67 thf(fact_8626_binomial__antimono,axiom,
% 0.32/0.67 ! [K: nat,K6: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ K @ K6 )
% 0.32/0.67 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.32/0.67 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.32/0.67 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_antimono
% 0.32/0.67 thf(fact_8627_binomial__le__pow2,axiom,
% 0.32/0.67 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_le_pow2
% 0.32/0.67 thf(fact_8628_choose__reduce__nat,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.67 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.67 => ( ( binomial @ N @ K )
% 0.32/0.67 = ( 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 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_reduce_nat
% 0.32/0.67 thf(fact_8629_times__binomial__minus1__eq,axiom,
% 0.32/0.67 ! [K: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.67 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.32/0.67 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % times_binomial_minus1_eq
% 0.32/0.67 thf(fact_8630_exp__le,axiom,
% 0.32/0.67 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_le
% 0.32/0.67 thf(fact_8631_binomial__less__binomial__Suc,axiom,
% 0.32/0.67 ! [K: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_less_binomial_Suc
% 0.32/0.67 thf(fact_8632_binomial__strict__mono,axiom,
% 0.32/0.67 ! [K: nat,K6: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ K @ K6 )
% 0.32/0.67 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 0.32/0.67 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_strict_mono
% 0.32/0.67 thf(fact_8633_binomial__strict__antimono,axiom,
% 0.32/0.67 ! [K: nat,K6: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ K @ K6 )
% 0.32/0.67 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.32/0.67 => ( ( ord_less_eq_nat @ K6 @ N )
% 0.32/0.67 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_strict_antimono
% 0.32/0.67 thf(fact_8634_central__binomial__odd,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.67 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % central_binomial_odd
% 0.32/0.67 thf(fact_8635_binomial__addition__formula,axiom,
% 0.32/0.67 ! [N: nat,K: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.67 => ( ( binomial @ N @ ( suc @ K ) )
% 0.32/0.67 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_addition_formula
% 0.32/0.67 thf(fact_8636_exp__half__le2,axiom,
% 0.32/0.67 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_half_le2
% 0.32/0.67 thf(fact_8637_tan__45,axiom,
% 0.32/0.67 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 = one_one_real ) ).
% 0.32/0.67
% 0.32/0.67 % tan_45
% 0.32/0.67 thf(fact_8638_choose__two,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.67 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_two
% 0.32/0.67 thf(fact_8639_lemma__tan__total,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ? [X3: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.32/0.67 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 & ( ord_less_real @ Y3 @ ( tan_real @ X3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % lemma_tan_total
% 0.32/0.67 thf(fact_8640_tan__gt__zero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_gt_zero
% 0.32/0.67 thf(fact_8641_tan__total,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ? [X3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.32/0.67 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 & ( ( tan_real @ X3 )
% 0.32/0.67 = Y3 )
% 0.32/0.67 & ! [Y5: real] :
% 0.32/0.67 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.32/0.67 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 & ( ( tan_real @ Y5 )
% 0.32/0.67 = Y3 ) )
% 0.32/0.67 => ( Y5 = X3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_total
% 0.32/0.67 thf(fact_8642_tan__monotone,axiom,
% 0.32/0.67 ! [Y3: real,X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_monotone
% 0.32/0.67 thf(fact_8643_tan__monotone_H,axiom,
% 0.32/0.67 ! [Y3: real,X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ X )
% 0.32/0.67 = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_monotone'
% 0.32/0.67 thf(fact_8644_tan__mono__lt__eq,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y3 ) )
% 0.32/0.67 = ( ord_less_real @ X @ Y3 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_mono_lt_eq
% 0.32/0.67 thf(fact_8645_lemma__tan__total1,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ? [X3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.32/0.67 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 & ( ( tan_real @ X3 )
% 0.32/0.67 = Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % lemma_tan_total1
% 0.32/0.67 thf(fact_8646_tan__minus__45,axiom,
% 0.32/0.67 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.67 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_minus_45
% 0.32/0.67 thf(fact_8647_tan__inverse,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
% 0.32/0.67 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_inverse
% 0.32/0.67 thf(fact_8648_exp__bound,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.67 => ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_bound
% 0.32/0.67 thf(fact_8649_tan__pos__pi2__le,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_pos_pi2_le
% 0.32/0.67 thf(fact_8650_tan__total__pos,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ? [X3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.32/0.67 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 & ( ( tan_real @ X3 )
% 0.32/0.67 = Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_total_pos
% 0.32/0.67 thf(fact_8651_tan__less__zero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.67 => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_less_zero
% 0.32/0.67 thf(fact_8652_tan__mono__le,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_mono_le
% 0.32/0.67 thf(fact_8653_tan__mono__le__eq,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y3 ) )
% 0.32/0.67 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_mono_le_eq
% 0.32/0.67 thf(fact_8654_tan__bound__pi2,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_bound_pi2
% 0.32/0.67 thf(fact_8655_arctan,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.32/0.67 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 & ( ( tan_real @ ( arctan @ Y3 ) )
% 0.32/0.67 = Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % arctan
% 0.32/0.67 thf(fact_8656_arctan__tan,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( arctan @ ( tan_real @ X ) )
% 0.32/0.67 = X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arctan_tan
% 0.32/0.67 thf(fact_8657_arctan__unique,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ( tan_real @ X )
% 0.32/0.67 = Y3 )
% 0.32/0.67 => ( ( arctan @ Y3 )
% 0.32/0.67 = X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arctan_unique
% 0.32/0.67 thf(fact_8658_real__exp__bound__lemma,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_exp_bound_lemma
% 0.32/0.67 thf(fact_8659_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.32/0.67 ! [N: nat,X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 0.32/0.67 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.67 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_ge_one_plus_x_over_n_power_n
% 0.32/0.67 thf(fact_8660_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.32/0.67 ! [X: real,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.67 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.67 => ( 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 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_ge_one_minus_x_over_n_power_n
% 0.32/0.67 thf(fact_8661_tan__total__pi4,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.67 => ? [Z5: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z5 )
% 0.32/0.67 & ( ord_less_real @ Z5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 & ( ( tan_real @ Z5 )
% 0.32/0.67 = X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_total_pi4
% 0.32/0.67 thf(fact_8662_Maclaurin__exp__le,axiom,
% 0.32/0.67 ! [X: real,N: nat] :
% 0.32/0.67 ? [T5: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.32/0.67 & ( ( exp_real @ X )
% 0.32/0.67 = ( plus_plus_real
% 0.32/0.67 @ ( groups6591440286371151544t_real
% 0.32/0.67 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.67 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Maclaurin_exp_le
% 0.32/0.67 thf(fact_8663_exp__lower__Taylor__quadratic,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( 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 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_lower_Taylor_quadratic
% 0.32/0.67 thf(fact_8664_tanh__real__altdef,axiom,
% 0.32/0.67 ( tanh_real
% 0.32/0.67 = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tanh_real_altdef
% 0.32/0.67 thf(fact_8665_central__binomial__lower__bound,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.67 => ( 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 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % central_binomial_lower_bound
% 0.32/0.67 thf(fact_8666_sin__tan,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( sin_real @ X )
% 0.32/0.67 = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_tan
% 0.32/0.67 thf(fact_8667_real__sqrt__eq__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ( sqrt @ X )
% 0.32/0.67 = ( sqrt @ Y3 ) )
% 0.32/0.67 = ( X = Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_eq_iff
% 0.32/0.67 thf(fact_8668_real__sqrt__zero,axiom,
% 0.32/0.67 ( ( sqrt @ zero_zero_real )
% 0.32/0.67 = zero_zero_real ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_zero
% 0.32/0.67 thf(fact_8669_real__sqrt__eq__zero__cancel__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ( sqrt @ X )
% 0.32/0.67 = zero_zero_real )
% 0.32/0.67 = ( X = zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_eq_zero_cancel_iff
% 0.32/0.67 thf(fact_8670_real__sqrt__less__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) )
% 0.32/0.67 = ( ord_less_real @ X @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_less_iff
% 0.32/0.67 thf(fact_8671_real__sqrt__le__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) )
% 0.32/0.67 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_le_iff
% 0.32/0.67 thf(fact_8672_real__sqrt__one,axiom,
% 0.32/0.67 ( ( sqrt @ one_one_real )
% 0.32/0.67 = one_one_real ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_one
% 0.32/0.67 thf(fact_8673_real__sqrt__eq__1__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ( sqrt @ X )
% 0.32/0.67 = one_one_real )
% 0.32/0.67 = ( X = one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_eq_1_iff
% 0.32/0.67 thf(fact_8674_real__sqrt__lt__0__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.32/0.67 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_lt_0_iff
% 0.32/0.67 thf(fact_8675_real__sqrt__gt__0__iff,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.32/0.67 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_gt_0_iff
% 0.32/0.67 thf(fact_8676_real__sqrt__ge__0__iff,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.32/0.67 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_ge_0_iff
% 0.32/0.67 thf(fact_8677_real__sqrt__le__0__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 0.32/0.67 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_le_0_iff
% 0.32/0.67 thf(fact_8678_real__sqrt__gt__1__iff,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.32/0.67 = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_gt_1_iff
% 0.32/0.67 thf(fact_8679_real__sqrt__lt__1__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 0.32/0.67 = ( ord_less_real @ X @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_lt_1_iff
% 0.32/0.67 thf(fact_8680_real__sqrt__le__1__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 0.32/0.67 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_le_1_iff
% 0.32/0.67 thf(fact_8681_real__sqrt__ge__1__iff,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.32/0.67 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_ge_1_iff
% 0.32/0.67 thf(fact_8682_real__sqrt__mult__self,axiom,
% 0.32/0.67 ! [A: real] :
% 0.32/0.67 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 0.32/0.67 = ( abs_abs_real @ A ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_mult_self
% 0.32/0.67 thf(fact_8683_real__sqrt__abs2,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( sqrt @ ( times_times_real @ X @ X ) )
% 0.32/0.67 = ( abs_abs_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_abs2
% 0.32/0.67 thf(fact_8684_real__sqrt__four,axiom,
% 0.32/0.67 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.32/0.67 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_four
% 0.32/0.67 thf(fact_8685_real__sqrt__abs,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.67 = ( abs_abs_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_abs
% 0.32/0.67 thf(fact_8686_real__sqrt__pow2__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.67 = X )
% 0.32/0.67 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_pow2_iff
% 0.32/0.67 thf(fact_8687_real__sqrt__pow2,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.67 = X ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_pow2
% 0.32/0.67 thf(fact_8688_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 0.32/0.67 ! [X: real,Y3: real,Xa: real,Ya: real] :
% 0.32/0.67 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.67 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_sum_squares_mult_squared_eq
% 0.32/0.67 thf(fact_8689_real__sqrt__minus,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( sqrt @ ( uminus_uminus_real @ X ) )
% 0.32/0.67 = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_minus
% 0.32/0.67 thf(fact_8690_real__sqrt__divide,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( sqrt @ ( divide_divide_real @ X @ Y3 ) )
% 0.32/0.67 = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_divide
% 0.32/0.67 thf(fact_8691_real__sqrt__mult,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( sqrt @ ( times_times_real @ X @ Y3 ) )
% 0.32/0.67 = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_mult
% 0.32/0.67 thf(fact_8692_real__sqrt__less__mono,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.67 => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_less_mono
% 0.32/0.67 thf(fact_8693_real__sqrt__le__mono,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.67 => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_le_mono
% 0.32/0.67 thf(fact_8694_real__sqrt__power,axiom,
% 0.32/0.67 ! [X: real,K: nat] :
% 0.32/0.67 ( ( sqrt @ ( power_power_real @ X @ K ) )
% 0.32/0.67 = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_power
% 0.32/0.67 thf(fact_8695_real__sqrt__gt__zero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_gt_zero
% 0.32/0.67 thf(fact_8696_real__sqrt__ge__zero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_ge_zero
% 0.32/0.67 thf(fact_8697_real__sqrt__eq__zero__cancel,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ( sqrt @ X )
% 0.32/0.67 = zero_zero_real )
% 0.32/0.67 => ( X = zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_eq_zero_cancel
% 0.32/0.67 thf(fact_8698_real__sqrt__ge__one,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/0.67 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_ge_one
% 0.32/0.67 thf(fact_8699_finite__nat__iff__bounded__le,axiom,
% 0.32/0.67 ( finite_finite_nat
% 0.32/0.67 = ( ^ [S5: set_nat] :
% 0.32/0.67 ? [K2: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % finite_nat_iff_bounded_le
% 0.32/0.67 thf(fact_8700_real__div__sqrt,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 0.32/0.67 = ( sqrt @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_div_sqrt
% 0.32/0.67 thf(fact_8701_sqrt__add__le__add__sqrt,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt_add_le_add_sqrt
% 0.32/0.67 thf(fact_8702_le__real__sqrt__sumsq,axiom,
% 0.32/0.67 ! [X: real,Y3: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % le_real_sqrt_sumsq
% 0.32/0.67 thf(fact_8703_sqrt2__less__2,axiom,
% 0.32/0.67 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt2_less_2
% 0.32/0.67 thf(fact_8704_sum__choose__lower,axiom,
% 0.32/0.67 ! [R: nat,N: nat] :
% 0.32/0.67 ( ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R @ K2 ) @ K2 )
% 0.32/0.67 @ ( set_ord_atMost_nat @ N ) )
% 0.32/0.67 = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).
% 0.32/0.67
% 0.32/0.67 % sum_choose_lower
% 0.32/0.67 thf(fact_8705_choose__rising__sum_I2_J,axiom,
% 0.32/0.67 ! [N: nat,M: nat] :
% 0.32/0.67 ( ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.32/0.67 @ ( set_ord_atMost_nat @ M ) )
% 0.32/0.67 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_rising_sum(2)
% 0.32/0.67 thf(fact_8706_choose__rising__sum_I1_J,axiom,
% 0.32/0.67 ! [N: nat,M: nat] :
% 0.32/0.67 ( ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 0.32/0.67 @ ( set_ord_atMost_nat @ M ) )
% 0.32/0.67 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_rising_sum(1)
% 0.32/0.67 thf(fact_8707_real__less__rsqrt,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.32/0.67 => ( ord_less_real @ X @ ( sqrt @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_less_rsqrt
% 0.32/0.67 thf(fact_8708_real__le__rsqrt,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.32/0.67 => ( ord_less_eq_real @ X @ ( sqrt @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_le_rsqrt
% 0.32/0.67 thf(fact_8709_sqrt__le__D,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y3 )
% 0.32/0.67 => ( ord_less_eq_real @ X @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt_le_D
% 0.32/0.67 thf(fact_8710_tan__60,axiom,
% 0.32/0.67 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.32/0.67 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_60
% 0.32/0.67 thf(fact_8711_sum__choose__diagonal,axiom,
% 0.32/0.67 ! [M: nat,N: nat] :
% 0.32/0.67 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.67 => ( ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ M ) )
% 0.32/0.67 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sum_choose_diagonal
% 0.32/0.67 thf(fact_8712_vandermonde,axiom,
% 0.32/0.67 ! [M: nat,N: nat,R: nat] :
% 0.32/0.67 ( ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K2 ) ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ R ) )
% 0.32/0.67 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).
% 0.32/0.67
% 0.32/0.67 % vandermonde
% 0.32/0.67 thf(fact_8713_real__sqrt__unique,axiom,
% 0.32/0.67 ! [Y3: real,X: real] :
% 0.32/0.67 ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.67 = X )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ( ( sqrt @ X )
% 0.32/0.67 = Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_unique
% 0.32/0.67 thf(fact_8714_real__le__lsqrt,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ord_less_eq_real @ ( sqrt @ X ) @ Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_le_lsqrt
% 0.32/0.67 thf(fact_8715_lemma__real__divide__sqrt__less,axiom,
% 0.32/0.67 ! [U: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ U )
% 0.32/0.67 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.32/0.67
% 0.32/0.67 % lemma_real_divide_sqrt_less
% 0.32/0.67 thf(fact_8716_real__sqrt__sum__squares__eq__cancel2,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 = Y3 )
% 0.32/0.67 => ( X = zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_sum_squares_eq_cancel2
% 0.32/0.67 thf(fact_8717_real__sqrt__sum__squares__eq__cancel,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 = X )
% 0.32/0.67 => ( Y3 = zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_sum_squares_eq_cancel
% 0.32/0.67 thf(fact_8718_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.32/0.67 ! [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 ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_sum_squares_triangle_ineq
% 0.32/0.67 thf(fact_8719_real__sqrt__sum__squares__ge2,axiom,
% 0.32/0.67 ! [Y3: real,X: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_sum_squares_ge2
% 0.32/0.67 thf(fact_8720_real__sqrt__sum__squares__ge1,axiom,
% 0.32/0.67 ! [X: real,Y3: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_sum_squares_ge1
% 0.32/0.67 thf(fact_8721_sqrt__ge__absD,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y3 ) )
% 0.32/0.67 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt_ge_absD
% 0.32/0.67 thf(fact_8722_cos__45,axiom,
% 0.32/0.67 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_45
% 0.32/0.67 thf(fact_8723_sin__45,axiom,
% 0.32/0.67 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_45
% 0.32/0.67 thf(fact_8724_choose__row__sum,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 0.32/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_row_sum
% 0.32/0.67 thf(fact_8725_real__less__lsqrt,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ( ( ord_less_real @ X @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ord_less_real @ ( sqrt @ X ) @ Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_less_lsqrt
% 0.32/0.67 thf(fact_8726_binomial,axiom,
% 0.32/0.67 ! [A: nat,B: nat,N: nat] :
% 0.32/0.67 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 0.32/0.67 = ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial
% 0.32/0.67 thf(fact_8727_sqrt__sum__squares__le__sum,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt_sum_squares_le_sum
% 0.32/0.67 thf(fact_8728_tan__30,axiom,
% 0.32/0.67 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.32/0.67 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_30
% 0.32/0.67 thf(fact_8729_sqrt__even__pow2,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.67 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.67 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt_even_pow2
% 0.32/0.67 thf(fact_8730_sqrt__sum__squares__le__sum__abs,axiom,
% 0.32/0.67 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt_sum_squares_le_sum_abs
% 0.32/0.67 thf(fact_8731_real__sqrt__ge__abs2,axiom,
% 0.32/0.67 ! [Y3: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_ge_abs2
% 0.32/0.67 thf(fact_8732_real__sqrt__ge__abs1,axiom,
% 0.32/0.67 ! [X: real,Y3: 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 @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_ge_abs1
% 0.32/0.67 thf(fact_8733_ln__sqrt,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ln_ln_real @ ( sqrt @ X ) )
% 0.32/0.67 = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % ln_sqrt
% 0.32/0.67 thf(fact_8734_cos__30,axiom,
% 0.32/0.67 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 0.32/0.67 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_30
% 0.32/0.67 thf(fact_8735_sin__60,axiom,
% 0.32/0.67 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 0.32/0.67 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_60
% 0.32/0.67 thf(fact_8736_complex__norm,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y3 ) )
% 0.32/0.67 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % complex_norm
% 0.32/0.67 thf(fact_8737_polynomial__product__nat,axiom,
% 0.32/0.67 ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 0.32/0.67 ( ! [I2: nat] :
% 0.32/0.67 ( ( ord_less_nat @ M @ I2 )
% 0.32/0.67 => ( ( A @ I2 )
% 0.32/0.67 = zero_zero_nat ) )
% 0.32/0.67 => ( ! [J2: nat] :
% 0.32/0.67 ( ( ord_less_nat @ N @ J2 )
% 0.32/0.67 => ( ( B @ J2 )
% 0.32/0.67 = zero_zero_nat ) )
% 0.32/0.67 => ( ( times_times_nat
% 0.32/0.67 @ ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ M ) )
% 0.32/0.67 @ ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ N ) ) )
% 0.32/0.67 = ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [R5: nat] :
% 0.32/0.67 ( times_times_nat
% 0.32/0.67 @ ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ R5 ) )
% 0.32/0.67 @ ( power_power_nat @ X @ R5 ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % polynomial_product_nat
% 0.32/0.67 thf(fact_8738_choose__square__sum,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ N ) )
% 0.32/0.67 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_square_sum
% 0.32/0.67 thf(fact_8739_arsinh__real__aux,axiom,
% 0.32/0.67 ! [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 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arsinh_real_aux
% 0.32/0.67 thf(fact_8740_real__sqrt__power__even,axiom,
% 0.32/0.67 ! [N: nat,X: real] :
% 0.32/0.67 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( power_power_real @ ( sqrt @ X ) @ N )
% 0.32/0.67 = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_power_even
% 0.32/0.67 thf(fact_8741_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.32/0.67 ! [X: real,Y3: 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 @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_sum_squares_mult_ge_zero
% 0.32/0.67 thf(fact_8742_arith__geo__mean__sqrt,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arith_geo_mean_sqrt
% 0.32/0.67 thf(fact_8743_cos__x__y__le__one,axiom,
% 0.32/0.67 ! [X: real,Y3: 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 @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.32/0.67
% 0.32/0.67 % cos_x_y_le_one
% 0.32/0.67 thf(fact_8744_real__sqrt__sum__squares__less,axiom,
% 0.32/0.67 ! [X: real,U: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.32/0.67 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_sum_squares_less
% 0.32/0.67 thf(fact_8745_arcosh__real__def,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/0.67 => ( ( arcosh_real @ X )
% 0.32/0.67 = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcosh_real_def
% 0.32/0.67 thf(fact_8746_cos__arctan,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( cos_real @ ( arctan @ X ) )
% 0.32/0.67 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_arctan
% 0.32/0.67 thf(fact_8747_sin__arctan,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( sin_real @ ( arctan @ X ) )
% 0.32/0.67 = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_arctan
% 0.32/0.67 thf(fact_8748_binomial__r__part__sum,axiom,
% 0.32/0.67 ! [M: nat] :
% 0.32/0.67 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.32/0.67 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % binomial_r_part_sum
% 0.32/0.67 thf(fact_8749_choose__linear__sum,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( groups3542108847815614940at_nat
% 0.32/0.67 @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
% 0.32/0.67 @ ( set_ord_atMost_nat @ N ) )
% 0.32/0.67 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % choose_linear_sum
% 0.32/0.67 thf(fact_8750_sqrt__sum__squares__half__less,axiom,
% 0.32/0.67 ! [X: real,U: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt_sum_squares_half_less
% 0.32/0.67 thf(fact_8751_sin__cos__sqrt,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 0.32/0.67 => ( ( sin_real @ X )
% 0.32/0.67 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_cos_sqrt
% 0.32/0.67 thf(fact_8752_arctan__half,axiom,
% 0.32/0.67 ( arctan
% 0.32/0.67 = ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arctan_half
% 0.32/0.67 thf(fact_8753_cos__tan,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( cos_real @ X )
% 0.32/0.67 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_tan
% 0.32/0.67 thf(fact_8754_arsinh__real__def,axiom,
% 0.32/0.67 ( arsinh_real
% 0.32/0.67 = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arsinh_real_def
% 0.32/0.67 thf(fact_8755_cos__arcsin,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.67 => ( ( cos_real @ ( arcsin @ X ) )
% 0.32/0.67 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_arcsin
% 0.32/0.67 thf(fact_8756_of__nat__id,axiom,
% 0.32/0.67 ( semiri1316708129612266289at_nat
% 0.32/0.67 = ( ^ [N2: nat] : N2 ) ) ).
% 0.32/0.67
% 0.32/0.67 % of_nat_id
% 0.32/0.67 thf(fact_8757_sin__arcsin,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.32/0.67 = Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_arcsin
% 0.32/0.67 thf(fact_8758_arcsin__1,axiom,
% 0.32/0.67 ( ( arcsin @ one_one_real )
% 0.32/0.67 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_1
% 0.32/0.67 thf(fact_8759_arcsin__minus__1,axiom,
% 0.32/0.67 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.67 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_minus_1
% 0.32/0.67 thf(fact_8760_real__scaleR__def,axiom,
% 0.32/0.67 real_V1485227260804924795R_real = times_times_real ).
% 0.32/0.67
% 0.32/0.67 % real_scaleR_def
% 0.32/0.67 thf(fact_8761_complex__scaleR,axiom,
% 0.32/0.67 ! [R: real,A: real,B: real] :
% 0.32/0.67 ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B ) )
% 0.32/0.67 = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % complex_scaleR
% 0.32/0.67 thf(fact_8762_arcsin__minus,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.67 => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 0.32/0.67 = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_minus
% 0.32/0.67 thf(fact_8763_arcsin__le__arcsin,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_le_arcsin
% 0.32/0.67 thf(fact_8764_arcsin__eq__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.32/0.67 => ( ( ( arcsin @ X )
% 0.32/0.67 = ( arcsin @ Y3 ) )
% 0.32/0.67 = ( X = Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_eq_iff
% 0.32/0.67 thf(fact_8765_arcsin__le__mono,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y3 ) )
% 0.32/0.67 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_le_mono
% 0.32/0.67 thf(fact_8766_arcsin__less__arcsin,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_less_arcsin
% 0.32/0.67 thf(fact_8767_arcsin__less__mono,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y3 ) )
% 0.32/0.67 = ( ord_less_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_less_mono
% 0.32/0.67 thf(fact_8768_cos__arcsin__nonzero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.67 => ( ( cos_real @ ( arcsin @ X ) )
% 0.32/0.67 != zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_arcsin_nonzero
% 0.32/0.67 thf(fact_8769_arcsin__lt__bounded,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.32/0.67 & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_lt_bounded
% 0.32/0.67 thf(fact_8770_arcsin__bounded,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_bounded
% 0.32/0.67 thf(fact_8771_arcsin__ubound,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_ubound
% 0.32/0.67 thf(fact_8772_arcsin__lbound,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_lbound
% 0.32/0.67 thf(fact_8773_arcsin__sin,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( arcsin @ ( sin_real @ X ) )
% 0.32/0.67 = X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_sin
% 0.32/0.67 thf(fact_8774_arcsin,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.32/0.67 = Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin
% 0.32/0.67 thf(fact_8775_arcsin__pi,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 0.32/0.67 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.32/0.67 = Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_pi
% 0.32/0.67 thf(fact_8776_arcsin__le__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y3 )
% 0.32/0.67 = ( ord_less_eq_real @ X @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arcsin_le_iff
% 0.32/0.67 thf(fact_8777_le__arcsin__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X ) )
% 0.32/0.67 = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % le_arcsin_iff
% 0.32/0.67 thf(fact_8778_sin__arccos__abs,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.32/0.67 => ( ( sin_real @ ( arccos @ Y3 ) )
% 0.32/0.67 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_arccos_abs
% 0.32/0.67 thf(fact_8779_sin__arccos,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.67 => ( ( sin_real @ ( arccos @ X ) )
% 0.32/0.67 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_arccos
% 0.32/0.67 thf(fact_8780_Maclaurin__sin__bound,axiom,
% 0.32/0.67 ! [X: real,N: nat] :
% 0.32/0.67 ( ord_less_eq_real
% 0.32/0.67 @ ( abs_abs_real
% 0.32/0.67 @ ( minus_minus_real @ ( sin_real @ X )
% 0.32/0.67 @ ( groups6591440286371151544t_real
% 0.32/0.67 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.67 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.32/0.67 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % Maclaurin_sin_bound
% 0.32/0.67 thf(fact_8781_arccos__1,axiom,
% 0.32/0.67 ( ( arccos @ one_one_real )
% 0.32/0.67 = zero_zero_real ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_1
% 0.32/0.67 thf(fact_8782_arccos__minus__1,axiom,
% 0.32/0.67 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.67 = pi ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_minus_1
% 0.32/0.67 thf(fact_8783_cos__arccos,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.32/0.67 = Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_arccos
% 0.32/0.67 thf(fact_8784_arccos__0,axiom,
% 0.32/0.67 ( ( arccos @ zero_zero_real )
% 0.32/0.67 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_0
% 0.32/0.67 thf(fact_8785_real__sqrt__inverse,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( sqrt @ ( inverse_inverse_real @ X ) )
% 0.32/0.67 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_sqrt_inverse
% 0.32/0.67 thf(fact_8786_divide__real__def,axiom,
% 0.32/0.67 ( divide_divide_real
% 0.32/0.67 = ( ^ [X2: real,Y: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % divide_real_def
% 0.32/0.67 thf(fact_8787_arccos__le__arccos,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_le_arccos
% 0.32/0.67 thf(fact_8788_arccos__eq__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.67 & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 0.32/0.67 => ( ( ( arccos @ X )
% 0.32/0.67 = ( arccos @ Y3 ) )
% 0.32/0.67 = ( X = Y3 ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_eq_iff
% 0.32/0.67 thf(fact_8789_arccos__le__mono,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y3 ) )
% 0.32/0.67 = ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_le_mono
% 0.32/0.67 thf(fact_8790_forall__pos__mono__1,axiom,
% 0.32/0.67 ! [P: real > $o,E: real] :
% 0.32/0.67 ( ! [D3: real,E2: real] :
% 0.32/0.67 ( ( ord_less_real @ D3 @ E2 )
% 0.32/0.67 => ( ( P @ D3 )
% 0.32/0.67 => ( P @ E2 ) ) )
% 0.32/0.67 => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.32/0.67 => ( P @ E ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % forall_pos_mono_1
% 0.32/0.67 thf(fact_8791_forall__pos__mono,axiom,
% 0.32/0.67 ! [P: real > $o,E: real] :
% 0.32/0.67 ( ! [D3: real,E2: real] :
% 0.32/0.67 ( ( ord_less_real @ D3 @ E2 )
% 0.32/0.67 => ( ( P @ D3 )
% 0.32/0.67 => ( P @ E2 ) ) )
% 0.32/0.67 => ( ! [N4: nat] :
% 0.32/0.67 ( ( N4 != zero_zero_nat )
% 0.32/0.67 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
% 0.32/0.67 => ( ( ord_less_real @ zero_zero_real @ E )
% 0.32/0.67 => ( P @ E ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % forall_pos_mono
% 0.32/0.67 thf(fact_8792_real__arch__inverse,axiom,
% 0.32/0.67 ! [E: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ E )
% 0.32/0.67 = ( ? [N2: nat] :
% 0.32/0.67 ( ( N2 != zero_zero_nat )
% 0.32/0.67 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.32/0.67 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_arch_inverse
% 0.32/0.67 thf(fact_8793_sqrt__divide__self__eq,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 0.32/0.67 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sqrt_divide_self_eq
% 0.32/0.67 thf(fact_8794_ln__inverse,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 0.32/0.67 = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % ln_inverse
% 0.32/0.67 thf(fact_8795_arccos__lbound,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_lbound
% 0.32/0.67 thf(fact_8796_arccos__less__arccos,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_less_arccos
% 0.32/0.67 thf(fact_8797_arccos__less__mono,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.32/0.67 => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y3 ) )
% 0.32/0.67 = ( ord_less_real @ Y3 @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_less_mono
% 0.32/0.67 thf(fact_8798_arccos__ubound,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_ubound
% 0.32/0.67 thf(fact_8799_arccos__cos,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ pi )
% 0.32/0.67 => ( ( arccos @ ( cos_real @ X ) )
% 0.32/0.67 = X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_cos
% 0.32/0.67 thf(fact_8800_cos__arccos__abs,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.32/0.67 => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.32/0.67 = Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % cos_arccos_abs
% 0.32/0.67 thf(fact_8801_arccos__cos__eq__abs,axiom,
% 0.32/0.67 ! [Theta: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 0.32/0.67 => ( ( arccos @ ( cos_real @ Theta ) )
% 0.32/0.67 = ( abs_abs_real @ Theta ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_cos_eq_abs
% 0.32/0.67 thf(fact_8802_arccos__lt__bounded,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.32/0.67 & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_lt_bounded
% 0.32/0.67 thf(fact_8803_arccos__bounded,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_bounded
% 0.32/0.67 thf(fact_8804_sin__arccos__nonzero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.67 => ( ( sin_real @ ( arccos @ X ) )
% 0.32/0.67 != zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % sin_arccos_nonzero
% 0.32/0.67 thf(fact_8805_arccos__cos2,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.32/0.67 => ( ( arccos @ ( cos_real @ X ) )
% 0.32/0.67 = ( uminus_uminus_real @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_cos2
% 0.32/0.67 thf(fact_8806_arccos__minus,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.67 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.67 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.32/0.67 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_minus
% 0.32/0.67 thf(fact_8807_exp__plus__inverse__exp,axiom,
% 0.32/0.67 ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % exp_plus_inverse_exp
% 0.32/0.67 thf(fact_8808_plus__inverse__ge__2,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % plus_inverse_ge_2
% 0.32/0.67 thf(fact_8809_real__inv__sqrt__pow2,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.67 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.67 = ( inverse_inverse_real @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_inv_sqrt_pow2
% 0.32/0.67 thf(fact_8810_arccos,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.32/0.67 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 0.32/0.67 & ( ( cos_real @ ( arccos @ Y3 ) )
% 0.32/0.67 = Y3 ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos
% 0.32/0.67 thf(fact_8811_arccos__minus__abs,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.67 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 0.32/0.67 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_minus_abs
% 0.32/0.67 thf(fact_8812_tan__cot,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.32/0.67 = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % tan_cot
% 0.32/0.67 thf(fact_8813_real__le__x__sinh,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.67 => ( 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 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_le_x_sinh
% 0.32/0.67 thf(fact_8814_real__le__abs__sinh,axiom,
% 0.32/0.67 ! [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 ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % real_le_abs_sinh
% 0.32/0.67 thf(fact_8815_arccos__le__pi2,axiom,
% 0.32/0.67 ! [Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.67 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.32/0.67 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_le_pi2
% 0.32/0.67 thf(fact_8816_arccos__cos__eq__abs__2pi,axiom,
% 0.32/0.67 ! [Theta: real] :
% 0.32/0.67 ~ ! [K3: int] :
% 0.32/0.67 ( ( arccos @ ( cos_real @ Theta ) )
% 0.32/0.67 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % arccos_cos_eq_abs_2pi
% 0.32/0.67 thf(fact_8817_cot__less__zero,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 0.32/0.67 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.67 => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 0.32/0.67
% 0.32/0.67 % cot_less_zero
% 0.32/0.67 thf(fact_8818_i__even__power,axiom,
% 0.32/0.67 ! [N: nat] :
% 0.32/0.67 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.67 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 0.32/0.67
% 0.32/0.67 % i_even_power
% 0.32/0.67 thf(fact_8819_sinh__real__less__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y3 ) )
% 0.32/0.67 = ( ord_less_real @ X @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % sinh_real_less_iff
% 0.32/0.67 thf(fact_8820_sinh__real__le__iff,axiom,
% 0.32/0.67 ! [X: real,Y3: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y3 ) )
% 0.32/0.67 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 0.32/0.67
% 0.32/0.67 % sinh_real_le_iff
% 0.32/0.67 thf(fact_8821_sinh__real__neg__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.32/0.67 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % sinh_real_neg_iff
% 0.32/0.67 thf(fact_8822_sinh__real__pos__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.32/0.67 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % sinh_real_pos_iff
% 0.32/0.67 thf(fact_8823_sinh__real__nonpos__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
% 0.32/0.67 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.32/0.67
% 0.32/0.67 % sinh_real_nonpos_iff
% 0.32/0.67 thf(fact_8824_sinh__real__nonneg__iff,axiom,
% 0.32/0.67 ! [X: real] :
% 0.32/0.67 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
% 0.32/0.67 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % sinh_real_nonneg_iff
% 0.32/0.67 thf(fact_8825_norm__ii,axiom,
% 0.32/0.67 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 0.32/0.67 = one_one_real ) ).
% 0.32/0.67
% 0.32/0.67 % norm_ii
% 0.32/0.67 thf(fact_8826_complex__i__mult__minus,axiom,
% 0.32/0.67 ! [X: complex] :
% 0.32/0.67 ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
% 0.32/0.67 = ( uminus1482373934393186551omplex @ X ) ) ).
% 0.32/0.67
% 0.32/0.67 % complex_i_mult_minus
% 0.32/0.67 thf(fact_8827_divide__i,axiom,
% 0.32/0.67 ! [X: complex] :
% 0.32/0.67 ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
% 0.32/0.67 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % divide_i
% 0.32/0.68 thf(fact_8828_i__squared,axiom,
% 0.32/0.68 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 0.32/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.68
% 0.32/0.68 % i_squared
% 0.32/0.68 thf(fact_8829_divide__numeral__i,axiom,
% 0.32/0.68 ! [Z: complex,N: num] :
% 0.32/0.68 ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 0.32/0.68 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % divide_numeral_i
% 0.32/0.68 thf(fact_8830_cot__npi,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 0.32/0.68 = zero_zero_real ) ).
% 0.32/0.68
% 0.32/0.68 % cot_npi
% 0.32/0.68 thf(fact_8831_cot__periodic,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.32/0.68 = ( cot_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % cot_periodic
% 0.32/0.68 thf(fact_8832_power2__i,axiom,
% 0.32/0.68 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.68
% 0.32/0.68 % power2_i
% 0.32/0.68 thf(fact_8833_divide__complex__def,axiom,
% 0.32/0.68 ( divide1717551699836669952omplex
% 0.32/0.68 = ( ^ [X2: complex,Y: complex] : ( times_times_complex @ X2 @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % divide_complex_def
% 0.32/0.68 thf(fact_8834_sinh__less__cosh__real,axiom,
% 0.32/0.68 ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % sinh_less_cosh_real
% 0.32/0.68 thf(fact_8835_sinh__le__cosh__real,axiom,
% 0.32/0.68 ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % sinh_le_cosh_real
% 0.32/0.68 thf(fact_8836_complex__i__not__numeral,axiom,
% 0.32/0.68 ! [W: num] :
% 0.32/0.68 ( imaginary_unit
% 0.32/0.68 != ( numera6690914467698888265omplex @ W ) ) ).
% 0.32/0.68
% 0.32/0.68 % complex_i_not_numeral
% 0.32/0.68 thf(fact_8837_cosh__real__pos,axiom,
% 0.32/0.68 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_real_pos
% 0.32/0.68 thf(fact_8838_cosh__real__nonpos__le__iff,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.68 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) )
% 0.32/0.68 = ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_real_nonpos_le_iff
% 0.32/0.68 thf(fact_8839_cosh__real__nonneg__le__iff,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) )
% 0.32/0.68 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_real_nonneg_le_iff
% 0.32/0.68 thf(fact_8840_cosh__real__nonneg,axiom,
% 0.32/0.68 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_real_nonneg
% 0.32/0.68 thf(fact_8841_cosh__real__ge__1,axiom,
% 0.32/0.68 ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_real_ge_1
% 0.32/0.68 thf(fact_8842_i__times__eq__iff,axiom,
% 0.32/0.68 ! [W: complex,Z: complex] :
% 0.32/0.68 ( ( ( times_times_complex @ imaginary_unit @ W )
% 0.32/0.68 = Z )
% 0.32/0.68 = ( W
% 0.32/0.68 = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % i_times_eq_iff
% 0.32/0.68 thf(fact_8843_complex__i__not__neg__numeral,axiom,
% 0.32/0.68 ! [W: num] :
% 0.32/0.68 ( imaginary_unit
% 0.32/0.68 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % complex_i_not_neg_numeral
% 0.32/0.68 thf(fact_8844_cosh__real__nonpos__less__iff,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.68 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.32/0.68 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) )
% 0.32/0.68 = ( ord_less_real @ Y3 @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_real_nonpos_less_iff
% 0.32/0.68 thf(fact_8845_cosh__real__nonneg__less__iff,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) )
% 0.32/0.68 = ( ord_less_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_real_nonneg_less_iff
% 0.32/0.68 thf(fact_8846_cosh__real__strict__mono,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ Y3 )
% 0.32/0.68 => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_real_strict_mono
% 0.32/0.68 thf(fact_8847_arcosh__cosh__real,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( arcosh_real @ ( cosh_real @ X ) )
% 0.32/0.68 = X ) ) ).
% 0.32/0.68
% 0.32/0.68 % arcosh_cosh_real
% 0.32/0.68 thf(fact_8848_imaginary__unit_Ocode,axiom,
% 0.32/0.68 ( imaginary_unit
% 0.32/0.68 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % imaginary_unit.code
% 0.32/0.68 thf(fact_8849_Complex__eq__i,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( ( complex2 @ X @ Y3 )
% 0.32/0.68 = imaginary_unit )
% 0.32/0.68 = ( ( X = zero_zero_real )
% 0.32/0.68 & ( Y3 = one_one_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Complex_eq_i
% 0.32/0.68 thf(fact_8850_i__mult__Complex,axiom,
% 0.32/0.68 ! [A: real,B: real] :
% 0.32/0.68 ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
% 0.32/0.68 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.32/0.68
% 0.32/0.68 % i_mult_Complex
% 0.32/0.68 thf(fact_8851_Complex__mult__i,axiom,
% 0.32/0.68 ! [A: real,B: real] :
% 0.32/0.68 ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
% 0.32/0.68 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.32/0.68
% 0.32/0.68 % Complex_mult_i
% 0.32/0.68 thf(fact_8852_complex__inverse,axiom,
% 0.32/0.68 ! [A: real,B: real] :
% 0.32/0.68 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 0.32/0.68 = ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % complex_inverse
% 0.32/0.68 thf(fact_8853_cosh__ln__real,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 0.32/0.68 = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cosh_ln_real
% 0.32/0.68 thf(fact_8854_cot__gt__zero,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.68 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cot_gt_zero
% 0.32/0.68 thf(fact_8855_sinh__ln__real,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 0.32/0.68 = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % sinh_ln_real
% 0.32/0.68 thf(fact_8856_tan__cot_H,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 0.32/0.68 = ( cot_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % tan_cot'
% 0.32/0.68 thf(fact_8857_Arg__minus__ii,axiom,
% 0.32/0.68 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 0.32/0.68 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Arg_minus_ii
% 0.32/0.68 thf(fact_8858_Arg__ii,axiom,
% 0.32/0.68 ( ( arg @ imaginary_unit )
% 0.32/0.68 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Arg_ii
% 0.32/0.68 thf(fact_8859_cis__minus__pi__half,axiom,
% 0.32/0.68 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.32/0.68 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 0.32/0.68
% 0.32/0.68 % cis_minus_pi_half
% 0.32/0.68 thf(fact_8860_log__base__10__eq1,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.68 = ( 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 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_base_10_eq1
% 0.32/0.68 thf(fact_8861_log__one,axiom,
% 0.32/0.68 ! [A: real] :
% 0.32/0.68 ( ( log @ A @ one_one_real )
% 0.32/0.68 = zero_zero_real ) ).
% 0.32/0.68
% 0.32/0.68 % log_one
% 0.32/0.68 thf(fact_8862_norm__cis,axiom,
% 0.32/0.68 ! [A: real] :
% 0.32/0.68 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 0.32/0.68 = one_one_real ) ).
% 0.32/0.68
% 0.32/0.68 % norm_cis
% 0.32/0.68 thf(fact_8863_zero__less__log__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.32/0.68 = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % zero_less_log_cancel_iff
% 0.32/0.68 thf(fact_8864_log__less__zero__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.32/0.68 = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_less_zero_cancel_iff
% 0.32/0.68 thf(fact_8865_one__less__log__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 0.32/0.68 = ( ord_less_real @ A @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % one_less_log_cancel_iff
% 0.32/0.68 thf(fact_8866_log__less__one__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 0.32/0.68 = ( ord_less_real @ X @ A ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_less_one_cancel_iff
% 0.32/0.68 thf(fact_8867_log__less__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y3 ) )
% 0.32/0.68 = ( ord_less_real @ X @ Y3 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_less_cancel_iff
% 0.32/0.68 thf(fact_8868_log__eq__one,axiom,
% 0.32/0.68 ! [A: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( log @ A @ A )
% 0.32/0.68 = one_one_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_eq_one
% 0.32/0.68 thf(fact_8869_zero__le__log__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 0.32/0.68 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % zero_le_log_cancel_iff
% 0.32/0.68 thf(fact_8870_log__le__zero__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 0.32/0.68 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_le_zero_cancel_iff
% 0.32/0.68 thf(fact_8871_one__le__log__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 0.32/0.68 = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % one_le_log_cancel_iff
% 0.32/0.68 thf(fact_8872_log__le__one__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 0.32/0.68 = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_le_one_cancel_iff
% 0.32/0.68 thf(fact_8873_log__le__cancel__iff,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y3 ) )
% 0.32/0.68 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_le_cancel_iff
% 0.32/0.68 thf(fact_8874_log__pow__cancel,axiom,
% 0.32/0.68 ! [A: real,B: nat] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 0.32/0.68 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_pow_cancel
% 0.32/0.68 thf(fact_8875_cis__pi__half,axiom,
% 0.32/0.68 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.68 = imaginary_unit ) ).
% 0.32/0.68
% 0.32/0.68 % cis_pi_half
% 0.32/0.68 thf(fact_8876_cis__2pi,axiom,
% 0.32/0.68 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.32/0.68 = one_one_complex ) ).
% 0.32/0.68
% 0.32/0.68 % cis_2pi
% 0.32/0.68 thf(fact_8877_cis__divide,axiom,
% 0.32/0.68 ! [A: real,B: real] :
% 0.32/0.68 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 0.32/0.68 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cis_divide
% 0.32/0.68 thf(fact_8878_log__def,axiom,
% 0.32/0.68 ( log
% 0.32/0.68 = ( ^ [A3: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_def
% 0.32/0.68 thf(fact_8879_cis__mult,axiom,
% 0.32/0.68 ! [A: real,B: real] :
% 0.32/0.68 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
% 0.32/0.68 = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cis_mult
% 0.32/0.68 thf(fact_8880_log__ln,axiom,
% 0.32/0.68 ( ln_ln_real
% 0.32/0.68 = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_ln
% 0.32/0.68 thf(fact_8881_log__base__change,axiom,
% 0.32/0.68 ! [A: real,B: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( log @ B @ X )
% 0.32/0.68 = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_base_change
% 0.32/0.68 thf(fact_8882_less__log__of__power,axiom,
% 0.32/0.68 ! [B: real,N: nat,M: real] :
% 0.32/0.68 ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % less_log_of_power
% 0.32/0.68 thf(fact_8883_log__of__power__eq,axiom,
% 0.32/0.68 ! [M: nat,B: real,N: nat] :
% 0.32/0.68 ( ( ( semiri5074537144036343181t_real @ M )
% 0.32/0.68 = ( power_power_real @ B @ N ) )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( semiri5074537144036343181t_real @ N )
% 0.32/0.68 = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_of_power_eq
% 0.32/0.68 thf(fact_8884_DeMoivre,axiom,
% 0.32/0.68 ! [A: real,N: nat] :
% 0.32/0.68 ( ( power_power_complex @ ( cis @ A ) @ N )
% 0.32/0.68 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % DeMoivre
% 0.32/0.68 thf(fact_8885_log__mult,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( log @ A @ ( times_times_real @ X @ Y3 ) )
% 0.32/0.68 = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_mult
% 0.32/0.68 thf(fact_8886_log__divide,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( log @ A @ ( divide_divide_real @ X @ Y3 ) )
% 0.32/0.68 = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_divide
% 0.32/0.68 thf(fact_8887_le__log__of__power,axiom,
% 0.32/0.68 ! [B: real,N: nat,M: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % le_log_of_power
% 0.32/0.68 thf(fact_8888_log__base__pow,axiom,
% 0.32/0.68 ! [A: real,N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( log @ ( power_power_real @ A @ N ) @ X )
% 0.32/0.68 = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_base_pow
% 0.32/0.68 thf(fact_8889_log__nat__power,axiom,
% 0.32/0.68 ! [X: real,B: real,N: nat] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( log @ B @ ( power_power_real @ X @ N ) )
% 0.32/0.68 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_nat_power
% 0.32/0.68 thf(fact_8890_log__inverse,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 0.32/0.68 = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_inverse
% 0.32/0.68 thf(fact_8891_log2__of__power__eq,axiom,
% 0.32/0.68 ! [M: nat,N: nat] :
% 0.32/0.68 ( ( M
% 0.32/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.68 => ( ( semiri5074537144036343181t_real @ N )
% 0.32/0.68 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log2_of_power_eq
% 0.32/0.68 thf(fact_8892_log__of__power__less,axiom,
% 0.32/0.68 ! [M: nat,B: real,N: nat] :
% 0.32/0.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.68 => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_of_power_less
% 0.32/0.68 thf(fact_8893_log__eq__div__ln__mult__log,axiom,
% 0.32/0.68 ! [A: real,B: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.68 => ( ( B != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( log @ A @ X )
% 0.32/0.68 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_eq_div_ln_mult_log
% 0.32/0.68 thf(fact_8894_Arg__bounded,axiom,
% 0.32/0.68 ! [Z: complex] :
% 0.32/0.68 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.32/0.68 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 0.32/0.68
% 0.32/0.68 % Arg_bounded
% 0.32/0.68 thf(fact_8895_log__of__power__le,axiom,
% 0.32/0.68 ! [M: nat,B: real,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.68 => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_of_power_le
% 0.32/0.68 thf(fact_8896_less__log2__of__power,axiom,
% 0.32/0.68 ! [N: nat,M: nat] :
% 0.32/0.68 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.32/0.68 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % less_log2_of_power
% 0.32/0.68 thf(fact_8897_le__log2__of__power,axiom,
% 0.32/0.68 ! [N: nat,M: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.32/0.68 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % le_log2_of_power
% 0.32/0.68 thf(fact_8898_log2__of__power__less,axiom,
% 0.32/0.68 ! [M: nat,N: nat] :
% 0.32/0.68 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.68 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.68 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log2_of_power_less
% 0.32/0.68 thf(fact_8899_log2__of__power__le,axiom,
% 0.32/0.68 ! [M: nat,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.68 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.68 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log2_of_power_le
% 0.32/0.68 thf(fact_8900_log__base__10__eq2,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 0.32/0.68 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_base_10_eq2
% 0.32/0.68 thf(fact_8901_ceiling__log__nat__eq__powr__iff,axiom,
% 0.32/0.68 ! [B: nat,K: nat,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.68 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.68 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.32/0.68 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.32/0.68 = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.32/0.68 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ceiling_log_nat_eq_powr_iff
% 0.32/0.68 thf(fact_8902_bij__betw__roots__unity,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( bij_betw_nat_complex
% 0.32/0.68 @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.32/0.68 @ ( set_ord_lessThan_nat @ N )
% 0.32/0.68 @ ( collect_complex
% 0.32/0.68 @ ^ [Z2: complex] :
% 0.32/0.68 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.68 = one_one_complex ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bij_betw_roots_unity
% 0.32/0.68 thf(fact_8903_ceiling__log__nat__eq__if,axiom,
% 0.32/0.68 ! [B: nat,N: nat,K: nat] :
% 0.32/0.68 ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.32/0.68 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.32/0.68 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.68 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.32/0.68 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ceiling_log_nat_eq_if
% 0.32/0.68 thf(fact_8904_ceiling__log2__div2,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.68 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.32/0.68 = ( 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 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ceiling_log2_div2
% 0.32/0.68 thf(fact_8905_ceiling__divide__eq__div__numeral,axiom,
% 0.32/0.68 ! [A: num,B: num] :
% 0.32/0.68 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ceiling_divide_eq_div_numeral
% 0.32/0.68 thf(fact_8906_ceiling__minus__divide__eq__div__numeral,axiom,
% 0.32/0.68 ! [A: num,B: num] :
% 0.32/0.68 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ceiling_minus_divide_eq_div_numeral
% 0.32/0.68 thf(fact_8907_ceiling__log__eq__powr__iff,axiom,
% 0.32/0.68 ! [X: real,B: real,K: nat] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 0.32/0.68 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.32/0.68 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 0.32/0.68 & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ceiling_log_eq_powr_iff
% 0.32/0.68 thf(fact_8908_floor__log__nat__eq__powr__iff,axiom,
% 0.32/0.68 ! [B: nat,K: nat,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.68 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.32/0.68 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.32/0.68 = ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.68 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.32/0.68 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_log_nat_eq_powr_iff
% 0.32/0.68 thf(fact_8909_floor__log__nat__eq__if,axiom,
% 0.32/0.68 ! [B: nat,N: nat,K: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 0.32/0.68 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.32/0.68 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.32/0.68 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.32/0.68 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_log_nat_eq_if
% 0.32/0.68 thf(fact_8910_arctan__def,axiom,
% 0.32/0.68 ( arctan
% 0.32/0.68 = ( ^ [Y: real] :
% 0.32/0.68 ( the_real
% 0.32/0.68 @ ^ [X2: real] :
% 0.32/0.68 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.32/0.68 & ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.68 & ( ( tan_real @ X2 )
% 0.32/0.68 = Y ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % arctan_def
% 0.32/0.68 thf(fact_8911_powr__gt__zero,axiom,
% 0.32/0.68 ! [X: real,A: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 0.32/0.68 = ( X != zero_zero_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_gt_zero
% 0.32/0.68 thf(fact_8912_powr__nonneg__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 0.32/0.68 = ( A = zero_zero_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_nonneg_iff
% 0.32/0.68 thf(fact_8913_powr__less__cancel__iff,axiom,
% 0.32/0.68 ! [X: real,A: real,B: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.32/0.68 = ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_less_cancel_iff
% 0.32/0.68 thf(fact_8914_powr__eq__one__iff,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ A )
% 0.32/0.68 => ( ( ( powr_real @ A @ X )
% 0.32/0.68 = one_one_real )
% 0.32/0.68 = ( X = zero_zero_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_eq_one_iff
% 0.32/0.68 thf(fact_8915_powr__one__gt__zero__iff,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ( powr_real @ X @ one_one_real )
% 0.32/0.68 = X )
% 0.32/0.68 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_one_gt_zero_iff
% 0.32/0.68 thf(fact_8916_powr__one,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( powr_real @ X @ one_one_real )
% 0.32/0.68 = X ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_one
% 0.32/0.68 thf(fact_8917_powr__le__cancel__iff,axiom,
% 0.32/0.68 ! [X: real,A: real,B: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.32/0.68 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_le_cancel_iff
% 0.32/0.68 thf(fact_8918_numeral__powr__numeral__real,axiom,
% 0.32/0.68 ! [M: num,N: num] :
% 0.32/0.68 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.32/0.68 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % numeral_powr_numeral_real
% 0.32/0.68 thf(fact_8919_log__powr__cancel,axiom,
% 0.32/0.68 ! [A: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 0.32/0.68 = Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_powr_cancel
% 0.32/0.68 thf(fact_8920_powr__log__cancel,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( powr_real @ A @ ( log @ A @ X ) )
% 0.32/0.68 = X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_log_cancel
% 0.32/0.68 thf(fact_8921_floor__divide__eq__div__numeral,axiom,
% 0.32/0.68 ! [A: num,B: num] :
% 0.32/0.68 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 0.32/0.68 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_divide_eq_div_numeral
% 0.32/0.68 thf(fact_8922_powr__numeral,axiom,
% 0.32/0.68 ! [X: real,N: num] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 0.32/0.68 = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_numeral
% 0.32/0.68 thf(fact_8923_floor__one__divide__eq__div__numeral,axiom,
% 0.32/0.68 ! [B: num] :
% 0.32/0.68 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 0.32/0.68 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_one_divide_eq_div_numeral
% 0.32/0.68 thf(fact_8924_floor__minus__divide__eq__div__numeral,axiom,
% 0.32/0.68 ! [A: num,B: num] :
% 0.32/0.68 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 0.32/0.68 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_minus_divide_eq_div_numeral
% 0.32/0.68 thf(fact_8925_square__powr__half,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.68 = ( abs_abs_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % square_powr_half
% 0.32/0.68 thf(fact_8926_floor__minus__one__divide__eq__div__numeral,axiom,
% 0.32/0.68 ! [B: num] :
% 0.32/0.68 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 0.32/0.68 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_minus_one_divide_eq_div_numeral
% 0.32/0.68 thf(fact_8927_powr__powr,axiom,
% 0.32/0.68 ! [X: real,A: real,B: real] :
% 0.32/0.68 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 0.32/0.68 = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_powr
% 0.32/0.68 thf(fact_8928_powr__less__mono2__neg,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ A @ zero_zero_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ Y3 )
% 0.32/0.68 => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_less_mono2_neg
% 0.32/0.68 thf(fact_8929_powr__non__neg,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 0.32/0.68
% 0.32/0.68 % powr_non_neg
% 0.32/0.68 thf(fact_8930_powr__mono2,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.68 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_mono2
% 0.32/0.68 thf(fact_8931_powr__ge__pzero,axiom,
% 0.32/0.68 ! [X: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y3 ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_ge_pzero
% 0.32/0.68 thf(fact_8932_powr__less__mono,axiom,
% 0.32/0.68 ! [A: real,B: real,X: real] :
% 0.32/0.68 ( ( ord_less_real @ A @ B )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.68 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_less_mono
% 0.32/0.68 thf(fact_8933_powr__less__cancel,axiom,
% 0.32/0.68 ! [X: real,A: real,B: real] :
% 0.32/0.68 ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.68 => ( ord_less_real @ A @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_less_cancel
% 0.32/0.68 thf(fact_8934_powr__mono,axiom,
% 0.32/0.68 ! [A: real,B: real,X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.68 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/0.68 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_mono
% 0.32/0.68 thf(fact_8935_powr__mono2_H,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.68 => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_mono2'
% 0.32/0.68 thf(fact_8936_powr__less__mono2,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ Y3 )
% 0.32/0.68 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_less_mono2
% 0.32/0.68 thf(fact_8937_gr__one__powr,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % gr_one_powr
% 0.32/0.68 thf(fact_8938_powr__inj,axiom,
% 0.32/0.68 ! [A: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( A != one_one_real )
% 0.32/0.68 => ( ( ( powr_real @ A @ X )
% 0.32/0.68 = ( powr_real @ A @ Y3 ) )
% 0.32/0.68 = ( X = Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_inj
% 0.32/0.68 thf(fact_8939_ge__one__powr__ge__zero,axiom,
% 0.32/0.68 ! [X: real,A: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ge_one_powr_ge_zero
% 0.32/0.68 thf(fact_8940_powr__mono__both,axiom,
% 0.32/0.68 ! [A: real,B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.68 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.68 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ B ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_mono_both
% 0.32/0.68 thf(fact_8941_powr__le1,axiom,
% 0.32/0.68 ! [A: real,X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.68 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_le1
% 0.32/0.68 thf(fact_8942_powr__divide,axiom,
% 0.32/0.68 ! [X: real,Y3: real,A: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( powr_real @ ( divide_divide_real @ X @ Y3 ) @ A )
% 0.32/0.68 = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_divide
% 0.32/0.68 thf(fact_8943_powr__mult,axiom,
% 0.32/0.68 ! [X: real,Y3: real,A: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( powr_real @ ( times_times_real @ X @ Y3 ) @ A )
% 0.32/0.68 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_mult
% 0.32/0.68 thf(fact_8944_inverse__powr,axiom,
% 0.32/0.68 ! [Y3: real,A: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
% 0.32/0.68 = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % inverse_powr
% 0.32/0.68 thf(fact_8945_divide__powr__uminus,axiom,
% 0.32/0.68 ! [A: real,B: real,C: real] :
% 0.32/0.68 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 0.32/0.68 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % divide_powr_uminus
% 0.32/0.68 thf(fact_8946_log__base__powr,axiom,
% 0.32/0.68 ! [A: real,B: real,X: real] :
% 0.32/0.68 ( ( A != zero_zero_real )
% 0.32/0.68 => ( ( log @ ( powr_real @ A @ B ) @ X )
% 0.32/0.68 = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_base_powr
% 0.32/0.68 thf(fact_8947_ln__powr,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( X != zero_zero_real )
% 0.32/0.68 => ( ( ln_ln_real @ ( powr_real @ X @ Y3 ) )
% 0.32/0.68 = ( times_times_real @ Y3 @ ( ln_ln_real @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ln_powr
% 0.32/0.68 thf(fact_8948_log__powr,axiom,
% 0.32/0.68 ! [X: real,B: real,Y3: real] :
% 0.32/0.68 ( ( X != zero_zero_real )
% 0.32/0.68 => ( ( log @ B @ ( powr_real @ X @ Y3 ) )
% 0.32/0.68 = ( times_times_real @ Y3 @ ( log @ B @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_powr
% 0.32/0.68 thf(fact_8949_ln__neg__is__const,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.68 => ( ( ln_ln_real @ X )
% 0.32/0.68 = ( the_real
% 0.32/0.68 @ ^ [X2: real] : $false ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ln_neg_is_const
% 0.32/0.68 thf(fact_8950_floor__log__eq__powr__iff,axiom,
% 0.32/0.68 ! [X: real,B: real,K: int] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 0.32/0.68 = K )
% 0.32/0.68 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 0.32/0.68 & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_log_eq_powr_iff
% 0.32/0.68 thf(fact_8951_powr__realpow,axiom,
% 0.32/0.68 ! [X: real,N: nat] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 0.32/0.68 = ( power_power_real @ X @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_realpow
% 0.32/0.68 thf(fact_8952_powr__less__iff,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X )
% 0.32/0.68 = ( ord_less_real @ Y3 @ ( log @ B @ X ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_less_iff
% 0.32/0.68 thf(fact_8953_less__powr__iff,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ ( powr_real @ B @ Y3 ) )
% 0.32/0.68 = ( ord_less_real @ ( log @ B @ X ) @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % less_powr_iff
% 0.32/0.68 thf(fact_8954_log__less__iff,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ ( log @ B @ X ) @ Y3 )
% 0.32/0.68 = ( ord_less_real @ X @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_less_iff
% 0.32/0.68 thf(fact_8955_less__log__iff,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ Y3 @ ( log @ B @ X ) )
% 0.32/0.68 = ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % less_log_iff
% 0.32/0.68 thf(fact_8956_floor__eq,axiom,
% 0.32/0.68 ! [N: int,X: real] :
% 0.32/0.68 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.32/0.68 => ( ( archim6058952711729229775r_real @ X )
% 0.32/0.68 = N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_eq
% 0.32/0.68 thf(fact_8957_real__of__int__floor__add__one__gt,axiom,
% 0.32/0.68 ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_of_int_floor_add_one_gt
% 0.32/0.68 thf(fact_8958_real__of__int__floor__add__one__ge,axiom,
% 0.32/0.68 ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_of_int_floor_add_one_ge
% 0.32/0.68 thf(fact_8959_real__of__int__floor__gt__diff__one,axiom,
% 0.32/0.68 ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_of_int_floor_gt_diff_one
% 0.32/0.68 thf(fact_8960_real__of__int__floor__ge__diff__one,axiom,
% 0.32/0.68 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_of_int_floor_ge_diff_one
% 0.32/0.68 thf(fact_8961_powr__neg__one,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 0.32/0.68 = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_neg_one
% 0.32/0.68 thf(fact_8962_powr__mult__base,axiom,
% 0.32/0.68 ! [X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( times_times_real @ X @ ( powr_real @ X @ Y3 ) )
% 0.32/0.68 = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_mult_base
% 0.32/0.68 thf(fact_8963_le__log__iff,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ Y3 @ ( log @ B @ X ) )
% 0.32/0.68 = ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % le_log_iff
% 0.32/0.68 thf(fact_8964_log__le__iff,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y3 )
% 0.32/0.68 = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_le_iff
% 0.32/0.68 thf(fact_8965_le__powr__iff,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y3 ) )
% 0.32/0.68 = ( ord_less_eq_real @ ( log @ B @ X ) @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % le_powr_iff
% 0.32/0.68 thf(fact_8966_powr__le__iff,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X )
% 0.32/0.68 = ( ord_less_eq_real @ Y3 @ ( log @ B @ X ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_le_iff
% 0.32/0.68 thf(fact_8967_floor__eq2,axiom,
% 0.32/0.68 ! [N: int,X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.32/0.68 => ( ( archim6058952711729229775r_real @ X )
% 0.32/0.68 = N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_eq2
% 0.32/0.68 thf(fact_8968_floor__divide__real__eq__div,axiom,
% 0.32/0.68 ! [B: int,A: real] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.32/0.68 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 0.32/0.68 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_divide_real_eq_div
% 0.32/0.68 thf(fact_8969_ln__powr__bound,axiom,
% 0.32/0.68 ! [X: real,A: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ln_powr_bound
% 0.32/0.68 thf(fact_8970_ln__powr__bound2,axiom,
% 0.32/0.68 ! [X: real,A: real] :
% 0.32/0.68 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ln_powr_bound2
% 0.32/0.68 thf(fact_8971_log__add__eq__powr,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.68 => ( ( B != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( plus_plus_real @ ( log @ B @ X ) @ Y3 )
% 0.32/0.68 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_add_eq_powr
% 0.32/0.68 thf(fact_8972_add__log__eq__powr,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.68 => ( ( B != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( plus_plus_real @ Y3 @ ( log @ B @ X ) )
% 0.32/0.68 = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y3 ) @ X ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % add_log_eq_powr
% 0.32/0.68 thf(fact_8973_minus__log__eq__powr,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.68 => ( ( B != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( minus_minus_real @ Y3 @ ( log @ B @ X ) )
% 0.32/0.68 = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y3 ) @ X ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % minus_log_eq_powr
% 0.32/0.68 thf(fact_8974_arccos__def,axiom,
% 0.32/0.68 ( arccos
% 0.32/0.68 = ( ^ [Y: real] :
% 0.32/0.68 ( the_real
% 0.32/0.68 @ ^ [X2: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.32/0.68 & ( ord_less_eq_real @ X2 @ pi )
% 0.32/0.68 & ( ( cos_real @ X2 )
% 0.32/0.68 = Y ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % arccos_def
% 0.32/0.68 thf(fact_8975_log__minus__eq__powr,axiom,
% 0.32/0.68 ! [B: real,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.68 => ( ( B != one_one_real )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( minus_minus_real @ ( log @ B @ X ) @ Y3 )
% 0.32/0.68 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_minus_eq_powr
% 0.32/0.68 thf(fact_8976_powr__half__sqrt,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.68 = ( sqrt @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_half_sqrt
% 0.32/0.68 thf(fact_8977_powr__neg__numeral,axiom,
% 0.32/0.68 ! [X: real,N: num] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.32/0.68 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_neg_numeral
% 0.32/0.68 thf(fact_8978_pi__half,axiom,
% 0.32/0.68 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.68 = ( the_real
% 0.32/0.68 @ ^ [X2: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.32/0.68 & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.68 & ( ( cos_real @ X2 )
% 0.32/0.68 = zero_zero_real ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % pi_half
% 0.32/0.68 thf(fact_8979_pi__def,axiom,
% 0.32/0.68 ( pi
% 0.32/0.68 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.32/0.68 @ ( the_real
% 0.32/0.68 @ ^ [X2: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.32/0.68 & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.32/0.68 & ( ( cos_real @ X2 )
% 0.32/0.68 = zero_zero_real ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % pi_def
% 0.32/0.68 thf(fact_8980_floor__log2__div2,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.68 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.32/0.68 = ( 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 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_log2_div2
% 0.32/0.68 thf(fact_8981_arcsin__def,axiom,
% 0.32/0.68 ( arcsin
% 0.32/0.68 = ( ^ [Y: real] :
% 0.32/0.68 ( the_real
% 0.32/0.68 @ ^ [X2: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.32/0.68 & ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.68 & ( ( sin_real @ X2 )
% 0.32/0.68 = Y ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % arcsin_def
% 0.32/0.68 thf(fact_8982_modulo__int__unfold,axiom,
% 0.32/0.68 ! [L2: int,K: int,N: nat,M: nat] :
% 0.32/0.68 ( ( ( ( ( sgn_sgn_int @ L2 )
% 0.32/0.68 = zero_zero_int )
% 0.32/0.68 | ( ( sgn_sgn_int @ K )
% 0.32/0.68 = zero_zero_int )
% 0.32/0.68 | ( N = zero_zero_nat ) )
% 0.32/0.68 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.68 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.32/0.68 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 0.32/0.68 = zero_zero_int )
% 0.32/0.68 | ( ( sgn_sgn_int @ K )
% 0.32/0.68 = zero_zero_int )
% 0.32/0.68 | ( N = zero_zero_nat ) )
% 0.32/0.68 => ( ( ( ( sgn_sgn_int @ K )
% 0.32/0.68 = ( sgn_sgn_int @ L2 ) )
% 0.32/0.68 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.68 = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.32/0.68 & ( ( ( sgn_sgn_int @ K )
% 0.32/0.68 != ( sgn_sgn_int @ L2 ) )
% 0.32/0.68 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.68 = ( times_times_int @ ( sgn_sgn_int @ L2 )
% 0.32/0.68 @ ( minus_minus_int
% 0.32/0.68 @ ( semiri1314217659103216013at_int
% 0.32/0.68 @ ( times_times_nat @ N
% 0.32/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.68 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.32/0.68 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % modulo_int_unfold
% 0.32/0.68 thf(fact_8983_sgn__mult__dvd__iff,axiom,
% 0.32/0.68 ! [R: int,L2: int,K: int] :
% 0.32/0.68 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L2 ) @ K )
% 0.32/0.68 = ( ( dvd_dvd_int @ L2 @ K )
% 0.32/0.68 & ( ( R = zero_zero_int )
% 0.32/0.68 => ( K = zero_zero_int ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % sgn_mult_dvd_iff
% 0.32/0.68 thf(fact_8984_mult__sgn__dvd__iff,axiom,
% 0.32/0.68 ! [L2: int,R: int,K: int] :
% 0.32/0.68 ( ( dvd_dvd_int @ ( times_times_int @ L2 @ ( sgn_sgn_int @ R ) ) @ K )
% 0.32/0.68 = ( ( dvd_dvd_int @ L2 @ K )
% 0.32/0.68 & ( ( R = zero_zero_int )
% 0.32/0.68 => ( K = zero_zero_int ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % mult_sgn_dvd_iff
% 0.32/0.68 thf(fact_8985_dvd__sgn__mult__iff,axiom,
% 0.32/0.68 ! [L2: int,R: int,K: int] :
% 0.32/0.68 ( ( dvd_dvd_int @ L2 @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
% 0.32/0.68 = ( ( dvd_dvd_int @ L2 @ K )
% 0.32/0.68 | ( R = zero_zero_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % dvd_sgn_mult_iff
% 0.32/0.68 thf(fact_8986_dvd__mult__sgn__iff,axiom,
% 0.32/0.68 ! [L2: int,K: int,R: int] :
% 0.32/0.68 ( ( dvd_dvd_int @ L2 @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
% 0.32/0.68 = ( ( dvd_dvd_int @ L2 @ K )
% 0.32/0.68 | ( R = zero_zero_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % dvd_mult_sgn_iff
% 0.32/0.68 thf(fact_8987_exp__pi__i_H,axiom,
% 0.32/0.68 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 0.32/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.68
% 0.32/0.68 % exp_pi_i'
% 0.32/0.68 thf(fact_8988_exp__pi__i,axiom,
% 0.32/0.68 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 0.32/0.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.32/0.68
% 0.32/0.68 % exp_pi_i
% 0.32/0.68 thf(fact_8989_exp__two__pi__i,axiom,
% 0.32/0.68 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 0.32/0.68 = one_one_complex ) ).
% 0.32/0.68
% 0.32/0.68 % exp_two_pi_i
% 0.32/0.68 thf(fact_8990_exp__two__pi__i_H,axiom,
% 0.32/0.68 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 0.32/0.68 = one_one_complex ) ).
% 0.32/0.68
% 0.32/0.68 % exp_two_pi_i'
% 0.32/0.68 thf(fact_8991_complex__exp__exists,axiom,
% 0.32/0.68 ! [Z: complex] :
% 0.32/0.68 ? [A5: complex,R2: real] :
% 0.32/0.68 ( Z
% 0.32/0.68 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( exp_complex @ A5 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % complex_exp_exists
% 0.32/0.68 thf(fact_8992_int__sgnE,axiom,
% 0.32/0.68 ! [K: int] :
% 0.32/0.68 ~ ! [N4: nat,L4: int] :
% 0.32/0.68 ( K
% 0.32/0.68 != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % int_sgnE
% 0.32/0.68 thf(fact_8993_div__eq__sgn__abs,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( ( sgn_sgn_int @ K )
% 0.32/0.68 = ( sgn_sgn_int @ L2 ) )
% 0.32/0.68 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.68 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % div_eq_sgn_abs
% 0.32/0.68 thf(fact_8994_sgn__mod,axiom,
% 0.32/0.68 ! [L2: int,K: int] :
% 0.32/0.68 ( ( L2 != zero_zero_int )
% 0.32/0.68 => ( ~ ( dvd_dvd_int @ L2 @ K )
% 0.32/0.68 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
% 0.32/0.68 = ( sgn_sgn_int @ L2 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % sgn_mod
% 0.32/0.68 thf(fact_8995_Complex__mult__complex__of__real,axiom,
% 0.32/0.68 ! [X: real,Y3: real,R: real] :
% 0.32/0.68 ( ( times_times_complex @ ( complex2 @ X @ Y3 ) @ ( real_V4546457046886955230omplex @ R ) )
% 0.32/0.68 = ( complex2 @ ( times_times_real @ X @ R ) @ ( times_times_real @ Y3 @ R ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Complex_mult_complex_of_real
% 0.32/0.68 thf(fact_8996_complex__of__real__mult__Complex,axiom,
% 0.32/0.68 ! [R: real,X: real,Y3: real] :
% 0.32/0.68 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y3 ) )
% 0.32/0.68 = ( complex2 @ ( times_times_real @ R @ X ) @ ( times_times_real @ R @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % complex_of_real_mult_Complex
% 0.32/0.68 thf(fact_8997_Complex__add__complex__of__real,axiom,
% 0.32/0.68 ! [X: real,Y3: real,R: real] :
% 0.32/0.68 ( ( plus_plus_complex @ ( complex2 @ X @ Y3 ) @ ( real_V4546457046886955230omplex @ R ) )
% 0.32/0.68 = ( complex2 @ ( plus_plus_real @ X @ R ) @ Y3 ) ) ).
% 0.32/0.68
% 0.32/0.68 % Complex_add_complex_of_real
% 0.32/0.68 thf(fact_8998_complex__of__real__add__Complex,axiom,
% 0.32/0.68 ! [R: real,X: real,Y3: real] :
% 0.32/0.68 ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y3 ) )
% 0.32/0.68 = ( complex2 @ ( plus_plus_real @ R @ X ) @ Y3 ) ) ).
% 0.32/0.68
% 0.32/0.68 % complex_of_real_add_Complex
% 0.32/0.68 thf(fact_8999_cis__conv__exp,axiom,
% 0.32/0.68 ( cis
% 0.32/0.68 = ( ^ [B4: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cis_conv_exp
% 0.32/0.68 thf(fact_9000_zsgn__def,axiom,
% 0.32/0.68 ( sgn_sgn_int
% 0.32/0.68 = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % zsgn_def
% 0.32/0.68 thf(fact_9001_div__sgn__abs__cancel,axiom,
% 0.32/0.68 ! [V: int,K: int,L2: int] :
% 0.32/0.68 ( ( V != zero_zero_int )
% 0.32/0.68 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
% 0.32/0.68 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % div_sgn_abs_cancel
% 0.32/0.68 thf(fact_9002_div__dvd__sgn__abs,axiom,
% 0.32/0.68 ! [L2: int,K: int] :
% 0.32/0.68 ( ( dvd_dvd_int @ L2 @ K )
% 0.32/0.68 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.68 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % div_dvd_sgn_abs
% 0.32/0.68 thf(fact_9003_i__complex__of__real,axiom,
% 0.32/0.68 ! [R: real] :
% 0.32/0.68 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R ) )
% 0.32/0.68 = ( complex2 @ zero_zero_real @ R ) ) ).
% 0.32/0.68
% 0.32/0.68 % i_complex_of_real
% 0.32/0.68 thf(fact_9004_complex__of__real__i,axiom,
% 0.32/0.68 ! [R: real] :
% 0.32/0.68 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ imaginary_unit )
% 0.32/0.68 = ( complex2 @ zero_zero_real @ R ) ) ).
% 0.32/0.68
% 0.32/0.68 % complex_of_real_i
% 0.32/0.68 thf(fact_9005_Complex__eq,axiom,
% 0.32/0.68 ( complex2
% 0.32/0.68 = ( ^ [A3: real,B4: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A3 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Complex_eq
% 0.32/0.68 thf(fact_9006_complex__split__polar,axiom,
% 0.32/0.68 ! [Z: complex] :
% 0.32/0.68 ? [R2: real,A5: real] :
% 0.32/0.68 ( Z
% 0.32/0.68 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % complex_split_polar
% 0.32/0.68 thf(fact_9007_eucl__rel__int__remainderI,axiom,
% 0.32/0.68 ! [R: int,L2: int,K: int,Q2: int] :
% 0.32/0.68 ( ( ( sgn_sgn_int @ R )
% 0.32/0.68 = ( sgn_sgn_int @ L2 ) )
% 0.32/0.68 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L2 ) )
% 0.32/0.68 => ( ( K
% 0.32/0.68 = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R ) )
% 0.32/0.68 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % eucl_rel_int_remainderI
% 0.32/0.68 thf(fact_9008_eucl__rel__int_Osimps,axiom,
% 0.32/0.68 ( eucl_rel_int
% 0.32/0.68 = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
% 0.32/0.68 ( ? [K2: int] :
% 0.32/0.68 ( ( A12 = K2 )
% 0.32/0.68 & ( A23 = zero_zero_int )
% 0.32/0.68 & ( A32
% 0.32/0.68 = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 0.32/0.68 | ? [L: int,K2: int,Q5: int] :
% 0.32/0.68 ( ( A12 = K2 )
% 0.32/0.68 & ( A23 = L )
% 0.32/0.68 & ( A32
% 0.32/0.68 = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
% 0.32/0.68 & ( L != zero_zero_int )
% 0.32/0.68 & ( K2
% 0.32/0.68 = ( times_times_int @ Q5 @ L ) ) )
% 0.32/0.68 | ? [R5: int,L: int,K2: int,Q5: int] :
% 0.32/0.68 ( ( A12 = K2 )
% 0.32/0.68 & ( A23 = L )
% 0.32/0.68 & ( A32
% 0.32/0.68 = ( product_Pair_int_int @ Q5 @ R5 ) )
% 0.32/0.68 & ( ( sgn_sgn_int @ R5 )
% 0.32/0.68 = ( sgn_sgn_int @ L ) )
% 0.32/0.68 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
% 0.32/0.68 & ( K2
% 0.32/0.68 = ( plus_plus_int @ ( times_times_int @ Q5 @ L ) @ R5 ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % eucl_rel_int.simps
% 0.32/0.68 thf(fact_9009_eucl__rel__int_Ocases,axiom,
% 0.32/0.68 ! [A1: int,A22: int,A33: product_prod_int_int] :
% 0.32/0.68 ( ( eucl_rel_int @ A1 @ A22 @ A33 )
% 0.32/0.68 => ( ( ( A22 = zero_zero_int )
% 0.32/0.68 => ( A33
% 0.32/0.68 != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 0.32/0.68 => ( ! [Q3: int] :
% 0.32/0.68 ( ( A33
% 0.32/0.68 = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 0.32/0.68 => ( ( A22 != zero_zero_int )
% 0.32/0.68 => ( A1
% 0.32/0.68 != ( times_times_int @ Q3 @ A22 ) ) ) )
% 0.32/0.68 => ~ ! [R2: int,Q3: int] :
% 0.32/0.68 ( ( A33
% 0.32/0.68 = ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.32/0.68 => ( ( ( sgn_sgn_int @ R2 )
% 0.32/0.68 = ( sgn_sgn_int @ A22 ) )
% 0.32/0.68 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A22 ) )
% 0.32/0.68 => ( A1
% 0.32/0.68 != ( plus_plus_int @ ( times_times_int @ Q3 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % eucl_rel_int.cases
% 0.32/0.68 thf(fact_9010_div__noneq__sgn__abs,axiom,
% 0.32/0.68 ! [L2: int,K: int] :
% 0.32/0.68 ( ( L2 != zero_zero_int )
% 0.32/0.68 => ( ( ( sgn_sgn_int @ K )
% 0.32/0.68 != ( sgn_sgn_int @ L2 ) )
% 0.32/0.68 => ( ( divide_divide_int @ K @ L2 )
% 0.32/0.68 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % div_noneq_sgn_abs
% 0.32/0.68 thf(fact_9011_cmod__unit__one,axiom,
% 0.32/0.68 ! [A: real] :
% 0.32/0.68 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 0.32/0.68 = one_one_real ) ).
% 0.32/0.68
% 0.32/0.68 % cmod_unit_one
% 0.32/0.68 thf(fact_9012_cmod__complex__polar,axiom,
% 0.32/0.68 ! [R: real,A: real] :
% 0.32/0.68 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 0.32/0.68 = ( abs_abs_real @ R ) ) ).
% 0.32/0.68
% 0.32/0.68 % cmod_complex_polar
% 0.32/0.68 thf(fact_9013_divide__int__unfold,axiom,
% 0.32/0.68 ! [L2: int,K: int,N: nat,M: nat] :
% 0.32/0.68 ( ( ( ( ( sgn_sgn_int @ L2 )
% 0.32/0.68 = zero_zero_int )
% 0.32/0.68 | ( ( sgn_sgn_int @ K )
% 0.32/0.68 = zero_zero_int )
% 0.32/0.68 | ( N = zero_zero_nat ) )
% 0.32/0.68 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.68 = zero_zero_int ) )
% 0.32/0.68 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 0.32/0.68 = zero_zero_int )
% 0.32/0.68 | ( ( sgn_sgn_int @ K )
% 0.32/0.68 = zero_zero_int )
% 0.32/0.68 | ( N = zero_zero_nat ) )
% 0.32/0.68 => ( ( ( ( sgn_sgn_int @ K )
% 0.32/0.68 = ( sgn_sgn_int @ L2 ) )
% 0.32/0.68 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.68 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.32/0.68 & ( ( ( sgn_sgn_int @ K )
% 0.32/0.68 != ( sgn_sgn_int @ L2 ) )
% 0.32/0.68 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.32/0.68 = ( uminus_uminus_int
% 0.32/0.68 @ ( semiri1314217659103216013at_int
% 0.32/0.68 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.32/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.68 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % divide_int_unfold
% 0.32/0.68 thf(fact_9014_csqrt__ii,axiom,
% 0.32/0.68 ( ( csqrt @ imaginary_unit )
% 0.32/0.68 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % csqrt_ii
% 0.32/0.68 thf(fact_9015_bij__betw__nth__root__unity,axiom,
% 0.32/0.68 ! [C: complex,N: nat] :
% 0.32/0.68 ( ( C != zero_zero_complex )
% 0.32/0.68 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.32/0.68 @ ( collect_complex
% 0.32/0.68 @ ^ [Z2: complex] :
% 0.32/0.68 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.68 = one_one_complex ) )
% 0.32/0.68 @ ( collect_complex
% 0.32/0.68 @ ^ [Z2: complex] :
% 0.32/0.68 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.68 = C ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bij_betw_nth_root_unity
% 0.32/0.68 thf(fact_9016_modulo__int__def,axiom,
% 0.32/0.68 ( modulo_modulo_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( if_int @ ( L = zero_zero_int ) @ K2
% 0.32/0.68 @ ( if_int
% 0.32/0.68 @ ( ( sgn_sgn_int @ K2 )
% 0.32/0.68 = ( sgn_sgn_int @ L ) )
% 0.32/0.68 @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
% 0.32/0.68 @ ( times_times_int @ ( sgn_sgn_int @ L )
% 0.32/0.68 @ ( minus_minus_int
% 0.32/0.68 @ ( times_times_int @ ( abs_abs_int @ L )
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ~ ( dvd_dvd_int @ L @ K2 ) ) )
% 0.32/0.68 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % modulo_int_def
% 0.32/0.68 thf(fact_9017_divide__int__def,axiom,
% 0.32/0.68 ( divide_divide_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
% 0.32/0.68 @ ( if_int
% 0.32/0.68 @ ( ( sgn_sgn_int @ K2 )
% 0.32/0.68 = ( sgn_sgn_int @ L ) )
% 0.32/0.68 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
% 0.32/0.68 @ ( uminus_uminus_int
% 0.32/0.68 @ ( semiri1314217659103216013at_int
% 0.32/0.68 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
% 0.32/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.68 @ ~ ( dvd_dvd_int @ L @ K2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % divide_int_def
% 0.32/0.68 thf(fact_9018_sgn__le__0__iff,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
% 0.32/0.68 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % sgn_le_0_iff
% 0.32/0.68 thf(fact_9019_zero__le__sgn__iff,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
% 0.32/0.68 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % zero_le_sgn_iff
% 0.32/0.68 thf(fact_9020_real__root__zero,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( root @ N @ zero_zero_real )
% 0.32/0.68 = zero_zero_real ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_zero
% 0.32/0.68 thf(fact_9021_nat__numeral,axiom,
% 0.32/0.68 ! [K: num] :
% 0.32/0.68 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.32/0.68 = ( numeral_numeral_nat @ K ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_numeral
% 0.32/0.68 thf(fact_9022_real__root__Suc__0,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 0.32/0.68 = X ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_Suc_0
% 0.32/0.68 thf(fact_9023_root__0,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( root @ zero_zero_nat @ X )
% 0.32/0.68 = zero_zero_real ) ).
% 0.32/0.68
% 0.32/0.68 % root_0
% 0.32/0.68 thf(fact_9024_real__root__eq__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ( root @ N @ X )
% 0.32/0.68 = ( root @ N @ Y3 ) )
% 0.32/0.68 = ( X = Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_eq_iff
% 0.32/0.68 thf(fact_9025_real__root__eq__0__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ( root @ N @ X )
% 0.32/0.68 = zero_zero_real )
% 0.32/0.68 = ( X = zero_zero_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_eq_0_iff
% 0.32/0.68 thf(fact_9026_real__root__less__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) )
% 0.32/0.68 = ( ord_less_real @ X @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_less_iff
% 0.32/0.68 thf(fact_9027_nat__0__iff,axiom,
% 0.32/0.68 ! [I: int] :
% 0.32/0.68 ( ( ( nat2 @ I )
% 0.32/0.68 = zero_zero_nat )
% 0.32/0.68 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_0_iff
% 0.32/0.68 thf(fact_9028_nat__le__0,axiom,
% 0.32/0.68 ! [Z: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.32/0.68 => ( ( nat2 @ Z )
% 0.32/0.68 = zero_zero_nat ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_le_0
% 0.32/0.68 thf(fact_9029_real__root__le__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) )
% 0.32/0.68 = ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_le_iff
% 0.32/0.68 thf(fact_9030_zless__nat__conj,axiom,
% 0.32/0.68 ! [W: int,Z: int] :
% 0.32/0.68 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.32/0.68 = ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.68 & ( ord_less_int @ W @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % zless_nat_conj
% 0.32/0.68 thf(fact_9031_real__root__one,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( root @ N @ one_one_real )
% 0.32/0.68 = one_one_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_one
% 0.32/0.68 thf(fact_9032_real__root__eq__1__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ( root @ N @ X )
% 0.32/0.68 = one_one_real )
% 0.32/0.68 = ( X = one_one_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_eq_1_iff
% 0.32/0.68 thf(fact_9033_nat__neg__numeral,axiom,
% 0.32/0.68 ! [K: num] :
% 0.32/0.68 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.68 = zero_zero_nat ) ).
% 0.32/0.68
% 0.32/0.68 % nat_neg_numeral
% 0.32/0.68 thf(fact_9034_int__nat__eq,axiom,
% 0.32/0.68 ! [Z: int] :
% 0.32/0.68 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.32/0.68 = Z ) )
% 0.32/0.68 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.32/0.68 = zero_zero_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % int_nat_eq
% 0.32/0.68 thf(fact_9035_real__root__lt__0__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.32/0.68 = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_lt_0_iff
% 0.32/0.68 thf(fact_9036_real__root__gt__0__iff,axiom,
% 0.32/0.68 ! [N: nat,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.32/0.68 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_gt_0_iff
% 0.32/0.68 thf(fact_9037_real__root__le__0__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 0.32/0.68 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_le_0_iff
% 0.32/0.68 thf(fact_9038_real__root__ge__0__iff,axiom,
% 0.32/0.68 ! [N: nat,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.32/0.68 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_ge_0_iff
% 0.32/0.68 thf(fact_9039_zero__less__nat__eq,axiom,
% 0.32/0.68 ! [Z: int] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 0.32/0.68 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.32/0.68
% 0.32/0.68 % zero_less_nat_eq
% 0.32/0.68 thf(fact_9040_real__root__gt__1__iff,axiom,
% 0.32/0.68 ! [N: nat,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.32/0.68 = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_gt_1_iff
% 0.32/0.68 thf(fact_9041_real__root__lt__1__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 0.32/0.68 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_lt_1_iff
% 0.32/0.68 thf(fact_9042_real__root__le__1__iff,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 0.32/0.68 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_le_1_iff
% 0.32/0.68 thf(fact_9043_real__root__ge__1__iff,axiom,
% 0.32/0.68 ! [N: nat,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.32/0.68 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_ge_1_iff
% 0.32/0.68 thf(fact_9044_diff__nat__numeral,axiom,
% 0.32/0.68 ! [V: num,V3: num] :
% 0.32/0.68 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 0.32/0.68 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % diff_nat_numeral
% 0.32/0.68 thf(fact_9045_nat__eq__numeral__power__cancel__iff,axiom,
% 0.32/0.68 ! [Y3: int,X: num,N: nat] :
% 0.32/0.68 ( ( ( nat2 @ Y3 )
% 0.32/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.32/0.68 = ( Y3
% 0.32/0.68 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_eq_numeral_power_cancel_iff
% 0.32/0.68 thf(fact_9046_numeral__power__eq__nat__cancel__iff,axiom,
% 0.32/0.68 ! [X: num,N: nat,Y3: int] :
% 0.32/0.68 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 0.32/0.68 = ( nat2 @ Y3 ) )
% 0.32/0.68 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 0.32/0.68 = Y3 ) ) ).
% 0.32/0.68
% 0.32/0.68 % numeral_power_eq_nat_cancel_iff
% 0.32/0.68 thf(fact_9047_power2__csqrt,axiom,
% 0.32/0.68 ! [Z: complex] :
% 0.32/0.68 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.68 = Z ) ).
% 0.32/0.68
% 0.32/0.68 % power2_csqrt
% 0.32/0.68 thf(fact_9048_nat__ceiling__le__eq,axiom,
% 0.32/0.68 ! [X: real,A: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 0.32/0.68 = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_ceiling_le_eq
% 0.32/0.68 thf(fact_9049_one__less__nat__eq,axiom,
% 0.32/0.68 ! [Z: int] :
% 0.32/0.68 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 0.32/0.68 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.32/0.68
% 0.32/0.68 % one_less_nat_eq
% 0.32/0.68 thf(fact_9050_real__root__pow__pos2,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.32/0.68 = X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_pow_pos2
% 0.32/0.68 thf(fact_9051_nat__numeral__diff__1,axiom,
% 0.32/0.68 ! [V: num] :
% 0.32/0.68 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.32/0.68 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_numeral_diff_1
% 0.32/0.68 thf(fact_9052_numeral__power__less__nat__cancel__iff,axiom,
% 0.32/0.68 ! [X: num,N: nat,A: int] :
% 0.32/0.68 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.32/0.68 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/0.68
% 0.32/0.68 % numeral_power_less_nat_cancel_iff
% 0.32/0.68 thf(fact_9053_nat__less__numeral__power__cancel__iff,axiom,
% 0.32/0.68 ! [A: int,X: num,N: nat] :
% 0.32/0.68 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.32/0.68 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_less_numeral_power_cancel_iff
% 0.32/0.68 thf(fact_9054_nat__le__numeral__power__cancel__iff,axiom,
% 0.32/0.68 ! [A: int,X: num,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 0.32/0.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_le_numeral_power_cancel_iff
% 0.32/0.68 thf(fact_9055_numeral__power__le__nat__cancel__iff,axiom,
% 0.32/0.68 ! [X: num,N: nat,A: int] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 0.32/0.68 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 0.32/0.68
% 0.32/0.68 % numeral_power_le_nat_cancel_iff
% 0.32/0.68 thf(fact_9056_sgn__root,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 0.32/0.68 = ( sgn_sgn_real @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % sgn_root
% 0.32/0.68 thf(fact_9057_real__root__commute,axiom,
% 0.32/0.68 ! [M: nat,N: nat,X: real] :
% 0.32/0.68 ( ( root @ M @ ( root @ N @ X ) )
% 0.32/0.68 = ( root @ N @ ( root @ M @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_commute
% 0.32/0.68 thf(fact_9058_real__root__mult__exp,axiom,
% 0.32/0.68 ! [M: nat,N: nat,X: real] :
% 0.32/0.68 ( ( root @ ( times_times_nat @ M @ N ) @ X )
% 0.32/0.68 = ( root @ M @ ( root @ N @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_mult_exp
% 0.32/0.68 thf(fact_9059_real__root__mult,axiom,
% 0.32/0.68 ! [N: nat,X: real,Y3: real] :
% 0.32/0.68 ( ( root @ N @ ( times_times_real @ X @ Y3 ) )
% 0.32/0.68 = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_mult
% 0.32/0.68 thf(fact_9060_real__root__minus,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( root @ N @ ( uminus_uminus_real @ X ) )
% 0.32/0.68 = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_minus
% 0.32/0.68 thf(fact_9061_real__root__divide,axiom,
% 0.32/0.68 ! [N: nat,X: real,Y3: real] :
% 0.32/0.68 ( ( root @ N @ ( divide_divide_real @ X @ Y3 ) )
% 0.32/0.68 = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_divide
% 0.32/0.68 thf(fact_9062_real__root__inverse,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( root @ N @ ( inverse_inverse_real @ X ) )
% 0.32/0.68 = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_inverse
% 0.32/0.68 thf(fact_9063_real__root__pos__pos__le,axiom,
% 0.32/0.68 ! [X: real,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_pos_pos_le
% 0.32/0.68 thf(fact_9064_nat__zero__as__int,axiom,
% 0.32/0.68 ( zero_zero_nat
% 0.32/0.68 = ( nat2 @ zero_zero_int ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_zero_as_int
% 0.32/0.68 thf(fact_9065_nat__numeral__as__int,axiom,
% 0.32/0.68 ( numeral_numeral_nat
% 0.32/0.68 = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_numeral_as_int
% 0.32/0.68 thf(fact_9066_real__sgn__eq,axiom,
% 0.32/0.68 ( sgn_sgn_real
% 0.32/0.68 = ( ^ [X2: real] : ( divide_divide_real @ X2 @ ( abs_abs_real @ X2 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_sgn_eq
% 0.32/0.68 thf(fact_9067_nat__mono,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ X @ Y3 )
% 0.32/0.68 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_mono
% 0.32/0.68 thf(fact_9068_eq__nat__nat__iff,axiom,
% 0.32/0.68 ! [Z: int,Z6: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.32/0.68 => ( ( ( nat2 @ Z )
% 0.32/0.68 = ( nat2 @ Z6 ) )
% 0.32/0.68 = ( Z = Z6 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % eq_nat_nat_iff
% 0.32/0.68 thf(fact_9069_all__nat,axiom,
% 0.32/0.68 ( ( ^ [P2: nat > $o] :
% 0.32/0.68 ! [X4: nat] : ( P2 @ X4 ) )
% 0.32/0.68 = ( ^ [P3: nat > $o] :
% 0.32/0.68 ! [X2: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.32/0.68 => ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % all_nat
% 0.32/0.68 thf(fact_9070_ex__nat,axiom,
% 0.32/0.68 ( ( ^ [P2: nat > $o] :
% 0.32/0.68 ? [X4: nat] : ( P2 @ X4 ) )
% 0.32/0.68 = ( ^ [P3: nat > $o] :
% 0.32/0.68 ? [X2: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.32/0.68 & ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ex_nat
% 0.32/0.68 thf(fact_9071_nat__one__as__int,axiom,
% 0.32/0.68 ( one_one_nat
% 0.32/0.68 = ( nat2 @ one_one_int ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_one_as_int
% 0.32/0.68 thf(fact_9072_root__sgn__power,axiom,
% 0.32/0.68 ! [N: nat,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 0.32/0.68 = Y3 ) ) ).
% 0.32/0.68
% 0.32/0.68 % root_sgn_power
% 0.32/0.68 thf(fact_9073_sgn__power__root,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 0.32/0.68 = X ) ) ).
% 0.32/0.68
% 0.32/0.68 % sgn_power_root
% 0.32/0.68 thf(fact_9074_sgn__eq,axiom,
% 0.32/0.68 ( sgn_sgn_complex
% 0.32/0.68 = ( ^ [Z2: complex] : ( divide1717551699836669952omplex @ Z2 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % sgn_eq
% 0.32/0.68 thf(fact_9075_unset__bit__nat__def,axiom,
% 0.32/0.68 ( bit_se4205575877204974255it_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % unset_bit_nat_def
% 0.32/0.68 thf(fact_9076_split__root,axiom,
% 0.32/0.68 ! [P: real > $o,N: nat,X: real] :
% 0.32/0.68 ( ( P @ ( root @ N @ X ) )
% 0.32/0.68 = ( ( ( N = zero_zero_nat )
% 0.32/0.68 => ( P @ zero_zero_real ) )
% 0.32/0.68 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ! [Y: real] :
% 0.32/0.68 ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.32/0.68 = X )
% 0.32/0.68 => ( P @ Y ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % split_root
% 0.32/0.68 thf(fact_9077_real__root__less__mono,axiom,
% 0.32/0.68 ! [N: nat,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ X @ Y3 )
% 0.32/0.68 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_less_mono
% 0.32/0.68 thf(fact_9078_real__root__le__mono,axiom,
% 0.32/0.68 ! [N: nat,X: real,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ X @ Y3 )
% 0.32/0.68 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_le_mono
% 0.32/0.68 thf(fact_9079_real__root__power,axiom,
% 0.32/0.68 ! [N: nat,X: real,K: nat] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 0.32/0.68 = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_power
% 0.32/0.68 thf(fact_9080_real__root__abs,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( root @ N @ ( abs_abs_real @ X ) )
% 0.32/0.68 = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_abs
% 0.32/0.68 thf(fact_9081_nat__mono__iff,axiom,
% 0.32/0.68 ! [Z: int,W: int] :
% 0.32/0.68 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.32/0.68 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_mono_iff
% 0.32/0.68 thf(fact_9082_zless__nat__eq__int__zless,axiom,
% 0.32/0.68 ! [M: nat,Z: int] :
% 0.32/0.68 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 0.32/0.68 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 0.32/0.68
% 0.32/0.68 % zless_nat_eq_int_zless
% 0.32/0.68 thf(fact_9083_nat__le__iff,axiom,
% 0.32/0.68 ! [X: int,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 0.32/0.68 = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_le_iff
% 0.32/0.68 thf(fact_9084_int__eq__iff,axiom,
% 0.32/0.68 ! [M: nat,Z: int] :
% 0.32/0.68 ( ( ( semiri1314217659103216013at_int @ M )
% 0.32/0.68 = Z )
% 0.32/0.68 = ( ( M
% 0.32/0.68 = ( nat2 @ Z ) )
% 0.32/0.68 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % int_eq_iff
% 0.32/0.68 thf(fact_9085_nat__0__le,axiom,
% 0.32/0.68 ! [Z: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.32/0.68 = Z ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_0_le
% 0.32/0.68 thf(fact_9086_nat__int__add,axiom,
% 0.32/0.68 ! [A: nat,B: nat] :
% 0.32/0.68 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 0.32/0.68 = ( plus_plus_nat @ A @ B ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_int_add
% 0.32/0.68 thf(fact_9087_int__minus,axiom,
% 0.32/0.68 ! [N: nat,M: nat] :
% 0.32/0.68 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 0.32/0.68 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % int_minus
% 0.32/0.68 thf(fact_9088_nat__abs__mult__distrib,axiom,
% 0.32/0.68 ! [W: int,Z: int] :
% 0.32/0.68 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 0.32/0.68 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_abs_mult_distrib
% 0.32/0.68 thf(fact_9089_nat__plus__as__int,axiom,
% 0.32/0.68 ( plus_plus_nat
% 0.32/0.68 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_plus_as_int
% 0.32/0.68 thf(fact_9090_nat__times__as__int,axiom,
% 0.32/0.68 ( times_times_nat
% 0.32/0.68 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_times_as_int
% 0.32/0.68 thf(fact_9091_real__nat__ceiling__ge,axiom,
% 0.32/0.68 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_nat_ceiling_ge
% 0.32/0.68 thf(fact_9092_nat__minus__as__int,axiom,
% 0.32/0.68 ( minus_minus_nat
% 0.32/0.68 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_minus_as_int
% 0.32/0.68 thf(fact_9093_nat__div__as__int,axiom,
% 0.32/0.68 ( divide_divide_nat
% 0.32/0.68 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_div_as_int
% 0.32/0.68 thf(fact_9094_nat__mod__as__int,axiom,
% 0.32/0.68 ( modulo_modulo_nat
% 0.32/0.68 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_mod_as_int
% 0.32/0.68 thf(fact_9095_real__root__gt__zero,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_gt_zero
% 0.32/0.68 thf(fact_9096_real__root__strict__decreasing,axiom,
% 0.32/0.68 ! [N: nat,N3: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_nat @ N @ N3 )
% 0.32/0.68 => ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.68 => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_strict_decreasing
% 0.32/0.68 thf(fact_9097_sqrt__def,axiom,
% 0.32/0.68 ( sqrt
% 0.32/0.68 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % sqrt_def
% 0.32/0.68 thf(fact_9098_sgn__real__def,axiom,
% 0.32/0.68 ( sgn_sgn_real
% 0.32/0.68 = ( ^ [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 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % sgn_real_def
% 0.32/0.68 thf(fact_9099_root__abs__power,axiom,
% 0.32/0.68 ! [N: nat,Y3: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 0.32/0.68 = ( abs_abs_real @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % root_abs_power
% 0.32/0.68 thf(fact_9100_nat__less__eq__zless,axiom,
% 0.32/0.68 ! [W: int,Z: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/0.68 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.32/0.68 = ( ord_less_int @ W @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_less_eq_zless
% 0.32/0.68 thf(fact_9101_nat__le__eq__zle,axiom,
% 0.32/0.68 ! [W: int,Z: int] :
% 0.32/0.68 ( ( ( ord_less_int @ zero_zero_int @ W )
% 0.32/0.68 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 0.32/0.68 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 0.32/0.68 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_le_eq_zle
% 0.32/0.68 thf(fact_9102_nat__eq__iff2,axiom,
% 0.32/0.68 ! [M: nat,W: int] :
% 0.32/0.68 ( ( M
% 0.32/0.68 = ( nat2 @ W ) )
% 0.32/0.68 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/0.68 => ( W
% 0.32/0.68 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.32/0.68 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/0.68 => ( M = zero_zero_nat ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_eq_iff2
% 0.32/0.68 thf(fact_9103_nat__eq__iff,axiom,
% 0.32/0.68 ! [W: int,M: nat] :
% 0.32/0.68 ( ( ( nat2 @ W )
% 0.32/0.68 = M )
% 0.32/0.68 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/0.68 => ( W
% 0.32/0.68 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.32/0.68 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/0.68 => ( M = zero_zero_nat ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_eq_iff
% 0.32/0.68 thf(fact_9104_split__nat,axiom,
% 0.32/0.68 ! [P: nat > $o,I: int] :
% 0.32/0.68 ( ( P @ ( nat2 @ I ) )
% 0.32/0.68 = ( ! [N2: nat] :
% 0.32/0.68 ( ( I
% 0.32/0.68 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.32/0.68 => ( P @ N2 ) )
% 0.32/0.68 & ( ( ord_less_int @ I @ zero_zero_int )
% 0.32/0.68 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % split_nat
% 0.32/0.68 thf(fact_9105_le__nat__iff,axiom,
% 0.32/0.68 ! [K: int,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.68 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.32/0.68 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % le_nat_iff
% 0.32/0.68 thf(fact_9106_nat__add__distrib,axiom,
% 0.32/0.68 ! [Z: int,Z6: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.32/0.68 => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
% 0.32/0.68 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_add_distrib
% 0.32/0.68 thf(fact_9107_nat__mult__distrib,axiom,
% 0.32/0.68 ! [Z: int,Z6: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 0.32/0.68 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_mult_distrib
% 0.32/0.68 thf(fact_9108_Suc__as__int,axiom,
% 0.32/0.68 ( suc
% 0.32/0.68 = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Suc_as_int
% 0.32/0.68 thf(fact_9109_nat__diff__distrib,axiom,
% 0.32/0.68 ! [Z6: int,Z: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 0.32/0.68 => ( ( ord_less_eq_int @ Z6 @ Z )
% 0.32/0.68 => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
% 0.32/0.68 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_diff_distrib
% 0.32/0.68 thf(fact_9110_nat__diff__distrib_H,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ( nat2 @ ( minus_minus_int @ X @ Y3 ) )
% 0.32/0.68 = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_diff_distrib'
% 0.32/0.68 thf(fact_9111_nat__abs__triangle__ineq,axiom,
% 0.32/0.68 ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_abs_triangle_ineq
% 0.32/0.68 thf(fact_9112_nat__div__distrib_H,axiom,
% 0.32/0.68 ! [Y3: int,X: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ( nat2 @ ( divide_divide_int @ X @ Y3 ) )
% 0.32/0.68 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_div_distrib'
% 0.32/0.68 thf(fact_9113_nat__div__distrib,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ( nat2 @ ( divide_divide_int @ X @ Y3 ) )
% 0.32/0.68 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_div_distrib
% 0.32/0.68 thf(fact_9114_nat__power__eq,axiom,
% 0.32/0.68 ! [Z: int,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 0.32/0.68 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_power_eq
% 0.32/0.68 thf(fact_9115_nat__floor__neg,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 0.32/0.68 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.32/0.68 = zero_zero_nat ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_floor_neg
% 0.32/0.68 thf(fact_9116_nat__mod__distrib,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ( nat2 @ ( modulo_modulo_int @ X @ Y3 ) )
% 0.32/0.68 = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_mod_distrib
% 0.32/0.68 thf(fact_9117_div__abs__eq__div__nat,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 0.32/0.68 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % div_abs_eq_div_nat
% 0.32/0.68 thf(fact_9118_floor__eq3,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.32/0.68 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.32/0.68 = N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_eq3
% 0.32/0.68 thf(fact_9119_le__nat__floor,axiom,
% 0.32/0.68 ! [X: nat,A: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 0.32/0.68 => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % le_nat_floor
% 0.32/0.68 thf(fact_9120_mod__abs__eq__div__nat,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 0.32/0.68 = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % mod_abs_eq_div_nat
% 0.32/0.68 thf(fact_9121_nat__take__bit__eq,axiom,
% 0.32/0.68 ! [K: int,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.68 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.32/0.68 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_take_bit_eq
% 0.32/0.68 thf(fact_9122_take__bit__nat__eq,axiom,
% 0.32/0.68 ! [K: int,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.68 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 0.32/0.68 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % take_bit_nat_eq
% 0.32/0.68 thf(fact_9123_bit__nat__iff,axiom,
% 0.32/0.68 ! [K: int,N: nat] :
% 0.32/0.68 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 0.32/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.68 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bit_nat_iff
% 0.32/0.68 thf(fact_9124_of__real__sqrt,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
% 0.32/0.68 = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % of_real_sqrt
% 0.32/0.68 thf(fact_9125_real__root__pos__pos,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_pos_pos
% 0.32/0.68 thf(fact_9126_real__root__strict__increasing,axiom,
% 0.32/0.68 ! [N: nat,N3: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_nat @ N @ N3 )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.68 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_strict_increasing
% 0.32/0.68 thf(fact_9127_real__root__decreasing,axiom,
% 0.32/0.68 ! [N: nat,N3: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.68 => ( ( ord_less_eq_real @ one_one_real @ X )
% 0.32/0.68 => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_decreasing
% 0.32/0.68 thf(fact_9128_real__root__pow__pos,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.32/0.68 = X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_pow_pos
% 0.32/0.68 thf(fact_9129_real__root__power__cancel,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.32/0.68 = X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_power_cancel
% 0.32/0.68 thf(fact_9130_real__root__pos__unique,axiom,
% 0.32/0.68 ! [N: nat,Y3: real,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.32/0.68 => ( ( ( power_power_real @ Y3 @ N )
% 0.32/0.68 = X )
% 0.32/0.68 => ( ( root @ N @ X )
% 0.32/0.68 = Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_pos_unique
% 0.32/0.68 thf(fact_9131_odd__real__root__power__cancel,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.68 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 0.32/0.68 = X ) ) ).
% 0.32/0.68
% 0.32/0.68 % odd_real_root_power_cancel
% 0.32/0.68 thf(fact_9132_odd__real__root__unique,axiom,
% 0.32/0.68 ! [N: nat,Y3: real,X: real] :
% 0.32/0.68 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.68 => ( ( ( power_power_real @ Y3 @ N )
% 0.32/0.68 = X )
% 0.32/0.68 => ( ( root @ N @ X )
% 0.32/0.68 = Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % odd_real_root_unique
% 0.32/0.68 thf(fact_9133_odd__real__root__pow,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.68 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 0.32/0.68 = X ) ) ).
% 0.32/0.68
% 0.32/0.68 % odd_real_root_pow
% 0.32/0.68 thf(fact_9134_nat__2,axiom,
% 0.32/0.68 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.68 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_2
% 0.32/0.68 thf(fact_9135_sgn__power__injE,axiom,
% 0.32/0.68 ! [A: real,N: nat,X: real,B: real] :
% 0.32/0.68 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 0.32/0.68 = X )
% 0.32/0.68 => ( ( X
% 0.32/0.68 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 0.32/0.68 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( A = B ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % sgn_power_injE
% 0.32/0.68 thf(fact_9136_Suc__nat__eq__nat__zadd1,axiom,
% 0.32/0.68 ! [Z: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( ( suc @ ( nat2 @ Z ) )
% 0.32/0.68 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Suc_nat_eq_nat_zadd1
% 0.32/0.68 thf(fact_9137_nat__less__iff,axiom,
% 0.32/0.68 ! [W: int,M: nat] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 0.32/0.68 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 0.32/0.68 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_less_iff
% 0.32/0.68 thf(fact_9138_nat__mult__distrib__neg,axiom,
% 0.32/0.68 ! [Z: int,Z6: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.32/0.68 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 0.32/0.68 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_mult_distrib_neg
% 0.32/0.68 thf(fact_9139_nat__abs__int__diff,axiom,
% 0.32/0.68 ! [A: nat,B: nat] :
% 0.32/0.68 ( ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.68 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.32/0.68 = ( minus_minus_nat @ B @ A ) ) )
% 0.32/0.68 & ( ~ ( ord_less_eq_nat @ A @ B )
% 0.32/0.68 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.32/0.68 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_abs_int_diff
% 0.32/0.68 thf(fact_9140_floor__eq4,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 0.32/0.68 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.32/0.68 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 0.32/0.68 = N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % floor_eq4
% 0.32/0.68 thf(fact_9141_diff__nat__eq__if,axiom,
% 0.32/0.68 ! [Z6: int,Z: int] :
% 0.32/0.68 ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 0.32/0.68 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 0.32/0.68 = ( nat2 @ Z ) ) )
% 0.32/0.68 & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 0.32/0.68 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 0.32/0.68 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % diff_nat_eq_if
% 0.32/0.68 thf(fact_9142_real__root__increasing,axiom,
% 0.32/0.68 ! [N: nat,N3: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_eq_nat @ N @ N3 )
% 0.32/0.68 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.68 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % real_root_increasing
% 0.32/0.68 thf(fact_9143_nat__dvd__iff,axiom,
% 0.32/0.68 ! [Z: int,M: nat] :
% 0.32/0.68 ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 0.32/0.68 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.32/0.68 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.32/0.68 => ( M = zero_zero_nat ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_dvd_iff
% 0.32/0.68 thf(fact_9144_cis__Arg__unique,axiom,
% 0.32/0.68 ! [Z: complex,X: real] :
% 0.32/0.68 ( ( ( sgn_sgn_complex @ Z )
% 0.32/0.68 = ( cis @ X ) )
% 0.32/0.68 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 0.32/0.68 => ( ( ord_less_eq_real @ X @ pi )
% 0.32/0.68 => ( ( arg @ Z )
% 0.32/0.68 = X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % cis_Arg_unique
% 0.32/0.68 thf(fact_9145_ln__root,axiom,
% 0.32/0.68 ! [N: nat,B: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.68 => ( ( ln_ln_real @ ( root @ N @ B ) )
% 0.32/0.68 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % ln_root
% 0.32/0.68 thf(fact_9146_log__root,axiom,
% 0.32/0.68 ! [N: nat,A: real,B: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ A )
% 0.32/0.68 => ( ( log @ B @ ( root @ N @ A ) )
% 0.32/0.68 = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_root
% 0.32/0.68 thf(fact_9147_log__base__root,axiom,
% 0.32/0.68 ! [N: nat,B: real,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ B )
% 0.32/0.68 => ( ( log @ ( root @ N @ B ) @ X )
% 0.32/0.68 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % log_base_root
% 0.32/0.68 thf(fact_9148_Arg__correct,axiom,
% 0.32/0.68 ! [Z: complex] :
% 0.32/0.68 ( ( Z != zero_zero_complex )
% 0.32/0.68 => ( ( ( sgn_sgn_complex @ Z )
% 0.32/0.68 = ( cis @ ( arg @ Z ) ) )
% 0.32/0.68 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.32/0.68 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Arg_correct
% 0.32/0.68 thf(fact_9149_root__powr__inverse,axiom,
% 0.32/0.68 ! [N: nat,X: real] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.68 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( root @ N @ X )
% 0.32/0.68 = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % root_powr_inverse
% 0.32/0.68 thf(fact_9150_even__nat__iff,axiom,
% 0.32/0.68 ! [K: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.68 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.32/0.68 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % even_nat_iff
% 0.32/0.68 thf(fact_9151_powr__real__of__int,axiom,
% 0.32/0.68 ! [X: real,N: int] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.32/0.68 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.32/0.68 = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 0.32/0.68 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.32/0.68 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.32/0.68 = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_real_of_int
% 0.32/0.68 thf(fact_9152_arctan__inverse,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( X != zero_zero_real )
% 0.32/0.68 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 0.32/0.68 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % arctan_inverse
% 0.32/0.68 thf(fact_9153_powr__int,axiom,
% 0.32/0.68 ! [X: real,I: int] :
% 0.32/0.68 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.68 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.32/0.68 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.32/0.68 = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 0.32/0.68 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.32/0.68 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 0.32/0.68 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % powr_int
% 0.32/0.68 thf(fact_9154_num_Osize__gen_I3_J,axiom,
% 0.32/0.68 ! [X32: num] :
% 0.32/0.68 ( ( size_num @ ( bit1 @ X32 ) )
% 0.32/0.68 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % num.size_gen(3)
% 0.32/0.68 thf(fact_9155_num_Osize__gen_I2_J,axiom,
% 0.32/0.68 ! [X22: num] :
% 0.32/0.68 ( ( size_num @ ( bit0 @ X22 ) )
% 0.32/0.68 = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % num.size_gen(2)
% 0.32/0.68 thf(fact_9156_mask__nat__positive__iff,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.32/0.68 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.32/0.68
% 0.32/0.68 % mask_nat_positive_iff
% 0.32/0.68 thf(fact_9157_nat__mask__eq,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 0.32/0.68 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.32/0.68
% 0.32/0.68 % nat_mask_eq
% 0.32/0.68 thf(fact_9158_less__eq__mask,axiom,
% 0.32/0.68 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.32/0.68
% 0.32/0.68 % less_eq_mask
% 0.32/0.68 thf(fact_9159_mask__nonnegative__int,axiom,
% 0.32/0.68 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.32/0.68
% 0.32/0.68 % mask_nonnegative_int
% 0.32/0.68 thf(fact_9160_not__mask__negative__int,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.32/0.68
% 0.32/0.68 % not_mask_negative_int
% 0.32/0.68 thf(fact_9161_less__mask,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.68 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % less_mask
% 0.32/0.68 thf(fact_9162_take__bit__eq__mask__iff,axiom,
% 0.32/0.68 ! [N: nat,K: int] :
% 0.32/0.68 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.32/0.68 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.32/0.68 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 0.32/0.68 = zero_zero_int ) ) ).
% 0.32/0.68
% 0.32/0.68 % take_bit_eq_mask_iff
% 0.32/0.68 thf(fact_9163_num_Osize__gen_I1_J,axiom,
% 0.32/0.68 ( ( size_num @ one )
% 0.32/0.68 = zero_zero_nat ) ).
% 0.32/0.68
% 0.32/0.68 % num.size_gen(1)
% 0.32/0.68 thf(fact_9164_Suc__mask__eq__exp,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.32/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.68
% 0.32/0.68 % Suc_mask_eq_exp
% 0.32/0.68 thf(fact_9165_mask__nat__less__exp,axiom,
% 0.32/0.68 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.68
% 0.32/0.68 % mask_nat_less_exp
% 0.32/0.68 thf(fact_9166_mask__nat__def,axiom,
% 0.32/0.68 ( bit_se2002935070580805687sk_nat
% 0.32/0.68 = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % mask_nat_def
% 0.32/0.68 thf(fact_9167_mask__half__int,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.68 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % mask_half_int
% 0.32/0.68 thf(fact_9168_mask__int__def,axiom,
% 0.32/0.68 ( bit_se2000444600071755411sk_int
% 0.32/0.68 = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % mask_int_def
% 0.32/0.68 thf(fact_9169_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.32/0.68 ! [N: nat,K: int] :
% 0.32/0.68 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.32/0.68 = ( bit_se2000444600071755411sk_int @ N ) )
% 0.32/0.68 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % take_bit_eq_mask_iff_exp_dvd
% 0.32/0.68 thf(fact_9170_Arg__def,axiom,
% 0.32/0.68 ( arg
% 0.32/0.68 = ( ^ [Z2: complex] :
% 0.32/0.68 ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
% 0.32/0.68 @ ( fChoice_real
% 0.32/0.68 @ ^ [A3: real] :
% 0.32/0.68 ( ( ( sgn_sgn_complex @ Z2 )
% 0.32/0.68 = ( cis @ A3 ) )
% 0.32/0.68 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 0.32/0.68 & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Arg_def
% 0.32/0.68 thf(fact_9171_and__int__unfold,axiom,
% 0.32/0.68 ( bit_se725231765392027082nd_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( if_int
% 0.32/0.68 @ ( ( K2 = zero_zero_int )
% 0.32/0.68 | ( L = zero_zero_int ) )
% 0.32/0.68 @ zero_zero_int
% 0.32/0.68 @ ( if_int
% 0.32/0.68 @ ( K2
% 0.32/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.68 @ L
% 0.32/0.68 @ ( if_int
% 0.32/0.68 @ ( L
% 0.32/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.68 @ K2
% 0.32/0.68 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_int_unfold
% 0.32/0.68 thf(fact_9172_or__int__unfold,axiom,
% 0.32/0.68 ( bit_se1409905431419307370or_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( if_int
% 0.32/0.68 @ ( ( K2
% 0.32/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.68 | ( L
% 0.32/0.68 = ( uminus_uminus_int @ one_one_int ) ) )
% 0.32/0.68 @ ( uminus_uminus_int @ one_one_int )
% 0.32/0.68 @ ( if_int @ ( K2 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_int_unfold
% 0.32/0.68 thf(fact_9173_and__nonnegative__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 0.32/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.68 | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_nonnegative_int_iff
% 0.32/0.68 thf(fact_9174_and__negative__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
% 0.32/0.68 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.68 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_negative_int_iff
% 0.32/0.68 thf(fact_9175_or__nonnegative__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
% 0.32/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.68 & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_nonnegative_int_iff
% 0.32/0.68 thf(fact_9176_or__negative__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
% 0.32/0.68 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.68 | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_negative_int_iff
% 0.32/0.68 thf(fact_9177_and__minus__numerals_I6_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.32/0.68 = one_one_int ) ).
% 0.32/0.68
% 0.32/0.68 % and_minus_numerals(6)
% 0.32/0.68 thf(fact_9178_and__minus__numerals_I2_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.68 = one_one_int ) ).
% 0.32/0.68
% 0.32/0.68 % and_minus_numerals(2)
% 0.32/0.68 thf(fact_9179_or__minus__numerals_I6_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 0.32/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_numerals(6)
% 0.32/0.68 thf(fact_9180_or__minus__numerals_I2_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_numerals(2)
% 0.32/0.68 thf(fact_9181_and__minus__numerals_I5_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.32/0.68 = zero_zero_int ) ).
% 0.32/0.68
% 0.32/0.68 % and_minus_numerals(5)
% 0.32/0.68 thf(fact_9182_and__minus__numerals_I1_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.68 = zero_zero_int ) ).
% 0.32/0.68
% 0.32/0.68 % and_minus_numerals(1)
% 0.32/0.68 thf(fact_9183_plus__and__or,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ ( bit_se1409905431419307370or_int @ X @ Y3 ) )
% 0.32/0.68 = ( plus_plus_int @ X @ Y3 ) ) ).
% 0.32/0.68
% 0.32/0.68 % plus_and_or
% 0.32/0.68 thf(fact_9184_bit__or__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int,N: nat] :
% 0.32/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N )
% 0.32/0.68 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.32/0.68 | ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bit_or_int_iff
% 0.32/0.68 thf(fact_9185_bit__and__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int,N: nat] :
% 0.32/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N )
% 0.32/0.68 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.32/0.68 & ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bit_and_int_iff
% 0.32/0.68 thf(fact_9186_OR__lower,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % OR_lower
% 0.32/0.68 thf(fact_9187_or__greater__eq,axiom,
% 0.32/0.68 ! [L2: int,K: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 0.32/0.68 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_greater_eq
% 0.32/0.68 thf(fact_9188_AND__lower,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % AND_lower
% 0.32/0.68 thf(fact_9189_AND__upper1,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ X ) ) ).
% 0.32/0.68
% 0.32/0.68 % AND_upper1
% 0.32/0.68 thf(fact_9190_AND__upper2,axiom,
% 0.32/0.68 ! [Y3: int,X: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Y3 ) ) ).
% 0.32/0.68
% 0.32/0.68 % AND_upper2
% 0.32/0.68 thf(fact_9191_AND__upper1_H,axiom,
% 0.32/0.68 ! [Y3: int,Z: int,Ya: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.32/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % AND_upper1'
% 0.32/0.68 thf(fact_9192_AND__upper2_H,axiom,
% 0.32/0.68 ! [Y3: int,Z: int,X: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.32/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % AND_upper2'
% 0.32/0.68 thf(fact_9193_pow_Osimps_I1_J,axiom,
% 0.32/0.68 ! [X: num] :
% 0.32/0.68 ( ( pow @ X @ one )
% 0.32/0.68 = X ) ).
% 0.32/0.68
% 0.32/0.68 % pow.simps(1)
% 0.32/0.68 thf(fact_9194_and__less__eq,axiom,
% 0.32/0.68 ! [L2: int,K: int] :
% 0.32/0.68 ( ( ord_less_int @ L2 @ zero_zero_int )
% 0.32/0.68 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_less_eq
% 0.32/0.68 thf(fact_9195_AND__upper1_H_H,axiom,
% 0.32/0.68 ! [Y3: int,Z: int,Ya: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ( ord_less_int @ Y3 @ Z )
% 0.32/0.68 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % AND_upper1''
% 0.32/0.68 thf(fact_9196_AND__upper2_H_H,axiom,
% 0.32/0.68 ! [Y3: int,Z: int,X: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ( ord_less_int @ Y3 @ Z )
% 0.32/0.68 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Z ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % AND_upper2''
% 0.32/0.68 thf(fact_9197_even__and__iff__int,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 0.32/0.68 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.32/0.68 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % even_and_iff_int
% 0.32/0.68 thf(fact_9198_OR__upper,axiom,
% 0.32/0.68 ! [X: int,N: nat,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.68 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.68 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % OR_upper
% 0.32/0.68 thf(fact_9199_or__int__rec,axiom,
% 0.32/0.68 ( bit_se1409905431419307370or_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( plus_plus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.32/0.68 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.32/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_int_rec
% 0.32/0.68 thf(fact_9200_and__int__rec,axiom,
% 0.32/0.68 ( bit_se725231765392027082nd_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( plus_plus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.32/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_int_rec
% 0.32/0.68 thf(fact_9201_and__int_Oelims,axiom,
% 0.32/0.68 ! [X: int,Xa: int,Y3: int] :
% 0.32/0.68 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.32/0.68 = Y3 )
% 0.32/0.68 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.32/0.68 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 = ( uminus_uminus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.32/0.68 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.32/0.68 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 = ( plus_plus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.32/0.68 @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_int.elims
% 0.32/0.68 thf(fact_9202_and__int_Osimps,axiom,
% 0.32/0.68 ( bit_se725231765392027082nd_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( if_int
% 0.32/0.68 @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.32/0.68 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.32/0.68 @ ( uminus_uminus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 0.32/0.68 @ ( plus_plus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.32/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_int.simps
% 0.32/0.68 thf(fact_9203_or__minus__numerals_I1_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_numerals(1)
% 0.32/0.68 thf(fact_9204_or__minus__numerals_I5_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 0.32/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_numerals(5)
% 0.32/0.68 thf(fact_9205_and__nat__numerals_I3_J,axiom,
% 0.32/0.68 ! [X: num] :
% 0.32/0.68 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = zero_zero_nat ) ).
% 0.32/0.68
% 0.32/0.68 % and_nat_numerals(3)
% 0.32/0.68 thf(fact_9206_and__nat__numerals_I1_J,axiom,
% 0.32/0.68 ! [Y3: num] :
% 0.32/0.68 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.32/0.68 = zero_zero_nat ) ).
% 0.32/0.68
% 0.32/0.68 % and_nat_numerals(1)
% 0.32/0.68 thf(fact_9207_or__nat__numerals_I2_J,axiom,
% 0.32/0.68 ! [Y3: num] :
% 0.32/0.68 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.32/0.68 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_nat_numerals(2)
% 0.32/0.68 thf(fact_9208_or__nat__numerals_I4_J,axiom,
% 0.32/0.68 ! [X: num] :
% 0.32/0.68 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_nat_numerals(4)
% 0.32/0.68 thf(fact_9209_or__nat__numerals_I1_J,axiom,
% 0.32/0.68 ! [Y3: num] :
% 0.32/0.68 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.32/0.68 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_nat_numerals(1)
% 0.32/0.68 thf(fact_9210_or__nat__numerals_I3_J,axiom,
% 0.32/0.68 ! [X: num] :
% 0.32/0.68 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_nat_numerals(3)
% 0.32/0.68 thf(fact_9211_and__nat__numerals_I2_J,axiom,
% 0.32/0.68 ! [Y3: num] :
% 0.32/0.68 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.32/0.68 = one_one_nat ) ).
% 0.32/0.68
% 0.32/0.68 % and_nat_numerals(2)
% 0.32/0.68 thf(fact_9212_and__nat__numerals_I4_J,axiom,
% 0.32/0.68 ! [X: num] :
% 0.32/0.68 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = one_one_nat ) ).
% 0.32/0.68
% 0.32/0.68 % and_nat_numerals(4)
% 0.32/0.68 thf(fact_9213_or__minus__numerals_I4_J,axiom,
% 0.32/0.68 ! [M: num,N: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_numerals(4)
% 0.32/0.68 thf(fact_9214_or__minus__numerals_I8_J,axiom,
% 0.32/0.68 ! [N: num,M: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_numerals(8)
% 0.32/0.68 thf(fact_9215_or__minus__numerals_I3_J,axiom,
% 0.32/0.68 ! [M: num,N: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_numerals(3)
% 0.32/0.68 thf(fact_9216_or__minus__numerals_I7_J,axiom,
% 0.32/0.68 ! [N: num,M: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_numerals(7)
% 0.32/0.68 thf(fact_9217_Suc__0__and__eq,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.68 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Suc_0_and_eq
% 0.32/0.68 thf(fact_9218_and__Suc__0__eq,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_Suc_0_eq
% 0.32/0.68 thf(fact_9219_or__not__num__neg_Osimps_I1_J,axiom,
% 0.32/0.68 ( ( bit_or_not_num_neg @ one @ one )
% 0.32/0.68 = one ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(1)
% 0.32/0.68 thf(fact_9220_or__not__num__neg_Osimps_I4_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 0.32/0.68 = ( bit0 @ one ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(4)
% 0.32/0.68 thf(fact_9221_or__not__num__neg_Osimps_I6_J,axiom,
% 0.32/0.68 ! [N: num,M: num] :
% 0.32/0.68 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 0.32/0.68 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(6)
% 0.32/0.68 thf(fact_9222_or__not__num__neg_Osimps_I7_J,axiom,
% 0.32/0.68 ! [N: num] :
% 0.32/0.68 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 0.32/0.68 = one ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(7)
% 0.32/0.68 thf(fact_9223_or__not__num__neg_Osimps_I3_J,axiom,
% 0.32/0.68 ! [M: num] :
% 0.32/0.68 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 0.32/0.68 = ( bit1 @ M ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(3)
% 0.32/0.68 thf(fact_9224_or__not__num__neg_Osimps_I5_J,axiom,
% 0.32/0.68 ! [N: num,M: num] :
% 0.32/0.68 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 0.32/0.68 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(5)
% 0.32/0.68 thf(fact_9225_or__not__num__neg_Osimps_I9_J,axiom,
% 0.32/0.68 ! [N: num,M: num] :
% 0.32/0.68 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 0.32/0.68 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(9)
% 0.32/0.68 thf(fact_9226_or__nat__def,axiom,
% 0.32/0.68 ( bit_se1412395901928357646or_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_nat_def
% 0.32/0.68 thf(fact_9227_and__nat__def,axiom,
% 0.32/0.68 ( bit_se727722235901077358nd_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_nat_def
% 0.32/0.68 thf(fact_9228_or__not__num__neg_Osimps_I2_J,axiom,
% 0.32/0.68 ! [M: num] :
% 0.32/0.68 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 0.32/0.68 = ( bit1 @ M ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(2)
% 0.32/0.68 thf(fact_9229_or__not__num__neg_Osimps_I8_J,axiom,
% 0.32/0.68 ! [N: num,M: num] :
% 0.32/0.68 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 0.32/0.68 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.simps(8)
% 0.32/0.68 thf(fact_9230_atLeastAtMostPlus1__int__conv,axiom,
% 0.32/0.68 ! [M: int,N: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.32/0.68 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.32/0.68 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % atLeastAtMostPlus1_int_conv
% 0.32/0.68 thf(fact_9231_simp__from__to,axiom,
% 0.32/0.68 ( set_or1266510415728281911st_int
% 0.32/0.68 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % simp_from_to
% 0.32/0.68 thf(fact_9232_or__not__num__neg_Oelims,axiom,
% 0.32/0.68 ! [X: num,Xa: num,Y3: num] :
% 0.32/0.68 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.32/0.68 = Y3 )
% 0.32/0.68 => ( ( ( X = one )
% 0.32/0.68 => ( ( Xa = one )
% 0.32/0.68 => ( Y3 != one ) ) )
% 0.32/0.68 => ( ( ( X = one )
% 0.32/0.68 => ! [M4: num] :
% 0.32/0.68 ( ( Xa
% 0.32/0.68 = ( bit0 @ M4 ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 != ( bit1 @ M4 ) ) ) )
% 0.32/0.68 => ( ( ( X = one )
% 0.32/0.68 => ! [M4: num] :
% 0.32/0.68 ( ( Xa
% 0.32/0.68 = ( bit1 @ M4 ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 != ( bit1 @ M4 ) ) ) )
% 0.32/0.68 => ( ( ? [N4: num] :
% 0.32/0.68 ( X
% 0.32/0.68 = ( bit0 @ N4 ) )
% 0.32/0.68 => ( ( Xa = one )
% 0.32/0.68 => ( Y3
% 0.32/0.68 != ( bit0 @ one ) ) ) )
% 0.32/0.68 => ( ! [N4: num] :
% 0.32/0.68 ( ( X
% 0.32/0.68 = ( bit0 @ N4 ) )
% 0.32/0.68 => ! [M4: num] :
% 0.32/0.68 ( ( Xa
% 0.32/0.68 = ( bit0 @ M4 ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 0.32/0.68 => ( ! [N4: num] :
% 0.32/0.68 ( ( X
% 0.32/0.68 = ( bit0 @ N4 ) )
% 0.32/0.68 => ! [M4: num] :
% 0.32/0.68 ( ( Xa
% 0.32/0.68 = ( bit1 @ M4 ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 0.32/0.68 => ( ( ? [N4: num] :
% 0.32/0.68 ( X
% 0.32/0.68 = ( bit1 @ N4 ) )
% 0.32/0.68 => ( ( Xa = one )
% 0.32/0.68 => ( Y3 != one ) ) )
% 0.32/0.68 => ( ! [N4: num] :
% 0.32/0.68 ( ( X
% 0.32/0.68 = ( bit1 @ N4 ) )
% 0.32/0.68 => ! [M4: num] :
% 0.32/0.68 ( ( Xa
% 0.32/0.68 = ( bit0 @ M4 ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 0.32/0.68 => ~ ! [N4: num] :
% 0.32/0.68 ( ( X
% 0.32/0.68 = ( bit1 @ N4 ) )
% 0.32/0.68 => ! [M4: num] :
% 0.32/0.68 ( ( Xa
% 0.32/0.68 = ( bit1 @ M4 ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_num_neg.elims
% 0.32/0.68 thf(fact_9233_and__nat__unfold,axiom,
% 0.32/0.68 ( bit_se727722235901077358nd_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.68 ( if_nat
% 0.32/0.68 @ ( ( M3 = zero_zero_nat )
% 0.32/0.68 | ( N2 = zero_zero_nat ) )
% 0.32/0.68 @ zero_zero_nat
% 0.32/0.68 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_nat_unfold
% 0.32/0.68 thf(fact_9234_Suc__0__or__eq,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.68 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Suc_0_or_eq
% 0.32/0.68 thf(fact_9235_or__Suc__0__eq,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_Suc_0_eq
% 0.32/0.68 thf(fact_9236_or__nat__rec,axiom,
% 0.32/0.68 ( bit_se1412395901928357646or_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.68 ( plus_plus_nat
% 0.32/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.68 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.32/0.68 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.68 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_nat_rec
% 0.32/0.68 thf(fact_9237_and__nat__rec,axiom,
% 0.32/0.68 ( bit_se727722235901077358nd_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.68 ( plus_plus_nat
% 0.32/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.68 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 0.32/0.68 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.68 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_nat_rec
% 0.32/0.68 thf(fact_9238_or__nat__unfold,axiom,
% 0.32/0.68 ( bit_se1412395901928357646or_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_nat_unfold
% 0.32/0.68 thf(fact_9239_and__int_Opelims,axiom,
% 0.32/0.68 ! [X: int,Xa: int,Y3: int] :
% 0.32/0.68 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 0.32/0.68 = Y3 )
% 0.32/0.68 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.32/0.68 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.32/0.68 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 = ( uminus_uminus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 0.32/0.68 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.32/0.68 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.32/0.68 => ( Y3
% 0.32/0.68 = ( plus_plus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 0.32/0.68 @ ( 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 ) ) ) ) ) ) ) ) )
% 0.32/0.68 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_int.pelims
% 0.32/0.68 thf(fact_9240_and__int_Opsimps,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
% 0.32/0.68 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.32/0.68 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.32/0.68 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 0.32/0.68 = ( uminus_uminus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
% 0.32/0.68 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.32/0.68 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.32/0.68 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 0.32/0.68 = ( plus_plus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.32/0.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.32/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_int.psimps
% 0.32/0.68 thf(fact_9241_cis__multiple__2pi,axiom,
% 0.32/0.68 ! [N: real] :
% 0.32/0.68 ( ( member_real @ N @ ring_1_Ints_real )
% 0.32/0.68 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.32/0.68 = one_one_complex ) ) ).
% 0.32/0.68
% 0.32/0.68 % cis_multiple_2pi
% 0.32/0.68 thf(fact_9242_set__encode__insert,axiom,
% 0.32/0.68 ! [A2: set_nat,N: nat] :
% 0.32/0.68 ( ( finite_finite_nat @ A2 )
% 0.32/0.68 => ( ~ ( member_nat @ N @ A2 )
% 0.32/0.68 => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 0.32/0.68 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % set_encode_insert
% 0.32/0.68 thf(fact_9243_atLeastAtMost__insertL,axiom,
% 0.32/0.68 ! [M: nat,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.68 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.32/0.68 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % atLeastAtMost_insertL
% 0.32/0.68 thf(fact_9244_atLeastAtMostSuc__conv,axiom,
% 0.32/0.68 ! [M: nat,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.32/0.68 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.32/0.68 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % atLeastAtMostSuc_conv
% 0.32/0.68 thf(fact_9245_Icc__eq__insert__lb__nat,axiom,
% 0.32/0.68 ! [M: nat,N: nat] :
% 0.32/0.68 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.68 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.32/0.68 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Icc_eq_insert_lb_nat
% 0.32/0.68 thf(fact_9246_lessThan__nat__numeral,axiom,
% 0.32/0.68 ! [K: num] :
% 0.32/0.68 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.32/0.68 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % lessThan_nat_numeral
% 0.32/0.68 thf(fact_9247_atMost__nat__numeral,axiom,
% 0.32/0.68 ! [K: num] :
% 0.32/0.68 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.32/0.68 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % atMost_nat_numeral
% 0.32/0.68 thf(fact_9248_sin__times__pi__eq__0,axiom,
% 0.32/0.68 ! [X: real] :
% 0.32/0.68 ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 0.32/0.68 = zero_zero_real )
% 0.32/0.68 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % sin_times_pi_eq_0
% 0.32/0.68 thf(fact_9249_atLeast1__atMost__eq__remove0,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.68 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % atLeast1_atMost_eq_remove0
% 0.32/0.68 thf(fact_9250_set__decode__plus__power__2,axiom,
% 0.32/0.68 ! [N: nat,Z: nat] :
% 0.32/0.68 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 0.32/0.68 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 0.32/0.68 = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % set_decode_plus_power_2
% 0.32/0.68 thf(fact_9251_sin__integer__2pi,axiom,
% 0.32/0.68 ! [N: real] :
% 0.32/0.68 ( ( member_real @ N @ ring_1_Ints_real )
% 0.32/0.68 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.32/0.68 = zero_zero_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % sin_integer_2pi
% 0.32/0.68 thf(fact_9252_cos__integer__2pi,axiom,
% 0.32/0.68 ! [N: real] :
% 0.32/0.68 ( ( member_real @ N @ ring_1_Ints_real )
% 0.32/0.68 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 0.32/0.68 = one_one_real ) ) ).
% 0.32/0.68
% 0.32/0.68 % cos_integer_2pi
% 0.32/0.68 thf(fact_9253_and__int_Opinduct,axiom,
% 0.32/0.68 ! [A0: int,A1: int,P: int > int > $o] :
% 0.32/0.68 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.32/0.68 => ( ! [K3: int,L4: int] :
% 0.32/0.68 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
% 0.32/0.68 => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.32/0.68 & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.32/0.68 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.32/0.68 => ( P @ K3 @ L4 ) ) )
% 0.32/0.68 => ( P @ A0 @ A1 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_int.pinduct
% 0.32/0.68 thf(fact_9254_upto_Opinduct,axiom,
% 0.32/0.68 ! [A0: int,A1: int,P: int > int > $o] :
% 0.32/0.68 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 0.32/0.68 => ( ! [I2: int,J2: int] :
% 0.32/0.68 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
% 0.32/0.68 => ( ( ( ord_less_eq_int @ I2 @ J2 )
% 0.32/0.68 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
% 0.32/0.68 => ( P @ I2 @ J2 ) ) )
% 0.32/0.68 => ( P @ A0 @ A1 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % upto.pinduct
% 0.32/0.68 thf(fact_9255_xor__Suc__0__eq,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.68 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_Suc_0_eq
% 0.32/0.68 thf(fact_9256_xor__nat__numerals_I1_J,axiom,
% 0.32/0.68 ! [Y3: num] :
% 0.32/0.68 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.32/0.68 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_nat_numerals(1)
% 0.32/0.68 thf(fact_9257_xor__nat__numerals_I2_J,axiom,
% 0.32/0.68 ! [Y3: num] :
% 0.32/0.68 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.32/0.68 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_nat_numerals(2)
% 0.32/0.68 thf(fact_9258_xor__nat__numerals_I3_J,axiom,
% 0.32/0.68 ! [X: num] :
% 0.32/0.68 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_nat_numerals(3)
% 0.32/0.68 thf(fact_9259_xor__nat__numerals_I4_J,axiom,
% 0.32/0.68 ! [X: num] :
% 0.32/0.68 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_nat_numerals(4)
% 0.32/0.68 thf(fact_9260_xor__nat__unfold,axiom,
% 0.32/0.68 ( bit_se6528837805403552850or_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_nat_unfold
% 0.32/0.68 thf(fact_9261_xor__nat__rec,axiom,
% 0.32/0.68 ( bit_se6528837805403552850or_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.68 ( plus_plus_nat
% 0.32/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.68 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.32/0.68 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.32/0.68 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_nat_rec
% 0.32/0.68 thf(fact_9262_Suc__0__xor__eq,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.68 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.68 @ ( zero_n2687167440665602831ol_nat
% 0.32/0.68 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % Suc_0_xor_eq
% 0.32/0.68 thf(fact_9263_horner__sum__of__bool__2__less,axiom,
% 0.32/0.68 ! [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 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % horner_sum_of_bool_2_less
% 0.32/0.68 thf(fact_9264_push__bit__nonnegative__int__iff,axiom,
% 0.32/0.68 ! [N: nat,K: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.32/0.68 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.68
% 0.32/0.68 % push_bit_nonnegative_int_iff
% 0.32/0.68 thf(fact_9265_push__bit__negative__int__iff,axiom,
% 0.32/0.68 ! [N: nat,K: int] :
% 0.32/0.68 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.68 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.68
% 0.32/0.68 % push_bit_negative_int_iff
% 0.32/0.68 thf(fact_9266_concat__bit__of__zero__1,axiom,
% 0.32/0.68 ! [N: nat,L2: int] :
% 0.32/0.68 ( ( bit_concat_bit @ N @ zero_zero_int @ L2 )
% 0.32/0.68 = ( bit_se545348938243370406it_int @ N @ L2 ) ) ).
% 0.32/0.68
% 0.32/0.68 % concat_bit_of_zero_1
% 0.32/0.68 thf(fact_9267_xor__nonnegative__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
% 0.32/0.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.32/0.68 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_nonnegative_int_iff
% 0.32/0.68 thf(fact_9268_xor__negative__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int] :
% 0.32/0.68 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
% 0.32/0.68 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.32/0.68 != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_negative_int_iff
% 0.32/0.68 thf(fact_9269_push__bit__of__Suc__0,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.68 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.32/0.68
% 0.32/0.68 % push_bit_of_Suc_0
% 0.32/0.68 thf(fact_9270_bit__xor__int__iff,axiom,
% 0.32/0.68 ! [K: int,L2: int,N: nat] :
% 0.32/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N )
% 0.32/0.68 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.32/0.68 != ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bit_xor_int_iff
% 0.32/0.68 thf(fact_9271_flip__bit__int__def,axiom,
% 0.32/0.68 ( bit_se2159334234014336723it_int
% 0.32/0.68 = ( ^ [N2: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % flip_bit_int_def
% 0.32/0.68 thf(fact_9272_push__bit__nat__eq,axiom,
% 0.32/0.68 ! [N: nat,K: int] :
% 0.32/0.68 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 0.32/0.68 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % push_bit_nat_eq
% 0.32/0.68 thf(fact_9273_XOR__lower,axiom,
% 0.32/0.68 ! [X: int,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.68 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y3 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % XOR_lower
% 0.32/0.68 thf(fact_9274_set__bit__nat__def,axiom,
% 0.32/0.68 ( bit_se7882103937844011126it_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % set_bit_nat_def
% 0.32/0.68 thf(fact_9275_flip__bit__nat__def,axiom,
% 0.32/0.68 ( bit_se2161824704523386999it_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % flip_bit_nat_def
% 0.32/0.68 thf(fact_9276_bit__push__bit__iff__int,axiom,
% 0.32/0.68 ! [M: nat,K: int,N: nat] :
% 0.32/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.32/0.68 = ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.68 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bit_push_bit_iff_int
% 0.32/0.68 thf(fact_9277_xor__nat__def,axiom,
% 0.32/0.68 ( bit_se6528837805403552850or_nat
% 0.32/0.68 = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_nat_def
% 0.32/0.68 thf(fact_9278_bit__push__bit__iff__nat,axiom,
% 0.32/0.68 ! [M: nat,Q2: nat,N: nat] :
% 0.32/0.68 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 0.32/0.68 = ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.68 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bit_push_bit_iff_nat
% 0.32/0.68 thf(fact_9279_concat__bit__eq,axiom,
% 0.32/0.68 ( bit_concat_bit
% 0.32/0.68 = ( ^ [N2: nat,K2: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % concat_bit_eq
% 0.32/0.68 thf(fact_9280_concat__bit__def,axiom,
% 0.32/0.68 ( bit_concat_bit
% 0.32/0.68 = ( ^ [N2: nat,K2: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % concat_bit_def
% 0.32/0.68 thf(fact_9281_set__bit__int__def,axiom,
% 0.32/0.68 ( bit_se7879613467334960850it_int
% 0.32/0.68 = ( ^ [N2: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % set_bit_int_def
% 0.32/0.68 thf(fact_9282_push__bit__int__def,axiom,
% 0.32/0.68 ( bit_se545348938243370406it_int
% 0.32/0.68 = ( ^ [N2: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % push_bit_int_def
% 0.32/0.68 thf(fact_9283_push__bit__nat__def,axiom,
% 0.32/0.68 ( bit_se547839408752420682it_nat
% 0.32/0.68 = ( ^ [N2: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % push_bit_nat_def
% 0.32/0.68 thf(fact_9284_push__bit__minus__one,axiom,
% 0.32/0.68 ! [N: nat] :
% 0.32/0.68 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.68 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % push_bit_minus_one
% 0.32/0.68 thf(fact_9285_XOR__upper,axiom,
% 0.32/0.68 ! [X: int,N: nat,Y3: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.68 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.68 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.32/0.68 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % XOR_upper
% 0.32/0.68 thf(fact_9286_xor__int__rec,axiom,
% 0.32/0.68 ( bit_se6526347334894502574or_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( plus_plus_int
% 0.32/0.68 @ ( zero_n2684676970156552555ol_int
% 0.32/0.68 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 0.32/0.68 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 0.32/0.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_int_rec
% 0.32/0.68 thf(fact_9287_xor__int__unfold,axiom,
% 0.32/0.68 ( bit_se6526347334894502574or_int
% 0.32/0.68 = ( ^ [K2: int,L: int] :
% 0.32/0.68 ( if_int
% 0.32/0.68 @ ( K2
% 0.32/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.68 @ ( bit_ri7919022796975470100ot_int @ L )
% 0.32/0.68 @ ( if_int
% 0.32/0.68 @ ( L
% 0.32/0.68 = ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.68 @ ( bit_ri7919022796975470100ot_int @ K2 )
% 0.32/0.68 @ ( if_int @ ( K2 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % xor_int_unfold
% 0.32/0.68 thf(fact_9288_not__nonnegative__int__iff,axiom,
% 0.32/0.68 ! [K: int] :
% 0.32/0.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.32/0.68 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.68
% 0.32/0.68 % not_nonnegative_int_iff
% 0.32/0.68 thf(fact_9289_not__negative__int__iff,axiom,
% 0.32/0.68 ! [K: int] :
% 0.32/0.68 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.32/0.68 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.68
% 0.32/0.68 % not_negative_int_iff
% 0.32/0.68 thf(fact_9290_and__minus__minus__numerals,axiom,
% 0.32/0.68 ! [M: num,N: num] :
% 0.32/0.68 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.68 = ( 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 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % and_minus_minus_numerals
% 0.32/0.68 thf(fact_9291_or__minus__minus__numerals,axiom,
% 0.32/0.68 ! [M: num,N: num] :
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.68 = ( 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 ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_minus_minus_numerals
% 0.32/0.68 thf(fact_9292_bit__not__int__iff,axiom,
% 0.32/0.68 ! [K: int,N: nat] :
% 0.32/0.68 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 0.32/0.68 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % bit_not_int_iff
% 0.32/0.68 thf(fact_9293_or__int__def,axiom,
% 0.32/0.68 ( bit_se1409905431419307370or_int
% 0.32/0.68 = ( ^ [K2: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_int_def
% 0.32/0.68 thf(fact_9294_not__int__def,axiom,
% 0.32/0.68 ( bit_ri7919022796975470100ot_int
% 0.32/0.68 = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 0.32/0.68
% 0.32/0.68 % not_int_def
% 0.32/0.68 thf(fact_9295_and__not__numerals_I1_J,axiom,
% 0.32/0.68 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.32/0.68 = zero_zero_int ) ).
% 0.32/0.68
% 0.32/0.68 % and_not_numerals(1)
% 0.32/0.68 thf(fact_9296_or__not__numerals_I1_J,axiom,
% 0.32/0.68 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.32/0.68 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.32/0.68
% 0.32/0.68 % or_not_numerals(1)
% 0.32/0.68 thf(fact_9297_unset__bit__int__def,axiom,
% 0.32/0.68 ( bit_se4203085406695923979it_int
% 0.32/0.68 = ( ^ [N2: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % unset_bit_int_def
% 0.32/0.69 thf(fact_9298_xor__int__def,axiom,
% 0.32/0.69 ( bit_se6526347334894502574or_int
% 0.32/0.69 = ( ^ [K2: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_int_def
% 0.32/0.69 thf(fact_9299_not__int__div__2,axiom,
% 0.32/0.69 ! [K: int] :
% 0.32/0.69 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.69 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % not_int_div_2
% 0.32/0.69 thf(fact_9300_even__not__iff__int,axiom,
% 0.32/0.69 ! [K: int] :
% 0.32/0.69 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.32/0.69 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % even_not_iff_int
% 0.32/0.69 thf(fact_9301_and__not__numerals_I2_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.69 = one_one_int ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_numerals(2)
% 0.32/0.69 thf(fact_9302_and__not__numerals_I4_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.32/0.69 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_numerals(4)
% 0.32/0.69 thf(fact_9303_or__not__numerals_I2_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.69 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_numerals(2)
% 0.32/0.69 thf(fact_9304_or__not__numerals_I4_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.32/0.69 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_numerals(4)
% 0.32/0.69 thf(fact_9305_bit__minus__int__iff,axiom,
% 0.32/0.69 ! [K: int,N: nat] :
% 0.32/0.69 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 0.32/0.69 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % bit_minus_int_iff
% 0.32/0.69 thf(fact_9306_numeral__or__not__num__eq,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 0.32/0.69 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % numeral_or_not_num_eq
% 0.32/0.69 thf(fact_9307_int__numeral__not__or__num__neg,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.69 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % int_numeral_not_or_num_neg
% 0.32/0.69 thf(fact_9308_int__numeral__or__not__num__neg,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % int_numeral_or_not_num_neg
% 0.32/0.69 thf(fact_9309_and__not__numerals_I5_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_numerals(5)
% 0.32/0.69 thf(fact_9310_and__not__numerals_I7_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.32/0.69 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_numerals(7)
% 0.32/0.69 thf(fact_9311_or__not__numerals_I3_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.69 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_numerals(3)
% 0.32/0.69 thf(fact_9312_and__not__numerals_I3_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.69 = zero_zero_int ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_numerals(3)
% 0.32/0.69 thf(fact_9313_or__not__numerals_I7_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 0.32/0.69 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_numerals(7)
% 0.32/0.69 thf(fact_9314_and__not__numerals_I9_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_numerals(9)
% 0.32/0.69 thf(fact_9315_and__not__numerals_I6_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_numerals(6)
% 0.32/0.69 thf(fact_9316_or__not__numerals_I6_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_numerals(6)
% 0.32/0.69 thf(fact_9317_or__not__numerals_I5_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_numerals(5)
% 0.32/0.69 thf(fact_9318_and__not__numerals_I8_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_numerals(8)
% 0.32/0.69 thf(fact_9319_or__not__numerals_I8_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_numerals(8)
% 0.32/0.69 thf(fact_9320_or__not__numerals_I9_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_numerals(9)
% 0.32/0.69 thf(fact_9321_not__int__rec,axiom,
% 0.32/0.69 ( bit_ri7919022796975470100ot_int
% 0.32/0.69 = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % not_int_rec
% 0.32/0.69 thf(fact_9322_vebt__maxt_Opelims,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Y3: option_nat] :
% 0.32/0.69 ( ( ( vEBT_vebt_maxt @ X )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
% 0.32/0.69 => ( ! [A5: $o,B3: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.69 => ( ( ( B3
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.69 & ( ~ B3
% 0.32/0.69 => ( ( A5
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.69 & ( ~ A5
% 0.32/0.69 => ( Y3 = none_nat ) ) ) ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B3 ) ) ) )
% 0.32/0.69 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.69 => ( ( Y3 = none_nat )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.32/0.69 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_nat @ Ma2 ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % vebt_maxt.pelims
% 0.32/0.69 thf(fact_9323_vebt__mint_Opelims,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Y3: option_nat] :
% 0.32/0.69 ( ( ( vEBT_vebt_mint @ X )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
% 0.32/0.69 => ( ! [A5: $o,B3: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ A5 @ B3 ) )
% 0.32/0.69 => ( ( ( A5
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( some_nat @ zero_zero_nat ) ) )
% 0.32/0.69 & ( ~ A5
% 0.32/0.69 => ( ( B3
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( some_nat @ one_one_nat ) ) )
% 0.32/0.69 & ( ~ B3
% 0.32/0.69 => ( Y3 = none_nat ) ) ) ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B3 ) ) ) )
% 0.32/0.69 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.32/0.69 => ( ( Y3 = none_nat )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.32/0.69 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_nat @ Mi2 ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % vebt_mint.pelims
% 0.32/0.69 thf(fact_9324_or__not__num__neg_Opelims,axiom,
% 0.32/0.69 ! [X: num,Xa: num,Y3: num] :
% 0.32/0.69 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3 = one )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ! [M4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ! [M4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [N4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( bit0 @ one ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ one ) ) ) ) )
% 0.32/0.69 => ( ! [N4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ! [M4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [N4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ! [M4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [N4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3 = one )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ one ) ) ) ) )
% 0.32/0.69 => ( ! [N4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ! [M4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 0.32/0.69 => ~ ! [N4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ! [M4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % or_not_num_neg.pelims
% 0.32/0.69 thf(fact_9325_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Y3: $o] :
% 0.32/0.69 ( ( ( vEBT_VEBT_minNull @ X )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.32/0.69 => ( ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 0.32/0.69 => ( ! [Uv2: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.32/0.69 => ( ~ Y3
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 0.32/0.69 => ( ! [Uu2: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.32/0.69 => ( ~ Y3
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 0.32/0.69 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 0.32/0.69 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 0.32/0.69 => ( ~ Y3
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.minNull.pelims(1)
% 0.32/0.69 thf(fact_9326_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT] :
% 0.32/0.69 ( ~ ( vEBT_VEBT_minNull @ X )
% 0.32/0.69 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.32/0.69 => ( ! [Uv2: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 0.32/0.69 => ( ! [Uu2: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 0.32/0.69 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.minNull.pelims(3)
% 0.32/0.69 thf(fact_9327_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT] :
% 0.32/0.69 ( ( vEBT_VEBT_minNull @ X )
% 0.32/0.69 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 0.32/0.69 => ( ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ $false @ $false ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 0.32/0.69 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.32/0.69 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.minNull.pelims(2)
% 0.32/0.69 thf(fact_9328_int__ge__less__than2__def,axiom,
% 0.32/0.69 ( int_ge_less_than2
% 0.32/0.69 = ( ^ [D2: int] :
% 0.32/0.69 ( collec213857154873943460nt_int
% 0.32/0.69 @ ( produc4947309494688390418_int_o
% 0.32/0.69 @ ^ [Z7: int,Z2: int] :
% 0.32/0.69 ( ( ord_less_eq_int @ D2 @ Z2 )
% 0.32/0.69 & ( ord_less_int @ Z7 @ Z2 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % int_ge_less_than2_def
% 0.32/0.69 thf(fact_9329_int__ge__less__than__def,axiom,
% 0.32/0.69 ( int_ge_less_than
% 0.32/0.69 = ( ^ [D2: int] :
% 0.32/0.69 ( collec213857154873943460nt_int
% 0.32/0.69 @ ( produc4947309494688390418_int_o
% 0.32/0.69 @ ^ [Z7: int,Z2: int] :
% 0.32/0.69 ( ( ord_less_eq_int @ D2 @ Z7 )
% 0.32/0.69 & ( ord_less_int @ Z7 @ Z2 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % int_ge_less_than_def
% 0.32/0.69 thf(fact_9330_Sum__Ico__nat,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( groups3542108847815614940at_nat
% 0.32/0.69 @ ^ [X2: nat] : X2
% 0.32/0.69 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.32/0.69 = ( 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 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Sum_Ico_nat
% 0.32/0.69 thf(fact_9331_Cauchy__iff2,axiom,
% 0.32/0.69 ( topolo4055970368930404560y_real
% 0.32/0.69 = ( ^ [X6: nat > real] :
% 0.32/0.69 ! [J3: nat] :
% 0.32/0.69 ? [M8: nat] :
% 0.32/0.69 ! [M3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ M8 @ M3 )
% 0.32/0.69 => ! [N2: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ M8 @ N2 )
% 0.32/0.69 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Cauchy_iff2
% 0.32/0.69 thf(fact_9332_all__nat__less__eq,axiom,
% 0.32/0.69 ! [N: nat,P: nat > $o] :
% 0.32/0.69 ( ( ! [M3: nat] :
% 0.32/0.69 ( ( ord_less_nat @ M3 @ N )
% 0.32/0.69 => ( P @ M3 ) ) )
% 0.32/0.69 = ( ! [X2: nat] :
% 0.32/0.69 ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.32/0.69 => ( P @ X2 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % all_nat_less_eq
% 0.32/0.69 thf(fact_9333_ex__nat__less__eq,axiom,
% 0.32/0.69 ! [N: nat,P: nat > $o] :
% 0.32/0.69 ( ( ? [M3: nat] :
% 0.32/0.69 ( ( ord_less_nat @ M3 @ N )
% 0.32/0.69 & ( P @ M3 ) ) )
% 0.32/0.69 = ( ? [X2: nat] :
% 0.32/0.69 ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.32/0.69 & ( P @ X2 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % ex_nat_less_eq
% 0.32/0.69 thf(fact_9334_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.32/0.69 ! [N3: set_nat,N: nat] :
% 0.32/0.69 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.32/0.69 => ( finite_finite_nat @ N3 ) ) ).
% 0.32/0.69
% 0.32/0.69 % subset_eq_atLeast0_lessThan_finite
% 0.32/0.69 thf(fact_9335_atLeastLessThanSuc,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.69 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.32/0.69 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.32/0.69 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.32/0.69 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.32/0.69 = bot_bot_set_nat ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % atLeastLessThanSuc
% 0.32/0.69 thf(fact_9336_atLeastLessThan__nat__numeral,axiom,
% 0.32/0.69 ! [M: nat,K: num] :
% 0.32/0.69 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.32/0.69 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.32/0.69 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.32/0.69 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.32/0.69 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.32/0.69 = bot_bot_set_nat ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % atLeastLessThan_nat_numeral
% 0.32/0.69 thf(fact_9337_atLeast1__lessThan__eq__remove0,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.32/0.69 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % atLeast1_lessThan_eq_remove0
% 0.32/0.69 thf(fact_9338_sum__power2,axiom,
% 0.32/0.69 ! [K: nat] :
% 0.32/0.69 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.32/0.69 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.32/0.69
% 0.32/0.69 % sum_power2
% 0.32/0.69 thf(fact_9339_Chebyshev__sum__upper__nat,axiom,
% 0.32/0.69 ! [N: nat,A: nat > nat,B: nat > nat] :
% 0.32/0.69 ( ! [I2: nat,J2: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.32/0.69 => ( ( ord_less_nat @ J2 @ N )
% 0.32/0.69 => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
% 0.32/0.69 => ( ! [I2: nat,J2: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.32/0.69 => ( ( ord_less_nat @ J2 @ N )
% 0.32/0.69 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
% 0.32/0.69 => ( ord_less_eq_nat
% 0.32/0.69 @ ( times_times_nat @ N
% 0.32/0.69 @ ( groups3542108847815614940at_nat
% 0.32/0.69 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
% 0.32/0.69 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.32/0.69 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Chebyshev_sum_upper_nat
% 0.32/0.69 thf(fact_9340_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.32/0.69 ! [L2: int,U: int] :
% 0.32/0.69 ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
% 0.32/0.69 = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% 0.32/0.69
% 0.32/0.69 % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.32/0.69 thf(fact_9341_valid__eq2,axiom,
% 0.32/0.69 ! [T: vEBT_VEBT,D: nat] :
% 0.32/0.69 ( ( vEBT_VEBT_valid @ T @ D )
% 0.32/0.69 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.32/0.69
% 0.32/0.69 % valid_eq2
% 0.32/0.69 thf(fact_9342_valid__eq1,axiom,
% 0.32/0.69 ! [T: vEBT_VEBT,D: nat] :
% 0.32/0.69 ( ( vEBT_invar_vebt @ T @ D )
% 0.32/0.69 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.32/0.69
% 0.32/0.69 % valid_eq1
% 0.32/0.69 thf(fact_9343_valid__eq,axiom,
% 0.32/0.69 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.32/0.69
% 0.32/0.69 % valid_eq
% 0.32/0.69 thf(fact_9344_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.32/0.69 ! [Uu: $o,Uv: $o,D: nat] :
% 0.32/0.69 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.32/0.69 = ( D = one_one_nat ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.valid'.simps(1)
% 0.32/0.69 thf(fact_9345_Code__Target__Int_Opositive__def,axiom,
% 0.32/0.69 code_Target_positive = numeral_numeral_int ).
% 0.32/0.69
% 0.32/0.69 % Code_Target_Int.positive_def
% 0.32/0.69 thf(fact_9346_divmod__step__integer__def,axiom,
% 0.32/0.69 ( unique4921790084139445826nteger
% 0.32/0.69 = ( ^ [L: num] :
% 0.32/0.69 ( produc6916734918728496179nteger
% 0.32/0.69 @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divmod_step_integer_def
% 0.32/0.69 thf(fact_9347_divmod__integer_H__def,axiom,
% 0.32/0.69 ( unique3479559517661332726nteger
% 0.32/0.69 = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divmod_integer'_def
% 0.32/0.69 thf(fact_9348_less__eq__integer__code_I1_J,axiom,
% 0.32/0.69 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 0.32/0.69
% 0.32/0.69 % less_eq_integer_code(1)
% 0.32/0.69 thf(fact_9349_times__integer__code_I2_J,axiom,
% 0.32/0.69 ! [L2: code_integer] :
% 0.32/0.69 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.32/0.69 = zero_z3403309356797280102nteger ) ).
% 0.32/0.69
% 0.32/0.69 % times_integer_code(2)
% 0.32/0.69 thf(fact_9350_times__integer__code_I1_J,axiom,
% 0.32/0.69 ! [K: code_integer] :
% 0.32/0.69 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 0.32/0.69 = zero_z3403309356797280102nteger ) ).
% 0.32/0.69
% 0.32/0.69 % times_integer_code(1)
% 0.32/0.69 thf(fact_9351_plus__integer__code_I1_J,axiom,
% 0.32/0.69 ! [K: code_integer] :
% 0.32/0.69 ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 0.32/0.69 = K ) ).
% 0.32/0.69
% 0.32/0.69 % plus_integer_code(1)
% 0.32/0.69 thf(fact_9352_plus__integer__code_I2_J,axiom,
% 0.32/0.69 ! [L2: code_integer] :
% 0.32/0.69 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.32/0.69 = L2 ) ).
% 0.32/0.69
% 0.32/0.69 % plus_integer_code(2)
% 0.32/0.69 thf(fact_9353_sgn__integer__code,axiom,
% 0.32/0.69 ( sgn_sgn_Code_integer
% 0.32/0.69 = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % sgn_integer_code
% 0.32/0.69 thf(fact_9354_minus__integer__code_I1_J,axiom,
% 0.32/0.69 ! [K: code_integer] :
% 0.32/0.69 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 0.32/0.69 = K ) ).
% 0.32/0.69
% 0.32/0.69 % minus_integer_code(1)
% 0.32/0.69 thf(fact_9355_minus__integer__code_I2_J,axiom,
% 0.32/0.69 ! [L2: code_integer] :
% 0.32/0.69 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.32/0.69 = ( uminus1351360451143612070nteger @ L2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_integer_code(2)
% 0.32/0.69 thf(fact_9356_one__natural_Orsp,axiom,
% 0.32/0.69 one_one_nat = one_one_nat ).
% 0.32/0.69
% 0.32/0.69 % one_natural.rsp
% 0.32/0.69 thf(fact_9357_integer__of__int__code,axiom,
% 0.32/0.69 ( code_integer_of_int
% 0.32/0.69 = ( ^ [K2: int] :
% 0.32/0.69 ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 0.32/0.69 @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.32/0.69 @ ( if_Code_integer
% 0.32/0.69 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.32/0.69 = zero_zero_int )
% 0.32/0.69 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % integer_of_int_code
% 0.32/0.69 thf(fact_9358_Code__Numeral_Opositive__def,axiom,
% 0.32/0.69 code_positive = numera6620942414471956472nteger ).
% 0.32/0.69
% 0.32/0.69 % Code_Numeral.positive_def
% 0.32/0.69 thf(fact_9359_divide__integer_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: int,X: int] :
% 0.32/0.69 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.32/0.69 = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divide_integer.abs_eq
% 0.32/0.69 thf(fact_9360_abs__integer__code,axiom,
% 0.32/0.69 ( abs_abs_Code_integer
% 0.32/0.69 = ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % abs_integer_code
% 0.32/0.69 thf(fact_9361_less__integer_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: int,X: int] :
% 0.32/0.69 ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.32/0.69 = ( ord_less_int @ Xa @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_integer.abs_eq
% 0.32/0.69 thf(fact_9362_less__integer__code_I1_J,axiom,
% 0.32/0.69 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.32/0.69
% 0.32/0.69 % less_integer_code(1)
% 0.32/0.69 thf(fact_9363_plus__integer_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: int,X: int] :
% 0.32/0.69 ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.32/0.69 = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % plus_integer.abs_eq
% 0.32/0.69 thf(fact_9364_times__integer_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: int,X: int] :
% 0.32/0.69 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.32/0.69 = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % times_integer.abs_eq
% 0.32/0.69 thf(fact_9365_less__eq__integer_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: int,X: int] :
% 0.32/0.69 ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.32/0.69 = ( ord_less_eq_int @ Xa @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_eq_integer.abs_eq
% 0.32/0.69 thf(fact_9366_minus__integer_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: int,X: int] :
% 0.32/0.69 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.32/0.69 = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_integer.abs_eq
% 0.32/0.69 thf(fact_9367_integer__of__num_I3_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 0.32/0.69 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 0.32/0.69
% 0.32/0.69 % integer_of_num(3)
% 0.32/0.69 thf(fact_9368_bit__cut__integer__def,axiom,
% 0.32/0.69 ( code_bit_cut_integer
% 0.32/0.69 = ( ^ [K2: code_integer] :
% 0.32/0.69 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.32/0.69 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bit_cut_integer_def
% 0.32/0.69 thf(fact_9369_integer__of__num__def,axiom,
% 0.32/0.69 code_integer_of_num = numera6620942414471956472nteger ).
% 0.32/0.69
% 0.32/0.69 % integer_of_num_def
% 0.32/0.69 thf(fact_9370_integer__of__num__triv_I1_J,axiom,
% 0.32/0.69 ( ( code_integer_of_num @ one )
% 0.32/0.69 = one_one_Code_integer ) ).
% 0.32/0.69
% 0.32/0.69 % integer_of_num_triv(1)
% 0.32/0.69 thf(fact_9371_integer__of__num_I2_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.32/0.69 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % integer_of_num(2)
% 0.32/0.69 thf(fact_9372_integer__of__num__triv_I2_J,axiom,
% 0.32/0.69 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 0.32/0.69 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % integer_of_num_triv(2)
% 0.32/0.69 thf(fact_9373_bit__cut__integer__code,axiom,
% 0.32/0.69 ( code_bit_cut_integer
% 0.32/0.69 = ( ^ [K2: code_integer] :
% 0.32/0.69 ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.32/0.69 @ ( produc9125791028180074456eger_o
% 0.32/0.69 @ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
% 0.32/0.69 @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bit_cut_integer_code
% 0.32/0.69 thf(fact_9374_divmod__integer__def,axiom,
% 0.32/0.69 ( code_divmod_integer
% 0.32/0.69 = ( ^ [K2: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L ) @ ( modulo364778990260209775nteger @ K2 @ L ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divmod_integer_def
% 0.32/0.69 thf(fact_9375_csqrt_Osimps_I1_J,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( re @ ( csqrt @ Z ) )
% 0.32/0.69 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt.simps(1)
% 0.32/0.69 thf(fact_9376_complex__Re__numeral,axiom,
% 0.32/0.69 ! [V: num] :
% 0.32/0.69 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 0.32/0.69 = ( numeral_numeral_real @ V ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_Re_numeral
% 0.32/0.69 thf(fact_9377_Re__divide__of__nat,axiom,
% 0.32/0.69 ! [Z: complex,N: nat] :
% 0.32/0.69 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.32/0.69 = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_divide_of_nat
% 0.32/0.69 thf(fact_9378_Re__divide__of__real,axiom,
% 0.32/0.69 ! [Z: complex,R: real] :
% 0.32/0.69 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
% 0.32/0.69 = ( divide_divide_real @ ( re @ Z ) @ R ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_divide_of_real
% 0.32/0.69 thf(fact_9379_Re__sgn,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( re @ ( sgn_sgn_complex @ Z ) )
% 0.32/0.69 = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_sgn
% 0.32/0.69 thf(fact_9380_Re__divide__numeral,axiom,
% 0.32/0.69 ! [Z: complex,W: num] :
% 0.32/0.69 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.69 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_divide_numeral
% 0.32/0.69 thf(fact_9381_complex__Re__le__cmod,axiom,
% 0.32/0.69 ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_Re_le_cmod
% 0.32/0.69 thf(fact_9382_one__complex_Osimps_I1_J,axiom,
% 0.32/0.69 ( ( re @ one_one_complex )
% 0.32/0.69 = one_one_real ) ).
% 0.32/0.69
% 0.32/0.69 % one_complex.simps(1)
% 0.32/0.69 thf(fact_9383_plus__complex_Osimps_I1_J,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( re @ ( plus_plus_complex @ X @ Y3 ) )
% 0.32/0.69 = ( plus_plus_real @ ( re @ X ) @ ( re @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % plus_complex.simps(1)
% 0.32/0.69 thf(fact_9384_scaleR__complex_Osimps_I1_J,axiom,
% 0.32/0.69 ! [R: real,X: complex] :
% 0.32/0.69 ( ( re @ ( real_V2046097035970521341omplex @ R @ X ) )
% 0.32/0.69 = ( times_times_real @ R @ ( re @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % scaleR_complex.simps(1)
% 0.32/0.69 thf(fact_9385_minus__complex_Osimps_I1_J,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( re @ ( minus_minus_complex @ X @ Y3 ) )
% 0.32/0.69 = ( minus_minus_real @ ( re @ X ) @ ( re @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_complex.simps(1)
% 0.32/0.69 thf(fact_9386_abs__Re__le__cmod,axiom,
% 0.32/0.69 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % abs_Re_le_cmod
% 0.32/0.69 thf(fact_9387_Re__csqrt,axiom,
% 0.32/0.69 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_csqrt
% 0.32/0.69 thf(fact_9388_cmod__plus__Re__le__0__iff,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 0.32/0.69 = ( ( re @ Z )
% 0.32/0.69 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % cmod_plus_Re_le_0_iff
% 0.32/0.69 thf(fact_9389_cos__n__Re__cis__pow__n,axiom,
% 0.32/0.69 ! [N: nat,A: real] :
% 0.32/0.69 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.32/0.69 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % cos_n_Re_cis_pow_n
% 0.32/0.69 thf(fact_9390_divmod__abs__def,axiom,
% 0.32/0.69 ( code_divmod_abs
% 0.32/0.69 = ( ^ [K2: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divmod_abs_def
% 0.32/0.69 thf(fact_9391_divmod__integer__code,axiom,
% 0.32/0.69 ( code_divmod_integer
% 0.32/0.69 = ( ^ [K2: code_integer,L: code_integer] :
% 0.32/0.69 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.32/0.69 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
% 0.32/0.69 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L )
% 0.32/0.69 @ ( produc6916734918728496179nteger
% 0.32/0.69 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S6 ) ) )
% 0.32/0.69 @ ( code_divmod_abs @ K2 @ L ) ) )
% 0.32/0.69 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.32/0.69 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.32/0.69 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L )
% 0.32/0.69 @ ( produc6916734918728496179nteger
% 0.32/0.69 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S6 ) ) )
% 0.32/0.69 @ ( code_divmod_abs @ K2 @ L ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divmod_integer_code
% 0.32/0.69 thf(fact_9392_csqrt_Ocode,axiom,
% 0.32/0.69 ( csqrt
% 0.32/0.69 = ( ^ [Z2: complex] :
% 0.32/0.69 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.69 @ ( times_times_real
% 0.32/0.69 @ ( if_real
% 0.32/0.69 @ ( ( im @ Z2 )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 @ one_one_real
% 0.32/0.69 @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
% 0.32/0.69 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt.code
% 0.32/0.69 thf(fact_9393_csqrt_Osimps_I2_J,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( im @ ( csqrt @ Z ) )
% 0.32/0.69 = ( times_times_real
% 0.32/0.69 @ ( if_real
% 0.32/0.69 @ ( ( im @ Z )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 @ one_one_real
% 0.32/0.69 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 0.32/0.69 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt.simps(2)
% 0.32/0.69 thf(fact_9394_complex__Im__numeral,axiom,
% 0.32/0.69 ! [V: num] :
% 0.32/0.69 ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 0.32/0.69 = zero_zero_real ) ).
% 0.32/0.69
% 0.32/0.69 % complex_Im_numeral
% 0.32/0.69 thf(fact_9395_Im__i__times,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
% 0.32/0.69 = ( re @ Z ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_i_times
% 0.32/0.69 thf(fact_9396_Im__divide__of__real,axiom,
% 0.32/0.69 ! [Z: complex,R: real] :
% 0.32/0.69 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
% 0.32/0.69 = ( divide_divide_real @ ( im @ Z ) @ R ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_divide_of_real
% 0.32/0.69 thf(fact_9397_Im__sgn,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( im @ ( sgn_sgn_complex @ Z ) )
% 0.32/0.69 = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_sgn
% 0.32/0.69 thf(fact_9398_Re__power__real,axiom,
% 0.32/0.69 ! [X: complex,N: nat] :
% 0.32/0.69 ( ( ( im @ X )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 => ( ( re @ ( power_power_complex @ X @ N ) )
% 0.32/0.69 = ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_power_real
% 0.32/0.69 thf(fact_9399_Re__i__times,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
% 0.32/0.69 = ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_i_times
% 0.32/0.69 thf(fact_9400_Im__divide__numeral,axiom,
% 0.32/0.69 ! [Z: complex,W: num] :
% 0.32/0.69 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.69 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_divide_numeral
% 0.32/0.69 thf(fact_9401_Im__divide__of__nat,axiom,
% 0.32/0.69 ! [Z: complex,N: nat] :
% 0.32/0.69 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 0.32/0.69 = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_divide_of_nat
% 0.32/0.69 thf(fact_9402_csqrt__of__real__nonneg,axiom,
% 0.32/0.69 ! [X: complex] :
% 0.32/0.69 ( ( ( im @ X )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
% 0.32/0.69 => ( ( csqrt @ X )
% 0.32/0.69 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt_of_real_nonneg
% 0.32/0.69 thf(fact_9403_csqrt__minus,axiom,
% 0.32/0.69 ! [X: complex] :
% 0.32/0.69 ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 0.32/0.69 | ( ( ( im @ X )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 0.32/0.69 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 0.32/0.69 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt_minus
% 0.32/0.69 thf(fact_9404_csqrt__of__real__nonpos,axiom,
% 0.32/0.69 ! [X: complex] :
% 0.32/0.69 ( ( ( im @ X )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 0.32/0.69 => ( ( csqrt @ X )
% 0.32/0.69 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt_of_real_nonpos
% 0.32/0.69 thf(fact_9405_imaginary__unit_Osimps_I2_J,axiom,
% 0.32/0.69 ( ( im @ imaginary_unit )
% 0.32/0.69 = one_one_real ) ).
% 0.32/0.69
% 0.32/0.69 % imaginary_unit.simps(2)
% 0.32/0.69 thf(fact_9406_plus__complex_Osimps_I2_J,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( im @ ( plus_plus_complex @ X @ Y3 ) )
% 0.32/0.69 = ( plus_plus_real @ ( im @ X ) @ ( im @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % plus_complex.simps(2)
% 0.32/0.69 thf(fact_9407_scaleR__complex_Osimps_I2_J,axiom,
% 0.32/0.69 ! [R: real,X: complex] :
% 0.32/0.69 ( ( im @ ( real_V2046097035970521341omplex @ R @ X ) )
% 0.32/0.69 = ( times_times_real @ R @ ( im @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % scaleR_complex.simps(2)
% 0.32/0.69 thf(fact_9408_minus__complex_Osimps_I2_J,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( im @ ( minus_minus_complex @ X @ Y3 ) )
% 0.32/0.69 = ( minus_minus_real @ ( im @ X ) @ ( im @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_complex.simps(2)
% 0.32/0.69 thf(fact_9409_abs__Im__le__cmod,axiom,
% 0.32/0.69 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % abs_Im_le_cmod
% 0.32/0.69 thf(fact_9410_times__complex_Osimps_I2_J,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( im @ ( times_times_complex @ X @ Y3 ) )
% 0.32/0.69 = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y3 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % times_complex.simps(2)
% 0.32/0.69 thf(fact_9411_cmod__Re__le__iff,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( ( im @ X )
% 0.32/0.69 = ( im @ Y3 ) )
% 0.32/0.69 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.32/0.69 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % cmod_Re_le_iff
% 0.32/0.69 thf(fact_9412_cmod__Im__le__iff,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( ( re @ X )
% 0.32/0.69 = ( re @ Y3 ) )
% 0.32/0.69 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.32/0.69 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % cmod_Im_le_iff
% 0.32/0.69 thf(fact_9413_times__complex_Osimps_I1_J,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( re @ ( times_times_complex @ X @ Y3 ) )
% 0.32/0.69 = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y3 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % times_complex.simps(1)
% 0.32/0.69 thf(fact_9414_plus__complex_Ocode,axiom,
% 0.32/0.69 ( plus_plus_complex
% 0.32/0.69 = ( ^ [X2: complex,Y: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % plus_complex.code
% 0.32/0.69 thf(fact_9415_scaleR__complex_Ocode,axiom,
% 0.32/0.69 ( real_V2046097035970521341omplex
% 0.32/0.69 = ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X2 ) ) @ ( times_times_real @ R5 @ ( im @ X2 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % scaleR_complex.code
% 0.32/0.69 thf(fact_9416_minus__complex_Ocode,axiom,
% 0.32/0.69 ( minus_minus_complex
% 0.32/0.69 = ( ^ [X2: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( minus_minus_real @ ( im @ X2 ) @ ( im @ Y ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_complex.code
% 0.32/0.69 thf(fact_9417_csqrt__principal,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 0.32/0.69 | ( ( ( re @ ( csqrt @ Z ) )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt_principal
% 0.32/0.69 thf(fact_9418_cmod__le,axiom,
% 0.32/0.69 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % cmod_le
% 0.32/0.69 thf(fact_9419_sin__n__Im__cis__pow__n,axiom,
% 0.32/0.69 ! [N: nat,A: real] :
% 0.32/0.69 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 0.32/0.69 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % sin_n_Im_cis_pow_n
% 0.32/0.69 thf(fact_9420_Re__exp,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( re @ ( exp_complex @ Z ) )
% 0.32/0.69 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_exp
% 0.32/0.69 thf(fact_9421_Im__exp,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( im @ ( exp_complex @ Z ) )
% 0.32/0.69 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_exp
% 0.32/0.69 thf(fact_9422_complex__eq,axiom,
% 0.32/0.69 ! [A: complex] :
% 0.32/0.69 ( A
% 0.32/0.69 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_eq
% 0.32/0.69 thf(fact_9423_times__complex_Ocode,axiom,
% 0.32/0.69 ( times_times_complex
% 0.32/0.69 = ( ^ [X2: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % times_complex.code
% 0.32/0.69 thf(fact_9424_exp__eq__polar,axiom,
% 0.32/0.69 ( exp_complex
% 0.32/0.69 = ( ^ [Z2: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z2 ) ) ) @ ( cis @ ( im @ Z2 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % exp_eq_polar
% 0.32/0.69 thf(fact_9425_cmod__power2,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.69 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % cmod_power2
% 0.32/0.69 thf(fact_9426_Im__power2,axiom,
% 0.32/0.69 ! [X: complex] :
% 0.32/0.69 ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.69 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_power2
% 0.32/0.69 thf(fact_9427_Re__power2,axiom,
% 0.32/0.69 ! [X: complex] :
% 0.32/0.69 ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.69 = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_power2
% 0.32/0.69 thf(fact_9428_complex__eq__0,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( Z = zero_zero_complex )
% 0.32/0.69 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.69 = zero_zero_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_eq_0
% 0.32/0.69 thf(fact_9429_norm__complex__def,axiom,
% 0.32/0.69 ( real_V1022390504157884413omplex
% 0.32/0.69 = ( ^ [Z2: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % norm_complex_def
% 0.32/0.69 thf(fact_9430_inverse__complex_Osimps_I1_J,axiom,
% 0.32/0.69 ! [X: complex] :
% 0.32/0.69 ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % inverse_complex.simps(1)
% 0.32/0.69 thf(fact_9431_complex__neq__0,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( Z != zero_zero_complex )
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_neq_0
% 0.32/0.69 thf(fact_9432_Re__divide,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( re @ ( divide1717551699836669952omplex @ X @ Y3 ) )
% 0.32/0.69 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_divide
% 0.32/0.69 thf(fact_9433_csqrt__unique,axiom,
% 0.32/0.69 ! [W: complex,Z: complex] :
% 0.32/0.69 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.32/0.69 = Z )
% 0.32/0.69 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 0.32/0.69 | ( ( ( re @ W )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 0.32/0.69 => ( ( csqrt @ Z )
% 0.32/0.69 = W ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt_unique
% 0.32/0.69 thf(fact_9434_csqrt__square,axiom,
% 0.32/0.69 ! [B: complex] :
% 0.32/0.69 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 0.32/0.69 | ( ( ( re @ B )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 0.32/0.69 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.69 = B ) ) ).
% 0.32/0.69
% 0.32/0.69 % csqrt_square
% 0.32/0.69 thf(fact_9435_inverse__complex_Osimps_I2_J,axiom,
% 0.32/0.69 ! [X: complex] :
% 0.32/0.69 ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % inverse_complex.simps(2)
% 0.32/0.69 thf(fact_9436_Im__divide,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( im @ ( divide1717551699836669952omplex @ X @ Y3 ) )
% 0.32/0.69 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_divide
% 0.32/0.69 thf(fact_9437_complex__abs__le__norm,axiom,
% 0.32/0.69 ! [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 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_abs_le_norm
% 0.32/0.69 thf(fact_9438_complex__unit__circle,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( Z != zero_zero_complex )
% 0.32/0.69 => ( ( 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 ) ) ) )
% 0.32/0.69 = one_one_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_unit_circle
% 0.32/0.69 thf(fact_9439_inverse__complex_Ocode,axiom,
% 0.32/0.69 ( invers8013647133539491842omplex
% 0.32/0.69 = ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % inverse_complex.code
% 0.32/0.69 thf(fact_9440_Complex__divide,axiom,
% 0.32/0.69 ( divide1717551699836669952omplex
% 0.32/0.69 = ( ^ [X2: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Complex_divide
% 0.32/0.69 thf(fact_9441_Im__Reals__divide,axiom,
% 0.32/0.69 ! [R: complex,Z: complex] :
% 0.32/0.69 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.32/0.69 => ( ( im @ ( divide1717551699836669952omplex @ R @ Z ) )
% 0.32/0.69 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_Reals_divide
% 0.32/0.69 thf(fact_9442_Re__Reals__divide,axiom,
% 0.32/0.69 ! [R: complex,Z: complex] :
% 0.32/0.69 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.32/0.69 => ( ( re @ ( divide1717551699836669952omplex @ R @ Z ) )
% 0.32/0.69 = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_Reals_divide
% 0.32/0.69 thf(fact_9443_complex__diff__cnj,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 0.32/0.69 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_diff_cnj
% 0.32/0.69 thf(fact_9444_complex__cnj__mult,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( cnj @ ( times_times_complex @ X @ Y3 ) )
% 0.32/0.69 = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_cnj_mult
% 0.32/0.69 thf(fact_9445_complex__cnj__divide,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( cnj @ ( divide1717551699836669952omplex @ X @ Y3 ) )
% 0.32/0.69 = ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_cnj_divide
% 0.32/0.69 thf(fact_9446_complex__cnj__add,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( cnj @ ( plus_plus_complex @ X @ Y3 ) )
% 0.32/0.69 = ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_cnj_add
% 0.32/0.69 thf(fact_9447_complex__cnj__numeral,axiom,
% 0.32/0.69 ! [W: num] :
% 0.32/0.69 ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
% 0.32/0.69 = ( numera6690914467698888265omplex @ W ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_cnj_numeral
% 0.32/0.69 thf(fact_9448_complex__cnj__diff,axiom,
% 0.32/0.69 ! [X: complex,Y3: complex] :
% 0.32/0.69 ( ( cnj @ ( minus_minus_complex @ X @ Y3 ) )
% 0.32/0.69 = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_cnj_diff
% 0.32/0.69 thf(fact_9449_complex__cnj__neg__numeral,axiom,
% 0.32/0.69 ! [W: num] :
% 0.32/0.69 ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 0.32/0.69 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_cnj_neg_numeral
% 0.32/0.69 thf(fact_9450_complex__In__mult__cnj__zero,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.32/0.69 = zero_zero_real ) ).
% 0.32/0.69
% 0.32/0.69 % complex_In_mult_cnj_zero
% 0.32/0.69 thf(fact_9451_Re__divide__Reals,axiom,
% 0.32/0.69 ! [R: complex,Z: complex] :
% 0.32/0.69 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.32/0.69 => ( ( re @ ( divide1717551699836669952omplex @ Z @ R ) )
% 0.32/0.69 = ( divide_divide_real @ ( re @ Z ) @ ( re @ R ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_divide_Reals
% 0.32/0.69 thf(fact_9452_real__eq__imaginary__iff,axiom,
% 0.32/0.69 ! [Y3: complex,X: complex] :
% 0.32/0.69 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 0.32/0.69 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 0.32/0.69 => ( ( X
% 0.32/0.69 = ( times_times_complex @ imaginary_unit @ Y3 ) )
% 0.32/0.69 = ( ( X = zero_zero_complex )
% 0.32/0.69 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % real_eq_imaginary_iff
% 0.32/0.69 thf(fact_9453_imaginary__eq__real__iff,axiom,
% 0.32/0.69 ! [Y3: complex,X: complex] :
% 0.32/0.69 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 0.32/0.69 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 0.32/0.69 => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
% 0.32/0.69 = X )
% 0.32/0.69 = ( ( X = zero_zero_complex )
% 0.32/0.69 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % imaginary_eq_real_iff
% 0.32/0.69 thf(fact_9454_Im__divide__Reals,axiom,
% 0.32/0.69 ! [R: complex,Z: complex] :
% 0.32/0.69 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 0.32/0.69 => ( ( im @ ( divide1717551699836669952omplex @ Z @ R ) )
% 0.32/0.69 = ( divide_divide_real @ ( im @ Z ) @ ( re @ R ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_divide_Reals
% 0.32/0.69 thf(fact_9455_Re__complex__div__eq__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.32/0.69 = zero_zero_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_complex_div_eq_0
% 0.32/0.69 thf(fact_9456_Im__complex__div__eq__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.32/0.69 = zero_zero_real )
% 0.32/0.69 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 0.32/0.69 = zero_zero_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_complex_div_eq_0
% 0.32/0.69 thf(fact_9457_complex__mod__sqrt__Re__mult__cnj,axiom,
% 0.32/0.69 ( real_V1022390504157884413omplex
% 0.32/0.69 = ( ^ [Z2: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_mod_sqrt_Re_mult_cnj
% 0.32/0.69 thf(fact_9458_Re__complex__div__lt__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.32/0.69 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_complex_div_lt_0
% 0.32/0.69 thf(fact_9459_Re__complex__div__gt__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.32/0.69 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_complex_div_gt_0
% 0.32/0.69 thf(fact_9460_Re__complex__div__le__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.32/0.69 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_complex_div_le_0
% 0.32/0.69 thf(fact_9461_Re__complex__div__ge__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.32/0.69 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Re_complex_div_ge_0
% 0.32/0.69 thf(fact_9462_Im__complex__div__lt__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.32/0.69 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_complex_div_lt_0
% 0.32/0.69 thf(fact_9463_Im__complex__div__gt__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.32/0.69 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_complex_div_gt_0
% 0.32/0.69 thf(fact_9464_Im__complex__div__le__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 0.32/0.69 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_complex_div_le_0
% 0.32/0.69 thf(fact_9465_Im__complex__div__ge__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.32/0.69 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Im_complex_div_ge_0
% 0.32/0.69 thf(fact_9466_complex__mod__mult__cnj,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 0.32/0.69 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_mod_mult_cnj
% 0.32/0.69 thf(fact_9467_complex__div__gt__0,axiom,
% 0.32/0.69 ! [A: complex,B: complex] :
% 0.32/0.69 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.32/0.69 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 0.32/0.69 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.32/0.69 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_div_gt_0
% 0.32/0.69 thf(fact_9468_complex__norm__square,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.69 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_norm_square
% 0.32/0.69 thf(fact_9469_complex__add__cnj,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 0.32/0.69 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_add_cnj
% 0.32/0.69 thf(fact_9470_complex__div__cnj,axiom,
% 0.32/0.69 ( divide1717551699836669952omplex
% 0.32/0.69 = ( ^ [A3: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_div_cnj
% 0.32/0.69 thf(fact_9471_cnj__add__mult__eq__Re,axiom,
% 0.32/0.69 ! [Z: complex,W: complex] :
% 0.32/0.69 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 0.32/0.69 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % cnj_add_mult_eq_Re
% 0.32/0.69 thf(fact_9472_complex__mult__cnj,axiom,
% 0.32/0.69 ! [Z: complex] :
% 0.32/0.69 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % complex_mult_cnj
% 0.32/0.69 thf(fact_9473_card__Collect__less__nat,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( finite_card_nat
% 0.32/0.69 @ ( collect_nat
% 0.32/0.69 @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 0.32/0.69 = N ) ).
% 0.32/0.69
% 0.32/0.69 % card_Collect_less_nat
% 0.32/0.69 thf(fact_9474_card__atLeastLessThan,axiom,
% 0.32/0.69 ! [L2: nat,U: nat] :
% 0.32/0.69 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
% 0.32/0.69 = ( minus_minus_nat @ U @ L2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_atLeastLessThan
% 0.32/0.69 thf(fact_9475_card__Collect__le__nat,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( finite_card_nat
% 0.32/0.69 @ ( collect_nat
% 0.32/0.69 @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 0.32/0.69 = ( suc @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_Collect_le_nat
% 0.32/0.69 thf(fact_9476_card__atLeastAtMost,axiom,
% 0.32/0.69 ! [L2: nat,U: nat] :
% 0.32/0.69 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 0.32/0.69 = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_atLeastAtMost
% 0.32/0.69 thf(fact_9477_card__atLeastLessThan__int,axiom,
% 0.32/0.69 ! [L2: int,U: int] :
% 0.32/0.69 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
% 0.32/0.69 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_atLeastLessThan_int
% 0.32/0.69 thf(fact_9478_card__atLeastAtMost__int,axiom,
% 0.32/0.69 ! [L2: int,U: int] :
% 0.32/0.69 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 0.32/0.69 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_atLeastAtMost_int
% 0.32/0.69 thf(fact_9479_nat_Odisc__eq__case_I2_J,axiom,
% 0.32/0.69 ! [Nat: nat] :
% 0.32/0.69 ( ( Nat != zero_zero_nat )
% 0.32/0.69 = ( case_nat_o @ $false
% 0.32/0.69 @ ^ [Uu3: nat] : $true
% 0.32/0.69 @ Nat ) ) ).
% 0.32/0.69
% 0.32/0.69 % nat.disc_eq_case(2)
% 0.32/0.69 thf(fact_9480_nat_Odisc__eq__case_I1_J,axiom,
% 0.32/0.69 ! [Nat: nat] :
% 0.32/0.69 ( ( Nat = zero_zero_nat )
% 0.32/0.69 = ( case_nat_o @ $true
% 0.32/0.69 @ ^ [Uu3: nat] : $false
% 0.32/0.69 @ Nat ) ) ).
% 0.32/0.69
% 0.32/0.69 % nat.disc_eq_case(1)
% 0.32/0.69 thf(fact_9481_card__less__Suc2,axiom,
% 0.32/0.69 ! [M5: set_nat,I: nat] :
% 0.32/0.69 ( ~ ( member_nat @ zero_zero_nat @ M5 )
% 0.32/0.69 => ( ( finite_card_nat
% 0.32/0.69 @ ( collect_nat
% 0.32/0.69 @ ^ [K2: nat] :
% 0.32/0.69 ( ( member_nat @ ( suc @ K2 ) @ M5 )
% 0.32/0.69 & ( ord_less_nat @ K2 @ I ) ) ) )
% 0.32/0.69 = ( finite_card_nat
% 0.32/0.69 @ ( collect_nat
% 0.32/0.69 @ ^ [K2: nat] :
% 0.32/0.69 ( ( member_nat @ K2 @ M5 )
% 0.32/0.69 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_less_Suc2
% 0.32/0.69 thf(fact_9482_card__less__Suc,axiom,
% 0.32/0.69 ! [M5: set_nat,I: nat] :
% 0.32/0.69 ( ( member_nat @ zero_zero_nat @ M5 )
% 0.32/0.69 => ( ( suc
% 0.32/0.69 @ ( finite_card_nat
% 0.32/0.69 @ ( collect_nat
% 0.32/0.69 @ ^ [K2: nat] :
% 0.32/0.69 ( ( member_nat @ ( suc @ K2 ) @ M5 )
% 0.32/0.69 & ( ord_less_nat @ K2 @ I ) ) ) ) )
% 0.32/0.69 = ( finite_card_nat
% 0.32/0.69 @ ( collect_nat
% 0.32/0.69 @ ^ [K2: nat] :
% 0.32/0.69 ( ( member_nat @ K2 @ M5 )
% 0.32/0.69 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_less_Suc
% 0.32/0.69 thf(fact_9483_card__less,axiom,
% 0.32/0.69 ! [M5: set_nat,I: nat] :
% 0.32/0.69 ( ( member_nat @ zero_zero_nat @ M5 )
% 0.32/0.69 => ( ( finite_card_nat
% 0.32/0.69 @ ( collect_nat
% 0.32/0.69 @ ^ [K2: nat] :
% 0.32/0.69 ( ( member_nat @ K2 @ M5 )
% 0.32/0.69 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
% 0.32/0.69 != zero_zero_nat ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_less
% 0.32/0.69 thf(fact_9484_subset__card__intvl__is__intvl,axiom,
% 0.32/0.69 ! [A2: set_nat,K: nat] :
% 0.32/0.69 ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 0.32/0.69 => ( A2
% 0.32/0.69 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % subset_card_intvl_is_intvl
% 0.32/0.69 thf(fact_9485_less__eq__nat_Osimps_I2_J,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.32/0.69 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_eq_nat.simps(2)
% 0.32/0.69 thf(fact_9486_max__Suc2,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.32/0.69 = ( case_nat_nat @ ( suc @ N )
% 0.32/0.69 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
% 0.32/0.69 @ M ) ) ).
% 0.32/0.69
% 0.32/0.69 % max_Suc2
% 0.32/0.69 thf(fact_9487_max__Suc1,axiom,
% 0.32/0.69 ! [N: nat,M: nat] :
% 0.32/0.69 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.32/0.69 = ( case_nat_nat @ ( suc @ N )
% 0.32/0.69 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
% 0.32/0.69 @ M ) ) ).
% 0.32/0.69
% 0.32/0.69 % max_Suc1
% 0.32/0.69 thf(fact_9488_subset__eq__atLeast0__lessThan__card,axiom,
% 0.32/0.69 ! [N3: set_nat,N: nat] :
% 0.32/0.69 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.32/0.69 => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % subset_eq_atLeast0_lessThan_card
% 0.32/0.69 thf(fact_9489_card__sum__le__nat__sum,axiom,
% 0.32/0.69 ! [S2: set_nat] :
% 0.32/0.69 ( ord_less_eq_nat
% 0.32/0.69 @ ( groups3542108847815614940at_nat
% 0.32/0.69 @ ^ [X2: nat] : X2
% 0.32/0.69 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 0.32/0.69 @ ( groups3542108847815614940at_nat
% 0.32/0.69 @ ^ [X2: nat] : X2
% 0.32/0.69 @ S2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_sum_le_nat_sum
% 0.32/0.69 thf(fact_9490_card__nth__roots,axiom,
% 0.32/0.69 ! [C: complex,N: nat] :
% 0.32/0.69 ( ( C != zero_zero_complex )
% 0.32/0.69 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.69 => ( ( finite_card_complex
% 0.32/0.69 @ ( collect_complex
% 0.32/0.69 @ ^ [Z2: complex] :
% 0.32/0.69 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.69 = C ) ) )
% 0.32/0.69 = N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_nth_roots
% 0.32/0.69 thf(fact_9491_card__roots__unity__eq,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.69 => ( ( finite_card_complex
% 0.32/0.69 @ ( collect_complex
% 0.32/0.69 @ ^ [Z2: complex] :
% 0.32/0.69 ( ( power_power_complex @ Z2 @ N )
% 0.32/0.69 = one_one_complex ) ) )
% 0.32/0.69 = N ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_roots_unity_eq
% 0.32/0.69 thf(fact_9492_diff__Suc,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.32/0.69 = ( case_nat_nat @ zero_zero_nat
% 0.32/0.69 @ ^ [K2: nat] : K2
% 0.32/0.69 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % diff_Suc
% 0.32/0.69 thf(fact_9493_floor__real__def,axiom,
% 0.32/0.69 ( archim6058952711729229775r_real
% 0.32/0.69 = ( ^ [X2: real] :
% 0.32/0.69 ( the_int
% 0.32/0.69 @ ^ [Z2: int] :
% 0.32/0.69 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X2 )
% 0.32/0.69 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % floor_real_def
% 0.32/0.69 thf(fact_9494_floor__rat__def,axiom,
% 0.32/0.69 ( archim3151403230148437115or_rat
% 0.32/0.69 = ( ^ [X2: rat] :
% 0.32/0.69 ( the_int
% 0.32/0.69 @ ^ [Z2: int] :
% 0.32/0.69 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X2 )
% 0.32/0.69 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % floor_rat_def
% 0.32/0.69 thf(fact_9495_less__eq__rat__def,axiom,
% 0.32/0.69 ( ord_less_eq_rat
% 0.32/0.69 = ( ^ [X2: rat,Y: rat] :
% 0.32/0.69 ( ( ord_less_rat @ X2 @ Y )
% 0.32/0.69 | ( X2 = Y ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_eq_rat_def
% 0.32/0.69 thf(fact_9496_obtain__pos__sum,axiom,
% 0.32/0.69 ! [R: rat] :
% 0.32/0.69 ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.32/0.69 => ~ ! [S3: rat] :
% 0.32/0.69 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 0.32/0.69 => ! [T5: rat] :
% 0.32/0.69 ( ( ord_less_rat @ zero_zero_rat @ T5 )
% 0.32/0.69 => ( R
% 0.32/0.69 != ( plus_plus_rat @ S3 @ T5 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % obtain_pos_sum
% 0.32/0.69 thf(fact_9497_sgn__rat__def,axiom,
% 0.32/0.69 ( sgn_sgn_rat
% 0.32/0.69 = ( ^ [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 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % sgn_rat_def
% 0.32/0.69 thf(fact_9498_abs__rat__def,axiom,
% 0.32/0.69 ( abs_abs_rat
% 0.32/0.69 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % abs_rat_def
% 0.32/0.69 thf(fact_9499_pred__def,axiom,
% 0.32/0.69 ( pred
% 0.32/0.69 = ( case_nat_nat @ zero_zero_nat
% 0.32/0.69 @ ^ [X24: nat] : X24 ) ) ).
% 0.32/0.69
% 0.32/0.69 % pred_def
% 0.32/0.69 thf(fact_9500_rat__inverse__code,axiom,
% 0.32/0.69 ! [P6: rat] :
% 0.32/0.69 ( ( quotient_of @ ( inverse_inverse_rat @ P6 ) )
% 0.32/0.69 = ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [A3: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B4 ) @ ( abs_abs_int @ A3 ) ) )
% 0.32/0.69 @ ( quotient_of @ P6 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % rat_inverse_code
% 0.32/0.69 thf(fact_9501_normalize__negative,axiom,
% 0.32/0.69 ! [Q2: int,P6: int] :
% 0.32/0.69 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 0.32/0.69 => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q2 ) )
% 0.32/0.69 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % normalize_negative
% 0.32/0.69 thf(fact_9502_quotient__of__number_I3_J,axiom,
% 0.32/0.69 ! [K: num] :
% 0.32/0.69 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.32/0.69 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.32/0.69
% 0.32/0.69 % quotient_of_number(3)
% 0.32/0.69 thf(fact_9503_quotient__of__number_I5_J,axiom,
% 0.32/0.69 ! [K: num] :
% 0.32/0.69 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.32/0.69 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.32/0.69
% 0.32/0.69 % quotient_of_number(5)
% 0.32/0.69 thf(fact_9504_divide__rat__def,axiom,
% 0.32/0.69 ( divide_divide_rat
% 0.32/0.69 = ( ^ [Q5: rat,R5: rat] : ( times_times_rat @ Q5 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divide_rat_def
% 0.32/0.69 thf(fact_9505_diff__rat__def,axiom,
% 0.32/0.69 ( minus_minus_rat
% 0.32/0.69 = ( ^ [Q5: rat,R5: rat] : ( plus_plus_rat @ Q5 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % diff_rat_def
% 0.32/0.69 thf(fact_9506_rat__divide__code,axiom,
% 0.32/0.69 ! [P6: rat,Q2: rat] :
% 0.32/0.69 ( ( quotient_of @ ( divide_divide_rat @ P6 @ Q2 ) )
% 0.32/0.69 = ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [A3: int,C3: int] :
% 0.32/0.69 ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B4 ) ) )
% 0.32/0.69 @ ( quotient_of @ Q2 ) )
% 0.32/0.69 @ ( quotient_of @ P6 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % rat_divide_code
% 0.32/0.69 thf(fact_9507_rat__times__code,axiom,
% 0.32/0.69 ! [P6: rat,Q2: rat] :
% 0.32/0.69 ( ( quotient_of @ ( times_times_rat @ P6 @ Q2 ) )
% 0.32/0.69 = ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [A3: int,C3: int] :
% 0.32/0.69 ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B4 ) @ ( times_times_int @ C3 @ D2 ) ) )
% 0.32/0.69 @ ( quotient_of @ Q2 ) )
% 0.32/0.69 @ ( quotient_of @ P6 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % rat_times_code
% 0.32/0.69 thf(fact_9508_quotient__of__div,axiom,
% 0.32/0.69 ! [R: rat,N: int,D: int] :
% 0.32/0.69 ( ( ( quotient_of @ R )
% 0.32/0.69 = ( product_Pair_int_int @ N @ D ) )
% 0.32/0.69 => ( R
% 0.32/0.69 = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % quotient_of_div
% 0.32/0.69 thf(fact_9509_rat__plus__code,axiom,
% 0.32/0.69 ! [P6: rat,Q2: rat] :
% 0.32/0.69 ( ( quotient_of @ ( plus_plus_rat @ P6 @ Q2 ) )
% 0.32/0.69 = ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [A3: int,C3: int] :
% 0.32/0.69 ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
% 0.32/0.69 @ ( quotient_of @ Q2 ) )
% 0.32/0.69 @ ( quotient_of @ P6 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % rat_plus_code
% 0.32/0.69 thf(fact_9510_rat__minus__code,axiom,
% 0.32/0.69 ! [P6: rat,Q2: rat] :
% 0.32/0.69 ( ( quotient_of @ ( minus_minus_rat @ P6 @ Q2 ) )
% 0.32/0.69 = ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [A3: int,C3: int] :
% 0.32/0.69 ( produc4245557441103728435nt_int
% 0.32/0.69 @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
% 0.32/0.69 @ ( quotient_of @ Q2 ) )
% 0.32/0.69 @ ( quotient_of @ P6 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % rat_minus_code
% 0.32/0.69 thf(fact_9511_quotient__of__denom__pos,axiom,
% 0.32/0.69 ! [R: rat,P6: int,Q2: int] :
% 0.32/0.69 ( ( ( quotient_of @ R )
% 0.32/0.69 = ( product_Pair_int_int @ P6 @ Q2 ) )
% 0.32/0.69 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % quotient_of_denom_pos
% 0.32/0.69 thf(fact_9512_normalize__denom__pos,axiom,
% 0.32/0.69 ! [R: product_prod_int_int,P6: int,Q2: int] :
% 0.32/0.69 ( ( ( normalize @ R )
% 0.32/0.69 = ( product_Pair_int_int @ P6 @ Q2 ) )
% 0.32/0.69 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % normalize_denom_pos
% 0.32/0.69 thf(fact_9513_normalize__crossproduct,axiom,
% 0.32/0.69 ! [Q2: int,S: int,P6: int,R: int] :
% 0.32/0.69 ( ( Q2 != zero_zero_int )
% 0.32/0.69 => ( ( S != zero_zero_int )
% 0.32/0.69 => ( ( ( normalize @ ( product_Pair_int_int @ P6 @ Q2 ) )
% 0.32/0.69 = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
% 0.32/0.69 => ( ( times_times_int @ P6 @ S )
% 0.32/0.69 = ( times_times_int @ R @ Q2 ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % normalize_crossproduct
% 0.32/0.69 thf(fact_9514_rat__less__code,axiom,
% 0.32/0.69 ( ord_less_rat
% 0.32/0.69 = ( ^ [P4: rat,Q5: rat] :
% 0.32/0.69 ( produc4947309494688390418_int_o
% 0.32/0.69 @ ^ [A3: int,C3: int] :
% 0.32/0.69 ( produc4947309494688390418_int_o
% 0.32/0.69 @ ^ [B4: int,D2: int] : ( ord_less_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
% 0.32/0.69 @ ( quotient_of @ Q5 ) )
% 0.32/0.69 @ ( quotient_of @ P4 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % rat_less_code
% 0.32/0.69 thf(fact_9515_rat__less__eq__code,axiom,
% 0.32/0.69 ( ord_less_eq_rat
% 0.32/0.69 = ( ^ [P4: rat,Q5: rat] :
% 0.32/0.69 ( produc4947309494688390418_int_o
% 0.32/0.69 @ ^ [A3: int,C3: int] :
% 0.32/0.69 ( produc4947309494688390418_int_o
% 0.32/0.69 @ ^ [B4: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
% 0.32/0.69 @ ( quotient_of @ Q5 ) )
% 0.32/0.69 @ ( quotient_of @ P4 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % rat_less_eq_code
% 0.32/0.69 thf(fact_9516_prod__decode__aux_Oelims,axiom,
% 0.32/0.69 ! [X: nat,Xa: nat,Y3: product_prod_nat_nat] :
% 0.32/0.69 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( ( ord_less_eq_nat @ Xa @ X )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 0.32/0.69 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % prod_decode_aux.elims
% 0.32/0.69 thf(fact_9517_prod__decode__aux_Osimps,axiom,
% 0.32/0.69 ( nat_prod_decode_aux
% 0.32/0.69 = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % prod_decode_aux.simps
% 0.32/0.69 thf(fact_9518_drop__bit__numeral__minus__bit1,axiom,
% 0.32/0.69 ! [L2: num,K: num] :
% 0.32/0.69 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.32/0.69 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_numeral_minus_bit1
% 0.32/0.69 thf(fact_9519_drop__bit__nonnegative__int__iff,axiom,
% 0.32/0.69 ! [N: nat,K: int] :
% 0.32/0.69 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 0.32/0.69 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_nonnegative_int_iff
% 0.32/0.69 thf(fact_9520_drop__bit__negative__int__iff,axiom,
% 0.32/0.69 ! [N: nat,K: int] :
% 0.32/0.69 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.32/0.69 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_negative_int_iff
% 0.32/0.69 thf(fact_9521_drop__bit__minus__one,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 0.32/0.69 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_minus_one
% 0.32/0.69 thf(fact_9522_drop__bit__Suc__minus__bit0,axiom,
% 0.32/0.69 ! [N: nat,K: num] :
% 0.32/0.69 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.32/0.69 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_Suc_minus_bit0
% 0.32/0.69 thf(fact_9523_drop__bit__numeral__minus__bit0,axiom,
% 0.32/0.69 ! [L2: num,K: num] :
% 0.32/0.69 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.32/0.69 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_numeral_minus_bit0
% 0.32/0.69 thf(fact_9524_drop__bit__Suc__minus__bit1,axiom,
% 0.32/0.69 ! [N: nat,K: num] :
% 0.32/0.69 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.32/0.69 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_Suc_minus_bit1
% 0.32/0.69 thf(fact_9525_drop__bit__push__bit__int,axiom,
% 0.32/0.69 ! [M: nat,N: nat,K: int] :
% 0.32/0.69 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.32/0.69 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_push_bit_int
% 0.32/0.69 thf(fact_9526_drop__bit__int__def,axiom,
% 0.32/0.69 ( bit_se8568078237143864401it_int
% 0.32/0.69 = ( ^ [N2: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_int_def
% 0.32/0.69 thf(fact_9527_prod__decode__aux_Opelims,axiom,
% 0.32/0.69 ! [X: nat,Xa: nat,Y3: product_prod_nat_nat] :
% 0.32/0.69 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.32/0.69 => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 0.32/0.69 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 0.32/0.69 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % prod_decode_aux.pelims
% 0.32/0.69 thf(fact_9528_Frct__code__post_I5_J,axiom,
% 0.32/0.69 ! [K: num] :
% 0.32/0.69 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.32/0.69 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Frct_code_post(5)
% 0.32/0.69 thf(fact_9529_Frct__code__post_I6_J,axiom,
% 0.32/0.69 ! [K: num,L2: num] :
% 0.32/0.69 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
% 0.32/0.69 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Frct_code_post(6)
% 0.32/0.69 thf(fact_9530_drop__bit__of__Suc__0,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.32/0.69 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_of_Suc_0
% 0.32/0.69 thf(fact_9531_drop__bit__nat__eq,axiom,
% 0.32/0.69 ! [N: nat,K: int] :
% 0.32/0.69 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 0.32/0.69 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_nat_eq
% 0.32/0.69 thf(fact_9532_drop__bit__nat__def,axiom,
% 0.32/0.69 ( bit_se8570568707652914677it_nat
% 0.32/0.69 = ( ^ [N2: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % drop_bit_nat_def
% 0.32/0.69 thf(fact_9533_Frct__code__post_I4_J,axiom,
% 0.32/0.69 ! [K: num] :
% 0.32/0.69 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.32/0.69 = ( numeral_numeral_rat @ K ) ) ).
% 0.32/0.69
% 0.32/0.69 % Frct_code_post(4)
% 0.32/0.69 thf(fact_9534_Suc__0__div__numeral,axiom,
% 0.32/0.69 ! [K: num] :
% 0.32/0.69 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.69 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Suc_0_div_numeral
% 0.32/0.69 thf(fact_9535_Suc__0__mod__numeral,axiom,
% 0.32/0.69 ! [K: num] :
% 0.32/0.69 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.69 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Suc_0_mod_numeral
% 0.32/0.69 thf(fact_9536_finite__enumerate,axiom,
% 0.32/0.69 ! [S2: set_nat] :
% 0.32/0.69 ( ( finite_finite_nat @ S2 )
% 0.32/0.69 => ? [R2: nat > nat] :
% 0.32/0.69 ( ( strict1292158309912662752at_nat @ R2 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 0.32/0.69 & ! [N7: nat] :
% 0.32/0.69 ( ( ord_less_nat @ N7 @ ( finite_card_nat @ S2 ) )
% 0.32/0.69 => ( member_nat @ ( R2 @ N7 ) @ S2 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % finite_enumerate
% 0.32/0.69 thf(fact_9537_fst__divmod__nat,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.32/0.69 = ( divide_divide_nat @ M @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % fst_divmod_nat
% 0.32/0.69 thf(fact_9538_snd__divmod__nat,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
% 0.32/0.69 = ( modulo_modulo_nat @ M @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % snd_divmod_nat
% 0.32/0.69 thf(fact_9539_fst__divmod__integer,axiom,
% 0.32/0.69 ! [K: code_integer,L2: code_integer] :
% 0.32/0.69 ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L2 ) )
% 0.32/0.69 = ( divide6298287555418463151nteger @ K @ L2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % fst_divmod_integer
% 0.32/0.69 thf(fact_9540_fst__divmod__abs,axiom,
% 0.32/0.69 ! [K: code_integer,L2: code_integer] :
% 0.32/0.69 ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L2 ) )
% 0.32/0.69 = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % fst_divmod_abs
% 0.32/0.69 thf(fact_9541_quotient__of__denom__pos_H,axiom,
% 0.32/0.69 ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % quotient_of_denom_pos'
% 0.32/0.69 thf(fact_9542_minus__one__mod__numeral,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.69 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_one_mod_numeral
% 0.32/0.69 thf(fact_9543_one__mod__minus__numeral,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % one_mod_minus_numeral
% 0.32/0.69 thf(fact_9544_numeral__mod__minus__numeral,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % numeral_mod_minus_numeral
% 0.32/0.69 thf(fact_9545_minus__numeral__mod__numeral,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.69 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_numeral_mod_numeral
% 0.32/0.69 thf(fact_9546_Divides_Oadjust__mod__def,axiom,
% 0.32/0.69 ( adjust_mod
% 0.32/0.69 = ( ^ [L: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R5 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Divides.adjust_mod_def
% 0.32/0.69 thf(fact_9547_bezw_Oelims,axiom,
% 0.32/0.69 ! [X: nat,Xa: nat,Y3: product_prod_int_int] :
% 0.32/0.69 ( ( ( bezw @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( ( Xa = zero_zero_nat )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.32/0.69 & ( ( Xa != zero_zero_nat )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bezw.elims
% 0.32/0.69 thf(fact_9548_bezw_Osimps,axiom,
% 0.32/0.69 ( bezw
% 0.32/0.69 = ( ^ [X2: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bezw.simps
% 0.32/0.69 thf(fact_9549_bezw_Opelims,axiom,
% 0.32/0.69 ! [X: nat,Xa: nat,Y3: product_prod_int_int] :
% 0.32/0.69 ( ( ( bezw @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 0.32/0.69 => ~ ( ( ( ( Xa = zero_zero_nat )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.32/0.69 & ( ( Xa != zero_zero_nat )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( 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 ) ) ) ) ) ) ) )
% 0.32/0.69 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bezw.pelims
% 0.32/0.69 thf(fact_9550_bezw__non__0,axiom,
% 0.32/0.69 ! [Y3: nat,X: nat] :
% 0.32/0.69 ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 0.32/0.69 => ( ( bezw @ X @ Y3 )
% 0.32/0.69 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bezw_non_0
% 0.32/0.69 thf(fact_9551_normalize__def,axiom,
% 0.32/0.69 ( normalize
% 0.32/0.69 = ( ^ [P4: product_prod_int_int] :
% 0.32/0.69 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P4 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) )
% 0.32/0.69 @ ( if_Pro3027730157355071871nt_int
% 0.32/0.69 @ ( ( product_snd_int_int @ P4 )
% 0.32/0.69 = zero_zero_int )
% 0.32/0.69 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.32/0.69 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % normalize_def
% 0.32/0.69 thf(fact_9552_divmod__integer__eq__cases,axiom,
% 0.32/0.69 ( code_divmod_integer
% 0.32/0.69 = ( ^ [K2: code_integer,L: code_integer] :
% 0.32/0.69 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.32/0.69 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 0.32/0.69 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
% 0.32/0.69 @ ( if_Pro6119634080678213985nteger
% 0.32/0.69 @ ( ( sgn_sgn_Code_integer @ K2 )
% 0.32/0.69 = ( sgn_sgn_Code_integer @ L ) )
% 0.32/0.69 @ ( code_divmod_abs @ K2 @ L )
% 0.32/0.69 @ ( produc6916734918728496179nteger
% 0.32/0.69 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S6 ) ) )
% 0.32/0.69 @ ( code_divmod_abs @ K2 @ L ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divmod_integer_eq_cases
% 0.32/0.69 thf(fact_9553_gcd__pos__int,axiom,
% 0.32/0.69 ! [M: int,N: int] :
% 0.32/0.69 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 0.32/0.69 = ( ( M != zero_zero_int )
% 0.32/0.69 | ( N != zero_zero_int ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_pos_int
% 0.32/0.69 thf(fact_9554_gcd__neg__numeral__1__int,axiom,
% 0.32/0.69 ! [N: num,X: int] :
% 0.32/0.69 ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
% 0.32/0.69 = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_neg_numeral_1_int
% 0.32/0.69 thf(fact_9555_gcd__neg__numeral__2__int,axiom,
% 0.32/0.69 ! [X: int,N: num] :
% 0.32/0.69 ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_neg_numeral_2_int
% 0.32/0.69 thf(fact_9556_gcd__ge__0__int,axiom,
% 0.32/0.69 ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y3 ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_ge_0_int
% 0.32/0.69 thf(fact_9557_bezout__int,axiom,
% 0.32/0.69 ! [X: int,Y3: int] :
% 0.32/0.69 ? [U4: int,V2: int] :
% 0.32/0.69 ( ( plus_plus_int @ ( times_times_int @ U4 @ X ) @ ( times_times_int @ V2 @ Y3 ) )
% 0.32/0.69 = ( gcd_gcd_int @ X @ Y3 ) ) ).
% 0.32/0.69
% 0.32/0.69 % bezout_int
% 0.32/0.69 thf(fact_9558_gcd__mult__distrib__int,axiom,
% 0.32/0.69 ! [K: int,M: int,N: int] :
% 0.32/0.69 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 0.32/0.69 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_mult_distrib_int
% 0.32/0.69 thf(fact_9559_gcd__le2__int,axiom,
% 0.32/0.69 ! [B: int,A: int] :
% 0.32/0.69 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.69 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_le2_int
% 0.32/0.69 thf(fact_9560_gcd__le1__int,axiom,
% 0.32/0.69 ! [A: int,B: int] :
% 0.32/0.69 ( ( ord_less_int @ zero_zero_int @ A )
% 0.32/0.69 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_le1_int
% 0.32/0.69 thf(fact_9561_gcd__cases__int,axiom,
% 0.32/0.69 ! [X: int,Y3: int,P: int > $o] :
% 0.32/0.69 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.69 => ( P @ ( gcd_gcd_int @ X @ Y3 ) ) ) )
% 0.32/0.69 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.32/0.69 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.32/0.69 => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.32/0.69 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.32/0.69 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.32/0.69 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y3 ) ) ) )
% 0.32/0.69 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 0.32/0.69 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.32/0.69 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.32/0.69 => ( P @ ( gcd_gcd_int @ X @ Y3 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_cases_int
% 0.32/0.69 thf(fact_9562_gcd__unique__int,axiom,
% 0.32/0.69 ! [D: int,A: int,B: int] :
% 0.32/0.69 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.32/0.69 & ( dvd_dvd_int @ D @ A )
% 0.32/0.69 & ( dvd_dvd_int @ D @ B )
% 0.32/0.69 & ! [E3: int] :
% 0.32/0.69 ( ( ( dvd_dvd_int @ E3 @ A )
% 0.32/0.69 & ( dvd_dvd_int @ E3 @ B ) )
% 0.32/0.69 => ( dvd_dvd_int @ E3 @ D ) ) )
% 0.32/0.69 = ( D
% 0.32/0.69 = ( gcd_gcd_int @ A @ B ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_unique_int
% 0.32/0.69 thf(fact_9563_gcd__non__0__int,axiom,
% 0.32/0.69 ! [Y3: int,X: int] :
% 0.32/0.69 ( ( ord_less_int @ zero_zero_int @ Y3 )
% 0.32/0.69 => ( ( gcd_gcd_int @ X @ Y3 )
% 0.32/0.69 = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X @ Y3 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_non_0_int
% 0.32/0.69 thf(fact_9564_nat__descend__induct,axiom,
% 0.32/0.69 ! [N: nat,P: nat > $o,M: nat] :
% 0.32/0.69 ( ! [K3: nat] :
% 0.32/0.69 ( ( ord_less_nat @ N @ K3 )
% 0.32/0.69 => ( P @ K3 ) )
% 0.32/0.69 => ( ! [K3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ K3 @ N )
% 0.32/0.69 => ( ! [I3: nat] :
% 0.32/0.69 ( ( ord_less_nat @ K3 @ I3 )
% 0.32/0.69 => ( P @ I3 ) )
% 0.32/0.69 => ( P @ K3 ) ) )
% 0.32/0.69 => ( P @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % nat_descend_induct
% 0.32/0.69 thf(fact_9565_card__greaterThanLessThan__int,axiom,
% 0.32/0.69 ! [L2: int,U: int] :
% 0.32/0.69 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
% 0.32/0.69 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_greaterThanLessThan_int
% 0.32/0.69 thf(fact_9566_gcd__1__nat,axiom,
% 0.32/0.69 ! [M: nat] :
% 0.32/0.69 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 0.32/0.69 = one_one_nat ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_1_nat
% 0.32/0.69 thf(fact_9567_gcd__pos__nat,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.32/0.69 = ( ( M != zero_zero_nat )
% 0.32/0.69 | ( N != zero_zero_nat ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_pos_nat
% 0.32/0.69 thf(fact_9568_gcd__mult__distrib__nat,axiom,
% 0.32/0.69 ! [K: nat,M: nat,N: nat] :
% 0.32/0.69 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 0.32/0.69 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_mult_distrib_nat
% 0.32/0.69 thf(fact_9569_gcd__le1__nat,axiom,
% 0.32/0.69 ! [A: nat,B: nat] :
% 0.32/0.69 ( ( A != zero_zero_nat )
% 0.32/0.69 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_le1_nat
% 0.32/0.69 thf(fact_9570_gcd__le2__nat,axiom,
% 0.32/0.69 ! [B: nat,A: nat] :
% 0.32/0.69 ( ( B != zero_zero_nat )
% 0.32/0.69 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_le2_nat
% 0.32/0.69 thf(fact_9571_gcd__diff2__nat,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.69 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.32/0.69 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_diff2_nat
% 0.32/0.69 thf(fact_9572_gcd__diff1__nat,axiom,
% 0.32/0.69 ! [N: nat,M: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ N @ M )
% 0.32/0.69 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.32/0.69 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % gcd_diff1_nat
% 0.32/0.69 thf(fact_9573_bezout__nat,axiom,
% 0.32/0.69 ! [A: nat,B: nat] :
% 0.32/0.69 ( ( A != zero_zero_nat )
% 0.32/0.69 => ? [X3: nat,Y4: nat] :
% 0.32/0.69 ( ( times_times_nat @ A @ X3 )
% 0.32/0.69 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bezout_nat
% 0.32/0.69 thf(fact_9574_bezout__gcd__nat_H,axiom,
% 0.32/0.69 ! [B: nat,A: nat] :
% 0.32/0.69 ? [X3: nat,Y4: nat] :
% 0.32/0.69 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X3 ) )
% 0.32/0.69 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
% 0.32/0.69 = ( gcd_gcd_nat @ A @ B ) ) )
% 0.32/0.69 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X3 ) )
% 0.32/0.69 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 0.32/0.69 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bezout_gcd_nat'
% 0.32/0.69 thf(fact_9575_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.32/0.69 ! [L2: int,U: int] :
% 0.32/0.69 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 0.32/0.69 = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% 0.32/0.69
% 0.32/0.69 % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.32/0.69 thf(fact_9576_bezw__aux,axiom,
% 0.32/0.69 ! [X: nat,Y3: nat] :
% 0.32/0.69 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y3 ) )
% 0.32/0.69 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % bezw_aux
% 0.32/0.69 thf(fact_9577_Code__Numeral_Onegative__def,axiom,
% 0.32/0.69 ( code_negative
% 0.32/0.69 = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% 0.32/0.69
% 0.32/0.69 % Code_Numeral.negative_def
% 0.32/0.69 thf(fact_9578_Code__Target__Int_Onegative__def,axiom,
% 0.32/0.69 ( code_Target_negative
% 0.32/0.69 = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 0.32/0.69
% 0.32/0.69 % Code_Target_Int.negative_def
% 0.32/0.69 thf(fact_9579_xor__minus__numerals_I1_J,axiom,
% 0.32/0.69 ! [N: num,K: int] :
% 0.32/0.69 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 0.32/0.69 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_minus_numerals(1)
% 0.32/0.69 thf(fact_9580_card__greaterThanLessThan,axiom,
% 0.32/0.69 ! [L2: nat,U: nat] :
% 0.32/0.69 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
% 0.32/0.69 = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_greaterThanLessThan
% 0.32/0.69 thf(fact_9581_xor__minus__numerals_I2_J,axiom,
% 0.32/0.69 ! [K: int,N: num] :
% 0.32/0.69 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_minus_numerals(2)
% 0.32/0.69 thf(fact_9582_tanh__real__bounds,axiom,
% 0.32/0.69 ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 0.32/0.69
% 0.32/0.69 % tanh_real_bounds
% 0.32/0.69 thf(fact_9583_sub__BitM__One__eq,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 0.32/0.69 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % sub_BitM_One_eq
% 0.32/0.69 thf(fact_9584_nat__of__integer__non__positive,axiom,
% 0.32/0.69 ! [K: code_integer] :
% 0.32/0.69 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.32/0.69 => ( ( code_nat_of_integer @ K )
% 0.32/0.69 = zero_zero_nat ) ) ).
% 0.32/0.69
% 0.32/0.69 % nat_of_integer_non_positive
% 0.32/0.69 thf(fact_9585_Suc__funpow,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( compow_nat_nat @ N @ suc )
% 0.32/0.69 = ( plus_plus_nat @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % Suc_funpow
% 0.32/0.69 thf(fact_9586_nat__of__integer__code__post_I3_J,axiom,
% 0.32/0.69 ! [K: num] :
% 0.32/0.69 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.32/0.69 = ( numeral_numeral_nat @ K ) ) ).
% 0.32/0.69
% 0.32/0.69 % nat_of_integer_code_post(3)
% 0.32/0.69 thf(fact_9587_nat__of__integer__code__post_I2_J,axiom,
% 0.32/0.69 ( ( code_nat_of_integer @ one_one_Code_integer )
% 0.32/0.69 = one_one_nat ) ).
% 0.32/0.69
% 0.32/0.69 % nat_of_integer_code_post(2)
% 0.32/0.69 thf(fact_9588_nat__of__integer__code,axiom,
% 0.32/0.69 ( code_nat_of_integer
% 0.32/0.69 = ( ^ [K2: code_integer] :
% 0.32/0.69 ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.32/0.69 @ ( produc1555791787009142072er_nat
% 0.32/0.69 @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
% 0.32/0.69 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % nat_of_integer_code
% 0.32/0.69 thf(fact_9589_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.32/0.69 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.32/0.69 @ ^ [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X2 )
% 0.32/0.69 @ ^ [X2: nat,Y: nat] : ( ord_less_nat @ Y @ X2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % max_nat.semilattice_neutr_order_axioms
% 0.32/0.69 thf(fact_9590_int__of__integer__code,axiom,
% 0.32/0.69 ( code_int_of_integer
% 0.32/0.69 = ( ^ [K2: code_integer] :
% 0.32/0.69 ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 0.32/0.69 @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.32/0.69 @ ( produc1553301316500091796er_int
% 0.32/0.69 @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
% 0.32/0.69 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % int_of_integer_code
% 0.32/0.69 thf(fact_9591_int__of__integer__numeral,axiom,
% 0.32/0.69 ! [K: num] :
% 0.32/0.69 ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 0.32/0.69 = ( numeral_numeral_int @ K ) ) ).
% 0.32/0.69
% 0.32/0.69 % int_of_integer_numeral
% 0.32/0.69 thf(fact_9592_plus__integer_Orep__eq,axiom,
% 0.32/0.69 ! [X: code_integer,Xa: code_integer] :
% 0.32/0.69 ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
% 0.32/0.69 = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % plus_integer.rep_eq
% 0.32/0.69 thf(fact_9593_times__integer_Orep__eq,axiom,
% 0.32/0.69 ! [X: code_integer,Xa: code_integer] :
% 0.32/0.69 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
% 0.32/0.69 = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % times_integer.rep_eq
% 0.32/0.69 thf(fact_9594_minus__integer_Orep__eq,axiom,
% 0.32/0.69 ! [X: code_integer,Xa: code_integer] :
% 0.32/0.69 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa ) )
% 0.32/0.69 = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_integer.rep_eq
% 0.32/0.69 thf(fact_9595_divide__integer_Orep__eq,axiom,
% 0.32/0.69 ! [X: code_integer,Xa: code_integer] :
% 0.32/0.69 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
% 0.32/0.69 = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % divide_integer.rep_eq
% 0.32/0.69 thf(fact_9596_less__integer_Orep__eq,axiom,
% 0.32/0.69 ( ord_le6747313008572928689nteger
% 0.32/0.69 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_integer.rep_eq
% 0.32/0.69 thf(fact_9597_integer__less__iff,axiom,
% 0.32/0.69 ( ord_le6747313008572928689nteger
% 0.32/0.69 = ( ^ [K2: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % integer_less_iff
% 0.32/0.69 thf(fact_9598_integer__less__eq__iff,axiom,
% 0.32/0.69 ( ord_le3102999989581377725nteger
% 0.32/0.69 = ( ^ [K2: code_integer,L: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % integer_less_eq_iff
% 0.32/0.69 thf(fact_9599_less__eq__integer_Orep__eq,axiom,
% 0.32/0.69 ( ord_le3102999989581377725nteger
% 0.32/0.69 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_eq_integer.rep_eq
% 0.32/0.69 thf(fact_9600_times__int_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.32/0.69 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.32/0.69 = ( abs_Integ
% 0.32/0.69 @ ( produc27273713700761075at_nat
% 0.32/0.69 @ ^ [X2: nat,Y: nat] :
% 0.32/0.69 ( produc2626176000494625587at_nat
% 0.32/0.69 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) )
% 0.32/0.69 @ Xa
% 0.32/0.69 @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % times_int.abs_eq
% 0.32/0.69 thf(fact_9601_nat_Oabs__eq,axiom,
% 0.32/0.69 ! [X: product_prod_nat_nat] :
% 0.32/0.69 ( ( nat2 @ ( abs_Integ @ X ) )
% 0.32/0.69 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % nat.abs_eq
% 0.32/0.69 thf(fact_9602_one__int__def,axiom,
% 0.32/0.69 ( one_one_int
% 0.32/0.69 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % one_int_def
% 0.32/0.69 thf(fact_9603_less__int_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.32/0.69 ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.32/0.69 = ( produc8739625826339149834_nat_o
% 0.32/0.69 @ ^ [X2: nat,Y: nat] :
% 0.32/0.69 ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) )
% 0.32/0.69 @ Xa
% 0.32/0.69 @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_int.abs_eq
% 0.32/0.69 thf(fact_9604_less__eq__int_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.32/0.69 ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.32/0.69 = ( produc8739625826339149834_nat_o
% 0.32/0.69 @ ^ [X2: nat,Y: nat] :
% 0.32/0.69 ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) )
% 0.32/0.69 @ Xa
% 0.32/0.69 @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_eq_int.abs_eq
% 0.32/0.69 thf(fact_9605_plus__int_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.32/0.69 ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.32/0.69 = ( abs_Integ
% 0.32/0.69 @ ( produc27273713700761075at_nat
% 0.32/0.69 @ ^ [X2: nat,Y: nat] :
% 0.32/0.69 ( produc2626176000494625587at_nat
% 0.32/0.69 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 0.32/0.69 @ Xa
% 0.32/0.69 @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % plus_int.abs_eq
% 0.32/0.69 thf(fact_9606_minus__int_Oabs__eq,axiom,
% 0.32/0.69 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 0.32/0.69 ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.32/0.69 = ( abs_Integ
% 0.32/0.69 @ ( produc27273713700761075at_nat
% 0.32/0.69 @ ^ [X2: nat,Y: nat] :
% 0.32/0.69 ( produc2626176000494625587at_nat
% 0.32/0.69 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) )
% 0.32/0.69 @ Xa
% 0.32/0.69 @ X ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % minus_int.abs_eq
% 0.32/0.69 thf(fact_9607_Gcd__remove0__nat,axiom,
% 0.32/0.69 ! [M5: set_nat] :
% 0.32/0.69 ( ( finite_finite_nat @ M5 )
% 0.32/0.69 => ( ( gcd_Gcd_nat @ M5 )
% 0.32/0.69 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Gcd_remove0_nat
% 0.32/0.69 thf(fact_9608_Gcd__nat__eq__one,axiom,
% 0.32/0.69 ! [N3: set_nat] :
% 0.32/0.69 ( ( member_nat @ one_one_nat @ N3 )
% 0.32/0.69 => ( ( gcd_Gcd_nat @ N3 )
% 0.32/0.69 = one_one_nat ) ) ).
% 0.32/0.69
% 0.32/0.69 % Gcd_nat_eq_one
% 0.32/0.69 thf(fact_9609_less__eq__int_Orep__eq,axiom,
% 0.32/0.69 ( ord_less_eq_int
% 0.32/0.69 = ( ^ [X2: int,Xa4: int] :
% 0.32/0.69 ( produc8739625826339149834_nat_o
% 0.32/0.69 @ ^ [Y: nat,Z2: nat] :
% 0.32/0.69 ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U3 @ Z2 ) ) )
% 0.32/0.69 @ ( rep_Integ @ X2 )
% 0.32/0.69 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_eq_int.rep_eq
% 0.32/0.69 thf(fact_9610_less__int_Orep__eq,axiom,
% 0.32/0.69 ( ord_less_int
% 0.32/0.69 = ( ^ [X2: int,Xa4: int] :
% 0.32/0.69 ( produc8739625826339149834_nat_o
% 0.32/0.69 @ ^ [Y: nat,Z2: nat] :
% 0.32/0.69 ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U3 @ Z2 ) ) )
% 0.32/0.69 @ ( rep_Integ @ X2 )
% 0.32/0.69 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % less_int.rep_eq
% 0.32/0.69 thf(fact_9611_prod__encode__def,axiom,
% 0.32/0.69 ( nat_prod_encode
% 0.32/0.69 = ( produc6842872674320459806at_nat
% 0.32/0.69 @ ^ [M3: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N2 ) ) @ M3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % prod_encode_def
% 0.32/0.69 thf(fact_9612_Gcd__int__greater__eq__0,axiom,
% 0.32/0.69 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 0.32/0.69
% 0.32/0.69 % Gcd_int_greater_eq_0
% 0.32/0.69 thf(fact_9613_le__prod__encode__1,axiom,
% 0.32/0.69 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % le_prod_encode_1
% 0.32/0.69 thf(fact_9614_le__prod__encode__2,axiom,
% 0.32/0.69 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % le_prod_encode_2
% 0.32/0.69 thf(fact_9615_nat_Orep__eq,axiom,
% 0.32/0.69 ( nat2
% 0.32/0.69 = ( ^ [X2: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X2 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % nat.rep_eq
% 0.32/0.69 thf(fact_9616_prod__encode__prod__decode__aux,axiom,
% 0.32/0.69 ! [K: nat,M: nat] :
% 0.32/0.69 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.32/0.69 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.32/0.69
% 0.32/0.69 % prod_encode_prod_decode_aux
% 0.32/0.69 thf(fact_9617_num__of__nat_Osimps_I2_J,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.69 => ( ( num_of_nat @ ( suc @ N ) )
% 0.32/0.69 = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.32/0.69 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.69 => ( ( num_of_nat @ ( suc @ N ) )
% 0.32/0.69 = one ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % num_of_nat.simps(2)
% 0.32/0.69 thf(fact_9618_num__of__nat__numeral__eq,axiom,
% 0.32/0.69 ! [Q2: num] :
% 0.32/0.69 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 0.32/0.69 = Q2 ) ).
% 0.32/0.69
% 0.32/0.69 % num_of_nat_numeral_eq
% 0.32/0.69 thf(fact_9619_num__of__nat_Osimps_I1_J,axiom,
% 0.32/0.69 ( ( num_of_nat @ zero_zero_nat )
% 0.32/0.69 = one ) ).
% 0.32/0.69
% 0.32/0.69 % num_of_nat.simps(1)
% 0.32/0.69 thf(fact_9620_numeral__num__of__nat,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.69 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.32/0.69 = N ) ) ).
% 0.32/0.69
% 0.32/0.69 % numeral_num_of_nat
% 0.32/0.69 thf(fact_9621_num__of__nat__One,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.32/0.69 => ( ( num_of_nat @ N )
% 0.32/0.69 = one ) ) ).
% 0.32/0.69
% 0.32/0.69 % num_of_nat_One
% 0.32/0.69 thf(fact_9622_num__of__nat__double,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.69 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.32/0.69 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % num_of_nat_double
% 0.32/0.69 thf(fact_9623_num__of__nat__plus__distrib,axiom,
% 0.32/0.69 ! [M: nat,N: nat] :
% 0.32/0.69 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.32/0.69 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.69 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.32/0.69 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % num_of_nat_plus_distrib
% 0.32/0.69 thf(fact_9624_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.32/0.69 ! [N: nat,J: nat,I: nat] :
% 0.32/0.69 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.32/0.69 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 0.32/0.69 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % nth_sorted_list_of_set_greaterThanLessThan
% 0.32/0.69 thf(fact_9625_pow_Osimps_I3_J,axiom,
% 0.32/0.69 ! [X: num,Y3: num] :
% 0.32/0.69 ( ( pow @ X @ ( bit1 @ Y3 ) )
% 0.32/0.69 = ( times_times_num @ ( sqr @ ( pow @ X @ Y3 ) ) @ X ) ) ).
% 0.32/0.69
% 0.32/0.69 % pow.simps(3)
% 0.32/0.69 thf(fact_9626_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.69 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 0.32/0.69 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.32/0.69 ( X
% 0.32/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.69 => ( Xa = one_one_nat ) )
% 0.32/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.69 => ( ( Deg2 = Xa )
% 0.32/0.69 & ! [X3: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 & ( case_o184042715313410164at_nat
% 0.32/0.69 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 @ ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 & ! [I4: nat] :
% 0.32/0.69 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.32/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.32/0.69 & ( ( Mi3 = Ma3 )
% 0.32/0.69 => ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 & ( ( Mi3 != Ma3 )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.32/0.69 & ! [X2: nat] :
% 0.32/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.32/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.32/0.69 @ Mima ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.valid'.elims(3)
% 0.32/0.69 thf(fact_9627_sqr_Osimps_I2_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( sqr @ ( bit0 @ N ) )
% 0.32/0.69 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % sqr.simps(2)
% 0.32/0.69 thf(fact_9628_sqr_Osimps_I1_J,axiom,
% 0.32/0.69 ( ( sqr @ one )
% 0.32/0.69 = one ) ).
% 0.32/0.69
% 0.32/0.69 % sqr.simps(1)
% 0.32/0.69 thf(fact_9629_sqr__conv__mult,axiom,
% 0.32/0.69 ( sqr
% 0.32/0.69 = ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % sqr_conv_mult
% 0.32/0.69 thf(fact_9630_pow_Osimps_I2_J,axiom,
% 0.32/0.69 ! [X: num,Y3: num] :
% 0.32/0.69 ( ( pow @ X @ ( bit0 @ Y3 ) )
% 0.32/0.69 = ( sqr @ ( pow @ X @ Y3 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % pow.simps(2)
% 0.32/0.69 thf(fact_9631_sqr_Osimps_I3_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( sqr @ ( bit1 @ N ) )
% 0.32/0.69 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % sqr.simps(3)
% 0.32/0.69 thf(fact_9632_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.32/0.69 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
% 0.32/0.69 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
% 0.32/0.69 = ( ( Deg = Deg3 )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.69 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.32/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 & ( case_o184042715313410164at_nat
% 0.32/0.69 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 @ ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.32/0.69 & ! [I4: nat] :
% 0.32/0.69 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
% 0.32/0.69 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.32/0.69 & ( ( Mi3 = Ma3 )
% 0.32/0.69 => ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 & ( ( Mi3 != Ma3 )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.32/0.69 & ! [X2: nat] :
% 0.32/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X2 )
% 0.32/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.32/0.69 @ Mima2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.valid'.simps(2)
% 0.32/0.69 thf(fact_9633_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.32/0.69 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.32/0.69 ( X
% 0.32/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( Xa != one_one_nat ) ) )
% 0.32/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 = ( ~ ( ( Deg2 = Xa )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 & ( case_o184042715313410164at_nat
% 0.32/0.69 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 @ ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 & ! [I4: nat] :
% 0.32/0.69 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.32/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.32/0.69 & ( ( Mi3 = Ma3 )
% 0.32/0.69 => ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 & ( ( Mi3 != Ma3 )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.32/0.69 & ! [X2: nat] :
% 0.32/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.32/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.32/0.69 @ Mima ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.valid'.elims(1)
% 0.32/0.69 thf(fact_9634_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.69 ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.32/0.69 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.32/0.69 ( X
% 0.32/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.69 => ( Xa != one_one_nat ) )
% 0.32/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.69 => ~ ( ( Deg2 = Xa )
% 0.32/0.69 & ! [X5: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 & ( case_o184042715313410164at_nat
% 0.32/0.69 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 @ ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 & ! [I4: nat] :
% 0.32/0.69 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.32/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.32/0.69 & ( ( Mi3 = Ma3 )
% 0.32/0.69 => ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 & ( ( Mi3 != Ma3 )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.32/0.69 & ! [X2: nat] :
% 0.32/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.32/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.32/0.69 @ Mima ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.valid'.elims(2)
% 0.32/0.69 thf(fact_9635_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 0.32/0.69 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.69 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( Xa = one_one_nat ) )
% 0.32/0.69 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 0.32/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( ( Deg2 = Xa )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 & ( case_o184042715313410164at_nat
% 0.32/0.69 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 @ ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 & ! [I4: nat] :
% 0.32/0.69 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.32/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.32/0.69 & ( ( Mi3 = Ma3 )
% 0.32/0.69 => ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 & ( ( Mi3 != Ma3 )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.32/0.69 & ! [X2: nat] :
% 0.32/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.32/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.32/0.69 @ Mima ) ) )
% 0.32/0.69 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.valid'.pelims(1)
% 0.32/0.69 thf(fact_9636_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.69 ( ( vEBT_VEBT_valid @ X @ Xa )
% 0.32/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.69 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 0.32/0.69 => ( Xa != one_one_nat ) ) )
% 0.32/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.32/0.69 => ~ ( ( Deg2 = Xa )
% 0.32/0.69 & ! [X5: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 & ( case_o184042715313410164at_nat
% 0.32/0.69 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 @ ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 & ! [I4: nat] :
% 0.32/0.69 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.32/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.32/0.69 & ( ( Mi3 = Ma3 )
% 0.32/0.69 => ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 & ( ( Mi3 != Ma3 )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.32/0.69 & ! [X2: nat] :
% 0.32/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.32/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.32/0.69 @ Mima ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.valid'.pelims(2)
% 0.32/0.69 thf(fact_9637_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.32/0.69 ! [X: vEBT_VEBT,Xa: nat] :
% 0.32/0.69 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 0.32/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 0.32/0.69 => ( ! [Uu2: $o,Uv2: $o] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.32/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 0.32/0.69 => ( Xa = one_one_nat ) ) )
% 0.32/0.69 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.32/0.69 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 0.32/0.69 => ( ( Deg2 = Xa )
% 0.32/0.69 & ! [X3: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.32/0.69 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.32/0.69 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 & ( case_o184042715313410164at_nat
% 0.32/0.69 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 0.32/0.69 & ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 @ ( produc6081775807080527818_nat_o
% 0.32/0.69 @ ^ [Mi3: nat,Ma3: nat] :
% 0.32/0.69 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.32/0.69 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 & ! [I4: nat] :
% 0.32/0.69 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.69 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 0.32/0.69 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.32/0.69 & ( ( Mi3 = Ma3 )
% 0.32/0.69 => ! [X2: vEBT_VEBT] :
% 0.32/0.69 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.32/0.69 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 0.32/0.69 & ( ( Mi3 != Ma3 )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.32/0.69 & ! [X2: nat] :
% 0.32/0.69 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.32/0.69 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 0.32/0.69 => ( ( ord_less_nat @ Mi3 @ X2 )
% 0.32/0.69 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 0.32/0.69 @ Mima ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % VEBT_internal.valid'.pelims(3)
% 0.32/0.69 thf(fact_9638_Sup__int__def,axiom,
% 0.32/0.69 ( complete_Sup_Sup_int
% 0.32/0.69 = ( ^ [X6: set_int] :
% 0.32/0.69 ( the_int
% 0.32/0.69 @ ^ [X2: int] :
% 0.32/0.69 ( ( member_int @ X2 @ X6 )
% 0.32/0.69 & ! [Y: int] :
% 0.32/0.69 ( ( member_int @ Y @ X6 )
% 0.32/0.69 => ( ord_less_eq_int @ Y @ X2 ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Sup_int_def
% 0.32/0.69 thf(fact_9639_take__bit__numeral__minus__numeral__int,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = ( case_option_int_num @ zero_zero_int
% 0.32/0.69 @ ^ [Q5: 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 @ Q5 ) ) )
% 0.32/0.69 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_numeral_minus_numeral_int
% 0.32/0.69 thf(fact_9640_and__minus__numerals_I7_J,axiom,
% 0.32/0.69 ! [N: num,M: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.32/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_minus_numerals(7)
% 0.32/0.69 thf(fact_9641_take__bit__num__simps_I1_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.32/0.69 = none_num ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_num_simps(1)
% 0.32/0.69 thf(fact_9642_take__bit__num__simps_I2_J,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 0.32/0.69 = ( some_num @ one ) ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_num_simps(2)
% 0.32/0.69 thf(fact_9643_take__bit__num__simps_I5_J,axiom,
% 0.32/0.69 ! [R: num] :
% 0.32/0.69 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
% 0.32/0.69 = ( some_num @ one ) ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_num_simps(5)
% 0.32/0.69 thf(fact_9644_take__bit__num__simps_I3_J,axiom,
% 0.32/0.69 ! [N: nat,M: num] :
% 0.32/0.69 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 0.32/0.69 = ( case_o6005452278849405969um_num @ none_num
% 0.32/0.69 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.32/0.69 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_num_simps(3)
% 0.32/0.69 thf(fact_9645_take__bit__num__simps_I4_J,axiom,
% 0.32/0.69 ! [N: nat,M: num] :
% 0.32/0.69 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 0.32/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_num_simps(4)
% 0.32/0.69 thf(fact_9646_take__bit__num__simps_I6_J,axiom,
% 0.32/0.69 ! [R: num,M: num] :
% 0.32/0.69 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
% 0.32/0.69 = ( case_o6005452278849405969um_num @ none_num
% 0.32/0.69 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.32/0.69 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_num_simps(6)
% 0.32/0.69 thf(fact_9647_take__bit__num__simps_I7_J,axiom,
% 0.32/0.69 ! [R: num,M: num] :
% 0.32/0.69 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
% 0.32/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_num_simps(7)
% 0.32/0.69 thf(fact_9648_and__minus__numerals_I4_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 0.32/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_minus_numerals(4)
% 0.32/0.69 thf(fact_9649_and__minus__numerals_I8_J,axiom,
% 0.32/0.69 ! [N: num,M: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 0.32/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_minus_numerals(8)
% 0.32/0.69 thf(fact_9650_and__minus__numerals_I3_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 0.32/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_minus_numerals(3)
% 0.32/0.69 thf(fact_9651_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.32/0.69 ! [N: nat,M: num] :
% 0.32/0.69 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 0.32/0.69 = ( case_nat_option_num @ none_num
% 0.32/0.69 @ ^ [N2: nat] :
% 0.32/0.69 ( case_o6005452278849405969um_num @ none_num
% 0.32/0.69 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.32/0.69 @ ( bit_take_bit_num @ N2 @ M ) )
% 0.32/0.69 @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % Code_Abstract_Nat.take_bit_num_code(2)
% 0.32/0.69 thf(fact_9652_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.32/0.69 ! [N: nat] :
% 0.32/0.69 ( ( bit_take_bit_num @ N @ one )
% 0.32/0.69 = ( case_nat_option_num @ none_num
% 0.32/0.69 @ ^ [N2: nat] : ( some_num @ one )
% 0.32/0.69 @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % Code_Abstract_Nat.take_bit_num_code(1)
% 0.32/0.69 thf(fact_9653_and__not__num_Osimps_I1_J,axiom,
% 0.32/0.69 ( ( bit_and_not_num @ one @ one )
% 0.32/0.69 = none_num ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(1)
% 0.32/0.69 thf(fact_9654_and__not__num_Osimps_I4_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 0.32/0.69 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(4)
% 0.32/0.69 thf(fact_9655_and__not__num_Osimps_I2_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 0.32/0.69 = ( some_num @ one ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(2)
% 0.32/0.69 thf(fact_9656_GreatestI__ex__nat,axiom,
% 0.32/0.69 ! [P: nat > $o,B: nat] :
% 0.32/0.69 ( ? [X_12: nat] : ( P @ X_12 )
% 0.32/0.69 => ( ! [Y4: nat] :
% 0.32/0.69 ( ( P @ Y4 )
% 0.32/0.69 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.32/0.69 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % GreatestI_ex_nat
% 0.32/0.69 thf(fact_9657_Greatest__le__nat,axiom,
% 0.32/0.69 ! [P: nat > $o,K: nat,B: nat] :
% 0.32/0.69 ( ( P @ K )
% 0.32/0.69 => ( ! [Y4: nat] :
% 0.32/0.69 ( ( P @ Y4 )
% 0.32/0.69 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.32/0.69 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Greatest_le_nat
% 0.32/0.69 thf(fact_9658_GreatestI__nat,axiom,
% 0.32/0.69 ! [P: nat > $o,K: nat,B: nat] :
% 0.32/0.69 ( ( P @ K )
% 0.32/0.69 => ( ! [Y4: nat] :
% 0.32/0.69 ( ( P @ Y4 )
% 0.32/0.69 => ( ord_less_eq_nat @ Y4 @ B ) )
% 0.32/0.69 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % GreatestI_nat
% 0.32/0.69 thf(fact_9659_and__not__num_Osimps_I3_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 0.32/0.69 = none_num ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(3)
% 0.32/0.69 thf(fact_9660_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.32/0.69 ! [N: nat,M: num] :
% 0.32/0.69 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 0.32/0.69 = ( case_nat_option_num @ none_num
% 0.32/0.69 @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
% 0.32/0.69 @ N ) ) ).
% 0.32/0.69
% 0.32/0.69 % Code_Abstract_Nat.take_bit_num_code(3)
% 0.32/0.69 thf(fact_9661_and__not__num_Osimps_I7_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 0.32/0.69 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(7)
% 0.32/0.69 thf(fact_9662_and__not__num__eq__Some__iff,axiom,
% 0.32/0.69 ! [M: num,N: num,Q2: num] :
% 0.32/0.69 ( ( ( bit_and_not_num @ M @ N )
% 0.32/0.69 = ( some_num @ Q2 ) )
% 0.32/0.69 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num_eq_Some_iff
% 0.32/0.69 thf(fact_9663_and__not__num_Osimps_I8_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.69 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.32/0.69 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.32/0.69 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(8)
% 0.32/0.69 thf(fact_9664_and__not__num__eq__None__iff,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( ( bit_and_not_num @ M @ N )
% 0.32/0.69 = none_num )
% 0.32/0.69 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = zero_zero_int ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num_eq_None_iff
% 0.32/0.69 thf(fact_9665_int__numeral__not__and__num,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % int_numeral_not_and_num
% 0.32/0.69 thf(fact_9666_int__numeral__and__not__num,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.69 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % int_numeral_and_not_num
% 0.32/0.69 thf(fact_9667_take__bit__num__def,axiom,
% 0.32/0.69 ( bit_take_bit_num
% 0.32/0.69 = ( ^ [N2: nat,M3: num] :
% 0.32/0.69 ( if_option_num
% 0.32/0.69 @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) )
% 0.32/0.69 = zero_zero_nat )
% 0.32/0.69 @ none_num
% 0.32/0.69 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % take_bit_num_def
% 0.32/0.69 thf(fact_9668_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.32/0.69 ! [N: nat,J: nat,I: nat] :
% 0.32/0.69 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 0.32/0.69 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 0.32/0.69 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % nth_sorted_list_of_set_greaterThanAtMost
% 0.32/0.69 thf(fact_9669_card__greaterThanAtMost,axiom,
% 0.32/0.69 ! [L2: nat,U: nat] :
% 0.32/0.69 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
% 0.32/0.69 = ( minus_minus_nat @ U @ L2 ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_greaterThanAtMost
% 0.32/0.69 thf(fact_9670_and__not__num_Oelims,axiom,
% 0.32/0.69 ! [X: num,Xa: num,Y3: option_num] :
% 0.32/0.69 ( ( ( bit_and_not_num @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3 != none_num ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ? [N4: num] :
% 0.32/0.69 ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ one ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ? [N4: num] :
% 0.32/0.69 ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( Y3 != none_num ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.32/0.69 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.32/0.69 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 0.32/0.69 => ~ ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.elims
% 0.32/0.69 thf(fact_9671_Bit__Operations_Otake__bit__num__code,axiom,
% 0.32/0.69 ( bit_take_bit_num
% 0.32/0.69 = ( ^ [N2: nat,M3: num] :
% 0.32/0.69 ( produc478579273971653890on_num
% 0.32/0.69 @ ^ [A3: nat,X2: num] :
% 0.32/0.69 ( case_nat_option_num @ none_num
% 0.32/0.69 @ ^ [O: nat] :
% 0.32/0.69 ( case_num_option_num @ ( some_num @ one )
% 0.32/0.69 @ ^ [P4: num] :
% 0.32/0.69 ( case_o6005452278849405969um_num @ none_num
% 0.32/0.69 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.32/0.69 @ ( bit_take_bit_num @ O @ P4 ) )
% 0.32/0.69 @ ^ [P4: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
% 0.32/0.69 @ X2 )
% 0.32/0.69 @ A3 )
% 0.32/0.69 @ ( product_Pair_nat_num @ N2 @ M3 ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % Bit_Operations.take_bit_num_code
% 0.32/0.69 thf(fact_9672_xor__num_Osimps_I8_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(8)
% 0.32/0.69 thf(fact_9673_card__greaterThanAtMost__int,axiom,
% 0.32/0.69 ! [L2: int,U: int] :
% 0.32/0.69 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
% 0.32/0.69 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % card_greaterThanAtMost_int
% 0.32/0.69 thf(fact_9674_xor__num_Osimps_I5_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(5)
% 0.32/0.69 thf(fact_9675_xor__num_Osimps_I9_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(9)
% 0.32/0.69 thf(fact_9676_and__not__num_Osimps_I5_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(5)
% 0.32/0.69 thf(fact_9677_and__not__num_Osimps_I6_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(6)
% 0.32/0.69 thf(fact_9678_and__not__num_Osimps_I9_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.simps(9)
% 0.32/0.69 thf(fact_9679_xor__num_Osimps_I1_J,axiom,
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.32/0.69 = none_num ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(1)
% 0.32/0.69 thf(fact_9680_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.32/0.69 ! [L2: int,U: int] :
% 0.32/0.69 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 0.32/0.69 = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% 0.32/0.69
% 0.32/0.69 % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.32/0.69 thf(fact_9681_xor__num_Oelims,axiom,
% 0.32/0.69 ! [X: num,Xa: num,Y3: option_num] :
% 0.32/0.69 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3 != none_num ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ ( bit1 @ N4 ) ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ ( bit0 @ N4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ~ ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.elims
% 0.32/0.69 thf(fact_9682_xor__num_Osimps_I7_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 0.32/0.69 = ( some_num @ ( bit0 @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(7)
% 0.32/0.69 thf(fact_9683_xor__num_Osimps_I4_J,axiom,
% 0.32/0.69 ! [M: num] :
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 0.32/0.69 = ( some_num @ ( bit1 @ M ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(4)
% 0.32/0.69 thf(fact_9684_xor__num_Osimps_I3_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 0.32/0.69 = ( some_num @ ( bit0 @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(3)
% 0.32/0.69 thf(fact_9685_xor__num_Osimps_I2_J,axiom,
% 0.32/0.69 ! [N: num] :
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 0.32/0.69 = ( some_num @ ( bit1 @ N ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(2)
% 0.32/0.69 thf(fact_9686_xor__num_Osimps_I6_J,axiom,
% 0.32/0.69 ! [M: num,N: num] :
% 0.32/0.69 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.simps(6)
% 0.32/0.69 thf(fact_9687_and__not__num_Opelims,axiom,
% 0.32/0.69 ! [X: num,Xa: num,Y3: option_num] :
% 0.32/0.69 ( ( ( bit_and_not_num @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3 = none_num )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ one ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( ( Y3 = none_num )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.32/0.69 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.32/0.69 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ~ ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % and_not_num.pelims
% 0.32/0.69 thf(fact_9688_xor__num_Opelims,axiom,
% 0.32/0.69 ! [X: num,Xa: num,Y3: option_num] :
% 0.32/0.69 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3 = none_num )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ ( bit1 @ N4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ ( bit0 @ N4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ ( bit1 @ M4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ ( bit0 @ M4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.69 => ~ ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( ( Y3
% 0.32/0.69 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
% 0.32/0.69 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.69
% 0.32/0.69 % xor_num.pelims
% 0.32/0.69 thf(fact_9689_and__num_Oelims,axiom,
% 0.32/0.69 ! [X: num,Xa: num,Y3: option_num] :
% 0.32/0.69 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.32/0.69 = Y3 )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ one ) ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ? [N4: num] :
% 0.32/0.69 ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3 != none_num ) ) )
% 0.32/0.69 => ( ( ( X = one )
% 0.32/0.69 => ( ? [N4: num] :
% 0.32/0.69 ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ one ) ) ) )
% 0.32/0.69 => ( ( ? [M4: num] :
% 0.32/0.69 ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3 != none_num ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit0 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit1 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 0.32/0.69 => ( ( ? [M4: num] :
% 0.32/0.69 ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ( ( Xa = one )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( some_num @ one ) ) ) )
% 0.32/0.69 => ( ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.69 = ( bit1 @ M4 ) )
% 0.32/0.69 => ! [N4: num] :
% 0.32/0.69 ( ( Xa
% 0.32/0.69 = ( bit0 @ N4 ) )
% 0.32/0.69 => ( Y3
% 0.32/0.69 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 0.32/0.69 => ~ ! [M4: num] :
% 0.32/0.69 ( ( X
% 0.32/0.70 = ( bit1 @ M4 ) )
% 0.32/0.70 => ! [N4: num] :
% 0.32/0.70 ( ( Xa
% 0.32/0.70 = ( bit1 @ N4 ) )
% 0.32/0.70 => ( Y3
% 0.32/0.70 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.32/0.70 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.32/0.70 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.elims
% 0.32/0.70 thf(fact_9690_xor__num__rel__dict,axiom,
% 0.32/0.70 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 0.32/0.70
% 0.32/0.70 % xor_num_rel_dict
% 0.32/0.70 thf(fact_9691_xor__num__dict,axiom,
% 0.32/0.70 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 0.32/0.70
% 0.32/0.70 % xor_num_dict
% 0.32/0.70 thf(fact_9692_and__num_Osimps_I1_J,axiom,
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 0.32/0.70 = ( some_num @ one ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(1)
% 0.32/0.70 thf(fact_9693_and__num_Osimps_I5_J,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.32/0.70 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(5)
% 0.32/0.70 thf(fact_9694_and__num_Osimps_I3_J,axiom,
% 0.32/0.70 ! [N: num] :
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 0.32/0.70 = ( some_num @ one ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(3)
% 0.32/0.70 thf(fact_9695_and__num_Osimps_I7_J,axiom,
% 0.32/0.70 ! [M: num] :
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 0.32/0.70 = ( some_num @ one ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(7)
% 0.32/0.70 thf(fact_9696_and__num_Osimps_I4_J,axiom,
% 0.32/0.70 ! [M: num] :
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.32/0.70 = none_num ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(4)
% 0.32/0.70 thf(fact_9697_and__num_Osimps_I2_J,axiom,
% 0.32/0.70 ! [N: num] :
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 0.32/0.70 = none_num ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(2)
% 0.32/0.70 thf(fact_9698_and__num_Osimps_I6_J,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.32/0.70 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(6)
% 0.32/0.70 thf(fact_9699_and__num_Osimps_I8_J,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.32/0.70 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(8)
% 0.32/0.70 thf(fact_9700_and__num_Osimps_I9_J,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.32/0.70 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.32/0.70 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.32/0.70 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.simps(9)
% 0.32/0.70 thf(fact_9701_and__num_Opelims,axiom,
% 0.32/0.70 ! [X: num,Xa: num,Y3: option_num] :
% 0.32/0.70 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.32/0.70 = Y3 )
% 0.32/0.70 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 0.32/0.70 => ( ( ( X = one )
% 0.32/0.70 => ( ( Xa = one )
% 0.32/0.70 => ( ( Y3
% 0.32/0.70 = ( some_num @ one ) )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 0.32/0.70 => ( ( ( X = one )
% 0.32/0.70 => ! [N4: num] :
% 0.32/0.70 ( ( Xa
% 0.32/0.70 = ( bit0 @ N4 ) )
% 0.32/0.70 => ( ( Y3 = none_num )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.70 => ( ( ( X = one )
% 0.32/0.70 => ! [N4: num] :
% 0.32/0.70 ( ( Xa
% 0.32/0.70 = ( bit1 @ N4 ) )
% 0.32/0.70 => ( ( Y3
% 0.32/0.70 = ( some_num @ one ) )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 0.32/0.70 => ( ! [M4: num] :
% 0.32/0.70 ( ( X
% 0.32/0.70 = ( bit0 @ M4 ) )
% 0.32/0.70 => ( ( Xa = one )
% 0.32/0.70 => ( ( Y3 = none_num )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 0.32/0.70 => ( ! [M4: num] :
% 0.32/0.70 ( ( X
% 0.32/0.70 = ( bit0 @ M4 ) )
% 0.32/0.70 => ! [N4: num] :
% 0.32/0.70 ( ( Xa
% 0.32/0.70 = ( bit0 @ N4 ) )
% 0.32/0.70 => ( ( Y3
% 0.32/0.70 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.70 => ( ! [M4: num] :
% 0.32/0.70 ( ( X
% 0.32/0.70 = ( bit0 @ M4 ) )
% 0.32/0.70 => ! [N4: num] :
% 0.32/0.70 ( ( Xa
% 0.32/0.70 = ( bit1 @ N4 ) )
% 0.32/0.70 => ( ( Y3
% 0.32/0.70 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 0.32/0.70 => ( ! [M4: num] :
% 0.32/0.70 ( ( X
% 0.32/0.70 = ( bit1 @ M4 ) )
% 0.32/0.70 => ( ( Xa = one )
% 0.32/0.70 => ( ( Y3
% 0.32/0.70 = ( some_num @ one ) )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 0.32/0.70 => ( ! [M4: num] :
% 0.32/0.70 ( ( X
% 0.32/0.70 = ( bit1 @ M4 ) )
% 0.32/0.70 => ! [N4: num] :
% 0.32/0.70 ( ( Xa
% 0.32/0.70 = ( bit0 @ N4 ) )
% 0.32/0.70 => ( ( Y3
% 0.32/0.70 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 0.32/0.70 => ~ ! [M4: num] :
% 0.32/0.70 ( ( X
% 0.32/0.70 = ( bit1 @ M4 ) )
% 0.32/0.70 => ! [N4: num] :
% 0.32/0.70 ( ( Xa
% 0.32/0.70 = ( bit1 @ N4 ) )
% 0.32/0.70 => ( ( Y3
% 0.32/0.70 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.32/0.70 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.32/0.70 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 0.32/0.70 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % and_num.pelims
% 0.32/0.70 thf(fact_9702_and__num__dict,axiom,
% 0.32/0.70 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 0.32/0.70
% 0.32/0.70 % and_num_dict
% 0.32/0.70 thf(fact_9703_and__num__rel__dict,axiom,
% 0.32/0.70 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 0.32/0.70
% 0.32/0.70 % and_num_rel_dict
% 0.32/0.70 thf(fact_9704_rat__floor__lemma,axiom,
% 0.32/0.70 ! [A: int,B: int] :
% 0.32/0.70 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 0.32/0.70 & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % rat_floor_lemma
% 0.32/0.70 thf(fact_9705_image__minus__const__atLeastLessThan__nat,axiom,
% 0.32/0.70 ! [C: nat,Y3: nat,X: nat] :
% 0.32/0.70 ( ( ( ord_less_nat @ C @ Y3 )
% 0.32/0.70 => ( ( image_nat_nat
% 0.32/0.70 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.32/0.70 @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
% 0.32/0.70 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_nat @ C @ Y3 )
% 0.32/0.70 => ( ( ( ord_less_nat @ X @ Y3 )
% 0.32/0.70 => ( ( image_nat_nat
% 0.32/0.70 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.32/0.70 @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
% 0.32/0.70 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.32/0.70 & ( ~ ( ord_less_nat @ X @ Y3 )
% 0.32/0.70 => ( ( image_nat_nat
% 0.32/0.70 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.32/0.70 @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
% 0.32/0.70 = bot_bot_set_nat ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % image_minus_const_atLeastLessThan_nat
% 0.32/0.70 thf(fact_9706_bij__betw__Suc,axiom,
% 0.32/0.70 ! [M5: set_nat,N3: set_nat] :
% 0.32/0.70 ( ( bij_betw_nat_nat @ suc @ M5 @ N3 )
% 0.32/0.70 = ( ( image_nat_nat @ suc @ M5 )
% 0.32/0.70 = N3 ) ) ).
% 0.32/0.70
% 0.32/0.70 % bij_betw_Suc
% 0.32/0.70 thf(fact_9707_mult__rat,axiom,
% 0.32/0.70 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.70 ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.32/0.70 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % mult_rat
% 0.32/0.70 thf(fact_9708_divide__rat,axiom,
% 0.32/0.70 ! [A: int,B: int,C: int,D: int] :
% 0.32/0.70 ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.32/0.70 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % divide_rat
% 0.32/0.70 thf(fact_9709_floor__Fract,axiom,
% 0.32/0.70 ! [A: int,B: int] :
% 0.32/0.70 ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
% 0.32/0.70 = ( divide_divide_int @ A @ B ) ) ).
% 0.32/0.70
% 0.32/0.70 % floor_Fract
% 0.32/0.70 thf(fact_9710_less__rat,axiom,
% 0.32/0.70 ! [B: int,D: int,A: int,C: int] :
% 0.32/0.70 ( ( B != zero_zero_int )
% 0.32/0.70 => ( ( D != zero_zero_int )
% 0.32/0.70 => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.32/0.70 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % less_rat
% 0.32/0.70 thf(fact_9711_add__rat,axiom,
% 0.32/0.70 ! [B: int,D: int,A: int,C: int] :
% 0.32/0.70 ( ( B != zero_zero_int )
% 0.32/0.70 => ( ( D != zero_zero_int )
% 0.32/0.70 => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.32/0.70 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % add_rat
% 0.32/0.70 thf(fact_9712_le__rat,axiom,
% 0.32/0.70 ! [B: int,D: int,A: int,C: int] :
% 0.32/0.70 ( ( B != zero_zero_int )
% 0.32/0.70 => ( ( D != zero_zero_int )
% 0.32/0.70 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.32/0.70 = ( 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 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % le_rat
% 0.32/0.70 thf(fact_9713_diff__rat,axiom,
% 0.32/0.70 ! [B: int,D: int,A: int,C: int] :
% 0.32/0.70 ( ( B != zero_zero_int )
% 0.32/0.70 => ( ( D != zero_zero_int )
% 0.32/0.70 => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.32/0.70 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % diff_rat
% 0.32/0.70 thf(fact_9714_sgn__rat,axiom,
% 0.32/0.70 ! [A: int,B: int] :
% 0.32/0.70 ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 0.32/0.70 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % sgn_rat
% 0.32/0.70 thf(fact_9715_Rat__induct__pos,axiom,
% 0.32/0.70 ! [P: rat > $o,Q2: rat] :
% 0.32/0.70 ( ! [A5: int,B3: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.32/0.70 => ( P @ ( fract @ A5 @ B3 ) ) )
% 0.32/0.70 => ( P @ Q2 ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rat_induct_pos
% 0.32/0.70 thf(fact_9716_mult__rat__cancel,axiom,
% 0.32/0.70 ! [C: int,A: int,B: int] :
% 0.32/0.70 ( ( C != zero_zero_int )
% 0.32/0.70 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.32/0.70 = ( fract @ A @ B ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % mult_rat_cancel
% 0.32/0.70 thf(fact_9717_eq__rat_I1_J,axiom,
% 0.32/0.70 ! [B: int,D: int,A: int,C: int] :
% 0.32/0.70 ( ( B != zero_zero_int )
% 0.32/0.70 => ( ( D != zero_zero_int )
% 0.32/0.70 => ( ( ( fract @ A @ B )
% 0.32/0.70 = ( fract @ C @ D ) )
% 0.32/0.70 = ( ( times_times_int @ A @ D )
% 0.32/0.70 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % eq_rat(1)
% 0.32/0.70 thf(fact_9718_Fract__coprime,axiom,
% 0.32/0.70 ! [A: int,B: int] :
% 0.32/0.70 ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
% 0.32/0.70 = ( fract @ A @ B ) ) ).
% 0.32/0.70
% 0.32/0.70 % Fract_coprime
% 0.32/0.70 thf(fact_9719_Fract__of__int__quotient,axiom,
% 0.32/0.70 ( fract
% 0.32/0.70 = ( ^ [K2: int,L: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K2 ) @ ( ring_1_of_int_rat @ L ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Fract_of_int_quotient
% 0.32/0.70 thf(fact_9720_rat__number__collapse_I3_J,axiom,
% 0.32/0.70 ! [W: num] :
% 0.32/0.70 ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 0.32/0.70 = ( numeral_numeral_rat @ W ) ) ).
% 0.32/0.70
% 0.32/0.70 % rat_number_collapse(3)
% 0.32/0.70 thf(fact_9721_rat__number__expand_I3_J,axiom,
% 0.32/0.70 ( numeral_numeral_rat
% 0.32/0.70 = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % rat_number_expand(3)
% 0.32/0.70 thf(fact_9722_image__Suc__lessThan,axiom,
% 0.32/0.70 ! [N: nat] :
% 0.32/0.70 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 0.32/0.70
% 0.32/0.70 % image_Suc_lessThan
% 0.32/0.70 thf(fact_9723_image__Suc__atMost,axiom,
% 0.32/0.70 ! [N: nat] :
% 0.32/0.70 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 0.32/0.70 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % image_Suc_atMost
% 0.32/0.70 thf(fact_9724_zero__less__Fract__iff,axiom,
% 0.32/0.70 ! [B: int,A: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.70 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.32/0.70 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % zero_less_Fract_iff
% 0.32/0.70 thf(fact_9725_Fract__less__zero__iff,axiom,
% 0.32/0.70 ! [B: int,A: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.70 => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.32/0.70 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Fract_less_zero_iff
% 0.32/0.70 thf(fact_9726_one__less__Fract__iff,axiom,
% 0.32/0.70 ! [B: int,A: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.70 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.32/0.70 = ( ord_less_int @ B @ A ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % one_less_Fract_iff
% 0.32/0.70 thf(fact_9727_Fract__less__one__iff,axiom,
% 0.32/0.70 ! [B: int,A: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.70 => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.32/0.70 = ( ord_less_int @ A @ B ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Fract_less_one_iff
% 0.32/0.70 thf(fact_9728_Fract__add__one,axiom,
% 0.32/0.70 ! [N: int,M: int] :
% 0.32/0.70 ( ( N != zero_zero_int )
% 0.32/0.70 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 0.32/0.70 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Fract_add_one
% 0.32/0.70 thf(fact_9729_zero__le__Fract__iff,axiom,
% 0.32/0.70 ! [B: int,A: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.32/0.70 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % zero_le_Fract_iff
% 0.32/0.70 thf(fact_9730_Fract__le__zero__iff,axiom,
% 0.32/0.70 ! [B: int,A: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.70 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.32/0.70 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Fract_le_zero_iff
% 0.32/0.70 thf(fact_9731_one__le__Fract__iff,axiom,
% 0.32/0.70 ! [B: int,A: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.70 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.32/0.70 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % one_le_Fract_iff
% 0.32/0.70 thf(fact_9732_Fract__le__one__iff,axiom,
% 0.32/0.70 ! [B: int,A: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ B )
% 0.32/0.70 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.32/0.70 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Fract_le_one_iff
% 0.32/0.70 thf(fact_9733_rat__number__expand_I5_J,axiom,
% 0.32/0.70 ! [K: num] :
% 0.32/0.70 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 0.32/0.70 = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.32/0.70
% 0.32/0.70 % rat_number_expand(5)
% 0.32/0.70 thf(fact_9734_rat__number__collapse_I4_J,axiom,
% 0.32/0.70 ! [W: num] :
% 0.32/0.70 ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 0.32/0.70 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % rat_number_collapse(4)
% 0.32/0.70 thf(fact_9735_positive__rat,axiom,
% 0.32/0.70 ! [A: int,B: int] :
% 0.32/0.70 ( ( positive @ ( fract @ A @ B ) )
% 0.32/0.70 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % positive_rat
% 0.32/0.70 thf(fact_9736_Inf__real__def,axiom,
% 0.32/0.70 ( comple4887499456419720421f_real
% 0.32/0.70 = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Inf_real_def
% 0.32/0.70 thf(fact_9737_Rat_Opositive__add,axiom,
% 0.32/0.70 ! [X: rat,Y3: rat] :
% 0.32/0.70 ( ( positive @ X )
% 0.32/0.70 => ( ( positive @ Y3 )
% 0.32/0.70 => ( positive @ ( plus_plus_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rat.positive_add
% 0.32/0.70 thf(fact_9738_Rat_Opositive__mult,axiom,
% 0.32/0.70 ! [X: rat,Y3: rat] :
% 0.32/0.70 ( ( positive @ X )
% 0.32/0.70 => ( ( positive @ Y3 )
% 0.32/0.70 => ( positive @ ( times_times_rat @ X @ Y3 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rat.positive_mult
% 0.32/0.70 thf(fact_9739_finite__int__iff__bounded,axiom,
% 0.32/0.70 ( finite_finite_int
% 0.32/0.70 = ( ^ [S5: set_int] :
% 0.32/0.70 ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % finite_int_iff_bounded
% 0.32/0.70 thf(fact_9740_finite__int__iff__bounded__le,axiom,
% 0.32/0.70 ( finite_finite_int
% 0.32/0.70 = ( ^ [S5: set_int] :
% 0.32/0.70 ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % finite_int_iff_bounded_le
% 0.32/0.70 thf(fact_9741_suminf__eq__SUP__real,axiom,
% 0.32/0.70 ! [X7: nat > real] :
% 0.32/0.70 ( ( summable_real @ X7 )
% 0.32/0.70 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X7 @ I2 ) )
% 0.32/0.70 => ( ( suminf_real @ X7 )
% 0.32/0.70 = ( comple1385675409528146559p_real
% 0.32/0.70 @ ( image_nat_real
% 0.32/0.70 @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X7 @ ( set_ord_lessThan_nat @ I4 ) )
% 0.32/0.70 @ top_top_set_nat ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % suminf_eq_SUP_real
% 0.32/0.70 thf(fact_9742_less__rat__def,axiom,
% 0.32/0.70 ( ord_less_rat
% 0.32/0.70 = ( ^ [X2: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X2 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % less_rat_def
% 0.32/0.70 thf(fact_9743_image__add__int__atLeastLessThan,axiom,
% 0.32/0.70 ! [L2: int,U: int] :
% 0.32/0.70 ( ( image_int_int
% 0.32/0.70 @ ^ [X2: int] : ( plus_plus_int @ X2 @ L2 )
% 0.32/0.70 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
% 0.32/0.70 = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% 0.32/0.70
% 0.32/0.70 % image_add_int_atLeastLessThan
% 0.32/0.70 thf(fact_9744_range__mod,axiom,
% 0.32/0.70 ! [N: nat] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( image_nat_nat
% 0.32/0.70 @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
% 0.32/0.70 @ top_top_set_nat )
% 0.32/0.70 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % range_mod
% 0.32/0.70 thf(fact_9745_image__atLeastZeroLessThan__int,axiom,
% 0.32/0.70 ! [U: int] :
% 0.32/0.70 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.32/0.70 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 0.32/0.70 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % image_atLeastZeroLessThan_int
% 0.32/0.70 thf(fact_9746_Rat_Opositive_Orep__eq,axiom,
% 0.32/0.70 ( positive
% 0.32/0.70 = ( ^ [X2: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X2 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rat.positive.rep_eq
% 0.32/0.70 thf(fact_9747_card__UNIV__unit,axiom,
% 0.32/0.70 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.32/0.70 = one_one_nat ) ).
% 0.32/0.70
% 0.32/0.70 % card_UNIV_unit
% 0.32/0.70 thf(fact_9748_card__UNIV__bool,axiom,
% 0.32/0.70 ( ( finite_card_o @ top_top_set_o )
% 0.32/0.70 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % card_UNIV_bool
% 0.32/0.70 thf(fact_9749_range__mult,axiom,
% 0.32/0.70 ! [A: real] :
% 0.32/0.70 ( ( ( A = zero_zero_real )
% 0.32/0.70 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.32/0.70 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.32/0.70 & ( ( A != zero_zero_real )
% 0.32/0.70 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.32/0.70 = top_top_set_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % range_mult
% 0.32/0.70 thf(fact_9750_root__def,axiom,
% 0.32/0.70 ( root
% 0.32/0.70 = ( ^ [N2: nat,X2: real] :
% 0.32/0.70 ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 0.32/0.70 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.32/0.70 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 0.32/0.70 @ X2 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % root_def
% 0.32/0.70 thf(fact_9751_card__UNIV__char,axiom,
% 0.32/0.70 ( ( finite_card_char @ top_top_set_char )
% 0.32/0.70 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % card_UNIV_char
% 0.32/0.70 thf(fact_9752_UNIV__char__of__nat,axiom,
% 0.32/0.70 ( top_top_set_char
% 0.32/0.70 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % UNIV_char_of_nat
% 0.32/0.70 thf(fact_9753_nat__of__char__less__256,axiom,
% 0.32/0.70 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % nat_of_char_less_256
% 0.32/0.70 thf(fact_9754_range__nat__of__char,axiom,
% 0.32/0.70 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.32/0.70 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % range_nat_of_char
% 0.32/0.70 thf(fact_9755_integer__of__char__code,axiom,
% 0.32/0.70 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.32/0.70 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.32/0.70 = ( 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 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % integer_of_char_code
% 0.32/0.70 thf(fact_9756_String_Ochar__of__ascii__of,axiom,
% 0.32/0.70 ! [C: char] :
% 0.32/0.70 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 0.32/0.70 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % String.char_of_ascii_of
% 0.32/0.70 thf(fact_9757_DERIV__real__root__generic,axiom,
% 0.32/0.70 ! [N: nat,X: real,D4: real] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( X != zero_zero_real )
% 0.32/0.70 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.70 => ( D4
% 0.32/0.70 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.32/0.70 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.70 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.70 => ( D4
% 0.32/0.70 = ( 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 ) ) ) ) ) ) ) ) )
% 0.32/0.70 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.70 => ( D4
% 0.32/0.70 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_real_root_generic
% 0.32/0.70 thf(fact_9758_DERIV__pos__imp__increasing,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_eq_real @ X3 @ B )
% 0.32/0.70 => ? [Y5: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.32/0.70 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_pos_imp_increasing
% 0.32/0.70 thf(fact_9759_DERIV__neg__imp__decreasing,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_eq_real @ X3 @ B )
% 0.32/0.70 => ? [Y5: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.32/0.70 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_neg_imp_decreasing
% 0.32/0.70 thf(fact_9760_DERIV__pos__inc__left,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.32/0.70 => ? [D3: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.70 & ! [H4: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.32/0.70 => ( ( ord_less_real @ H4 @ D3 )
% 0.32/0.70 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_pos_inc_left
% 0.32/0.70 thf(fact_9761_DERIV__neg__dec__left,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.32/0.70 => ? [D3: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.70 & ! [H4: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.32/0.70 => ( ( ord_less_real @ H4 @ D3 )
% 0.32/0.70 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_neg_dec_left
% 0.32/0.70 thf(fact_9762_DERIV__pos__inc__right,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.32/0.70 => ? [D3: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.70 & ! [H4: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.32/0.70 => ( ( ord_less_real @ H4 @ D3 )
% 0.32/0.70 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_pos_inc_right
% 0.32/0.70 thf(fact_9763_DERIV__neg__dec__right,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.32/0.70 => ? [D3: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.70 & ! [H4: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.32/0.70 => ( ( ord_less_real @ H4 @ D3 )
% 0.32/0.70 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_neg_dec_right
% 0.32/0.70 thf(fact_9764_DERIV__isconst3,axiom,
% 0.32/0.70 ! [A: real,B: real,X: real,Y3: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.32/0.70 => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.32/0.70 => ( ( F @ X )
% 0.32/0.70 = ( F @ Y3 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_isconst3
% 0.32/0.70 thf(fact_9765_has__real__derivative__neg__dec__right,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real,S2: set_real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.32/0.70 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.32/0.70 => ? [D3: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.70 & ! [H4: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.32/0.70 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
% 0.32/0.70 => ( ( ord_less_real @ H4 @ D3 )
% 0.32/0.70 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % has_real_derivative_neg_dec_right
% 0.32/0.70 thf(fact_9766_has__real__derivative__pos__inc__right,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real,S2: set_real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.32/0.70 => ? [D3: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.70 & ! [H4: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.32/0.70 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
% 0.32/0.70 => ( ( ord_less_real @ H4 @ D3 )
% 0.32/0.70 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % has_real_derivative_pos_inc_right
% 0.32/0.70 thf(fact_9767_has__real__derivative__neg__dec__left,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real,S2: set_real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.32/0.70 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.32/0.70 => ? [D3: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.70 & ! [H4: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.32/0.70 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
% 0.32/0.70 => ( ( ord_less_real @ H4 @ D3 )
% 0.32/0.70 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % has_real_derivative_neg_dec_left
% 0.32/0.70 thf(fact_9768_has__real__derivative__pos__inc__left,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real,S2: set_real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.32/0.70 => ? [D3: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.32/0.70 & ! [H4: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.32/0.70 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
% 0.32/0.70 => ( ( ord_less_real @ H4 @ D3 )
% 0.32/0.70 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % has_real_derivative_pos_inc_left
% 0.32/0.70 thf(fact_9769_DERIV__isconst__all,axiom,
% 0.32/0.70 ! [F: real > real,X: real,Y3: real] :
% 0.32/0.70 ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 => ( ( F @ X )
% 0.32/0.70 = ( F @ Y3 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_isconst_all
% 0.32/0.70 thf(fact_9770_DERIV__mirror,axiom,
% 0.32/0.70 ! [F: real > real,Y3: real,X: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
% 0.32/0.70 = ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X2: real] : ( F @ ( uminus_uminus_real @ X2 ) )
% 0.32/0.70 @ ( uminus_uminus_real @ Y3 )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_mirror
% 0.32/0.70 thf(fact_9771_DERIV__const__ratio__const,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real,K: real] :
% 0.32/0.70 ( ( A != B )
% 0.32/0.70 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.32/0.70 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_const_ratio_const
% 0.32/0.70 thf(fact_9772_DERIV__const__ratio__const2,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real,K: real] :
% 0.32/0.70 ( ( A != B )
% 0.32/0.70 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 0.32/0.70 = K ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_const_ratio_const2
% 0.32/0.70 thf(fact_9773_deriv__nonneg__imp__mono,axiom,
% 0.32/0.70 ! [A: real,B: real,G: real > real,G2: real > real] :
% 0.32/0.70 ( ! [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.32/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
% 0.32/0.70 => ( ( ord_less_eq_real @ A @ B )
% 0.32/0.70 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % deriv_nonneg_imp_mono
% 0.32/0.70 thf(fact_9774_DERIV__nonpos__imp__nonincreasing,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_eq_real @ X3 @ B )
% 0.32/0.70 => ? [Y5: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 0.32/0.70 => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_nonpos_imp_nonincreasing
% 0.32/0.70 thf(fact_9775_DERIV__nonneg__imp__nondecreasing,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_eq_real @ X3 @ B )
% 0.32/0.70 => ? [Y5: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 0.32/0.70 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_nonneg_imp_nondecreasing
% 0.32/0.70 thf(fact_9776_MVT2,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real,F3: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_eq_real @ X3 @ B )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ? [Z5: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ Z5 )
% 0.32/0.70 & ( ord_less_real @ Z5 @ B )
% 0.32/0.70 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.32/0.70 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F3 @ Z5 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % MVT2
% 0.32/0.70 thf(fact_9777_DERIV__local__const,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real,D: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.32/0.70 => ( ! [Y4: real] :
% 0.32/0.70 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.32/0.70 => ( ( F @ X )
% 0.32/0.70 = ( F @ Y4 ) ) )
% 0.32/0.70 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_local_const
% 0.32/0.70 thf(fact_9778_DERIV__ln,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_ln
% 0.32/0.70 thf(fact_9779_DERIV__const__average,axiom,
% 0.32/0.70 ! [A: real,B: real,V: real > real,K: real] :
% 0.32/0.70 ( ( A != B )
% 0.32/0.70 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.32/0.70 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_const_average
% 0.32/0.70 thf(fact_9780_DERIV__local__max,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real,D: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.32/0.70 => ( ! [Y4: real] :
% 0.32/0.70 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.32/0.70 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
% 0.32/0.70 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_local_max
% 0.32/0.70 thf(fact_9781_DERIV__local__min,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,X: real,D: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.32/0.70 => ( ! [Y4: real] :
% 0.32/0.70 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 0.32/0.70 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
% 0.32/0.70 => ( L2 = zero_zero_real ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_local_min
% 0.32/0.70 thf(fact_9782_DERIV__ln__divide,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_ln_divide
% 0.32/0.70 thf(fact_9783_DERIV__pow,axiom,
% 0.32/0.70 ! [N: nat,X: real,S: set_real] :
% 0.32/0.70 ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X2: real] : ( power_power_real @ X2 @ N )
% 0.32/0.70 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_pow
% 0.32/0.70 thf(fact_9784_DERIV__fun__pow,axiom,
% 0.32/0.70 ! [G: real > real,M: real,X: real,N: nat] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N )
% 0.32/0.70 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_fun_pow
% 0.32/0.70 thf(fact_9785_has__real__derivative__powr,axiom,
% 0.32/0.70 ! [Z: real,R: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.32/0.70 => ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [Z2: real] : ( powr_real @ Z2 @ R )
% 0.32/0.70 @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % has_real_derivative_powr
% 0.32/0.70 thf(fact_9786_DERIV__log,axiom,
% 0.32/0.70 ! [X: real,B: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.70 => ( 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 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_log
% 0.32/0.70 thf(fact_9787_DERIV__fun__powr,axiom,
% 0.32/0.70 ! [G: real > real,M: real,X: real,R: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R )
% 0.32/0.70 @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_fun_powr
% 0.32/0.70 thf(fact_9788_DERIV__powr,axiom,
% 0.32/0.70 ! [G: real > real,M: real,X: real,F: real > real,R: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 0.32/0.70 => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
% 0.32/0.70 @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_powr
% 0.32/0.70 thf(fact_9789_DERIV__real__sqrt,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_real_sqrt
% 0.32/0.70 thf(fact_9790_DERIV__series_H,axiom,
% 0.32/0.70 ! [F: real > nat > real,F3: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
% 0.32/0.70 ( ! [N4: nat] :
% 0.32/0.70 ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X2: real] : ( F @ X2 @ N4 )
% 0.32/0.70 @ ( F3 @ X0 @ N4 )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.32/0.70 => ( summable_real @ ( F @ X3 ) ) )
% 0.32/0.70 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.32/0.70 => ( ( summable_real @ ( F3 @ X0 ) )
% 0.32/0.70 => ( ( summable_real @ L5 )
% 0.32/0.70 => ( ! [N4: nat,X3: real,Y4: real] :
% 0.32/0.70 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.32/0.70 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.32/0.70 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N4 ) @ ( F @ Y4 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) ) ) ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
% 0.32/0.70 @ ( suminf_real @ ( F3 @ X0 ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_series'
% 0.32/0.70 thf(fact_9791_DERIV__arctan,axiom,
% 0.32/0.70 ! [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 ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_arctan
% 0.32/0.70 thf(fact_9792_arsinh__real__has__field__derivative,axiom,
% 0.32/0.70 ! [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 ) ) ).
% 0.32/0.70
% 0.32/0.70 % arsinh_real_has_field_derivative
% 0.32/0.70 thf(fact_9793_DERIV__real__sqrt__generic,axiom,
% 0.32/0.70 ! [X: real,D4: real] :
% 0.32/0.70 ( ( X != zero_zero_real )
% 0.32/0.70 => ( ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.70 => ( D4
% 0.32/0.70 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.32/0.70 => ( ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.70 => ( D4
% 0.32/0.70 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_real_sqrt_generic
% 0.32/0.70 thf(fact_9794_arcosh__real__has__field__derivative,axiom,
% 0.32/0.70 ! [X: real,A2: set_real] :
% 0.32/0.70 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.70 => ( 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 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % arcosh_real_has_field_derivative
% 0.32/0.70 thf(fact_9795_artanh__real__has__field__derivative,axiom,
% 0.32/0.70 ! [X: real,A2: set_real] :
% 0.32/0.70 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.70 => ( 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 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % artanh_real_has_field_derivative
% 0.32/0.70 thf(fact_9796_DERIV__power__series_H,axiom,
% 0.32/0.70 ! [R4: real,F: nat > real,X0: real] :
% 0.32/0.70 ( ! [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R4 ) @ R4 ) )
% 0.32/0.70 => ( summable_real
% 0.32/0.70 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) ) )
% 0.32/0.70 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R4 ) @ R4 ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ R4 )
% 0.32/0.70 => ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( suminf_real
% 0.32/0.70 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) )
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_power_series'
% 0.32/0.70 thf(fact_9797_DERIV__real__root,axiom,
% 0.32/0.70 ! [N: nat,X: real] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ X )
% 0.32/0.70 => ( 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 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_real_root
% 0.32/0.70 thf(fact_9798_DERIV__arccos,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.70 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.70 => ( 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 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_arccos
% 0.32/0.70 thf(fact_9799_DERIV__arcsin,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.70 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.70 => ( 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 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_arcsin
% 0.32/0.70 thf(fact_9800_Maclaurin__all__le,axiom,
% 0.32/0.70 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.32/0.70 ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.32/0.70 & ( ( F @ X )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Maclaurin_all_le
% 0.32/0.70 thf(fact_9801_Maclaurin__all__le__objl,axiom,
% 0.32/0.70 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 0.32/0.70 ( ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 & ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.32/0.70 & ( ( F @ X )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Maclaurin_all_le_objl
% 0.32/0.70 thf(fact_9802_DERIV__odd__real__root,axiom,
% 0.32/0.70 ! [N: nat,X: real] :
% 0.32/0.70 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.70 => ( ( X != zero_zero_real )
% 0.32/0.70 => ( 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 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_odd_real_root
% 0.32/0.70 thf(fact_9803_Maclaurin,axiom,
% 0.32/0.70 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.32/0.70 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ! [M4: nat,T5: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M4 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.32/0.70 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.32/0.70 & ( ord_less_real @ T5 @ H2 )
% 0.32/0.70 & ( ( F @ H2 )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Maclaurin
% 0.32/0.70 thf(fact_9804_Maclaurin2,axiom,
% 0.32/0.70 ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.32/0.70 => ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ! [M4: nat,T5: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M4 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.32/0.70 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ T5 )
% 0.32/0.70 & ( ord_less_eq_real @ T5 @ H2 )
% 0.32/0.70 & ( ( F @ H2 )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Maclaurin2
% 0.32/0.70 thf(fact_9805_Maclaurin__minus,axiom,
% 0.32/0.70 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ H2 @ zero_zero_real )
% 0.32/0.70 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ! [M4: nat,T5: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M4 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ H2 @ T5 )
% 0.32/0.70 & ( ord_less_eq_real @ T5 @ zero_zero_real ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_real @ H2 @ T5 )
% 0.32/0.70 & ( ord_less_real @ T5 @ zero_zero_real )
% 0.32/0.70 & ( ( F @ H2 )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Maclaurin_minus
% 0.32/0.70 thf(fact_9806_Maclaurin__all__lt,axiom,
% 0.32/0.70 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.32/0.70 ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( X != zero_zero_real )
% 0.32/0.70 => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 0.32/0.70 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.32/0.70 & ( ( F @ X )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Maclaurin_all_lt
% 0.32/0.70 thf(fact_9807_Maclaurin__bi__le,axiom,
% 0.32/0.70 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 0.32/0.70 ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ! [M4: nat,T5: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M4 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 0.32/0.70 & ( ( F @ X )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Maclaurin_bi_le
% 0.32/0.70 thf(fact_9808_Taylor,axiom,
% 0.32/0.70 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ! [M4: nat,T5: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M4 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ A @ T5 )
% 0.32/0.70 & ( ord_less_eq_real @ T5 @ B ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.32/0.70 => ( ( ord_less_eq_real @ A @ C )
% 0.32/0.70 => ( ( ord_less_eq_real @ C @ B )
% 0.32/0.70 => ( ( ord_less_eq_real @ A @ X )
% 0.32/0.70 => ( ( ord_less_eq_real @ X @ B )
% 0.32/0.70 => ( ( X != C )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ( ord_less_real @ X @ C )
% 0.32/0.70 => ( ( ord_less_real @ X @ T5 )
% 0.32/0.70 & ( ord_less_real @ T5 @ C ) ) )
% 0.32/0.70 & ( ~ ( ord_less_real @ X @ C )
% 0.32/0.70 => ( ( ord_less_real @ C @ T5 )
% 0.32/0.70 & ( ord_less_real @ T5 @ X ) ) )
% 0.32/0.70 & ( ( F @ X )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Taylor
% 0.32/0.70 thf(fact_9809_Taylor__up,axiom,
% 0.32/0.70 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ! [M4: nat,T5: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M4 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ A @ T5 )
% 0.32/0.70 & ( ord_less_eq_real @ T5 @ B ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.32/0.70 => ( ( ord_less_eq_real @ A @ C )
% 0.32/0.70 => ( ( ord_less_real @ C @ B )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_real @ C @ T5 )
% 0.32/0.70 & ( ord_less_real @ T5 @ B )
% 0.32/0.70 & ( ( F @ B )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Taylor_up
% 0.32/0.70 thf(fact_9810_Taylor__down,axiom,
% 0.32/0.70 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( ( Diff @ zero_zero_nat )
% 0.32/0.70 = F )
% 0.32/0.70 => ( ! [M4: nat,T5: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M4 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ A @ T5 )
% 0.32/0.70 & ( ord_less_eq_real @ T5 @ B ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.32/0.70 => ( ( ord_less_real @ A @ C )
% 0.32/0.70 => ( ( ord_less_eq_real @ C @ B )
% 0.32/0.70 => ? [T5: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ T5 )
% 0.32/0.70 & ( ord_less_real @ T5 @ C )
% 0.32/0.70 & ( ( F @ A )
% 0.32/0.70 = ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ N ) )
% 0.32/0.70 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Taylor_down
% 0.32/0.70 thf(fact_9811_Maclaurin__lemma2,axiom,
% 0.32/0.70 ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
% 0.32/0.70 ( ! [M4: nat,T5: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M4 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 0.32/0.70 & ( ord_less_eq_real @ T5 @ H2 ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 0.32/0.70 => ( ( N
% 0.32/0.70 = ( suc @ K ) )
% 0.32/0.70 => ! [M2: nat,T6: real] :
% 0.32/0.70 ( ( ( ord_less_nat @ M2 @ N )
% 0.32/0.70 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.32/0.70 & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.32/0.70 => ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [U3: real] :
% 0.32/0.70 ( minus_minus_real @ ( Diff @ M2 @ U3 )
% 0.32/0.70 @ ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U3 @ P4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
% 0.32/0.70 @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U3 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
% 0.32/0.70 @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T6 )
% 0.32/0.70 @ ( plus_plus_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T6 @ P4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
% 0.32/0.70 @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Maclaurin_lemma2
% 0.32/0.70 thf(fact_9812_DERIV__arctan__series,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.70 => ( has_fi5821293074295781190e_real
% 0.32/0.70 @ ^ [X9: real] :
% 0.32/0.70 ( suminf_real
% 0.32/0.70 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_arctan_series
% 0.32/0.70 thf(fact_9813_DERIV__even__real__root,axiom,
% 0.32/0.70 ! [N: nat,X: real] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.70 => ( ( ord_less_real @ X @ zero_zero_real )
% 0.32/0.70 => ( 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 ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_even_real_root
% 0.32/0.70 thf(fact_9814_isCont__Lb__Ub,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 & ( ord_less_eq_real @ X3 @ B ) )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 0.32/0.70 => ? [L6: real,M9: real] :
% 0.32/0.70 ( ! [X5: real] :
% 0.32/0.70 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.32/0.70 & ( ord_less_eq_real @ X5 @ B ) )
% 0.32/0.70 => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
% 0.32/0.70 & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
% 0.32/0.70 & ! [Y5: real] :
% 0.32/0.70 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 0.32/0.70 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 0.32/0.70 => ? [X3: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 & ( ord_less_eq_real @ X3 @ B )
% 0.32/0.70 & ( ( F @ X3 )
% 0.32/0.70 = Y5 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_Lb_Ub
% 0.32/0.70 thf(fact_9815_LIM__fun__gt__zero,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,C: real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 0.32/0.70 => ? [R2: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.32/0.70 & ! [X5: real] :
% 0.32/0.70 ( ( ( X5 != C )
% 0.32/0.70 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 0.32/0.70 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIM_fun_gt_zero
% 0.32/0.70 thf(fact_9816_LIM__fun__not__zero,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,C: real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.32/0.70 => ( ( L2 != zero_zero_real )
% 0.32/0.70 => ? [R2: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.32/0.70 & ! [X5: real] :
% 0.32/0.70 ( ( ( X5 != C )
% 0.32/0.70 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 0.32/0.70 => ( ( F @ X5 )
% 0.32/0.70 != zero_zero_real ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIM_fun_not_zero
% 0.32/0.70 thf(fact_9817_LIM__fun__less__zero,axiom,
% 0.32/0.70 ! [F: real > real,L2: real,C: real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.32/0.70 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 0.32/0.70 => ? [R2: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.32/0.70 & ! [X5: real] :
% 0.32/0.70 ( ( ( X5 != C )
% 0.32/0.70 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 0.32/0.70 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIM_fun_less_zero
% 0.32/0.70 thf(fact_9818_isCont__real__sqrt,axiom,
% 0.32/0.70 ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_real_sqrt
% 0.32/0.70 thf(fact_9819_isCont__real__root,axiom,
% 0.32/0.70 ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_real_root
% 0.32/0.70 thf(fact_9820_continuous__frac,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ~ ( member_real @ X @ ring_1_Ints_real )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ archim2898591450579166408c_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % continuous_frac
% 0.32/0.70 thf(fact_9821_isCont__inverse__function2,axiom,
% 0.32/0.70 ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X )
% 0.32/0.70 => ( ( ord_less_real @ X @ B )
% 0.32/0.70 => ( ! [Z5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ Z5 )
% 0.32/0.70 => ( ( ord_less_eq_real @ Z5 @ B )
% 0.32/0.70 => ( ( G @ ( F @ Z5 ) )
% 0.32/0.70 = Z5 ) ) )
% 0.32/0.70 => ( ! [Z5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ Z5 )
% 0.32/0.70 => ( ( ord_less_eq_real @ Z5 @ B )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_inverse_function2
% 0.32/0.70 thf(fact_9822_isCont__arcosh,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_arcosh
% 0.32/0.70 thf(fact_9823_LIM__cos__div__sin,axiom,
% 0.32/0.70 ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIM_cos_div_sin
% 0.32/0.70 thf(fact_9824_continuous__floor,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ~ ( member_real @ X @ ring_1_Ints_real )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % continuous_floor
% 0.32/0.70 thf(fact_9825_DERIV__inverse__function,axiom,
% 0.32/0.70 ! [F: real > real,D4: real,G: real > real,X: real,A: real,B: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 0.32/0.70 => ( ( D4 != zero_zero_real )
% 0.32/0.70 => ( ( ord_less_real @ A @ X )
% 0.32/0.70 => ( ( ord_less_real @ X @ B )
% 0.32/0.70 => ( ! [Y4: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ Y4 )
% 0.32/0.70 => ( ( ord_less_real @ Y4 @ B )
% 0.32/0.70 => ( ( F @ ( G @ Y4 ) )
% 0.32/0.70 = Y4 ) ) )
% 0.32/0.70 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_inverse_function
% 0.32/0.70 thf(fact_9826_isCont__arccos,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.70 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_arccos
% 0.32/0.70 thf(fact_9827_isCont__arcsin,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.70 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_arcsin
% 0.32/0.70 thf(fact_9828_LIM__less__bound,axiom,
% 0.32/0.70 ! [B: real,X: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ B @ X )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 0.32/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 0.32/0.70 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 0.32/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIM_less_bound
% 0.32/0.70 thf(fact_9829_isCont__artanh,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 0.32/0.70 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_artanh
% 0.32/0.70 thf(fact_9830_isCont__inverse__function,axiom,
% 0.32/0.70 ! [D: real,X: real,G: real > real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ D )
% 0.32/0.70 => ( ! [Z5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
% 0.32/0.70 => ( ( G @ ( F @ Z5 ) )
% 0.32/0.70 = Z5 ) )
% 0.32/0.70 => ( ! [Z5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % isCont_inverse_function
% 0.32/0.70 thf(fact_9831_GMVT_H,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ! [Z5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ Z5 )
% 0.32/0.70 => ( ( ord_less_eq_real @ Z5 @ B )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
% 0.32/0.70 => ( ! [Z5: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ Z5 )
% 0.32/0.70 => ( ( ord_less_eq_real @ Z5 @ B )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ G ) ) )
% 0.32/0.70 => ( ! [Z5: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ Z5 )
% 0.32/0.70 => ( ( ord_less_real @ Z5 @ B )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ( ! [Z5: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ Z5 )
% 0.32/0.70 => ( ( ord_less_real @ Z5 @ B )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ? [C2: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ C2 )
% 0.32/0.70 & ( ord_less_real @ C2 @ B )
% 0.32/0.70 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
% 0.32/0.70 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F3 @ C2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % GMVT'
% 0.32/0.70 thf(fact_9832_summable__Leibniz_I3_J,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ( topolo6980174941875973593q_real @ A )
% 0.32/0.70 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.32/0.70 => ! [N7: nat] :
% 0.32/0.70 ( member_real
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.32/0.70 @ ( set_or1222579329274155063t_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) )
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz(3)
% 0.32/0.70 thf(fact_9833_summable__Leibniz_I2_J,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ( topolo6980174941875973593q_real @ A )
% 0.32/0.70 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.32/0.70 => ! [N7: nat] :
% 0.32/0.70 ( member_real
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.32/0.70 @ ( set_or1222579329274155063t_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz(2)
% 0.32/0.70 thf(fact_9834_mult__nat__left__at__top,axiom,
% 0.32/0.70 ! [C: nat] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.70 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % mult_nat_left_at_top
% 0.32/0.70 thf(fact_9835_mult__nat__right__at__top,axiom,
% 0.32/0.70 ! [C: nat] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.32/0.70 => ( filterlim_nat_nat
% 0.32/0.70 @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
% 0.32/0.70 @ at_top_nat
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % mult_nat_right_at_top
% 0.32/0.70 thf(fact_9836_monoseq__convergent,axiom,
% 0.32/0.70 ! [X7: nat > real,B5: real] :
% 0.32/0.70 ( ( topolo6980174941875973593q_real @ X7 )
% 0.32/0.70 => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X7 @ I2 ) ) @ B5 )
% 0.32/0.70 => ~ ! [L6: real] :
% 0.32/0.70 ~ ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % monoseq_convergent
% 0.32/0.70 thf(fact_9837_LIMSEQ__root,axiom,
% 0.32/0.70 ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.32/0.70 @ at_top_nat ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_root
% 0.32/0.70 thf(fact_9838_nested__sequence__unique,axiom,
% 0.32/0.70 ! [F: nat > real,G: nat > real] :
% 0.32/0.70 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 0.32/0.70 => ( ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_nat )
% 0.32/0.70 => ? [L4: real] :
% 0.32/0.70 ( ! [N7: nat] : ( ord_less_eq_real @ ( F @ N7 ) @ L4 )
% 0.32/0.70 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 0.32/0.70 & ! [N7: nat] : ( ord_less_eq_real @ L4 @ ( G @ N7 ) )
% 0.32/0.70 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % nested_sequence_unique
% 0.32/0.70 thf(fact_9839_LIMSEQ__inverse__zero,axiom,
% 0.32/0.70 ! [X7: nat > real] :
% 0.32/0.70 ( ! [R2: real] :
% 0.32/0.70 ? [N8: nat] :
% 0.32/0.70 ! [N4: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ N8 @ N4 )
% 0.32/0.70 => ( ord_less_real @ R2 @ ( X7 @ N4 ) ) )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( inverse_inverse_real @ ( X7 @ N2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_inverse_zero
% 0.32/0.70 thf(fact_9840_lim__inverse__n_H,axiom,
% 0.32/0.70 ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_nat ) ).
% 0.32/0.70
% 0.32/0.70 % lim_inverse_n'
% 0.32/0.70 thf(fact_9841_LIMSEQ__root__const,axiom,
% 0.32/0.70 ! [C: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ C )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( root @ N2 @ C )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_root_const
% 0.32/0.70 thf(fact_9842_LIMSEQ__inverse__real__of__nat__add,axiom,
% 0.32/0.70 ! [R: real] :
% 0.32/0.70 ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ R )
% 0.32/0.70 @ at_top_nat ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_inverse_real_of_nat_add
% 0.32/0.70 thf(fact_9843_increasing__LIMSEQ,axiom,
% 0.32/0.70 ! [F: nat > real,L2: real] :
% 0.32/0.70 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
% 0.32/0.70 => ( ! [E2: real] :
% 0.32/0.70 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.32/0.70 => ? [N7: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N7 ) @ E2 ) ) )
% 0.32/0.70 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % increasing_LIMSEQ
% 0.32/0.70 thf(fact_9844_LIMSEQ__realpow__zero,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.70 => ( ( ord_less_real @ X @ one_one_real )
% 0.32/0.70 => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_realpow_zero
% 0.32/0.70 thf(fact_9845_LIMSEQ__divide__realpow__zero,axiom,
% 0.32/0.70 ! [X: real,A: real] :
% 0.32/0.70 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_divide_realpow_zero
% 0.32/0.70 thf(fact_9846_LIMSEQ__abs__realpow__zero2,axiom,
% 0.32/0.70 ! [C: real] :
% 0.32/0.70 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.32/0.70 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_abs_realpow_zero2
% 0.32/0.70 thf(fact_9847_LIMSEQ__abs__realpow__zero,axiom,
% 0.32/0.70 ! [C: real] :
% 0.32/0.70 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.32/0.70 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_abs_realpow_zero
% 0.32/0.70 thf(fact_9848_LIMSEQ__inverse__realpow__zero,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_real @ one_one_real @ X )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_inverse_realpow_zero
% 0.32/0.70 thf(fact_9849_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 0.32/0.70 ! [R: real] :
% 0.32/0.70 ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ R )
% 0.32/0.70 @ at_top_nat ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_inverse_real_of_nat_add_minus
% 0.32/0.70 thf(fact_9850_tendsto__exp__limit__sequentially,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.32/0.70 @ at_top_nat ) ).
% 0.32/0.70
% 0.32/0.70 % tendsto_exp_limit_sequentially
% 0.32/0.70 thf(fact_9851_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.32/0.70 ! [R: real] :
% 0.32/0.70 ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ R )
% 0.32/0.70 @ at_top_nat ) ).
% 0.32/0.70
% 0.32/0.70 % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.32/0.70 thf(fact_9852_summable__Leibniz_I1_J,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ( topolo6980174941875973593q_real @ A )
% 0.32/0.70 => ( summable_real
% 0.32/0.70 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz(1)
% 0.32/0.70 thf(fact_9853_summable,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.32/0.70 => ( summable_real
% 0.32/0.70 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable
% 0.32/0.70 thf(fact_9854_cos__diff__limit__1,axiom,
% 0.32/0.70 ! [Theta: nat > real,Theta2: real] :
% 0.32/0.70 ( ( filterlim_nat_real
% 0.32/0.70 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.32/0.70 @ at_top_nat )
% 0.32/0.70 => ~ ! [K3: nat > int] :
% 0.32/0.70 ~ ( filterlim_nat_real
% 0.32/0.70 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ Theta2 )
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % cos_diff_limit_1
% 0.32/0.70 thf(fact_9855_cos__limit__1,axiom,
% 0.32/0.70 ! [Theta: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real
% 0.32/0.70 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.32/0.70 @ at_top_nat )
% 0.32/0.70 => ? [K3: nat > int] :
% 0.32/0.70 ( filterlim_nat_real
% 0.32/0.70 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % cos_limit_1
% 0.32/0.70 thf(fact_9856_summable__Leibniz_I4_J,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ( topolo6980174941875973593q_real @ A )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] :
% 0.32/0.70 ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.32/0.70 @ at_top_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz(4)
% 0.32/0.70 thf(fact_9857_zeroseq__arctan__series,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % zeroseq_arctan_series
% 0.32/0.70 thf(fact_9858_summable__Leibniz_H_I2_J,axiom,
% 0.32/0.70 ! [A: nat > real,N: nat] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.32/0.70 => ( ord_less_eq_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz'(2)
% 0.32/0.70 thf(fact_9859_summable__Leibniz_H_I3_J,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] :
% 0.32/0.70 ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.32/0.70 @ at_top_nat ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz'(3)
% 0.32/0.70 thf(fact_9860_sums__alternating__upper__lower,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.32/0.70 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ? [L4: real] :
% 0.32/0.70 ( ! [N7: nat] :
% 0.32/0.70 ( ord_less_eq_real
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
% 0.32/0.70 @ L4 )
% 0.32/0.70 & ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] :
% 0.32/0.70 ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ L4 )
% 0.32/0.70 @ at_top_nat )
% 0.32/0.70 & ! [N7: nat] :
% 0.32/0.70 ( ord_less_eq_real @ L4
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) )
% 0.32/0.70 & ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] :
% 0.32/0.70 ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ L4 )
% 0.32/0.70 @ at_top_nat ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % sums_alternating_upper_lower
% 0.32/0.70 thf(fact_9861_summable__Leibniz_I5_J,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ( topolo6980174941875973593q_real @ A )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] :
% 0.32/0.70 ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.32/0.70 @ at_top_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz(5)
% 0.32/0.70 thf(fact_9862_summable__Leibniz_H_I5_J,axiom,
% 0.32/0.70 ! [A: nat > real] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.32/0.70 => ( filterlim_nat_real
% 0.32/0.70 @ ^ [N2: nat] :
% 0.32/0.70 ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 0.32/0.70 @ at_top_nat ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz'(5)
% 0.32/0.70 thf(fact_9863_summable__Leibniz_H_I4_J,axiom,
% 0.32/0.70 ! [A: nat > real,N: nat] :
% 0.32/0.70 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 0.32/0.70 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 0.32/0.70 => ( ord_less_eq_real
% 0.32/0.70 @ ( suminf_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 0.32/0.70 @ ( groups6591440286371151544t_real
% 0.32/0.70 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 0.32/0.70 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_Leibniz'(4)
% 0.32/0.70 thf(fact_9864_eventually__sequentially__seg,axiom,
% 0.32/0.70 ! [P: nat > $o,K: nat] :
% 0.32/0.70 ( ( eventually_nat
% 0.32/0.70 @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
% 0.32/0.70 @ at_top_nat )
% 0.32/0.70 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % eventually_sequentially_seg
% 0.32/0.70 thf(fact_9865_le__sequentially,axiom,
% 0.32/0.70 ! [F4: filter_nat] :
% 0.32/0.70 ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
% 0.32/0.70 = ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F4 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % le_sequentially
% 0.32/0.70 thf(fact_9866_eventually__sequentiallyI,axiom,
% 0.32/0.70 ! [C: nat,P: nat > $o] :
% 0.32/0.70 ( ! [X3: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ C @ X3 )
% 0.32/0.70 => ( P @ X3 ) )
% 0.32/0.70 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % eventually_sequentiallyI
% 0.32/0.70 thf(fact_9867_eventually__sequentially,axiom,
% 0.32/0.70 ! [P: nat > $o] :
% 0.32/0.70 ( ( eventually_nat @ P @ at_top_nat )
% 0.32/0.70 = ( ? [N5: nat] :
% 0.32/0.70 ! [N2: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ N5 @ N2 )
% 0.32/0.70 => ( P @ N2 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % eventually_sequentially
% 0.32/0.70 thf(fact_9868_sequentially__offset,axiom,
% 0.32/0.70 ! [P: nat > $o,K: nat] :
% 0.32/0.70 ( ( eventually_nat @ P @ at_top_nat )
% 0.32/0.70 => ( eventually_nat
% 0.32/0.70 @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
% 0.32/0.70 @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % sequentially_offset
% 0.32/0.70 thf(fact_9869_real__bounded__linear,axiom,
% 0.32/0.70 ( real_V5970128139526366754l_real
% 0.32/0.70 = ( ^ [F5: real > real] :
% 0.32/0.70 ? [C3: real] :
% 0.32/0.70 ( F5
% 0.32/0.70 = ( ^ [X2: real] : ( times_times_real @ X2 @ C3 ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % real_bounded_linear
% 0.32/0.70 thf(fact_9870_sqrt__at__top,axiom,
% 0.32/0.70 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 0.32/0.70
% 0.32/0.70 % sqrt_at_top
% 0.32/0.70 thf(fact_9871_lhopital__left__at__top__at__top,axiom,
% 0.32/0.70 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ at_top_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ at_top_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_left_at_top_at_top
% 0.32/0.70 thf(fact_9872_lhopital__at__top__at__top,axiom,
% 0.32/0.70 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ at_top_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ at_top_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_at_top_at_top
% 0.32/0.70 thf(fact_9873_eventually__at__left__real,axiom,
% 0.32/0.70 ! [B: real,A: real] :
% 0.32/0.70 ( ( ord_less_real @ B @ A )
% 0.32/0.70 => ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ B @ A ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % eventually_at_left_real
% 0.32/0.70 thf(fact_9874_lhopital__left__at__top,axiom,
% 0.32/0.70 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y3: real] :
% 0.32/0.70 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_left_at_top
% 0.32/0.70 thf(fact_9875_lhopital__at__top,axiom,
% 0.32/0.70 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y3: real] :
% 0.32/0.70 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_at_top
% 0.32/0.70 thf(fact_9876_lhospital__at__top__at__top,axiom,
% 0.32/0.70 ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 0.32/0.70 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ at_top_real )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ at_top_real )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ at_top_real )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ X )
% 0.32/0.70 @ at_top_real )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ X )
% 0.32/0.70 @ at_top_real ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhospital_at_top_at_top
% 0.32/0.70 thf(fact_9877_tanh__real__at__top,axiom,
% 0.32/0.70 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 0.32/0.70
% 0.32/0.70 % tanh_real_at_top
% 0.32/0.70 thf(fact_9878_artanh__real__at__left__1,axiom,
% 0.32/0.70 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % artanh_real_at_left_1
% 0.32/0.70 thf(fact_9879_ln__x__over__x__tendsto__0,axiom,
% 0.32/0.70 ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_real ) ).
% 0.32/0.70
% 0.32/0.70 % ln_x_over_x_tendsto_0
% 0.32/0.70 thf(fact_9880_tendsto__power__div__exp__0,axiom,
% 0.32/0.70 ! [K: nat] :
% 0.32/0.70 ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.32/0.70 @ at_top_real ) ).
% 0.32/0.70
% 0.32/0.70 % tendsto_power_div_exp_0
% 0.32/0.70 thf(fact_9881_lhopital__left,axiom,
% 0.32/0.70 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ F4
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ F4
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_left
% 0.32/0.70 thf(fact_9882_lhopital,axiom,
% 0.32/0.70 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ F4
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ F4
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital
% 0.32/0.70 thf(fact_9883_tendsto__exp__limit__at__top,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( filterlim_real_real
% 0.32/0.70 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y ) ) @ Y )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.32/0.70 @ at_top_real ) ).
% 0.32/0.70
% 0.32/0.70 % tendsto_exp_limit_at_top
% 0.32/0.70 thf(fact_9884_filterlim__tan__at__left,axiom,
% 0.32/0.70 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 ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % filterlim_tan_at_left
% 0.32/0.70 thf(fact_9885_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.32/0.70 ! [B: real,F: real > real,Flim: real] :
% 0.32/0.70 ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ B @ X3 )
% 0.32/0.70 => ? [Y5: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.32/0.70 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_neg_imp_decreasing_at_top
% 0.32/0.70 thf(fact_9886_tendsto__arctan__at__top,axiom,
% 0.32/0.70 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 0.32/0.70
% 0.32/0.70 % tendsto_arctan_at_top
% 0.32/0.70 thf(fact_9887_at__top__le__at__infinity,axiom,
% 0.32/0.70 ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 0.32/0.70
% 0.32/0.70 % at_top_le_at_infinity
% 0.32/0.70 thf(fact_9888_dist__complex__def,axiom,
% 0.32/0.70 ( real_V3694042436643373181omplex
% 0.32/0.70 = ( ^ [X2: complex,Y: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % dist_complex_def
% 0.32/0.70 thf(fact_9889_dist__real__def,axiom,
% 0.32/0.70 ( real_V975177566351809787t_real
% 0.32/0.70 = ( ^ [X2: real,Y: real] : ( abs_abs_real @ ( minus_minus_real @ X2 @ Y ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % dist_real_def
% 0.32/0.70 thf(fact_9890_filterlim__pow__at__bot__even,axiom,
% 0.32/0.70 ! [N: nat,F: real > real,F4: filter_real] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 0.32/0.70 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
% 0.32/0.70 @ at_top_real
% 0.32/0.70 @ F4 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % filterlim_pow_at_bot_even
% 0.32/0.70 thf(fact_9891_at__bot__le__at__infinity,axiom,
% 0.32/0.70 ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 0.32/0.70
% 0.32/0.70 % at_bot_le_at_infinity
% 0.32/0.70 thf(fact_9892_tanh__real__at__bot,axiom,
% 0.32/0.70 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 0.32/0.70
% 0.32/0.70 % tanh_real_at_bot
% 0.32/0.70 thf(fact_9893_lhopital__at__top__at__bot,axiom,
% 0.32/0.70 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ at_bot_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ at_bot_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_at_top_at_bot
% 0.32/0.70 thf(fact_9894_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.32/0.70 ! [B: real,F: real > real,Flim: real] :
% 0.32/0.70 ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ X3 @ B )
% 0.32/0.70 => ? [Y5: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.32/0.70 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_pos_imp_increasing_at_bot
% 0.32/0.70 thf(fact_9895_filterlim__pow__at__bot__odd,axiom,
% 0.32/0.70 ! [N: nat,F: real > real,F4: filter_real] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 0.32/0.70 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
% 0.32/0.70 @ at_bot_real
% 0.32/0.70 @ F4 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % filterlim_pow_at_bot_odd
% 0.32/0.70 thf(fact_9896_lhopital__left__at__top__at__bot,axiom,
% 0.32/0.70 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ at_bot_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ at_bot_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_left_at_top_at_bot
% 0.32/0.70 thf(fact_9897_tendsto__arctan__at__bot,axiom,
% 0.32/0.70 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 0.32/0.70
% 0.32/0.70 % tendsto_arctan_at_bot
% 0.32/0.70 thf(fact_9898_Bseq__eq__bounded,axiom,
% 0.32/0.70 ! [F: nat > real,A: real,B: real] :
% 0.32/0.70 ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.32/0.70 => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % Bseq_eq_bounded
% 0.32/0.70 thf(fact_9899_Bseq__realpow,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.70 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.70 => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Bseq_realpow
% 0.32/0.70 thf(fact_9900_tendsto__exp__limit__at__right,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( filterlim_real_real
% 0.32/0.70 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % tendsto_exp_limit_at_right
% 0.32/0.70 thf(fact_9901_filterlim__tan__at__right,axiom,
% 0.32/0.70 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 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % filterlim_tan_at_right
% 0.32/0.70 thf(fact_9902_eventually__at__right__to__0,axiom,
% 0.32/0.70 ! [P: real > $o,A: real] :
% 0.32/0.70 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 = ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % eventually_at_right_to_0
% 0.32/0.70 thf(fact_9903_eventually__at__right__real,axiom,
% 0.32/0.70 ! [A: real,B: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % eventually_at_right_real
% 0.32/0.70 thf(fact_9904_tendsto__arcosh__at__left__1,axiom,
% 0.32/0.70 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % tendsto_arcosh_at_left_1
% 0.32/0.70 thf(fact_9905_artanh__real__at__right__1,axiom,
% 0.32/0.70 filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % artanh_real_at_right_1
% 0.32/0.70 thf(fact_9906_lhopital__right__at__top__at__top,axiom,
% 0.32/0.70 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ at_top_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ at_top_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_right_at_top_at_top
% 0.32/0.70 thf(fact_9907_lhopital__right,axiom,
% 0.32/0.70 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ F4
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ F4
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_right
% 0.32/0.70 thf(fact_9908_lhopital__right__0,axiom,
% 0.32/0.70 ! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 0.32/0.70 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G0 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ F4
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
% 0.32/0.70 @ F4
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_right_0
% 0.32/0.70 thf(fact_9909_lhopital__right__at__top__at__bot,axiom,
% 0.32/0.70 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 0.32/0.70 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ at_bot_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ at_bot_real
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_right_at_top_at_bot
% 0.32/0.70 thf(fact_9910_lhopital__right__0__at__top,axiom,
% 0.32/0.70 ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 0.32/0.70 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ X )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ X )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_right_0_at_top
% 0.32/0.70 thf(fact_9911_lhopital__right__at__top,axiom,
% 0.32/0.70 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y3: real] :
% 0.32/0.70 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] :
% 0.32/0.70 ( ( G2 @ X2 )
% 0.32/0.70 != zero_zero_real )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( eventually_real
% 0.32/0.70 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 0.32/0.70 => ( filterlim_real_real
% 0.32/0.70 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 0.32/0.70 @ ( topolo2815343760600316023s_real @ Y3 )
% 0.32/0.70 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % lhopital_right_at_top
% 0.32/0.70 thf(fact_9912_greaterThan__Suc,axiom,
% 0.32/0.70 ! [K: nat] :
% 0.32/0.70 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.32/0.70 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % greaterThan_Suc
% 0.32/0.70 thf(fact_9913_atLeast__Suc,axiom,
% 0.32/0.70 ! [K: nat] :
% 0.32/0.70 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.32/0.70 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % atLeast_Suc
% 0.32/0.70 thf(fact_9914_decseq__bounded,axiom,
% 0.32/0.70 ! [X7: nat > real,B5: real] :
% 0.32/0.70 ( ( order_9091379641038594480t_real @ X7 )
% 0.32/0.70 => ( ! [I2: nat] : ( ord_less_eq_real @ B5 @ ( X7 @ I2 ) )
% 0.32/0.70 => ( bfun_nat_real @ X7 @ at_top_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % decseq_bounded
% 0.32/0.70 thf(fact_9915_decseq__convergent,axiom,
% 0.32/0.70 ! [X7: nat > real,B5: real] :
% 0.32/0.70 ( ( order_9091379641038594480t_real @ X7 )
% 0.32/0.70 => ( ! [I2: nat] : ( ord_less_eq_real @ B5 @ ( X7 @ I2 ) )
% 0.32/0.70 => ~ ! [L6: real] :
% 0.32/0.70 ( ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.32/0.70 => ~ ! [I3: nat] : ( ord_less_eq_real @ L6 @ ( X7 @ I3 ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % decseq_convergent
% 0.32/0.70 thf(fact_9916_GMVT,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real,G: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 & ( ord_less_eq_real @ X3 @ B ) )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 & ( ord_less_real @ X3 @ B ) )
% 0.32/0.70 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ( ord_less_eq_real @ A @ X3 )
% 0.32/0.70 & ( ord_less_eq_real @ X3 @ B ) )
% 0.32/0.70 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 & ( ord_less_real @ X3 @ B ) )
% 0.32/0.70 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 0.32/0.70 => ? [G_c: real,F_c: real,C2: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 0.32/0.70 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_real @ A @ C2 )
% 0.32/0.70 & ( ord_less_real @ C2 @ B )
% 0.32/0.70 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 0.32/0.70 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % GMVT
% 0.32/0.70 thf(fact_9917_real__differentiable__def,axiom,
% 0.32/0.70 ! [F: real > real,X: real,S: set_real] :
% 0.32/0.70 ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.32/0.70 = ( ? [D6: real] : ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % real_differentiable_def
% 0.32/0.70 thf(fact_9918_real__differentiableE,axiom,
% 0.32/0.70 ! [F: real > real,X: real,S: set_real] :
% 0.32/0.70 ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X @ S ) )
% 0.32/0.70 => ~ ! [Df: real] :
% 0.32/0.70 ~ ( has_fi5821293074295781190e_real @ F @ Df @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % real_differentiableE
% 0.32/0.70 thf(fact_9919_MVT,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_real @ X3 @ B )
% 0.32/0.70 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ? [L4: real,Z5: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ Z5 )
% 0.32/0.70 & ( ord_less_real @ Z5 @ B )
% 0.32/0.70 & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) )
% 0.32/0.70 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.32/0.70 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % MVT
% 0.32/0.70 thf(fact_9920_continuous__on__arcosh_H,axiom,
% 0.32/0.70 ! [A2: set_real,F: real > real] :
% 0.32/0.70 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ A2 )
% 0.32/0.70 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 0.32/0.70 => ( topolo5044208981011980120l_real @ A2
% 0.32/0.70 @ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % continuous_on_arcosh'
% 0.32/0.70 thf(fact_9921_continuous__image__closed__interval,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_eq_real @ A @ B )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ? [C2: real,D3: real] :
% 0.32/0.70 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 0.32/0.70 = ( set_or1222579329274155063t_real @ C2 @ D3 ) )
% 0.32/0.70 & ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % continuous_image_closed_interval
% 0.32/0.70 thf(fact_9922_continuous__on__arcosh,axiom,
% 0.32/0.70 ! [A2: set_real] :
% 0.32/0.70 ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.32/0.70 => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % continuous_on_arcosh
% 0.32/0.70 thf(fact_9923_continuous__on__arccos_H,axiom,
% 0.32/0.70 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 0.32/0.70
% 0.32/0.70 % continuous_on_arccos'
% 0.32/0.70 thf(fact_9924_continuous__on__arcsin_H,axiom,
% 0.32/0.70 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 0.32/0.70
% 0.32/0.70 % continuous_on_arcsin'
% 0.32/0.70 thf(fact_9925_continuous__on__artanh_H,axiom,
% 0.32/0.70 ! [A2: set_real,F: real > real] :
% 0.32/0.70 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ A2 )
% 0.32/0.70 => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 0.32/0.70 => ( topolo5044208981011980120l_real @ A2
% 0.32/0.70 @ ^ [X2: real] : ( artanh_real @ ( F @ X2 ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % continuous_on_artanh'
% 0.32/0.70 thf(fact_9926_Rolle__deriv,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ( ( F @ A )
% 0.32/0.70 = ( F @ B ) )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_real @ X3 @ B )
% 0.32/0.70 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ? [Z5: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ Z5 )
% 0.32/0.70 & ( ord_less_real @ Z5 @ B )
% 0.32/0.70 & ( ( F3 @ Z5 )
% 0.32/0.70 = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rolle_deriv
% 0.32/0.70 thf(fact_9927_mvt,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_real @ X3 @ B )
% 0.32/0.70 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ~ ! [Xi: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ Xi )
% 0.32/0.70 => ( ( ord_less_real @ Xi @ B )
% 0.32/0.70 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.32/0.70 != ( F3 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % mvt
% 0.32/0.70 thf(fact_9928_DERIV__pos__imp__increasing__open,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_real @ X3 @ B )
% 0.32/0.70 => ? [Y5: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_pos_imp_increasing_open
% 0.32/0.70 thf(fact_9929_DERIV__neg__imp__decreasing__open,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_real @ X3 @ B )
% 0.32/0.70 => ? [Y5: real] :
% 0.32/0.70 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_neg_imp_decreasing_open
% 0.32/0.70 thf(fact_9930_DERIV__isconst__end,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_real @ X3 @ B )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ( ( F @ B )
% 0.32/0.70 = ( F @ A ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_isconst_end
% 0.32/0.70 thf(fact_9931_continuous__on__artanh,axiom,
% 0.32/0.70 ! [A2: set_real] :
% 0.32/0.70 ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.32/0.70 => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % continuous_on_artanh
% 0.32/0.70 thf(fact_9932_DERIV__isconst2,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real,X: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_real @ X3 @ B )
% 0.32/0.70 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ( ( ord_less_eq_real @ A @ X )
% 0.32/0.70 => ( ( ord_less_eq_real @ X @ B )
% 0.32/0.70 => ( ( F @ X )
% 0.32/0.70 = ( F @ A ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % DERIV_isconst2
% 0.32/0.70 thf(fact_9933_Rolle,axiom,
% 0.32/0.70 ! [A: real,B: real,F: real > real] :
% 0.32/0.70 ( ( ord_less_real @ A @ B )
% 0.32/0.70 => ( ( ( F @ A )
% 0.32/0.70 = ( F @ B ) )
% 0.32/0.70 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.32/0.70 => ( ! [X3: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ X3 )
% 0.32/0.70 => ( ( ord_less_real @ X3 @ B )
% 0.32/0.70 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 0.32/0.70 => ? [Z5: real] :
% 0.32/0.70 ( ( ord_less_real @ A @ Z5 )
% 0.32/0.70 & ( ord_less_real @ Z5 @ B )
% 0.32/0.70 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rolle
% 0.32/0.70 thf(fact_9934_uniformity__complex__def,axiom,
% 0.32/0.70 ( topolo896644834953643431omplex
% 0.32/0.70 = ( comple8358262395181532106omplex
% 0.32/0.70 @ ( image_5971271580939081552omplex
% 0.32/0.70 @ ^ [E3: real] :
% 0.32/0.70 ( princi3496590319149328850omplex
% 0.32/0.70 @ ( collec8663557070575231912omplex
% 0.32/0.70 @ ( produc6771430404735790350plex_o
% 0.32/0.70 @ ^ [X2: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X2 @ Y ) @ E3 ) ) ) )
% 0.32/0.70 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % uniformity_complex_def
% 0.32/0.70 thf(fact_9935_uniformity__real__def,axiom,
% 0.32/0.70 ( topolo1511823702728130853y_real
% 0.32/0.70 = ( comple2936214249959783750l_real
% 0.32/0.70 @ ( image_2178119161166701260l_real
% 0.32/0.70 @ ^ [E3: real] :
% 0.32/0.70 ( princi6114159922880469582l_real
% 0.32/0.70 @ ( collec3799799289383736868l_real
% 0.32/0.70 @ ( produc5414030515140494994real_o
% 0.32/0.70 @ ^ [X2: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X2 @ Y ) @ E3 ) ) ) )
% 0.32/0.70 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % uniformity_real_def
% 0.32/0.70 thf(fact_9936_mono__Suc,axiom,
% 0.32/0.70 order_mono_nat_nat @ suc ).
% 0.32/0.70
% 0.32/0.70 % mono_Suc
% 0.32/0.70 thf(fact_9937_mono__times__nat,axiom,
% 0.32/0.70 ! [N: nat] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % mono_times_nat
% 0.32/0.70 thf(fact_9938_incseq__bounded,axiom,
% 0.32/0.70 ! [X7: nat > real,B5: real] :
% 0.32/0.70 ( ( order_mono_nat_real @ X7 )
% 0.32/0.70 => ( ! [I2: nat] : ( ord_less_eq_real @ ( X7 @ I2 ) @ B5 )
% 0.32/0.70 => ( bfun_nat_real @ X7 @ at_top_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % incseq_bounded
% 0.32/0.70 thf(fact_9939_incseq__convergent,axiom,
% 0.32/0.70 ! [X7: nat > real,B5: real] :
% 0.32/0.70 ( ( order_mono_nat_real @ X7 )
% 0.32/0.70 => ( ! [I2: nat] : ( ord_less_eq_real @ ( X7 @ I2 ) @ B5 )
% 0.32/0.70 => ~ ! [L6: real] :
% 0.32/0.70 ( ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.32/0.70 => ~ ! [I3: nat] : ( ord_less_eq_real @ ( X7 @ I3 ) @ L6 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % incseq_convergent
% 0.32/0.70 thf(fact_9940_mono__ge2__power__minus__self,axiom,
% 0.32/0.70 ! [K: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.32/0.70 => ( order_mono_nat_nat
% 0.32/0.70 @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % mono_ge2_power_minus_self
% 0.32/0.70 thf(fact_9941_inj__sgn__power,axiom,
% 0.32/0.70 ! [N: nat] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( inj_on_real_real
% 0.32/0.70 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.32/0.70 @ top_top_set_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % inj_sgn_power
% 0.32/0.70 thf(fact_9942_log__inj,axiom,
% 0.32/0.70 ! [B: real] :
% 0.32/0.70 ( ( ord_less_real @ one_one_real @ B )
% 0.32/0.70 => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % log_inj
% 0.32/0.70 thf(fact_9943_inj__on__diff__nat,axiom,
% 0.32/0.70 ! [N3: set_nat,K: nat] :
% 0.32/0.70 ( ! [N4: nat] :
% 0.32/0.70 ( ( member_nat @ N4 @ N3 )
% 0.32/0.70 => ( ord_less_eq_nat @ K @ N4 ) )
% 0.32/0.70 => ( inj_on_nat_nat
% 0.32/0.70 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 0.32/0.70 @ N3 ) ) ).
% 0.32/0.70
% 0.32/0.70 % inj_on_diff_nat
% 0.32/0.70 thf(fact_9944_inj__Suc,axiom,
% 0.32/0.70 ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).
% 0.32/0.70
% 0.32/0.70 % inj_Suc
% 0.32/0.70 thf(fact_9945_summable__reindex,axiom,
% 0.32/0.70 ! [F: nat > real,G: nat > nat] :
% 0.32/0.70 ( ( summable_real @ F )
% 0.32/0.70 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.32/0.70 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.32/0.70 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % summable_reindex
% 0.32/0.70 thf(fact_9946_suminf__reindex__mono,axiom,
% 0.32/0.70 ! [F: nat > real,G: nat > nat] :
% 0.32/0.70 ( ( summable_real @ F )
% 0.32/0.70 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.32/0.70 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.32/0.70 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % suminf_reindex_mono
% 0.32/0.70 thf(fact_9947_inj__on__char__of__nat,axiom,
% 0.32/0.70 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % inj_on_char_of_nat
% 0.32/0.70 thf(fact_9948_suminf__reindex,axiom,
% 0.32/0.70 ! [F: nat > real,G: nat > nat] :
% 0.32/0.70 ( ( summable_real @ F )
% 0.32/0.70 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 0.32/0.70 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.32/0.70 => ( ! [X3: nat] :
% 0.32/0.70 ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 0.32/0.70 => ( ( F @ X3 )
% 0.32/0.70 = zero_zero_real ) )
% 0.32/0.70 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 0.32/0.70 = ( suminf_real @ F ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % suminf_reindex
% 0.32/0.70 thf(fact_9949_nonneg__incseq__Bseq__subseq__iff,axiom,
% 0.32/0.70 ! [F: nat > real,G: nat > nat] :
% 0.32/0.70 ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 0.32/0.70 => ( ( order_mono_nat_real @ F )
% 0.32/0.70 => ( ( order_5726023648592871131at_nat @ G )
% 0.32/0.70 => ( ( bfun_nat_real
% 0.32/0.70 @ ^ [X2: nat] : ( F @ ( G @ X2 ) )
% 0.32/0.70 @ at_top_nat )
% 0.32/0.70 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % nonneg_incseq_Bseq_subseq_iff
% 0.32/0.70 thf(fact_9950_strict__mono__imp__increasing,axiom,
% 0.32/0.70 ! [F: nat > nat,N: nat] :
% 0.32/0.70 ( ( order_5726023648592871131at_nat @ F )
% 0.32/0.70 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % strict_mono_imp_increasing
% 0.32/0.70 thf(fact_9951_powr__real__of__int_H,axiom,
% 0.32/0.70 ! [X: real,N: int] :
% 0.32/0.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.70 => ( ( ( X != zero_zero_real )
% 0.32/0.70 | ( ord_less_int @ zero_zero_int @ N ) )
% 0.32/0.70 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 0.32/0.70 = ( power_int_real @ X @ N ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % powr_real_of_int'
% 0.32/0.70 thf(fact_9952_Rats__eq__int__div__nat,axiom,
% 0.32/0.70 ( field_5140801741446780682s_real
% 0.32/0.70 = ( collect_real
% 0.32/0.70 @ ^ [Uu3: real] :
% 0.32/0.70 ? [I4: int,N2: nat] :
% 0.32/0.70 ( ( Uu3
% 0.32/0.70 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.32/0.70 & ( N2 != zero_zero_nat ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rats_eq_int_div_nat
% 0.32/0.70 thf(fact_9953_Rats__abs__iff,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
% 0.32/0.70 = ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rats_abs_iff
% 0.32/0.70 thf(fact_9954_Rats__no__top__le,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ? [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.32/0.70 & ( ord_less_eq_real @ X @ X3 ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rats_no_top_le
% 0.32/0.70 thf(fact_9955_Rats__dense__in__real,axiom,
% 0.32/0.70 ! [X: real,Y3: real] :
% 0.32/0.70 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.70 => ? [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.32/0.70 & ( ord_less_real @ X @ X3 )
% 0.32/0.70 & ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rats_dense_in_real
% 0.32/0.70 thf(fact_9956_Rats__no__bot__less,axiom,
% 0.32/0.70 ! [X: real] :
% 0.32/0.70 ? [X3: real] :
% 0.32/0.70 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 0.32/0.70 & ( ord_less_real @ X3 @ X ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rats_no_bot_less
% 0.32/0.70 thf(fact_9957_atLeastLessThan__add__Un,axiom,
% 0.32/0.70 ! [I: nat,J: nat,K: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.70 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.32/0.70 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % atLeastLessThan_add_Un
% 0.32/0.70 thf(fact_9958_Rats__eq__int__div__int,axiom,
% 0.32/0.70 ( field_5140801741446780682s_real
% 0.32/0.70 = ( collect_real
% 0.32/0.70 @ ^ [Uu3: real] :
% 0.32/0.70 ? [I4: int,J3: int] :
% 0.32/0.70 ( ( Uu3
% 0.32/0.70 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 0.32/0.70 & ( J3 != zero_zero_int ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % Rats_eq_int_div_int
% 0.32/0.70 thf(fact_9959_pos__deriv__imp__strict__mono,axiom,
% 0.32/0.70 ! [F: real > real,F3: real > real] :
% 0.32/0.70 ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.32/0.70 => ( ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( F3 @ X3 ) )
% 0.32/0.70 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % pos_deriv_imp_strict_mono
% 0.32/0.70 thf(fact_9960_sup__enat__def,axiom,
% 0.32/0.70 sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 0.32/0.70
% 0.32/0.70 % sup_enat_def
% 0.32/0.70 thf(fact_9961_sup__nat__def,axiom,
% 0.32/0.70 sup_sup_nat = ord_max_nat ).
% 0.32/0.70
% 0.32/0.70 % sup_nat_def
% 0.32/0.70 thf(fact_9962_num__of__integer__code,axiom,
% 0.32/0.70 ( code_num_of_integer
% 0.32/0.70 = ( ^ [K2: code_integer] :
% 0.32/0.70 ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
% 0.32/0.70 @ ( produc7336495610019696514er_num
% 0.32/0.70 @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
% 0.32/0.70 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % num_of_integer_code
% 0.32/0.70 thf(fact_9963_min__Suc__Suc,axiom,
% 0.32/0.70 ! [M: nat,N: nat] :
% 0.32/0.70 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.32/0.70 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % min_Suc_Suc
% 0.32/0.70 thf(fact_9964_min__0R,axiom,
% 0.32/0.70 ! [N: nat] :
% 0.32/0.70 ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.32/0.70 = zero_zero_nat ) ).
% 0.32/0.70
% 0.32/0.70 % min_0R
% 0.32/0.70 thf(fact_9965_min__0L,axiom,
% 0.32/0.70 ! [N: nat] :
% 0.32/0.70 ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.32/0.70 = zero_zero_nat ) ).
% 0.32/0.70
% 0.32/0.70 % min_0L
% 0.32/0.70 thf(fact_9966_min__Suc__numeral,axiom,
% 0.32/0.70 ! [N: nat,K: num] :
% 0.32/0.70 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.32/0.70 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % min_Suc_numeral
% 0.32/0.70 thf(fact_9967_min__numeral__Suc,axiom,
% 0.32/0.70 ! [K: num,N: nat] :
% 0.32/0.70 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.32/0.70 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % min_numeral_Suc
% 0.32/0.70 thf(fact_9968_min__diff,axiom,
% 0.32/0.70 ! [M: nat,I: nat,N: nat] :
% 0.32/0.70 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 0.32/0.70 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 0.32/0.70
% 0.32/0.70 % min_diff
% 0.32/0.70 thf(fact_9969_nat__mult__min__left,axiom,
% 0.32/0.70 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.70 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 0.32/0.70 = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % nat_mult_min_left
% 0.32/0.70 thf(fact_9970_nat__mult__min__right,axiom,
% 0.32/0.70 ! [M: nat,N: nat,Q2: nat] :
% 0.32/0.70 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 0.32/0.70 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % nat_mult_min_right
% 0.32/0.70 thf(fact_9971_concat__bit__assoc__sym,axiom,
% 0.32/0.70 ! [M: nat,N: nat,K: int,L2: int,R: int] :
% 0.32/0.70 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L2 ) @ R )
% 0.32/0.70 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L2 @ R ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % concat_bit_assoc_sym
% 0.32/0.70 thf(fact_9972_inf__nat__def,axiom,
% 0.32/0.70 inf_inf_nat = ord_min_nat ).
% 0.32/0.70
% 0.32/0.70 % inf_nat_def
% 0.32/0.70 thf(fact_9973_take__bit__concat__bit__eq,axiom,
% 0.32/0.70 ! [M: nat,N: nat,K: int,L2: int] :
% 0.32/0.70 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L2 ) )
% 0.32/0.70 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L2 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % take_bit_concat_bit_eq
% 0.32/0.70 thf(fact_9974_min__Suc1,axiom,
% 0.32/0.70 ! [N: nat,M: nat] :
% 0.32/0.70 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 0.32/0.70 = ( case_nat_nat @ zero_zero_nat
% 0.32/0.70 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
% 0.32/0.70 @ M ) ) ).
% 0.32/0.70
% 0.32/0.70 % min_Suc1
% 0.32/0.70 thf(fact_9975_min__Suc2,axiom,
% 0.32/0.70 ! [M: nat,N: nat] :
% 0.32/0.70 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 0.32/0.70 = ( case_nat_nat @ zero_zero_nat
% 0.32/0.70 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
% 0.32/0.70 @ M ) ) ).
% 0.32/0.70
% 0.32/0.70 % min_Suc2
% 0.32/0.70 thf(fact_9976_min__enat__simps_I2_J,axiom,
% 0.32/0.70 ! [Q2: extended_enat] :
% 0.32/0.70 ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 0.32/0.70 = zero_z5237406670263579293d_enat ) ).
% 0.32/0.70
% 0.32/0.70 % min_enat_simps(2)
% 0.32/0.70 thf(fact_9977_min__enat__simps_I3_J,axiom,
% 0.32/0.70 ! [Q2: extended_enat] :
% 0.32/0.70 ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 0.32/0.70 = zero_z5237406670263579293d_enat ) ).
% 0.32/0.70
% 0.32/0.70 % min_enat_simps(3)
% 0.32/0.70 thf(fact_9978_inf__enat__def,axiom,
% 0.32/0.70 inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 0.32/0.70
% 0.32/0.70 % inf_enat_def
% 0.32/0.70 thf(fact_9979_card__le__Suc__Max,axiom,
% 0.32/0.70 ! [S2: set_nat] :
% 0.32/0.70 ( ( finite_finite_nat @ S2 )
% 0.32/0.70 => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % card_le_Suc_Max
% 0.32/0.70 thf(fact_9980_divide__nat__def,axiom,
% 0.32/0.70 ( divide_divide_nat
% 0.32/0.70 = ( ^ [M3: nat,N2: nat] :
% 0.32/0.70 ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 0.32/0.70 @ ( lattic8265883725875713057ax_nat
% 0.32/0.70 @ ( collect_nat
% 0.32/0.70 @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N2 ) @ M3 ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % divide_nat_def
% 0.32/0.70 thf(fact_9981_gcd__is__Max__divisors__nat,axiom,
% 0.32/0.70 ! [N: nat,M: nat] :
% 0.32/0.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.70 => ( ( gcd_gcd_nat @ M @ N )
% 0.32/0.70 = ( lattic8265883725875713057ax_nat
% 0.32/0.70 @ ( collect_nat
% 0.32/0.70 @ ^ [D2: nat] :
% 0.32/0.70 ( ( dvd_dvd_nat @ D2 @ M )
% 0.32/0.70 & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % gcd_is_Max_divisors_nat
% 0.32/0.70 thf(fact_9982_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.32/0.70 ! [I: nat,J: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.32/0.70 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.32/0.70 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % sorted_list_of_set_greaterThanAtMost
% 0.32/0.70 thf(fact_9983_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.32/0.70 ! [I: nat,J: nat] :
% 0.32/0.70 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.32/0.70 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.32/0.70 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % sorted_list_of_set_greaterThanLessThan
% 0.32/0.70 thf(fact_9984_upto__aux__rec,axiom,
% 0.32/0.70 ( upto_aux
% 0.32/0.70 = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_aux_rec
% 0.32/0.70 thf(fact_9985_upto_Opelims,axiom,
% 0.32/0.70 ! [X: int,Xa: int,Y3: list_int] :
% 0.32/0.70 ( ( ( upto @ X @ Xa )
% 0.32/0.70 = Y3 )
% 0.32/0.70 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 0.32/0.70 => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
% 0.32/0.70 => ( Y3
% 0.32/0.70 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 0.32/0.70 => ( Y3 = nil_int ) ) )
% 0.32/0.70 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto.pelims
% 0.32/0.70 thf(fact_9986_upto_Opsimps,axiom,
% 0.32/0.70 ! [I: int,J: int] :
% 0.32/0.70 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.32/0.70 => ( ( ( ord_less_eq_int @ I @ J )
% 0.32/0.70 => ( ( upto @ I @ J )
% 0.32/0.70 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_eq_int @ I @ J )
% 0.32/0.70 => ( ( upto @ I @ J )
% 0.32/0.70 = nil_int ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto.psimps
% 0.32/0.70 thf(fact_9987_upto__Nil,axiom,
% 0.32/0.70 ! [I: int,J: int] :
% 0.32/0.70 ( ( ( upto @ I @ J )
% 0.32/0.70 = nil_int )
% 0.32/0.70 = ( ord_less_int @ J @ I ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_Nil
% 0.32/0.70 thf(fact_9988_upto__Nil2,axiom,
% 0.32/0.70 ! [I: int,J: int] :
% 0.32/0.70 ( ( nil_int
% 0.32/0.70 = ( upto @ I @ J ) )
% 0.32/0.70 = ( ord_less_int @ J @ I ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_Nil2
% 0.32/0.70 thf(fact_9989_upto__empty,axiom,
% 0.32/0.70 ! [J: int,I: int] :
% 0.32/0.70 ( ( ord_less_int @ J @ I )
% 0.32/0.70 => ( ( upto @ I @ J )
% 0.32/0.70 = nil_int ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_empty
% 0.32/0.70 thf(fact_9990_nth__upto,axiom,
% 0.32/0.70 ! [I: int,K: nat,J: int] :
% 0.32/0.70 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.32/0.70 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.32/0.70 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % nth_upto
% 0.32/0.70 thf(fact_9991_length__upto,axiom,
% 0.32/0.70 ! [I: int,J: int] :
% 0.32/0.70 ( ( size_size_list_int @ ( upto @ I @ J ) )
% 0.32/0.70 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % length_upto
% 0.32/0.70 thf(fact_9992_upto__rec__numeral_I1_J,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.70 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.70 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.70 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.70 = nil_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_rec_numeral(1)
% 0.32/0.70 thf(fact_9993_upto__rec__numeral_I2_J,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.70 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.70 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.70 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.70 = nil_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_rec_numeral(2)
% 0.32/0.70 thf(fact_9994_upto__rec__numeral_I3_J,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.70 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.70 = ( 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 ) ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.70 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.32/0.70 = nil_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_rec_numeral(3)
% 0.32/0.70 thf(fact_9995_upto__rec__numeral_I4_J,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.70 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.70 = ( 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 ) ) ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.70 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.32/0.70 = nil_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_rec_numeral(4)
% 0.32/0.70 thf(fact_9996_upto__split2,axiom,
% 0.32/0.70 ! [I: int,J: int,K: int] :
% 0.32/0.70 ( ( ord_less_eq_int @ I @ J )
% 0.32/0.70 => ( ( ord_less_eq_int @ J @ K )
% 0.32/0.70 => ( ( upto @ I @ K )
% 0.32/0.70 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_split2
% 0.32/0.70 thf(fact_9997_upto__split1,axiom,
% 0.32/0.70 ! [I: int,J: int,K: int] :
% 0.32/0.70 ( ( ord_less_eq_int @ I @ J )
% 0.32/0.70 => ( ( ord_less_eq_int @ J @ K )
% 0.32/0.70 => ( ( upto @ I @ K )
% 0.32/0.70 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_split1
% 0.32/0.70 thf(fact_9998_atLeastLessThan__upto,axiom,
% 0.32/0.70 ( set_or4662586982721622107an_int
% 0.32/0.70 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % atLeastLessThan_upto
% 0.32/0.70 thf(fact_9999_greaterThanAtMost__upto,axiom,
% 0.32/0.70 ( set_or6656581121297822940st_int
% 0.32/0.70 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % greaterThanAtMost_upto
% 0.32/0.70 thf(fact_10000_upto_Osimps,axiom,
% 0.32/0.70 ( upto
% 0.32/0.70 = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto.simps
% 0.32/0.70 thf(fact_10001_upto_Oelims,axiom,
% 0.32/0.70 ! [X: int,Xa: int,Y3: list_int] :
% 0.32/0.70 ( ( ( upto @ X @ Xa )
% 0.32/0.70 = Y3 )
% 0.32/0.70 => ( ( ( ord_less_eq_int @ X @ Xa )
% 0.32/0.70 => ( Y3
% 0.32/0.70 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 0.32/0.70 => ( Y3 = nil_int ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto.elims
% 0.32/0.70 thf(fact_10002_upto__rec1,axiom,
% 0.32/0.70 ! [I: int,J: int] :
% 0.32/0.70 ( ( ord_less_eq_int @ I @ J )
% 0.32/0.70 => ( ( upto @ I @ J )
% 0.32/0.70 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_rec1
% 0.32/0.70 thf(fact_10003_upto__rec2,axiom,
% 0.32/0.70 ! [I: int,J: int] :
% 0.32/0.70 ( ( ord_less_eq_int @ I @ J )
% 0.32/0.70 => ( ( upto @ I @ J )
% 0.32/0.70 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_rec2
% 0.32/0.70 thf(fact_10004_greaterThanLessThan__upto,axiom,
% 0.32/0.70 ( set_or5832277885323065728an_int
% 0.32/0.70 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % greaterThanLessThan_upto
% 0.32/0.70 thf(fact_10005_upto__split3,axiom,
% 0.32/0.70 ! [I: int,J: int,K: int] :
% 0.32/0.70 ( ( ord_less_eq_int @ I @ J )
% 0.32/0.70 => ( ( ord_less_eq_int @ J @ K )
% 0.32/0.70 => ( ( upto @ I @ K )
% 0.32/0.70 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upto_split3
% 0.32/0.70 thf(fact_10006_hd__upt,axiom,
% 0.32/0.70 ! [I: nat,J: nat] :
% 0.32/0.70 ( ( ord_less_nat @ I @ J )
% 0.32/0.70 => ( ( hd_nat @ ( upt @ I @ J ) )
% 0.32/0.70 = I ) ) ).
% 0.32/0.70
% 0.32/0.70 % hd_upt
% 0.32/0.70 thf(fact_10007_upt__conv__Nil,axiom,
% 0.32/0.70 ! [J: nat,I: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ J @ I )
% 0.32/0.70 => ( ( upt @ I @ J )
% 0.32/0.70 = nil_nat ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_conv_Nil
% 0.32/0.70 thf(fact_10008_drop__upt,axiom,
% 0.32/0.70 ! [M: nat,I: nat,J: nat] :
% 0.32/0.70 ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 0.32/0.70 = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 0.32/0.70
% 0.32/0.70 % drop_upt
% 0.32/0.70 thf(fact_10009_length__upt,axiom,
% 0.32/0.70 ! [I: nat,J: nat] :
% 0.32/0.70 ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 0.32/0.70 = ( minus_minus_nat @ J @ I ) ) ).
% 0.32/0.70
% 0.32/0.70 % length_upt
% 0.32/0.70 thf(fact_10010_upt__eq__Nil__conv,axiom,
% 0.32/0.70 ! [I: nat,J: nat] :
% 0.32/0.70 ( ( ( upt @ I @ J )
% 0.32/0.70 = nil_nat )
% 0.32/0.70 = ( ( J = zero_zero_nat )
% 0.32/0.70 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_eq_Nil_conv
% 0.32/0.70 thf(fact_10011_take__upt,axiom,
% 0.32/0.70 ! [I: nat,M: nat,N: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 0.32/0.70 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 0.32/0.70 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % take_upt
% 0.32/0.70 thf(fact_10012_nth__upt,axiom,
% 0.32/0.70 ! [I: nat,K: nat,J: nat] :
% 0.32/0.70 ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 0.32/0.70 => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 0.32/0.70 = ( plus_plus_nat @ I @ K ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % nth_upt
% 0.32/0.70 thf(fact_10013_upt__rec__numeral,axiom,
% 0.32/0.70 ! [M: num,N: num] :
% 0.32/0.70 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.70 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.70 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.70 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.32/0.70 = nil_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_rec_numeral
% 0.32/0.70 thf(fact_10014_map__add__upt,axiom,
% 0.32/0.70 ! [N: nat,M: nat] :
% 0.32/0.70 ( ( map_nat_nat
% 0.32/0.70 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 0.32/0.70 @ ( upt @ zero_zero_nat @ M ) )
% 0.32/0.70 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % map_add_upt
% 0.32/0.70 thf(fact_10015_upt__add__eq__append,axiom,
% 0.32/0.70 ! [I: nat,J: nat,K: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.70 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 0.32/0.70 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_add_eq_append
% 0.32/0.70 thf(fact_10016_upt__conv__Cons,axiom,
% 0.32/0.70 ! [I: nat,J: nat] :
% 0.32/0.70 ( ( ord_less_nat @ I @ J )
% 0.32/0.70 => ( ( upt @ I @ J )
% 0.32/0.70 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_conv_Cons
% 0.32/0.70 thf(fact_10017_upt__rec,axiom,
% 0.32/0.70 ( upt
% 0.32/0.70 = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_rec
% 0.32/0.70 thf(fact_10018_upt__Suc,axiom,
% 0.32/0.70 ! [I: nat,J: nat] :
% 0.32/0.70 ( ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.70 => ( ( upt @ I @ ( suc @ J ) )
% 0.32/0.70 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.32/0.70 & ( ~ ( ord_less_eq_nat @ I @ J )
% 0.32/0.70 => ( ( upt @ I @ ( suc @ J ) )
% 0.32/0.70 = nil_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_Suc
% 0.32/0.70 thf(fact_10019_upt__Suc__append,axiom,
% 0.32/0.70 ! [I: nat,J: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ I @ J )
% 0.32/0.70 => ( ( upt @ I @ ( suc @ J ) )
% 0.32/0.70 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_Suc_append
% 0.32/0.70 thf(fact_10020_map__decr__upt,axiom,
% 0.32/0.70 ! [M: nat,N: nat] :
% 0.32/0.70 ( ( map_nat_nat
% 0.32/0.70 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.32/0.70 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.32/0.70 = ( upt @ M @ N ) ) ).
% 0.32/0.70
% 0.32/0.70 % map_decr_upt
% 0.32/0.70 thf(fact_10021_upt__eq__Cons__conv,axiom,
% 0.32/0.70 ! [I: nat,J: nat,X: nat,Xs2: list_nat] :
% 0.32/0.70 ( ( ( upt @ I @ J )
% 0.32/0.70 = ( cons_nat @ X @ Xs2 ) )
% 0.32/0.70 = ( ( ord_less_nat @ I @ J )
% 0.32/0.70 & ( I = X )
% 0.32/0.70 & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 0.32/0.70 = Xs2 ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % upt_eq_Cons_conv
% 0.32/0.70 thf(fact_10022_sum__list__upt,axiom,
% 0.32/0.70 ! [M: nat,N: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.32/0.70 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 0.32/0.70 = ( groups3542108847815614940at_nat
% 0.32/0.70 @ ^ [X2: nat] : X2
% 0.32/0.70 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % sum_list_upt
% 0.32/0.70 thf(fact_10023_card__length__sum__list__rec,axiom,
% 0.32/0.70 ! [M: nat,N3: nat] :
% 0.32/0.70 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.32/0.70 => ( ( finite_card_list_nat
% 0.32/0.70 @ ( collect_list_nat
% 0.32/0.70 @ ^ [L: list_nat] :
% 0.32/0.70 ( ( ( size_size_list_nat @ L )
% 0.32/0.70 = M )
% 0.32/0.70 & ( ( groups4561878855575611511st_nat @ L )
% 0.32/0.70 = N3 ) ) ) )
% 0.32/0.70 = ( plus_plus_nat
% 0.32/0.70 @ ( finite_card_list_nat
% 0.32/0.70 @ ( collect_list_nat
% 0.32/0.70 @ ^ [L: list_nat] :
% 0.32/0.70 ( ( ( size_size_list_nat @ L )
% 0.32/0.70 = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.32/0.70 & ( ( groups4561878855575611511st_nat @ L )
% 0.32/0.70 = N3 ) ) ) )
% 0.32/0.70 @ ( finite_card_list_nat
% 0.32/0.70 @ ( collect_list_nat
% 0.32/0.70 @ ^ [L: list_nat] :
% 0.32/0.70 ( ( ( size_size_list_nat @ L )
% 0.32/0.70 = M )
% 0.32/0.70 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L ) @ one_one_nat )
% 0.32/0.70 = N3 ) ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % card_length_sum_list_rec
% 0.32/0.70 thf(fact_10024_card__length__sum__list,axiom,
% 0.32/0.70 ! [M: nat,N3: nat] :
% 0.32/0.70 ( ( finite_card_list_nat
% 0.32/0.70 @ ( collect_list_nat
% 0.32/0.70 @ ^ [L: list_nat] :
% 0.32/0.70 ( ( ( size_size_list_nat @ L )
% 0.32/0.70 = M )
% 0.32/0.70 & ( ( groups4561878855575611511st_nat @ L )
% 0.32/0.70 = N3 ) ) ) )
% 0.32/0.70 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ N3 ) ) ).
% 0.32/0.70
% 0.32/0.70 % card_length_sum_list
% 0.32/0.70 thf(fact_10025_sorted__upt,axiom,
% 0.32/0.70 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 0.32/0.70
% 0.32/0.70 % sorted_upt
% 0.32/0.70 thf(fact_10026_sorted__wrt__upt,axiom,
% 0.32/0.70 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 0.32/0.70
% 0.32/0.70 % sorted_wrt_upt
% 0.32/0.70 thf(fact_10027_sorted__wrt__less__idx,axiom,
% 0.32/0.70 ! [Ns: list_nat,I: nat] :
% 0.32/0.70 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 0.32/0.70 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 0.32/0.70 => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % sorted_wrt_less_idx
% 0.32/0.70 thf(fact_10028_VEBT_Osize_I3_J,axiom,
% 0.32/0.70 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.32/0.70 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.32/0.70 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % VEBT.size(3)
% 0.32/0.70 thf(fact_10029_VEBT_Osize__gen_I1_J,axiom,
% 0.32/0.70 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.32/0.70 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.32/0.70 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % VEBT.size_gen(1)
% 0.32/0.70 thf(fact_10030_sorted__upto,axiom,
% 0.32/0.70 ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 0.32/0.70
% 0.32/0.70 % sorted_upto
% 0.32/0.70 thf(fact_10031_sorted__wrt__upto,axiom,
% 0.32/0.70 ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 0.32/0.70
% 0.32/0.70 % sorted_wrt_upto
% 0.32/0.70 thf(fact_10032_quotient__of__def,axiom,
% 0.32/0.70 ( quotient_of
% 0.32/0.70 = ( ^ [X2: rat] :
% 0.32/0.70 ( the_Pr4378521158711661632nt_int
% 0.32/0.70 @ ^ [Pair: product_prod_int_int] :
% 0.32/0.70 ( ( X2
% 0.32/0.70 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 0.32/0.70 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 0.32/0.70 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % quotient_of_def
% 0.32/0.70 thf(fact_10033_coprime__abs__right__iff,axiom,
% 0.32/0.70 ! [K: int,L2: int] :
% 0.32/0.70 ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L2 ) )
% 0.32/0.70 = ( algebr932160517623751201me_int @ K @ L2 ) ) ).
% 0.32/0.70
% 0.32/0.70 % coprime_abs_right_iff
% 0.32/0.70 thf(fact_10034_coprime__abs__left__iff,axiom,
% 0.32/0.70 ! [K: int,L2: int] :
% 0.32/0.70 ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L2 )
% 0.32/0.70 = ( algebr932160517623751201me_int @ K @ L2 ) ) ).
% 0.32/0.70
% 0.32/0.70 % coprime_abs_left_iff
% 0.32/0.70 thf(fact_10035_normalize__stable,axiom,
% 0.32/0.70 ! [Q2: int,P6: int] :
% 0.32/0.70 ( ( ord_less_int @ zero_zero_int @ Q2 )
% 0.32/0.70 => ( ( algebr932160517623751201me_int @ P6 @ Q2 )
% 0.32/0.70 => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q2 ) )
% 0.32/0.70 = ( product_Pair_int_int @ P6 @ Q2 ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % normalize_stable
% 0.32/0.70 thf(fact_10036_coprime__common__divisor__int,axiom,
% 0.32/0.70 ! [A: int,B: int,X: int] :
% 0.32/0.70 ( ( algebr932160517623751201me_int @ A @ B )
% 0.32/0.70 => ( ( dvd_dvd_int @ X @ A )
% 0.32/0.70 => ( ( dvd_dvd_int @ X @ B )
% 0.32/0.70 => ( ( abs_abs_int @ X )
% 0.32/0.70 = one_one_int ) ) ) ) ).
% 0.32/0.70
% 0.32/0.70 % coprime_common_divisor_int
% 0.32/0.70 thf(fact_10037_coprime__crossproduct__int,axiom,
% 0.32/0.70 ! [A: int,D: int,B: int,C: int] :
% 0.32/0.70 ( ( algebr932160517623751201me_int @ A @ D )
% 0.32/0.70 => ( ( algebr932160517623751201me_int @ B @ C )
% 0.32/0.70 => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 0.32/0.70 = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
% 0.32/0.71 = ( ( ( abs_abs_int @ A )
% 0.32/0.71 = ( abs_abs_int @ B ) )
% 0.32/0.71 & ( ( abs_abs_int @ C )
% 0.32/0.71 = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_crossproduct_int
% 0.32/0.71 thf(fact_10038_Rat__induct,axiom,
% 0.32/0.71 ! [P: rat > $o,Q2: rat] :
% 0.32/0.71 ( ! [A5: int,B3: int] :
% 0.32/0.71 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.32/0.71 => ( ( algebr932160517623751201me_int @ A5 @ B3 )
% 0.32/0.71 => ( P @ ( fract @ A5 @ B3 ) ) ) )
% 0.32/0.71 => ( P @ Q2 ) ) ).
% 0.32/0.71
% 0.32/0.71 % Rat_induct
% 0.32/0.71 thf(fact_10039_Rat__cases,axiom,
% 0.32/0.71 ! [Q2: rat] :
% 0.32/0.71 ~ ! [A5: int,B3: int] :
% 0.32/0.71 ( ( Q2
% 0.32/0.71 = ( fract @ A5 @ B3 ) )
% 0.32/0.71 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.32/0.71 => ~ ( algebr932160517623751201me_int @ A5 @ B3 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Rat_cases
% 0.32/0.71 thf(fact_10040_Rat__cases__nonzero,axiom,
% 0.32/0.71 ! [Q2: rat] :
% 0.32/0.71 ( ! [A5: int,B3: int] :
% 0.32/0.71 ( ( Q2
% 0.32/0.71 = ( fract @ A5 @ B3 ) )
% 0.32/0.71 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.32/0.71 => ( ( A5 != zero_zero_int )
% 0.32/0.71 => ~ ( algebr932160517623751201me_int @ A5 @ B3 ) ) ) )
% 0.32/0.71 => ( Q2 = zero_zero_rat ) ) ).
% 0.32/0.71
% 0.32/0.71 % Rat_cases_nonzero
% 0.32/0.71 thf(fact_10041_quotient__of__unique,axiom,
% 0.32/0.71 ! [R: rat] :
% 0.32/0.71 ? [X3: product_prod_int_int] :
% 0.32/0.71 ( ( R
% 0.32/0.71 = ( fract @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 0.32/0.71 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X3 ) )
% 0.32/0.71 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) )
% 0.32/0.71 & ! [Y5: product_prod_int_int] :
% 0.32/0.71 ( ( ( R
% 0.32/0.71 = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.32/0.71 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 0.32/0.71 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.32/0.71 => ( Y5 = X3 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % quotient_of_unique
% 0.32/0.71 thf(fact_10042_vanishes__mult__bounded,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ? [A8: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ A8 )
% 0.32/0.71 & ! [N4: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N4 ) ) @ A8 ) )
% 0.32/0.71 => ( ( vanishes @ Y7 )
% 0.32/0.71 => ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishes_mult_bounded
% 0.32/0.71 thf(fact_10043_coprime__int__iff,axiom,
% 0.32/0.71 ! [M: nat,N: nat] :
% 0.32/0.71 ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.32/0.71 = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_int_iff
% 0.32/0.71 thf(fact_10044_vanishes__const,axiom,
% 0.32/0.71 ! [C: rat] :
% 0.32/0.71 ( ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : C )
% 0.32/0.71 = ( C = zero_zero_rat ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishes_const
% 0.32/0.71 thf(fact_10045_coprime__nat__abs__right__iff,axiom,
% 0.32/0.71 ! [N: nat,K: int] :
% 0.32/0.71 ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.32/0.71 = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_nat_abs_right_iff
% 0.32/0.71 thf(fact_10046_coprime__nat__abs__left__iff,axiom,
% 0.32/0.71 ! [K: int,N: nat] :
% 0.32/0.71 ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 0.32/0.71 = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_nat_abs_left_iff
% 0.32/0.71 thf(fact_10047_coprime__Suc__0__left,axiom,
% 0.32/0.71 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_Suc_0_left
% 0.32/0.71 thf(fact_10048_coprime__Suc__0__right,axiom,
% 0.32/0.71 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_Suc_0_right
% 0.32/0.71 thf(fact_10049_coprime__crossproduct__nat,axiom,
% 0.32/0.71 ! [A: nat,D: nat,B: nat,C: nat] :
% 0.32/0.71 ( ( algebr934650988132801477me_nat @ A @ D )
% 0.32/0.71 => ( ( algebr934650988132801477me_nat @ B @ C )
% 0.32/0.71 => ( ( ( times_times_nat @ A @ C )
% 0.32/0.71 = ( times_times_nat @ B @ D ) )
% 0.32/0.71 = ( ( A = B )
% 0.32/0.71 & ( C = D ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_crossproduct_nat
% 0.32/0.71 thf(fact_10050_coprime__common__divisor__nat,axiom,
% 0.32/0.71 ! [A: nat,B: nat,X: nat] :
% 0.32/0.71 ( ( algebr934650988132801477me_nat @ A @ B )
% 0.32/0.71 => ( ( dvd_dvd_nat @ X @ A )
% 0.32/0.71 => ( ( dvd_dvd_nat @ X @ B )
% 0.32/0.71 => ( X = one_one_nat ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_common_divisor_nat
% 0.32/0.71 thf(fact_10051_vanishes__diff,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( vanishes @ X7 )
% 0.32/0.71 => ( ( vanishes @ Y7 )
% 0.32/0.71 => ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishes_diff
% 0.32/0.71 thf(fact_10052_vanishes__minus,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( vanishes @ X7 )
% 0.32/0.71 => ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X7 @ N2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishes_minus
% 0.32/0.71 thf(fact_10053_vanishes__add,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( vanishes @ X7 )
% 0.32/0.71 => ( ( vanishes @ Y7 )
% 0.32/0.71 => ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishes_add
% 0.32/0.71 thf(fact_10054_coprime__diff__one__left__nat,axiom,
% 0.32/0.71 ! [N: nat] :
% 0.32/0.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.71 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_diff_one_left_nat
% 0.32/0.71 thf(fact_10055_coprime__diff__one__right__nat,axiom,
% 0.32/0.71 ! [N: nat] :
% 0.32/0.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.32/0.71 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % coprime_diff_one_right_nat
% 0.32/0.71 thf(fact_10056_Rats__abs__nat__div__natE,axiom,
% 0.32/0.71 ! [X: real] :
% 0.32/0.71 ( ( member_real @ X @ field_5140801741446780682s_real )
% 0.32/0.71 => ~ ! [M4: nat,N4: nat] :
% 0.32/0.71 ( ( N4 != zero_zero_nat )
% 0.32/0.71 => ( ( ( abs_abs_real @ X )
% 0.32/0.71 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
% 0.32/0.71 => ~ ( algebr934650988132801477me_nat @ M4 @ N4 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Rats_abs_nat_div_natE
% 0.32/0.71 thf(fact_10057_vanishes__def,axiom,
% 0.32/0.71 ( vanishes
% 0.32/0.71 = ( ^ [X6: nat > rat] :
% 0.32/0.71 ! [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 => ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishes_def
% 0.32/0.71 thf(fact_10058_vanishesI,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ! [R2: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.32/0.71 => ? [K4: nat] :
% 0.32/0.71 ! [N4: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K4 @ N4 )
% 0.32/0.71 => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N4 ) ) @ R2 ) ) )
% 0.32/0.71 => ( vanishes @ X7 ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishesI
% 0.32/0.71 thf(fact_10059_vanishesD,axiom,
% 0.32/0.71 ! [X7: nat > rat,R: rat] :
% 0.32/0.71 ( ( vanishes @ X7 )
% 0.32/0.71 => ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.32/0.71 => ? [K3: nat] :
% 0.32/0.71 ! [N7: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K3 @ N7 )
% 0.32/0.71 => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N7 ) ) @ R ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishesD
% 0.32/0.71 thf(fact_10060_rcis__inverse,axiom,
% 0.32/0.71 ! [R: real,A: real] :
% 0.32/0.71 ( ( invers8013647133539491842omplex @ ( rcis @ R @ A ) )
% 0.32/0.71 = ( rcis @ ( divide_divide_real @ one_one_real @ R ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % rcis_inverse
% 0.32/0.71 thf(fact_10061_Re__rcis,axiom,
% 0.32/0.71 ! [R: real,A: real] :
% 0.32/0.71 ( ( re @ ( rcis @ R @ A ) )
% 0.32/0.71 = ( times_times_real @ R @ ( cos_real @ A ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Re_rcis
% 0.32/0.71 thf(fact_10062_Im__rcis,axiom,
% 0.32/0.71 ! [R: real,A: real] :
% 0.32/0.71 ( ( im @ ( rcis @ R @ A ) )
% 0.32/0.71 = ( times_times_real @ R @ ( sin_real @ A ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Im_rcis
% 0.32/0.71 thf(fact_10063_cis__rcis__eq,axiom,
% 0.32/0.71 ( cis
% 0.32/0.71 = ( rcis @ one_one_real ) ) ).
% 0.32/0.71
% 0.32/0.71 % cis_rcis_eq
% 0.32/0.71 thf(fact_10064_rcis__mult,axiom,
% 0.32/0.71 ! [R1: real,A: real,R22: real,B: real] :
% 0.32/0.71 ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 0.32/0.71 = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % rcis_mult
% 0.32/0.71 thf(fact_10065_rcis__divide,axiom,
% 0.32/0.71 ! [R1: real,A: real,R22: real,B: real] :
% 0.32/0.71 ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 0.32/0.71 = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % rcis_divide
% 0.32/0.71 thf(fact_10066_rcis__def,axiom,
% 0.32/0.71 ( rcis
% 0.32/0.71 = ( ^ [R5: real,A3: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A3 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % rcis_def
% 0.32/0.71 thf(fact_10067_DeMoivre2,axiom,
% 0.32/0.71 ! [R: real,A: real,N: nat] :
% 0.32/0.71 ( ( power_power_complex @ ( rcis @ R @ A ) @ N )
% 0.32/0.71 = ( rcis @ ( power_power_real @ R @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % DeMoivre2
% 0.32/0.71 thf(fact_10068_cauchy__def,axiom,
% 0.32/0.71 ( cauchy
% 0.32/0.71 = ( ^ [X6: nat > rat] :
% 0.32/0.71 ! [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 => ? [K2: nat] :
% 0.32/0.71 ! [M3: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ M3 )
% 0.32/0.71 => ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_def
% 0.32/0.71 thf(fact_10069_cauchyI,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ! [R2: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.32/0.71 => ? [K4: nat] :
% 0.32/0.71 ! [M4: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K4 @ M4 )
% 0.32/0.71 => ! [N4: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K4 @ N4 )
% 0.32/0.71 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) ) @ R2 ) ) ) )
% 0.32/0.71 => ( cauchy @ X7 ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchyI
% 0.32/0.71 thf(fact_10070_cauchy__mult,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( cauchy
% 0.32/0.71 @ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_mult
% 0.32/0.71 thf(fact_10071_cauchy__diff,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( cauchy
% 0.32/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_diff
% 0.32/0.71 thf(fact_10072_cauchy__const,axiom,
% 0.32/0.71 ! [X: rat] :
% 0.32/0.71 ( cauchy
% 0.32/0.71 @ ^ [N2: nat] : X ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_const
% 0.32/0.71 thf(fact_10073_cauchy__add,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( cauchy
% 0.32/0.71 @ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_add
% 0.32/0.71 thf(fact_10074_cauchy__minus,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( cauchy
% 0.32/0.71 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X7 @ N2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_minus
% 0.32/0.71 thf(fact_10075_cauchy__inverse,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ~ ( vanishes @ X7 )
% 0.32/0.71 => ( cauchy
% 0.32/0.71 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X7 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_inverse
% 0.32/0.71 thf(fact_10076_cauchy__imp__bounded,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ? [B3: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.32/0.71 & ! [N7: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N7 ) ) @ B3 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_imp_bounded
% 0.32/0.71 thf(fact_10077_vanishes__diff__inverse,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ~ ( vanishes @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ~ ( vanishes @ Y7 )
% 0.32/0.71 => ( ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) )
% 0.32/0.71 => ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X7 @ N2 ) ) @ ( inverse_inverse_rat @ ( Y7 @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % vanishes_diff_inverse
% 0.32/0.71 thf(fact_10078_cauchy__not__vanishes__cases,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ~ ( vanishes @ X7 )
% 0.32/0.71 => ? [B3: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.32/0.71 & ? [K3: nat] :
% 0.32/0.71 ( ! [N7: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K3 @ N7 )
% 0.32/0.71 => ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X7 @ N7 ) ) ) )
% 0.32/0.71 | ! [N7: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K3 @ N7 )
% 0.32/0.71 => ( ord_less_rat @ B3 @ ( X7 @ N7 ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_not_vanishes_cases
% 0.32/0.71 thf(fact_10079_cauchy__not__vanishes,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ~ ( vanishes @ X7 )
% 0.32/0.71 => ? [B3: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.32/0.71 & ? [K3: nat] :
% 0.32/0.71 ! [N7: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K3 @ N7 )
% 0.32/0.71 => ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X7 @ N7 ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchy_not_vanishes
% 0.32/0.71 thf(fact_10080_cauchyD,axiom,
% 0.32/0.71 ! [X7: nat > rat,R: rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( ord_less_rat @ zero_zero_rat @ R )
% 0.32/0.71 => ? [K3: nat] :
% 0.32/0.71 ! [M2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K3 @ M2 )
% 0.32/0.71 => ! [N7: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K3 @ N7 )
% 0.32/0.71 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M2 ) @ ( X7 @ N7 ) ) ) @ R ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cauchyD
% 0.32/0.71 thf(fact_10081_le__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ( ord_less_eq_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
% 0.32/0.71 = ( ! [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 => ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % le_Real
% 0.32/0.71 thf(fact_10082_inverse__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( ( vanishes @ X7 )
% 0.32/0.71 => ( ( inverse_inverse_real @ ( real2 @ X7 ) )
% 0.32/0.71 = zero_zero_real ) )
% 0.32/0.71 & ( ~ ( vanishes @ X7 )
% 0.32/0.71 => ( ( inverse_inverse_real @ ( real2 @ X7 ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X7 @ N2 ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % inverse_Real
% 0.32/0.71 thf(fact_10083_Real__induct,axiom,
% 0.32/0.71 ! [P: real > $o,X: real] :
% 0.32/0.71 ( ! [X10: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X10 )
% 0.32/0.71 => ( P @ ( real2 @ X10 ) ) )
% 0.32/0.71 => ( P @ X ) ) ).
% 0.32/0.71
% 0.32/0.71 % Real_induct
% 0.32/0.71 thf(fact_10084_of__int__Real,axiom,
% 0.32/0.71 ( ring_1_of_int_real
% 0.32/0.71 = ( ^ [X2: int] :
% 0.32/0.71 ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( ring_1_of_int_rat @ X2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % of_int_Real
% 0.32/0.71 thf(fact_10085_one__real__def,axiom,
% 0.32/0.71 ( one_one_real
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : one_one_rat ) ) ).
% 0.32/0.71
% 0.32/0.71 % one_real_def
% 0.32/0.71 thf(fact_10086_zero__real__def,axiom,
% 0.32/0.71 ( zero_zero_real
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat ) ) ).
% 0.32/0.71
% 0.32/0.71 % zero_real_def
% 0.32/0.71 thf(fact_10087_of__nat__Real,axiom,
% 0.32/0.71 ( semiri5074537144036343181t_real
% 0.32/0.71 = ( ^ [X2: nat] :
% 0.32/0.71 ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( semiri681578069525770553at_rat @ X2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % of_nat_Real
% 0.32/0.71 thf(fact_10088_minus__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( uminus_uminus_real @ ( real2 @ X7 ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X7 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % minus_Real
% 0.32/0.71 thf(fact_10089_add__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ( plus_plus_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % add_Real
% 0.32/0.71 thf(fact_10090_mult__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ( times_times_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % mult_Real
% 0.32/0.71 thf(fact_10091_diff__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ( minus_minus_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % diff_Real
% 0.32/0.71 thf(fact_10092_eq__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ( ( real2 @ X7 )
% 0.32/0.71 = ( real2 @ Y7 ) )
% 0.32/0.71 = ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % eq_Real
% 0.32/0.71 thf(fact_10093_not__positive__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( ~ ( positive2 @ ( real2 @ X7 ) ) )
% 0.32/0.71 = ( ! [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 => ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_eq_rat @ ( X7 @ N2 ) @ R5 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % not_positive_Real
% 0.32/0.71 thf(fact_10094_positive__Real,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( positive2 @ ( real2 @ X7 ) )
% 0.32/0.71 = ( ? [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 & ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_rat @ R5 @ ( X7 @ N2 ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % positive_Real
% 0.32/0.71 thf(fact_10095_Real_Opositive__minus,axiom,
% 0.32/0.71 ! [X: real] :
% 0.32/0.71 ( ~ ( positive2 @ X )
% 0.32/0.71 => ( ( X != zero_zero_real )
% 0.32/0.71 => ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Real.positive_minus
% 0.32/0.71 thf(fact_10096_Real_Opositive__mult,axiom,
% 0.32/0.71 ! [X: real,Y3: real] :
% 0.32/0.71 ( ( positive2 @ X )
% 0.32/0.71 => ( ( positive2 @ Y3 )
% 0.32/0.71 => ( positive2 @ ( times_times_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Real.positive_mult
% 0.32/0.71 thf(fact_10097_Real_Opositive__add,axiom,
% 0.32/0.71 ! [X: real,Y3: real] :
% 0.32/0.71 ( ( positive2 @ X )
% 0.32/0.71 => ( ( positive2 @ Y3 )
% 0.32/0.71 => ( positive2 @ ( plus_plus_real @ X @ Y3 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Real.positive_add
% 0.32/0.71 thf(fact_10098_Real_Opositive__zero,axiom,
% 0.32/0.71 ~ ( positive2 @ zero_zero_real ) ).
% 0.32/0.71
% 0.32/0.71 % Real.positive_zero
% 0.32/0.71 thf(fact_10099_less__real__def,axiom,
% 0.32/0.71 ( ord_less_real
% 0.32/0.71 = ( ^ [X2: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % less_real_def
% 0.32/0.71 thf(fact_10100_Real_Opositive_Orep__eq,axiom,
% 0.32/0.71 ( positive2
% 0.32/0.71 = ( ^ [X2: real] :
% 0.32/0.71 ? [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 & ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_rat @ R5 @ ( rep_real @ X2 @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Real.positive.rep_eq
% 0.32/0.71 thf(fact_10101_inverse__real_Oabs__eq,axiom,
% 0.32/0.71 ! [X: nat > rat] :
% 0.32/0.71 ( ( realrel @ X @ X )
% 0.32/0.71 => ( ( inverse_inverse_real @ ( real2 @ X ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ( if_nat_rat @ ( vanishes @ X )
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat
% 0.32/0.71 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % inverse_real.abs_eq
% 0.32/0.71 thf(fact_10102_realrel__refl,axiom,
% 0.32/0.71 ! [X7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( realrel @ X7 @ X7 ) ) ).
% 0.32/0.71
% 0.32/0.71 % realrel_refl
% 0.32/0.71 thf(fact_10103_zero__real_Orsp,axiom,
% 0.32/0.71 ( realrel
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat ) ).
% 0.32/0.71
% 0.32/0.71 % zero_real.rsp
% 0.32/0.71 thf(fact_10104_one__real_Orsp,axiom,
% 0.32/0.71 ( realrel
% 0.32/0.71 @ ^ [N2: nat] : one_one_rat
% 0.32/0.71 @ ^ [N2: nat] : one_one_rat ) ).
% 0.32/0.71
% 0.32/0.71 % one_real.rsp
% 0.32/0.71 thf(fact_10105_real_Oabs__induct,axiom,
% 0.32/0.71 ! [P: real > $o,X: real] :
% 0.32/0.71 ( ! [Y4: nat > rat] :
% 0.32/0.71 ( ( realrel @ Y4 @ Y4 )
% 0.32/0.71 => ( P @ ( real2 @ Y4 ) ) )
% 0.32/0.71 => ( P @ X ) ) ).
% 0.32/0.71
% 0.32/0.71 % real.abs_induct
% 0.32/0.71 thf(fact_10106_uminus__real_Oabs__eq,axiom,
% 0.32/0.71 ! [X: nat > rat] :
% 0.32/0.71 ( ( realrel @ X @ X )
% 0.32/0.71 => ( ( uminus_uminus_real @ ( real2 @ X ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % uminus_real.abs_eq
% 0.32/0.71 thf(fact_10107_plus__real_Oabs__eq,axiom,
% 0.32/0.71 ! [Xa: nat > rat,X: nat > rat] :
% 0.32/0.71 ( ( realrel @ Xa @ Xa )
% 0.32/0.71 => ( ( realrel @ X @ X )
% 0.32/0.71 => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( plus_plus_rat @ ( Xa @ N2 ) @ ( X @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % plus_real.abs_eq
% 0.32/0.71 thf(fact_10108_times__real_Oabs__eq,axiom,
% 0.32/0.71 ! [Xa: nat > rat,X: nat > rat] :
% 0.32/0.71 ( ( realrel @ Xa @ Xa )
% 0.32/0.71 => ( ( realrel @ X @ X )
% 0.32/0.71 => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 0.32/0.71 = ( real2
% 0.32/0.71 @ ^ [N2: nat] : ( times_times_rat @ ( Xa @ N2 ) @ ( X @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % times_real.abs_eq
% 0.32/0.71 thf(fact_10109_realrelI,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y7: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) )
% 0.32/0.71 => ( realrel @ X7 @ Y7 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % realrelI
% 0.32/0.71 thf(fact_10110_realrel__def,axiom,
% 0.32/0.71 ( realrel
% 0.32/0.71 = ( ^ [X6: nat > rat,Y8: nat > rat] :
% 0.32/0.71 ( ( cauchy @ X6 )
% 0.32/0.71 & ( cauchy @ Y8 )
% 0.32/0.71 & ( vanishes
% 0.32/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % realrel_def
% 0.32/0.71 thf(fact_10111_Real_Opositive_Oabs__eq,axiom,
% 0.32/0.71 ! [X: nat > rat] :
% 0.32/0.71 ( ( realrel @ X @ X )
% 0.32/0.71 => ( ( positive2 @ ( real2 @ X ) )
% 0.32/0.71 = ( ? [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 & ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_rat @ R5 @ ( X @ N2 ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Real.positive.abs_eq
% 0.32/0.71 thf(fact_10112_inverse__real__def,axiom,
% 0.32/0.71 ( inverse_inverse_real
% 0.32/0.71 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 0.32/0.71 @ ^ [X6: nat > rat] :
% 0.32/0.71 ( if_nat_rat @ ( vanishes @ X6 )
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat
% 0.32/0.71 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % inverse_real_def
% 0.32/0.71 thf(fact_10113_cr__real__def,axiom,
% 0.32/0.71 ( cr_real
% 0.32/0.71 = ( ^ [X2: nat > rat,Y: real] :
% 0.32/0.71 ( ( realrel @ X2 @ X2 )
% 0.32/0.71 & ( ( real2 @ X2 )
% 0.32/0.71 = Y ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cr_real_def
% 0.32/0.71 thf(fact_10114_uminus__real__def,axiom,
% 0.32/0.71 ( uminus_uminus_real
% 0.32/0.71 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 0.32/0.71 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % uminus_real_def
% 0.32/0.71 thf(fact_10115_times__real__def,axiom,
% 0.32/0.71 ( times_times_real
% 0.32/0.71 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 0.32/0.71 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % times_real_def
% 0.32/0.71 thf(fact_10116_plus__real__def,axiom,
% 0.32/0.71 ( plus_plus_real
% 0.32/0.71 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 0.32/0.71 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % plus_real_def
% 0.32/0.71 thf(fact_10117_Real_Opositive_Orsp,axiom,
% 0.32/0.71 ( bNF_re728719798268516973at_o_o @ realrel
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [X6: nat > rat] :
% 0.32/0.71 ? [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 & ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 0.32/0.71 @ ^ [X6: nat > rat] :
% 0.32/0.71 ? [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 & ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Real.positive.rsp
% 0.32/0.71 thf(fact_10118_uminus__real_Orsp,axiom,
% 0.32/0.71 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 0.32/0.71 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
% 0.32/0.71 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % uminus_real.rsp
% 0.32/0.71 thf(fact_10119_plus__real_Orsp,axiom,
% 0.32/0.71 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 0.32/0.71 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 0.32/0.71 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % plus_real.rsp
% 0.32/0.71 thf(fact_10120_times__real_Orsp,axiom,
% 0.32/0.71 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 0.32/0.71 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 0.32/0.71 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % times_real.rsp
% 0.32/0.71 thf(fact_10121_less__natural_Orsp,axiom,
% 0.32/0.71 ( bNF_re578469030762574527_nat_o
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re4705727531993890431at_o_o
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ord_less_nat
% 0.32/0.71 @ ord_less_nat ) ).
% 0.32/0.71
% 0.32/0.71 % less_natural.rsp
% 0.32/0.71 thf(fact_10122_less__integer_Orsp,axiom,
% 0.32/0.71 ( bNF_re3403563459893282935_int_o
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re5089333283451836215nt_o_o
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ord_less_int
% 0.32/0.71 @ ord_less_int ) ).
% 0.32/0.71
% 0.32/0.71 % less_integer.rsp
% 0.32/0.71 thf(fact_10123_sub_Orsp,axiom,
% 0.32/0.71 ( bNF_re8402795839162346335um_int
% 0.32/0.71 @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re1822329894187522285nt_int
% 0.32/0.71 @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) )
% 0.32/0.71 @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % sub.rsp
% 0.32/0.71 thf(fact_10124_divide__natural_Orsp,axiom,
% 0.32/0.71 ( bNF_re1345281282404953727at_nat
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re5653821019739307937at_nat
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.71 @ divide_divide_nat
% 0.32/0.71 @ divide_divide_nat ) ).
% 0.32/0.71
% 0.32/0.71 % divide_natural.rsp
% 0.32/0.71 thf(fact_10125_divide__integer_Orsp,axiom,
% 0.32/0.71 ( bNF_re711492959462206631nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re4712519889275205905nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.71 @ divide_divide_int
% 0.32/0.71 @ divide_divide_int ) ).
% 0.32/0.71
% 0.32/0.71 % divide_integer.rsp
% 0.32/0.71 thf(fact_10126_minus__integer_Orsp,axiom,
% 0.32/0.71 ( bNF_re711492959462206631nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re4712519889275205905nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.71 @ minus_minus_int
% 0.32/0.71 @ minus_minus_int ) ).
% 0.32/0.71
% 0.32/0.71 % minus_integer.rsp
% 0.32/0.71 thf(fact_10127_minus__natural_Orsp,axiom,
% 0.32/0.71 ( bNF_re1345281282404953727at_nat
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re5653821019739307937at_nat
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.71 @ minus_minus_nat
% 0.32/0.71 @ minus_minus_nat ) ).
% 0.32/0.71
% 0.32/0.71 % minus_natural.rsp
% 0.32/0.71 thf(fact_10128_times__natural_Orsp,axiom,
% 0.32/0.71 ( bNF_re1345281282404953727at_nat
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re5653821019739307937at_nat
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.71 @ times_times_nat
% 0.32/0.71 @ times_times_nat ) ).
% 0.32/0.71
% 0.32/0.71 % times_natural.rsp
% 0.32/0.71 thf(fact_10129_times__integer_Orsp,axiom,
% 0.32/0.71 ( bNF_re711492959462206631nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re4712519889275205905nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.71 @ times_times_int
% 0.32/0.71 @ times_times_int ) ).
% 0.32/0.71
% 0.32/0.71 % times_integer.rsp
% 0.32/0.71 thf(fact_10130_dup_Orsp,axiom,
% 0.32/0.71 ( bNF_re4712519889275205905nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 )
% 0.32/0.71 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ).
% 0.32/0.71
% 0.32/0.71 % dup.rsp
% 0.32/0.71 thf(fact_10131_plus__integer_Orsp,axiom,
% 0.32/0.71 ( bNF_re711492959462206631nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re4712519889275205905nt_int
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 0.32/0.71 @ plus_plus_int
% 0.32/0.71 @ plus_plus_int ) ).
% 0.32/0.71
% 0.32/0.71 % plus_integer.rsp
% 0.32/0.71 thf(fact_10132_plus__natural_Orsp,axiom,
% 0.32/0.71 ( bNF_re1345281282404953727at_nat
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re5653821019739307937at_nat
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 0.32/0.71 @ plus_plus_nat
% 0.32/0.71 @ plus_plus_nat ) ).
% 0.32/0.71
% 0.32/0.71 % plus_natural.rsp
% 0.32/0.71 thf(fact_10133_less__eq__natural_Orsp,axiom,
% 0.32/0.71 ( bNF_re578469030762574527_nat_o
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re4705727531993890431at_o_o
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ord_less_eq_nat
% 0.32/0.71 @ ord_less_eq_nat ) ).
% 0.32/0.71
% 0.32/0.71 % less_eq_natural.rsp
% 0.32/0.71 thf(fact_10134_less__eq__integer_Orsp,axiom,
% 0.32/0.71 ( bNF_re3403563459893282935_int_o
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ( bNF_re5089333283451836215nt_o_o
% 0.32/0.71 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ord_less_eq_int
% 0.32/0.71 @ ord_less_eq_int ) ).
% 0.32/0.71
% 0.32/0.71 % less_eq_integer.rsp
% 0.32/0.71 thf(fact_10135_inverse__real_Orsp,axiom,
% 0.32/0.71 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 0.32/0.71 @ ^ [X6: nat > rat] :
% 0.32/0.71 ( if_nat_rat @ ( vanishes @ X6 )
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat
% 0.32/0.71 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
% 0.32/0.71 @ ^ [X6: nat > rat] :
% 0.32/0.71 ( if_nat_rat @ ( vanishes @ X6 )
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat
% 0.32/0.71 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % inverse_real.rsp
% 0.32/0.71 thf(fact_10136_Real_Opositive_Otransfer,axiom,
% 0.32/0.71 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [X6: nat > rat] :
% 0.32/0.71 ? [R5: rat] :
% 0.32/0.71 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.32/0.71 & ? [K2: nat] :
% 0.32/0.71 ! [N2: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.32/0.71 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 0.32/0.71 @ positive2 ) ).
% 0.32/0.71
% 0.32/0.71 % Real.positive.transfer
% 0.32/0.71 thf(fact_10137_plus__rat_Otransfer,axiom,
% 0.32/0.71 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.32/0.71 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
% 0.32/0.71 @ plus_plus_rat ) ).
% 0.32/0.71
% 0.32/0.71 % plus_rat.transfer
% 0.32/0.71 thf(fact_10138_real_Orel__eq__transfer,axiom,
% 0.32/0.71 ( bNF_re4521903465945308077real_o @ pcr_real
% 0.32/0.71 @ ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ realrel
% 0.32/0.71 @ ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) ) ).
% 0.32/0.71
% 0.32/0.71 % real.rel_eq_transfer
% 0.32/0.71 thf(fact_10139_zero__real_Otransfer,axiom,
% 0.32/0.71 ( pcr_real
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat
% 0.32/0.71 @ zero_zero_real ) ).
% 0.32/0.71
% 0.32/0.71 % zero_real.transfer
% 0.32/0.71 thf(fact_10140_real_Opcr__cr__eq,axiom,
% 0.32/0.71 pcr_real = cr_real ).
% 0.32/0.71
% 0.32/0.71 % real.pcr_cr_eq
% 0.32/0.71 thf(fact_10141_one__real_Otransfer,axiom,
% 0.32/0.71 ( pcr_real
% 0.32/0.71 @ ^ [N2: nat] : one_one_rat
% 0.32/0.71 @ one_one_real ) ).
% 0.32/0.71
% 0.32/0.71 % one_real.transfer
% 0.32/0.71 thf(fact_10142_cr__real__eq,axiom,
% 0.32/0.71 ( pcr_real
% 0.32/0.71 = ( ^ [X2: nat > rat,Y: real] :
% 0.32/0.71 ( ( cauchy @ X2 )
% 0.32/0.71 & ( ( real2 @ X2 )
% 0.32/0.71 = Y ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % cr_real_eq
% 0.32/0.71 thf(fact_10143_uminus__real_Otransfer,axiom,
% 0.32/0.71 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 0.32/0.71 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
% 0.32/0.71 @ uminus_uminus_real ) ).
% 0.32/0.71
% 0.32/0.71 % uminus_real.transfer
% 0.32/0.71 thf(fact_10144_plus__real_Otransfer,axiom,
% 0.32/0.71 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 0.32/0.71 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 0.32/0.71 @ plus_plus_real ) ).
% 0.32/0.71
% 0.32/0.71 % plus_real.transfer
% 0.32/0.71 thf(fact_10145_times__real_Otransfer,axiom,
% 0.32/0.71 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 0.32/0.71 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 0.32/0.71 @ times_times_real ) ).
% 0.32/0.71
% 0.32/0.71 % times_real.transfer
% 0.32/0.71 thf(fact_10146_inverse__real_Otransfer,axiom,
% 0.32/0.71 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 0.32/0.71 @ ^ [X6: nat > rat] :
% 0.32/0.71 ( if_nat_rat @ ( vanishes @ X6 )
% 0.32/0.71 @ ^ [N2: nat] : zero_zero_rat
% 0.32/0.71 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
% 0.32/0.71 @ inverse_inverse_real ) ).
% 0.32/0.71
% 0.32/0.71 % inverse_real.transfer
% 0.32/0.71 thf(fact_10147_times__rat_Otransfer,axiom,
% 0.32/0.71 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 0.32/0.71 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
% 0.32/0.71 @ times_times_rat ) ).
% 0.32/0.71
% 0.32/0.71 % times_rat.transfer
% 0.32/0.71 thf(fact_10148_Rat_Opositive_Otransfer,axiom,
% 0.32/0.71 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
% 0.32/0.71 @ positive ) ).
% 0.32/0.71
% 0.32/0.71 % Rat.positive.transfer
% 0.32/0.71 thf(fact_10149_times__int_Otransfer,axiom,
% 0.32/0.71 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) )
% 0.32/0.71 @ times_times_int ) ).
% 0.32/0.71
% 0.32/0.71 % times_int.transfer
% 0.32/0.71 thf(fact_10150_nat_Otransfer,axiom,
% 0.32/0.71 ( bNF_re4555766996558763186at_nat @ pcr_int
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.32/0.71 @ nat2 ) ).
% 0.32/0.71
% 0.32/0.71 % nat.transfer
% 0.32/0.71 thf(fact_10151_one__int_Otransfer,axiom,
% 0.32/0.71 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 0.32/0.71
% 0.32/0.71 % one_int.transfer
% 0.32/0.71 thf(fact_10152_less__int_Otransfer,axiom,
% 0.32/0.71 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.32/0.71 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ( produc8739625826339149834_nat_o
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc6081775807080527818_nat_o
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
% 0.32/0.71 @ ord_less_int ) ).
% 0.32/0.71
% 0.32/0.71 % less_int.transfer
% 0.32/0.71 thf(fact_10153_less__eq__int_Otransfer,axiom,
% 0.32/0.71 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.32/0.71 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ( produc8739625826339149834_nat_o
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc6081775807080527818_nat_o
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
% 0.32/0.71 @ ord_less_eq_int ) ).
% 0.32/0.71
% 0.32/0.71 % less_eq_int.transfer
% 0.32/0.71 thf(fact_10154_plus__int_Otransfer,axiom,
% 0.32/0.71 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 0.32/0.71 @ plus_plus_int ) ).
% 0.32/0.71
% 0.32/0.71 % plus_int.transfer
% 0.32/0.71 thf(fact_10155_minus__int_Otransfer,axiom,
% 0.32/0.71 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) )
% 0.32/0.71 @ minus_minus_int ) ).
% 0.32/0.71
% 0.32/0.71 % minus_int.transfer
% 0.32/0.71 thf(fact_10156_times__int_Orsp,axiom,
% 0.32/0.71 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % times_int.rsp
% 0.32/0.71 thf(fact_10157_intrel__iff,axiom,
% 0.32/0.71 ! [X: nat,Y3: nat,U: nat,V: nat] :
% 0.32/0.71 ( ( intrel @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
% 0.32/0.71 = ( ( plus_plus_nat @ X @ V )
% 0.32/0.71 = ( plus_plus_nat @ U @ Y3 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % intrel_iff
% 0.32/0.71 thf(fact_10158_nat_Orsp,axiom,
% 0.32/0.71 ( bNF_re8246922863344978751at_nat @ intrel
% 0.32/0.71 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 0.32/0.71 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 0.32/0.71 @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 0.32/0.71
% 0.32/0.71 % nat.rsp
% 0.32/0.71 thf(fact_10159_le__RealI,axiom,
% 0.32/0.71 ! [Y7: nat > rat,X: real] :
% 0.32/0.71 ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ! [N4: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) )
% 0.32/0.71 => ( ord_less_eq_real @ X @ ( real2 @ Y7 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % le_RealI
% 0.32/0.71 thf(fact_10160_Real__leI,axiom,
% 0.32/0.71 ! [X7: nat > rat,Y3: real] :
% 0.32/0.71 ( ( cauchy @ X7 )
% 0.32/0.71 => ( ! [N4: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X7 @ N4 ) ) @ Y3 )
% 0.32/0.71 => ( ord_less_eq_real @ ( real2 @ X7 ) @ Y3 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Real_leI
% 0.32/0.71 thf(fact_10161_less__RealD,axiom,
% 0.32/0.71 ! [Y7: nat > rat,X: real] :
% 0.32/0.71 ( ( cauchy @ Y7 )
% 0.32/0.71 => ( ( ord_less_real @ X @ ( real2 @ Y7 ) )
% 0.32/0.71 => ? [N4: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % less_RealD
% 0.32/0.71 thf(fact_10162_of__rat__dense,axiom,
% 0.32/0.71 ! [X: real,Y3: real] :
% 0.32/0.71 ( ( ord_less_real @ X @ Y3 )
% 0.32/0.71 => ? [Q3: rat] :
% 0.32/0.71 ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q3 ) )
% 0.32/0.71 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y3 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % of_rat_dense
% 0.32/0.71 thf(fact_10163_of__rat__Real,axiom,
% 0.32/0.71 ( field_7254667332652039916t_real
% 0.32/0.71 = ( ^ [X2: rat] :
% 0.32/0.71 ( real2
% 0.32/0.71 @ ^ [N2: nat] : X2 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % of_rat_Real
% 0.32/0.71 thf(fact_10164_one__int_Orsp,axiom,
% 0.32/0.71 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.32/0.71
% 0.32/0.71 % one_int.rsp
% 0.32/0.71 thf(fact_10165_intrel__def,axiom,
% 0.32/0.71 ( intrel
% 0.32/0.71 = ( produc8739625826339149834_nat_o
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc6081775807080527818_nat_o
% 0.32/0.71 @ ^ [U3: nat,V4: nat] :
% 0.32/0.71 ( ( plus_plus_nat @ X2 @ V4 )
% 0.32/0.71 = ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % intrel_def
% 0.32/0.71 thf(fact_10166_less__int_Orsp,axiom,
% 0.32/0.71 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.32/0.71 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ( produc8739625826339149834_nat_o
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc6081775807080527818_nat_o
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
% 0.32/0.71 @ ( produc8739625826339149834_nat_o
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc6081775807080527818_nat_o
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % less_int.rsp
% 0.32/0.71 thf(fact_10167_less__eq__int_Orsp,axiom,
% 0.32/0.71 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.32/0.71 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 0.32/0.71 @ ( produc8739625826339149834_nat_o
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc6081775807080527818_nat_o
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
% 0.32/0.71 @ ( produc8739625826339149834_nat_o
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc6081775807080527818_nat_o
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % less_eq_int.rsp
% 0.32/0.71 thf(fact_10168_plus__int_Orsp,axiom,
% 0.32/0.71 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % plus_int.rsp
% 0.32/0.71 thf(fact_10169_minus__int_Orsp,axiom,
% 0.32/0.71 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) )
% 0.32/0.71 @ ( produc27273713700761075at_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] :
% 0.32/0.71 ( produc2626176000494625587at_nat
% 0.32/0.71 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % minus_int.rsp
% 0.32/0.71 thf(fact_10170_le__enumerate,axiom,
% 0.32/0.71 ! [S2: set_nat,N: nat] :
% 0.32/0.71 ( ~ ( finite_finite_nat @ S2 )
% 0.32/0.71 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % le_enumerate
% 0.32/0.71 thf(fact_10171_finite__le__enumerate,axiom,
% 0.32/0.71 ! [S2: set_nat,N: nat] :
% 0.32/0.71 ( ( finite_finite_nat @ S2 )
% 0.32/0.71 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
% 0.32/0.71 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % finite_le_enumerate
% 0.32/0.71 thf(fact_10172_Least__eq__0,axiom,
% 0.32/0.71 ! [P: nat > $o] :
% 0.32/0.71 ( ( P @ zero_zero_nat )
% 0.32/0.71 => ( ( ord_Least_nat @ P )
% 0.32/0.71 = zero_zero_nat ) ) ).
% 0.32/0.71
% 0.32/0.71 % Least_eq_0
% 0.32/0.71 thf(fact_10173_Least__Suc,axiom,
% 0.32/0.71 ! [P: nat > $o,N: nat] :
% 0.32/0.71 ( ( P @ N )
% 0.32/0.71 => ( ~ ( P @ zero_zero_nat )
% 0.32/0.71 => ( ( ord_Least_nat @ P )
% 0.32/0.71 = ( suc
% 0.32/0.71 @ ( ord_Least_nat
% 0.32/0.71 @ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Least_Suc
% 0.32/0.71 thf(fact_10174_Least__Suc2,axiom,
% 0.32/0.71 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 0.32/0.71 ( ( P @ N )
% 0.32/0.71 => ( ( Q @ M )
% 0.32/0.71 => ( ~ ( P @ zero_zero_nat )
% 0.32/0.71 => ( ! [K3: nat] :
% 0.32/0.71 ( ( P @ ( suc @ K3 ) )
% 0.32/0.71 = ( Q @ K3 ) )
% 0.32/0.71 => ( ( ord_Least_nat @ P )
% 0.32/0.71 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Least_Suc2
% 0.32/0.71 thf(fact_10175_Sup__real__def,axiom,
% 0.32/0.71 ( comple1385675409528146559p_real
% 0.32/0.71 = ( ^ [X6: set_real] :
% 0.32/0.71 ( ord_Least_real
% 0.32/0.71 @ ^ [Z2: real] :
% 0.32/0.71 ! [X2: real] :
% 0.32/0.71 ( ( member_real @ X2 @ X6 )
% 0.32/0.71 => ( ord_less_eq_real @ X2 @ Z2 ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Sup_real_def
% 0.32/0.71 thf(fact_10176_has__field__derivative__iff__has__vector__derivative,axiom,
% 0.32/0.71 has_fi5821293074295781190e_real = has_ve631408500373753343e_real ).
% 0.32/0.71
% 0.32/0.71 % has_field_derivative_iff_has_vector_derivative
% 0.32/0.71 thf(fact_10177_has__vector__derivative__id,axiom,
% 0.32/0.71 ! [Net: filter_real] :
% 0.32/0.71 ( has_ve631408500373753343e_real
% 0.32/0.71 @ ^ [X2: real] : X2
% 0.32/0.71 @ one_one_real
% 0.32/0.71 @ Net ) ).
% 0.32/0.71
% 0.32/0.71 % has_vector_derivative_id
% 0.32/0.71 thf(fact_10178_plus__rat_Orsp,axiom,
% 0.32/0.71 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.32/0.71 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
% 0.32/0.71 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % plus_rat.rsp
% 0.32/0.71 thf(fact_10179_ratrel__iff,axiom,
% 0.32/0.71 ( ratrel
% 0.32/0.71 = ( ^ [X2: product_prod_int_int,Y: product_prod_int_int] :
% 0.32/0.71 ( ( ( product_snd_int_int @ X2 )
% 0.32/0.71 != zero_zero_int )
% 0.32/0.71 & ( ( product_snd_int_int @ Y )
% 0.32/0.71 != zero_zero_int )
% 0.32/0.71 & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) )
% 0.32/0.71 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % ratrel_iff
% 0.32/0.71 thf(fact_10180_ratrel__def,axiom,
% 0.32/0.71 ( ratrel
% 0.32/0.71 = ( ^ [X2: product_prod_int_int,Y: product_prod_int_int] :
% 0.32/0.71 ( ( ( product_snd_int_int @ X2 )
% 0.32/0.71 != zero_zero_int )
% 0.32/0.71 & ( ( product_snd_int_int @ Y )
% 0.32/0.71 != zero_zero_int )
% 0.32/0.71 & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) )
% 0.32/0.71 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % ratrel_def
% 0.32/0.71 thf(fact_10181_times__rat_Orsp,axiom,
% 0.32/0.71 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 0.32/0.71 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
% 0.32/0.71 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % times_rat.rsp
% 0.32/0.71 thf(fact_10182_Rat_Opositive_Orsp,axiom,
% 0.32/0.71 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 0.32/0.71 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 0.32/0.71 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
% 0.32/0.71 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Rat.positive.rsp
% 0.32/0.71 thf(fact_10183_plus__rat_Oabs__eq,axiom,
% 0.32/0.71 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 0.32/0.71 ( ( ratrel @ Xa @ Xa )
% 0.32/0.71 => ( ( ratrel @ X @ X )
% 0.32/0.71 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.32/0.71 = ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % plus_rat.abs_eq
% 0.32/0.71 thf(fact_10184_Rat_Opositive_Oabs__eq,axiom,
% 0.32/0.71 ! [X: product_prod_int_int] :
% 0.32/0.71 ( ( ratrel @ X @ X )
% 0.32/0.71 => ( ( positive @ ( abs_Rat @ X ) )
% 0.32/0.71 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Rat.positive.abs_eq
% 0.32/0.71 thf(fact_10185_times__rat_Oabs__eq,axiom,
% 0.32/0.71 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 0.32/0.71 ( ( ratrel @ Xa @ Xa )
% 0.32/0.71 => ( ( ratrel @ X @ X )
% 0.32/0.71 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.32/0.71 = ( 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 ) ) ) ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % times_rat.abs_eq
% 0.32/0.71 thf(fact_10186_last__upt,axiom,
% 0.32/0.71 ! [I: nat,J: nat] :
% 0.32/0.71 ( ( ord_less_nat @ I @ J )
% 0.32/0.71 => ( ( last_nat @ ( upt @ I @ J ) )
% 0.32/0.71 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % last_upt
% 0.32/0.71 thf(fact_10187_set__decode__div__2,axiom,
% 0.32/0.71 ! [X: nat] :
% 0.32/0.71 ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.32/0.71 = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % set_decode_div_2
% 0.32/0.71 thf(fact_10188_set__encode__vimage__Suc,axiom,
% 0.32/0.71 ! [A2: set_nat] :
% 0.32/0.71 ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
% 0.32/0.71 = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % set_encode_vimage_Suc
% 0.32/0.71 thf(fact_10189_Bseq__monoseq__convergent_H__dec,axiom,
% 0.32/0.71 ! [F: nat > real,M5: nat] :
% 0.32/0.71 ( ( bfun_nat_real
% 0.32/0.71 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
% 0.32/0.71 @ at_top_nat )
% 0.32/0.71 => ( ! [M4: nat,N4: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ M5 @ M4 )
% 0.32/0.71 => ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.71 => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ M4 ) ) ) )
% 0.32/0.71 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Bseq_monoseq_convergent'_dec
% 0.32/0.71 thf(fact_10190_Bseq__mono__convergent,axiom,
% 0.32/0.71 ! [X7: nat > real] :
% 0.32/0.71 ( ( bfun_nat_real @ X7 @ at_top_nat )
% 0.32/0.71 => ( ! [M4: nat,N4: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.71 => ( ord_less_eq_real @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 0.32/0.71 => ( topolo7531315842566124627t_real @ X7 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Bseq_mono_convergent
% 0.32/0.71 thf(fact_10191_convergent__realpow,axiom,
% 0.32/0.71 ! [X: real] :
% 0.32/0.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.32/0.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 0.32/0.71 => ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % convergent_realpow
% 0.32/0.71 thf(fact_10192_Bseq__monoseq__convergent_H__inc,axiom,
% 0.32/0.71 ! [F: nat > real,M5: nat] :
% 0.32/0.71 ( ( bfun_nat_real
% 0.32/0.71 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
% 0.32/0.71 @ at_top_nat )
% 0.32/0.71 => ( ! [M4: nat,N4: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ M5 @ M4 )
% 0.32/0.71 => ( ( ord_less_eq_nat @ M4 @ N4 )
% 0.32/0.71 => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N4 ) ) ) )
% 0.32/0.71 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % Bseq_monoseq_convergent'_inc
% 0.32/0.71 thf(fact_10193_filtermap__at__right__shift,axiom,
% 0.32/0.71 ! [D: real,A: real] :
% 0.32/0.71 ( ( filtermap_real_real
% 0.32/0.71 @ ^ [X2: real] : ( minus_minus_real @ X2 @ D )
% 0.32/0.71 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 0.32/0.71 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % filtermap_at_right_shift
% 0.32/0.71 thf(fact_10194_at__right__to__0,axiom,
% 0.32/0.71 ! [A: real] :
% 0.32/0.71 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 0.32/0.71 = ( filtermap_real_real
% 0.32/0.71 @ ^ [X2: real] : ( plus_plus_real @ X2 @ A )
% 0.32/0.71 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % at_right_to_0
% 0.32/0.71 thf(fact_10195_pair__lessI2,axiom,
% 0.32/0.71 ! [A: nat,B: nat,S: nat,T: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.71 => ( ( ord_less_nat @ S @ T )
% 0.32/0.71 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % pair_lessI2
% 0.32/0.71 thf(fact_10196_pair__less__iff1,axiom,
% 0.32/0.71 ! [X: nat,Y3: nat,Z: nat] :
% 0.32/0.71 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
% 0.32/0.71 = ( ord_less_nat @ Y3 @ Z ) ) ).
% 0.32/0.71
% 0.32/0.71 % pair_less_iff1
% 0.32/0.71 thf(fact_10197_pair__lessI1,axiom,
% 0.32/0.71 ! [A: nat,B: nat,S: nat,T: nat] :
% 0.32/0.71 ( ( ord_less_nat @ A @ B )
% 0.32/0.71 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% 0.32/0.71
% 0.32/0.71 % pair_lessI1
% 0.32/0.71 thf(fact_10198_bot__nat__0_Oordering__top__axioms,axiom,
% 0.32/0.71 ( ordering_top_nat
% 0.32/0.71 @ ^ [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X2 )
% 0.32/0.71 @ ^ [X2: nat,Y: nat] : ( ord_less_nat @ Y @ X2 )
% 0.32/0.71 @ zero_zero_nat ) ).
% 0.32/0.71
% 0.32/0.71 % bot_nat_0.ordering_top_axioms
% 0.32/0.71 thf(fact_10199_pair__leqI2,axiom,
% 0.32/0.71 ! [A: nat,B: nat,S: nat,T: nat] :
% 0.32/0.71 ( ( ord_less_eq_nat @ A @ B )
% 0.32/0.71 => ( ( ord_less_eq_nat @ S @ T )
% 0.32/0.71 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % pair_leqI2
% 0.32/0.71 thf(fact_10200_pair__leqI1,axiom,
% 0.32/0.71 ! [A: nat,B: nat,S: nat,T: nat] :
% 0.32/0.71 ( ( ord_less_nat @ A @ B )
% 0.32/0.71 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% 0.32/0.71
% 0.32/0.71 % pair_leqI1
% 0.32/0.71 thf(fact_10201_euclidean__size__int__def,axiom,
% 0.32/0.71 ( euclid4774559944035922753ze_int
% 0.32/0.71 = ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% 0.32/0.71
% 0.32/0.71 % euclidean_size_int_def
% 0.32/0.71 thf(fact_10202_abs__division__segment,axiom,
% 0.32/0.71 ! [K: int] :
% 0.32/0.71 ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
% 0.32/0.71 = one_one_int ) ).
% 0.32/0.71
% 0.32/0.71 % abs_division_segment
% 0.32/0.71 thf(fact_10203_division__segment__nat__def,axiom,
% 0.32/0.71 ( euclid3398187327856392827nt_nat
% 0.32/0.71 = ( ^ [N2: nat] : one_one_nat ) ) ).
% 0.32/0.71
% 0.32/0.71 % division_segment_nat_def
% 0.32/0.71 thf(fact_10204_division__segment__eq__sgn,axiom,
% 0.32/0.71 ! [K: int] :
% 0.32/0.71 ( ( K != zero_zero_int )
% 0.32/0.71 => ( ( euclid3395696857347342551nt_int @ K )
% 0.32/0.71 = ( sgn_sgn_int @ K ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % division_segment_eq_sgn
% 0.32/0.71 thf(fact_10205_division__segment__int__def,axiom,
% 0.32/0.71 ( euclid3395696857347342551nt_int
% 0.32/0.71 = ( ^ [K2: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % division_segment_int_def
% 0.32/0.71 thf(fact_10206_less__eq__enat__def,axiom,
% 0.32/0.71 ( ord_le2932123472753598470d_enat
% 0.32/0.71 = ( ^ [M3: extended_enat] :
% 0.32/0.71 ( extended_case_enat_o
% 0.32/0.71 @ ^ [N1: nat] :
% 0.32/0.71 ( extended_case_enat_o
% 0.32/0.71 @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
% 0.32/0.71 @ $false
% 0.32/0.71 @ M3 )
% 0.32/0.71 @ $true ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % less_eq_enat_def
% 0.32/0.71 thf(fact_10207_less__enat__def,axiom,
% 0.32/0.71 ( ord_le72135733267957522d_enat
% 0.32/0.71 = ( ^ [M3: extended_enat,N2: extended_enat] :
% 0.32/0.71 ( extended_case_enat_o
% 0.32/0.71 @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
% 0.32/0.71 @ $false
% 0.32/0.71 @ M3 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 % less_enat_def
% 0.32/0.71 thf(fact_10208_transp__realrel,axiom,
% 0.32/0.71 transp_nat_rat @ realrel ).
% 0.32/0.71
% 0.32/0.71 % transp_realrel
% 0.32/0.71
% 0.32/0.71 % Helper facts (46)
% 0.32/0.71 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.32/0.71 ! [X: int,Y3: int] :
% 0.32/0.71 ( ( if_int @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.32/0.71 ! [X: int,Y3: int] :
% 0.32/0.71 ( ( if_int @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.32/0.71 ! [X: nat,Y3: nat] :
% 0.32/0.71 ( ( if_nat @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.32/0.71 ! [X: nat,Y3: nat] :
% 0.32/0.71 ( ( if_nat @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.32/0.71 ! [X: num,Y3: num] :
% 0.32/0.71 ( ( if_num @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.32/0.71 ! [X: num,Y3: num] :
% 0.32/0.71 ( ( if_num @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.32/0.71 ! [X: rat,Y3: rat] :
% 0.32/0.71 ( ( if_rat @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.32/0.71 ! [X: rat,Y3: rat] :
% 0.32/0.71 ( ( if_rat @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.32/0.71 ! [X: real,Y3: real] :
% 0.32/0.71 ( ( if_real @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.32/0.71 ! [X: real,Y3: real] :
% 0.32/0.71 ( ( if_real @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.32/0.71 ! [P: real > $o] :
% 0.32/0.71 ( ( P @ ( fChoice_real @ P ) )
% 0.32/0.71 = ( ? [X6: real] : ( P @ X6 ) ) ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.32/0.71 ! [X: complex,Y3: complex] :
% 0.32/0.71 ( ( if_complex @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.32/0.71 ! [X: complex,Y3: complex] :
% 0.32/0.71 ( ( if_complex @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.32/0.71 ! [X: extended_enat,Y3: extended_enat] :
% 0.32/0.71 ( ( if_Extended_enat @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.32/0.71 ! [X: extended_enat,Y3: extended_enat] :
% 0.32/0.71 ( ( if_Extended_enat @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.32/0.71 ! [X: code_integer,Y3: code_integer] :
% 0.32/0.71 ( ( if_Code_integer @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.32/0.71 ! [X: code_integer,Y3: code_integer] :
% 0.32/0.71 ( ( if_Code_integer @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.32/0.71 ! [X: set_int,Y3: set_int] :
% 0.32/0.71 ( ( if_set_int @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.32/0.71 ! [X: set_int,Y3: set_int] :
% 0.32/0.71 ( ( if_set_int @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.32/0.71 ! [X: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.32/0.71 ( ( if_VEBT_VEBT @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.32/0.71 ! [X: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.32/0.71 ( ( if_VEBT_VEBT @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.32/0.71 ! [X: list_int,Y3: list_int] :
% 0.32/0.71 ( ( if_list_int @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.32/0.71 ! [X: list_int,Y3: list_int] :
% 0.32/0.71 ( ( if_list_int @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.32/0.71 ! [X: list_nat,Y3: list_nat] :
% 0.32/0.71 ( ( if_list_nat @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.32/0.71 ! [X: list_nat,Y3: list_nat] :
% 0.32/0.71 ( ( if_list_nat @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.32/0.71 ! [X: int > int,Y3: int > int] :
% 0.32/0.71 ( ( if_int_int @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.32/0.71 ! [X: int > int,Y3: int > int] :
% 0.32/0.71 ( ( if_int_int @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 0.32/0.71 ! [X: nat > rat,Y3: nat > rat] :
% 0.32/0.71 ( ( if_nat_rat @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 0.32/0.71 ! [X: nat > rat,Y3: nat > rat] :
% 0.32/0.71 ( ( if_nat_rat @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.32/0.71 ! [X: option_nat,Y3: option_nat] :
% 0.32/0.71 ( ( if_option_nat @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.32/0.71 ! [X: option_nat,Y3: option_nat] :
% 0.32/0.71 ( ( if_option_nat @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.32/0.71 ! [X: option_num,Y3: option_num] :
% 0.32/0.71 ( ( if_option_num @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.32/0.71 ! [X: option_num,Y3: option_num] :
% 0.32/0.71 ( ( if_option_num @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.32/0.71 ! [X: product_prod_int_int,Y3: product_prod_int_int] :
% 0.32/0.71 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.32/0.71 ! [X: product_prod_int_int,Y3: product_prod_int_int] :
% 0.32/0.71 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.32/0.71 ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.32/0.71 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.32/0.71 ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.32/0.71 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 0.32/0.71 ! [X: nat > int > int,Y3: nat > int > int] :
% 0.32/0.71 ( ( if_nat_int_int @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 0.32/0.71 ! [X: nat > int > int,Y3: nat > int > int] :
% 0.32/0.71 ( ( if_nat_int_int @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 0.32/0.71 ! [X: nat > nat > nat,Y3: nat > nat > nat] :
% 0.32/0.71 ( ( if_nat_nat_nat @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 0.32/0.71 ! [X: nat > nat > nat,Y3: nat > nat > nat] :
% 0.32/0.71 ( ( if_nat_nat_nat @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.32/0.71 ! [X: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.32/0.71 ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.32/0.71 ! [X: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.32/0.71 ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.32/0.71 ! [P: $o] :
% 0.32/0.71 ( ( P = $true )
% 0.32/0.71 | ( P = $false ) ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.32/0.71 ! [X: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.32/0.71 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y3 )
% 0.32/0.71 = Y3 ) ).
% 0.32/0.71
% 0.32/0.71 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.32/0.71 ! [X: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.32/0.71 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y3 )
% 0.32/0.71 = X ) ).
% 0.32/0.71
% 0.32/0.71 % Conjectures (1)
% 0.32/0.71 thf(conj_0,conjecture,
% 0.32/0.71 vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ).
% 0.32/0.71
% 0.32/0.71 %------------------------------------------------------------------------------
% 0.32/0.71 ------------------------
% 0.33/3.51 ---- Embedded in THF ---
% 0.33/3.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.33/3.51 %%% Generated on Thu Feb 26 20:02:57 EST 2026
% 0.33/3.51 %%% using '$$dhol' embedding, version 1.3.0.
% 0.33/3.51 %%% Logic specification used:
% 0.33/3.51 %%% thf(spec, logic, $$dhol).
% 0.33/3.51
% 0.33/3.51 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2DTF31377.p
% See solution above
% 0.33/3.55 ------------------------
% 13.19/7.55 ---- Cleaned THF ---
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 13.19/7.55 produc5542196010084753463at_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 13.19/7.55 produc5491161045314408544at_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_Mt__Num__Onum_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 13.19/7.55 produc1193250871479095198on_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 13.19/7.55 produc8306885398267862888on_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 13.19/7.55 produc6121120109295599847at_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J_J,type,
% 13.19/7.55 produc7036089656553540234on_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J_J,type,
% 13.19/7.55 produc2233624965454879586on_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_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,
% 13.19/7.55 list_P8469869581646625389at_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 set_Pr8693737435421807431at_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 produc859450856879609959at_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J_J,type,
% 13.19/7.55 set_fi4554929511873752355omplex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 13.19/7.55 list_P7413028617227757229T_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Num__Onum_J_Mt__Option__Ooption_It__Num__Onum_J_J,type,
% 13.19/7.55 produc3447558737645232053on_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 13.19/7.55 produc4953844613479565601on_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J_J,type,
% 13.19/7.55 set_fi7789364187291644575l_real: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 13.19/7.55 filter6041513312241820739omplex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 13.19/7.55 list_P7037539587688870467BT_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 13.19/7.55 list_P4547456442757143711BT_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 13.19/7.55 list_P5647936690300460905T_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.55 produc8243902056947475879T_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 13.19/7.55 set_Pr5085853215250843933omplex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 13.19/7.55 produc8923325533196201883nteger: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 13.19/7.55 list_P3126845725202233233VEBT_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 13.19/7.55 list_P7495141550334521929T_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 13.19/7.55 filter2146258269922977983l_real: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 13.19/7.55 option4927543243414619207at_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 13.19/7.55 set_Pr6218003697084177305l_real: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
% 13.19/7.55 list_P3744719386663036955um_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
% 13.19/7.55 list_P1726324292696863441at_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 13.19/7.55 produc9072475918466114483BT_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 13.19/7.55 produc4894624898956917775BT_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
% 13.19/7.55 set_Pr958786334691620121nt_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 13.19/7.55 produc4411394909380815293omplex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 13.19/7.55 list_P7333126701944960589_nat_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 13.19/7.55 list_P6285523579766656935_o_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 13.19/7.55 list_P3795440434834930179_o_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
% 13.19/7.55 set_list_VEBT_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 13.19/7.55 produc334124729049499915VEBT_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.55 produc2504756804600209347T_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
% 13.19/7.55 produc6271795597528267376eger_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 13.19/7.55 produc2422161461964618553l_real: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 13.19/7.55 product_prod_num_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 13.19/7.55 product_prod_nat_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.55 product_prod_nat_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 13.19/7.55 product_prod_int_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 13.19/7.55 list_P4002435161011370285od_o_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
% 13.19/7.55 set_list_complex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
% 13.19/7.55 set_set_complex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 13.19/7.55 list_list_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.55 list_VEBT_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 13.19/7.55 set_list_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
% 13.19/7.55 set_list_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 13.19/7.55 product_prod_o_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 13.19/7.55 product_prod_o_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.55 set_VEBT_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 13.19/7.55 set_set_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 13.19/7.55 set_set_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 13.19/7.55 set_Code_integer: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
% 13.19/7.55 set_Product_unit: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
% 13.19/7.55 list_complex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
% 13.19/7.55 set_list_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 13.19/7.55 product_prod_o_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
% 13.19/7.55 set_complex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Filter__Ofilter_It__Real__Oreal_J,type,
% 13.19/7.55 filter_real: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
% 13.19/7.55 option_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
% 13.19/7.55 option_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
% 13.19/7.55 filter_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
% 13.19/7.55 set_char: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
% 13.19/7.55 list_real: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
% 13.19/7.55 set_real: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
% 13.19/7.55 list_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
% 13.19/7.55 list_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
% 13.19/7.55 list_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__VEBT____Definitions__OVEBT,type,
% 13.19/7.55 vEBT_VEBT: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
% 13.19/7.55 set_rat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
% 13.19/7.55 set_num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 13.19/7.55 set_nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
% 13.19/7.55 set_int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Code____Numeral__Ointeger,type,
% 13.19/7.55 code_integer: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Extended____Nat__Oenat,type,
% 13.19/7.55 extended_enat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__List__Olist_I_Eo_J,type,
% 13.19/7.55 list_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Complex__Ocomplex,type,
% 13.19/7.55 complex: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Set__Oset_I_Eo_J,type,
% 13.19/7.55 set_o: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__String__Ochar,type,
% 13.19/7.55 char: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Real__Oreal,type,
% 13.19/7.55 real: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Rat__Orat,type,
% 13.19/7.55 rat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Num__Onum,type,
% 13.19/7.55 num: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Nat__Onat,type,
% 13.19/7.55 nat: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(ty_n_t__Int__Oint,type,
% 13.19/7.55 int: $tType ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
% 13.19/7.55 archim7802044766580827645g_real: real > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
% 13.19/7.55 archim3151403230148437115or_rat: rat > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
% 13.19/7.55 archim6058952711729229775r_real: real > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
% 13.19/7.55 archim2898591450579166408c_real: real > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
% 13.19/7.55 archim7778729529865785530nd_rat: rat > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
% 13.19/7.55 archim8280529875227126926d_real: real > int ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re4521903465945308077real_o: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > $o ) > ( real > $o ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( real > real > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
% 13.19/7.55 bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001_Eo_001_Eo,type,
% 13.19/7.55 bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_BNF__Def_Orel__fun_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
% 13.19/7.55 bNF_re4712519889275205905nt_int: ( int > int > $o ) > ( int > int > $o ) > ( int > int ) > ( int > int ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re1345281282404953727at_nat: ( nat > nat > $o ) > ( ( nat > nat ) > ( nat > nat ) > $o ) > ( nat > nat > nat ) > ( nat > nat > nat ) > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001_Eo_001_Eo,type,
% 13.19/7.55 bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_BNF__Def_Orel__fun_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.55 bNF_re5653821019739307937at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > ( nat > nat ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re8402795839162346335um_int: ( num > num > $o ) > ( ( num > int ) > ( num > int ) > $o ) > ( num > num > int ) > ( num > num > int ) > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_BNF__Def_Orel__fun_001t__Num__Onum_001t__Num__Onum_001t__Int__Oint_001t__Int__Oint,type,
% 13.19/7.55 bNF_re1822329894187522285nt_int: ( num > num > $o ) > ( int > int > $o ) > ( num > int ) > ( num > int ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re1494630372529172596at_o_o: ( product_prod_int_int > rat > $o ) > ( $o > $o > $o ) > ( product_prod_int_int > $o ) > ( rat > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re6644619430987730960nt_o_o: ( product_prod_nat_nat > int > $o ) > ( $o > $o > $o ) > ( product_prod_nat_nat > $o ) > ( int > $o ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 bNF_re4555766996558763186at_nat: ( product_prod_nat_nat > int > $o ) > ( nat > nat > $o ) > ( product_prod_nat_nat > nat ) > ( int > nat ) > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 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 ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Binomial_Obinomial,type,
% 13.19/7.55 binomial: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 13.19/7.55 bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 13.19/7.55 bit_and_not_num: num > num > option_num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 13.19/7.55 bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 13.19/7.55 bit_concat_bit: nat > int > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 13.19/7.55 bit_or_not_num_neg: num > num > num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 13.19/7.55 bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
% 13.19/7.55 bit_ri7919022796975470100ot_int: int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Int__Oint,type,
% 13.19/7.55 bit_ri631733984087533419it_int: nat > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
% 13.19/7.55 bit_se725231765392027082nd_int: int > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
% 13.19/7.55 bit_se727722235901077358nd_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Int__Oint,type,
% 13.19/7.55 bit_se8568078237143864401it_int: nat > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
% 13.19/7.55 bit_se8570568707652914677it_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 bit_se1345352211410354436nteger: nat > code_integer > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Int__Oint,type,
% 13.19/7.55 bit_se2159334234014336723it_int: nat > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
% 13.19/7.55 bit_se2161824704523386999it_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
% 13.19/7.55 bit_se2000444600071755411sk_int: nat > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
% 13.19/7.55 bit_se2002935070580805687sk_nat: nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
% 13.19/7.55 bit_se1409905431419307370or_int: int > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
% 13.19/7.55 bit_se1412395901928357646or_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Int__Oint,type,
% 13.19/7.55 bit_se545348938243370406it_int: nat > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
% 13.19/7.55 bit_se547839408752420682it_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
% 13.19/7.55 bit_se7879613467334960850it_int: nat > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Nat__Onat,type,
% 13.19/7.55 bit_se7882103937844011126it_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 bit_se1745604003318907178nteger: nat > code_integer > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Int__Oint,type,
% 13.19/7.55 bit_se2923211474154528505it_int: nat > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
% 13.19/7.55 bit_se2925701944663578781it_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
% 13.19/7.55 bit_se4203085406695923979it_int: nat > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
% 13.19/7.55 bit_se4205575877204974255it_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Int__Oint,type,
% 13.19/7.55 bit_se6526347334894502574or_int: int > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
% 13.19/7.55 bit_se6528837805403552850or_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 bit_se9216721137139052372nteger: code_integer > nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Int__Oint,type,
% 13.19/7.55 bit_se1146084159140164899it_int: int > nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
% 13.19/7.55 bit_se1148574629649215175it_nat: nat > nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 13.19/7.55 bit_take_bit_num: nat > num > option_num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
% 13.19/7.55 bit_un1837492267222099188nd_num: num > num > option_num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
% 13.19/7.55 bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
% 13.19/7.55 bit_un6178654185764691216or_num: num > num > option_num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
% 13.19/7.55 bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 13.19/7.55 bit_un7362597486090784418nd_num: num > num > option_num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
% 13.19/7.55 bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 13.19/7.55 bit_un2480387367778600638or_num: num > num > option_num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
% 13.19/7.55 bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 13.19/7.55 code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 13.19/7.55 code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 13.19/7.55 code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 13.19/7.55 code_int_of_integer: code_integer > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 13.19/7.55 code_integer_of_int: int > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Ointeger__of__num,type,
% 13.19/7.55 code_integer_of_num: num > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 13.19/7.55 code_nat_of_integer: code_integer > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Onegative,type,
% 13.19/7.55 code_negative: num > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Onum__of__integer,type,
% 13.19/7.55 code_num_of_integer: code_integer > num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Numeral_Opositive,type,
% 13.19/7.55 code_positive: num > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Target__Int_Onegative,type,
% 13.19/7.55 code_Target_negative: num > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Code__Target__Int_Opositive,type,
% 13.19/7.55 code_Target_positive: num > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 13.19/7.55 comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 13.19/7.55 comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
% 13.19/7.55 comple4887499456419720421f_real: set_real > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
% 13.19/7.55 complete_Sup_Sup_int: set_int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
% 13.19/7.55 comple1385675409528146559p_real: set_real > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_OArg,type,
% 13.19/7.55 arg: complex > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_Ocis,type,
% 13.19/7.55 cis: real > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_Ocnj,type,
% 13.19/7.55 cnj: complex > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_Ocomplex_OComplex,type,
% 13.19/7.55 complex2: real > real > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_Ocomplex_OIm,type,
% 13.19/7.55 im: complex > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_Ocomplex_ORe,type,
% 13.19/7.55 re: complex > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_Ocsqrt,type,
% 13.19/7.55 csqrt: complex > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_Oimaginary__unit,type,
% 13.19/7.55 imaginary_unit: complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Complex_Orcis,type,
% 13.19/7.55 rcis: real > real > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.55 differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.55 has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
% 13.19/7.55 has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Deriv_Ohas__vector__derivative_001t__Real__Oreal,type,
% 13.19/7.55 has_ve631408500373753343e_real: ( real > real ) > real > filter_real > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Oadjust__div,type,
% 13.19/7.55 adjust_div: product_prod_int_int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Oadjust__mod,type,
% 13.19/7.55 adjust_mod: int > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Odivmod__nat,type,
% 13.19/7.55 divmod_nat: nat > nat > product_prod_nat_nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Oeucl__rel__int,type,
% 13.19/7.55 eucl_rel_int: int > int > product_prod_int_int > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
% 13.19/7.55 unique6319869463603278526ux_int: product_prod_int_int > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
% 13.19/7.55 unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
% 13.19/7.55 unique5052692396658037445od_int: num > num > product_prod_int_int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
% 13.19/7.55 unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
% 13.19/7.55 unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
% 13.19/7.55 unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
% 13.19/7.55 euclid4774559944035922753ze_int: int > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
% 13.19/7.55 euclid3395696857347342551nt_int: int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Nat__Onat,type,
% 13.19/7.55 euclid3398187327856392827nt_nat: nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
% 13.19/7.55 extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 comm_s8582702949713902594nteger: code_integer > nat > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
% 13.19/7.55 comm_s2602460028002588243omplex: complex > nat > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
% 13.19/7.55 comm_s4660882817536571857er_int: int > nat > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
% 13.19/7.55 comm_s4663373288045622133er_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
% 13.19/7.55 comm_s4028243227959126397er_rat: rat > nat > rat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
% 13.19/7.55 comm_s7457072308508201937r_real: real > nat > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 semiri3624122377584611663nteger: nat > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 13.19/7.55 semiri5044797733671781792omplex: nat > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 13.19/7.55 semiri1406184849735516958ct_int: nat > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 13.19/7.55 semiri1408675320244567234ct_nat: nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 13.19/7.55 semiri773545260158071498ct_rat: nat > rat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 13.19/7.55 semiri2265585572941072030t_real: nat > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
% 13.19/7.55 invers8013647133539491842omplex: complex > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 13.19/7.55 inverse_inverse_rat: rat > rat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 13.19/7.55 inverse_inverse_real: real > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 13.19/7.55 at_bot_real: filter_real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 13.19/7.55 at_top_nat: filter_nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 13.19/7.55 at_top_real: filter_real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 13.19/7.55 eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 13.19/7.55 eventually_real: ( real > $o ) > filter_real > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.55 filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 13.19/7.55 filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.55 filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Ofiltermap_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.55 filtermap_real_real: ( real > real ) > filter_real > filter_real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 13.19/7.55 princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 13.19/7.55 princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ocard_001_Eo,type,
% 13.19/7.55 finite_card_o: set_o > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 13.19/7.55 finite_card_complex: set_complex > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
% 13.19/7.55 finite_card_int: set_int > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 13.19/7.55 finite_card_list_nat: set_list_nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 13.19/7.55 finite_card_nat: set_nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
% 13.19/7.55 finite410649719033368117t_unit: set_Product_unit > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ocard_001t__String__Ochar,type,
% 13.19/7.55 finite_card_char: set_char > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
% 13.19/7.55 finite_finite_o: set_o > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 13.19/7.55 finite3207457112153483333omplex: set_complex > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
% 13.19/7.55 finite_finite_int: set_int > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
% 13.19/7.55 finite_finite_list_o: set_list_o > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 13.19/7.55 finite8712137658972009173omplex: set_list_complex > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
% 13.19/7.55 finite3922522038869484883st_int: set_list_int > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
% 13.19/7.55 finite8100373058378681591st_nat: set_list_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.55 finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 13.19/7.55 finite_finite_nat: set_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
% 13.19/7.55 finite_finite_num: set_num > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
% 13.19/7.55 finite_finite_rat: set_rat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
% 13.19/7.55 finite_finite_real: set_real > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 13.19/7.55 finite6551019134538273531omplex: set_set_complex > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
% 13.19/7.55 finite6197958912794628473et_int: set_set_int > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 13.19/7.55 finite1152437895449049373et_nat: set_set_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Finite__Set_Ofinite_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.55 finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
% 13.19/7.55 bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 13.19/7.55 bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.55 bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Ocomp_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 13.19/7.55 comp_C3531382070062128313er_num: ( code_integer > code_integer ) > ( num > code_integer ) > num > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Int__Oint_001t__Num__Onum,type,
% 13.19/7.55 comp_int_int_num: ( int > int ) > ( num > int ) > num > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
% 13.19/7.55 comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.55 comp_int_real_real: ( int > real ) > ( real > int ) > real > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Real__Oreal_001t__Nat__Onat,type,
% 13.19/7.55 comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.55 inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Ochar,type,
% 13.19/7.55 inj_on_nat_char: ( nat > char ) > set_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.55 inj_on_real_real: ( real > real ) > set_real > $o ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 map_fu1532550112467129777l_real: ( real > nat > rat ) > ( ( ( nat > rat ) > nat > rat ) > real > real ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > real > real > real ).
% 13.19/7.55
% 13.19/7.55 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,
% 13.19/7.55 map_fu7146612038024189824t_real: ( real > nat > rat ) > ( ( nat > rat ) > real ) > ( ( nat > rat ) > nat > rat ) > real > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Ostrict__mono__on_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.55 strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun_Othe__inv__into_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.55 the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun__Def_Opair__leq,type,
% 13.19/7.55 fun_pair_leq: set_Pr8693737435421807431at_nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Fun__Def_Opair__less,type,
% 13.19/7.55 fun_pair_less: set_Pr8693737435421807431at_nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_GCD_OGcd__class_OGcd_001t__Int__Oint,type,
% 13.19/7.55 gcd_Gcd_int: set_int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
% 13.19/7.55 gcd_Gcd_nat: set_nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_GCD_Obezw,type,
% 13.19/7.55 bezw: nat > nat > product_prod_int_int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_GCD_Obezw__rel,type,
% 13.19/7.55 bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Int__Oint,type,
% 13.19/7.55 gcd_gcd_int: int > int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
% 13.19/7.55 gcd_gcd_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oabs__class_Oabs_001t__Code____Numeral__Ointeger,type,
% 13.19/7.55 abs_abs_Code_integer: code_integer > code_integer ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oabs__class_Oabs_001t__Complex__Ocomplex,type,
% 13.19/7.55 abs_abs_complex: complex > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
% 13.19/7.55 abs_abs_int: int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
% 13.19/7.55 abs_abs_rat: rat > rat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
% 13.19/7.55 abs_abs_real: real > real ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001_062_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_M_Eo_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 13.19/7.55 one_one_nat: nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
% 13.19/7.55 one_one_rat: rat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Osgn__class_Osgn_001t__Complex__Ocomplex,type,
% 13.19/7.55 sgn_sgn_complex: complex > complex ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
% 13.19/7.55 sgn_sgn_int: int > int ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
% 13.19/7.55 sgn_sgn_rat: rat > rat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Extended____Nat__Oenat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
% 13.19/7.55 times_times_nat: nat > nat > nat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
% 13.19/7.55 times_times_num: num > num > num ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Rat__Orat,type,
% 13.19/7.55 times_times_rat: rat > rat > rat ).
% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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% 13.19/7.55 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Code____Numeral__Ointeger,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Real__Oreal_J,type,
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% 13.19/7.55
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
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% 13.19/7.55
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% 13.19/7.55
% 13.19/7.55 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Complex__Ocomplex_001t__Nat__Onat,type,
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% 13.19/7.55
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56 thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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% 13.19/7.56
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% 13.19/7.56
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% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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% 13.19/7.56
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% 13.19/7.56 set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 13.19/7.56 set_real2: list_real > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist__update_001_Eo,type,
% 13.19/7.56 list_update_o: list_o > nat > $o > list_o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
% 13.19/7.56 list_update_complex: list_complex > nat > complex > list_complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 13.19/7.56 list_update_int: list_int > nat > int > list_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 13.19/7.56 list_update_nat: list_nat > nat > nat > list_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist__update_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,
% 13.19/7.56 list_u5003261594476800725at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat > list_P8469869581646625389at_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 13.19/7.56 list_update_real: list_real > nat > real > list_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001_Eo,type,
% 13.19/7.56 nth_o: list_o > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
% 13.19/7.56 nth_complex: list_complex > nat > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 13.19/7.56 nth_int: list_int > nat > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
% 13.19/7.56 nth_list_nat: list_list_nat > nat > list_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 13.19/7.56 nth_nat: list_nat > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Num__Onum,type,
% 13.19/7.56 nth_num: list_num > nat > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 13.19/7.56 nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 13.19/7.56 nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 13.19/7.56 nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.56 nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
% 13.19/7.56 nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 13.19/7.56 nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_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,
% 13.19/7.56 nth_Pr6744343527793145070at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 13.19/7.56 nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 13.19/7.56 nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 13.19/7.56 nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.56 nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 13.19/7.56 nth_real: list_real > nat > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 13.19/7.56 product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 13.19/7.56 product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 13.19/7.56 product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 13.19/7.56 product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Num__Onum,type,
% 13.19/7.56 product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
% 13.19/7.56 product_num_num: list_num > list_num > list_P3744719386663036955um_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 13.19/7.56 product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 13.19/7.56 produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 13.19/7.56 produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 13.19/7.56 replicate_int: nat > int > list_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 13.19/7.56 replicate_nat: nat > nat > list_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 13.19/7.56 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 13.19/7.56 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 13.19/7.56 take_nat: nat > list_nat > list_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oupt,type,
% 13.19/7.56 upt: nat > nat > list_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oupto,type,
% 13.19/7.56 upto: int > int > list_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oupto__aux,type,
% 13.19/7.56 upto_aux: int > int > list_int > list_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_List_Oupto__rel,type,
% 13.19/7.56 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_OSuc,type,
% 13.19/7.56 suc: nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.56 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 13.19/7.56 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 13.19/7.56 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 13.19/7.56 case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Onat_Opred,type,
% 13.19/7.56 pred: nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 semiri4939895301339042750nteger: nat > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 13.19/7.56 semiri8010041392384452111omplex: nat > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 13.19/7.56 semiri4216267220026989637d_enat: nat > extended_enat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 13.19/7.56 semiri1314217659103216013at_int: nat > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 13.19/7.56 semiri1316708129612266289at_nat: nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 13.19/7.56 semiri681578069525770553at_rat: nat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 13.19/7.56 semiri5074537144036343181t_real: nat > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
% 13.19/7.56 semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
% 13.19/7.56 semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
% 13.19/7.56 semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
% 13.19/7.56 semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
% 13.19/7.56 semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 13.19/7.56 size_size_list_o: list_o > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 13.19/7.56 size_s3451745648224563538omplex: list_complex > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 13.19/7.56 size_size_list_int: list_int > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 13.19/7.56 size_s3023201423986296836st_nat: list_list_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 13.19/7.56 size_size_list_nat: list_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 13.19/7.56 size_size_list_num: list_num > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 13.19/7.56 size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 13.19/7.56 size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 13.19/7.56 size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 13.19/7.56 size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 13.19/7.56 size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 13.19/7.56 size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_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,
% 13.19/7.56 size_s3679842834875189465at_nat: list_P8469869581646625389at_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 13.19/7.56 size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 13.19/7.56 size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 13.19/7.56 size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 13.19/7.56 size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 13.19/7.56 size_size_list_real: list_real > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.56 size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 13.19/7.56 size_size_num: num > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 13.19/7.56 size_size_option_nat: option_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 13.19/7.56 size_size_option_num: option_num > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 13.19/7.56 size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 13.19/7.56 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 13.19/7.56 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 13.19/7.56 nat_prod_encode: product_prod_nat_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 13.19/7.56 nat_set_decode: nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 13.19/7.56 nat_set_encode: set_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Nat__Bijection_Otriangle,type,
% 13.19/7.56 nat_triangle: nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_NthRoot_Oroot,type,
% 13.19/7.56 root: nat > real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_NthRoot_Osqrt,type,
% 13.19/7.56 sqrt: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_OBitM,type,
% 13.19/7.56 bitM: num > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oinc,type,
% 13.19/7.56 inc: num > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 neg_nu8804712462038260780nteger: code_integer > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
% 13.19/7.56 neg_nu7009210354673126013omplex: complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
% 13.19/7.56 neg_numeral_dbl_int: int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
% 13.19/7.56 neg_numeral_dbl_rat: rat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
% 13.19/7.56 neg_numeral_dbl_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 13.19/7.56 neg_nu6511756317524482435omplex: complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 13.19/7.56 neg_nu3811975205180677377ec_int: int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 13.19/7.56 neg_nu3179335615603231917ec_rat: rat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 13.19/7.56 neg_nu6075765906172075777c_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 neg_nu5831290666863070958nteger: code_integer > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 13.19/7.56 neg_nu8557863876264182079omplex: complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 13.19/7.56 neg_nu5851722552734809277nc_int: int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 13.19/7.56 neg_nu5219082963157363817nc_rat: rat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 13.19/7.56 neg_nu8295874005876285629c_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 13.19/7.56 neg_numeral_sub_int: num > num > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onum_OBit0,type,
% 13.19/7.56 bit0: num > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onum_OBit1,type,
% 13.19/7.56 bit1: num > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onum_OOne,type,
% 13.19/7.56 one: num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 13.19/7.56 case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onum_Osize__num,type,
% 13.19/7.56 size_num: num > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onum__of__nat,type,
% 13.19/7.56 num_of_nat: nat > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 numera6620942414471956472nteger: num > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 13.19/7.56 numera6690914467698888265omplex: num > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 13.19/7.56 numera1916890842035813515d_enat: num > extended_enat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 13.19/7.56 numeral_numeral_int: num > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 13.19/7.56 numeral_numeral_nat: num > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 13.19/7.56 numeral_numeral_rat: num > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 13.19/7.56 numeral_numeral_real: num > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Opow,type,
% 13.19/7.56 pow: num > num > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Opred__numeral,type,
% 13.19/7.56 pred_numeral: num > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Num_Osqr,type,
% 13.19/7.56 sqr: num > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
% 13.19/7.56 none_nat: option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 13.19/7.56 none_num: option_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.56 none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
% 13.19/7.56 some_nat: nat > option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 13.19/7.56 some_num: num > option_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.56 some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.56 case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Ocase__option_001t__Int__Oint_001t__Num__Onum,type,
% 13.19/7.56 case_option_int_num: int > ( num > int ) > option_num > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 13.19/7.56 case_option_num_num: num > ( num > num ) > option_num > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 13.19/7.56 case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 13.19/7.56 map_option_num_num: ( num > num ) > option_num > option_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
% 13.19/7.56 size_option_nat: ( nat > nat ) > option_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
% 13.19/7.56 size_option_num: ( num > nat ) > option_num > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.56 size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
% 13.19/7.56 the_nat: option_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
% 13.19/7.56 the_num: option_num > num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.56 the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 13.19/7.56 bot_bot_nat_o: nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
% 13.19/7.56 bot_bo4199563552545308370d_enat: extended_enat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 13.19/7.56 bot_bot_nat: nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 13.19/7.56 bot_bot_set_complex: set_complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 13.19/7.56 bot_bot_set_int: set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 13.19/7.56 bot_bot_set_nat: set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
% 13.19/7.56 bot_bot_set_num: set_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
% 13.19/7.56 bot_bot_set_rat: set_rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
% 13.19/7.56 bot_bot_set_real: set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 13.19/7.56 bot_bot_set_set_int: set_set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
% 13.19/7.56 ord_Least_nat: ( nat > $o ) > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_OLeast_001t__Real__Oreal,type,
% 13.19/7.56 ord_Least_real: ( real > $o ) > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 13.19/7.56 ord_less_complex_o: ( complex > $o ) > ( complex > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Int__Oint_M_Eo_J,type,
% 13.19/7.56 ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
% 13.19/7.56 ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001_062_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_M_Eo_J,type,
% 13.19/7.56 ord_le7432393201274343594_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Real__Oreal_M_Eo_J,type,
% 13.19/7.56 ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Extended____Nat__Oenat,type,
% 13.19/7.56 ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
% 13.19/7.56 ord_less_int: int > int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 13.19/7.56 ord_less_nat: nat > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
% 13.19/7.56 ord_less_num: num > num > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
% 13.19/7.56 ord_less_rat: rat > rat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
% 13.19/7.56 ord_less_real: real > real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
% 13.19/7.56 ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 13.19/7.56 ord_less_set_complex: set_complex > set_complex > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
% 13.19/7.56 ord_less_set_int: set_int > set_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 13.19/7.56 ord_less_set_nat: set_nat > set_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
% 13.19/7.56 ord_less_set_num: set_num > set_num > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__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,
% 13.19/7.56 ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Rat__Orat_J,type,
% 13.19/7.56 ord_less_set_rat: set_rat > set_rat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Real__Oreal_J,type,
% 13.19/7.56 ord_less_set_real: set_real > set_real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 13.19/7.56 ord_less_set_set_int: set_set_int > set_set_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
% 13.19/7.56 ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
% 13.19/7.56 ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
% 13.19/7.56 ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_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_M_Eo_J,type,
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% 13.19/7.56
% 13.19/7.56 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,
% 13.19/7.56 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 13.19/7.56
% 13.19/7.56 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,
% 13.19/7.56 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 13.19/7.56 produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 13.19/7.56 produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 13.19/7.56
% 13.19/7.56 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,
% 13.19/7.56 produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 13.19/7.56
% 13.19/7.56 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,
% 13.19/7.56 produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 13.19/7.56
% 13.19/7.56 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,
% 13.19/7.56 produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 13.19/7.56 produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 13.19/7.56 produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 13.19/7.56 produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.56 produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 13.19/7.56
% 13.19/7.56 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,
% 13.19/7.56 produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 13.19/7.56 produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 13.19/7.56 produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 13.19/7.56 produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 13.19/7.56 produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 13.19/7.56 product_fst_int_int: product_prod_int_int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.56 product_fst_nat_nat: product_prod_nat_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 13.19/7.56 product_snd_int_int: product_prod_int_int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.56 product_snd_nat_nat: product_prod_nat_nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_OAbs__Rat,type,
% 13.19/7.56 abs_Rat: product_prod_int_int > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_OFract,type,
% 13.19/7.56 fract: int > int > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_OFrct,type,
% 13.19/7.56 frct: product_prod_int_int > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_ORep__Rat,type,
% 13.19/7.56 rep_Rat: rat > product_prod_int_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 13.19/7.56 field_5140801741446780682s_real: set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 13.19/7.56 field_7254667332652039916t_real: rat > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_Onormalize,type,
% 13.19/7.56 normalize: product_prod_int_int > product_prod_int_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_Opcr__rat,type,
% 13.19/7.56 pcr_rat: product_prod_int_int > rat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_Opositive,type,
% 13.19/7.56 positive: rat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_Oquotient__of,type,
% 13.19/7.56 quotient_of: rat > product_prod_int_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rat_Oratrel,type,
% 13.19/7.56 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real_OReal,type,
% 13.19/7.56 real2: ( nat > rat ) > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real_Ocauchy,type,
% 13.19/7.56 cauchy: ( nat > rat ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real_Ocr__real,type,
% 13.19/7.56 cr_real: ( nat > rat ) > real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real_Opcr__real,type,
% 13.19/7.56 pcr_real: ( nat > rat ) > real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real_Opositive,type,
% 13.19/7.56 positive2: real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real_Orealrel,type,
% 13.19/7.56 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real_Orep__real,type,
% 13.19/7.56 rep_real: real > nat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real_Ovanishes,type,
% 13.19/7.56 vanishes: ( nat > rat ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
% 13.19/7.56 real_V2521375963428798218omplex: set_complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.56 real_V5970128139526366754l_real: ( real > real ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 13.19/7.56 real_V3694042436643373181omplex: complex > complex > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 13.19/7.56 real_V975177566351809787t_real: real > real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 13.19/7.56 real_V1022390504157884413omplex: complex > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 13.19/7.56 real_V7735802525324610683m_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 13.19/7.56 real_V4546457046886955230omplex: real > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
% 13.19/7.56 real_V2046097035970521341omplex: real > complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
% 13.19/7.56 real_V1485227260804924795R_real: real > real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Relation_Otransp_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
% 13.19/7.56 transp_nat_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 13.19/7.56 algebr932160517623751201me_int: int > int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 13.19/7.56 algebr934650988132801477me_nat: nat > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 13.19/7.56 divide1717551699836669952omplex: complex > complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 13.19/7.56 divide_divide_int: int > int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 13.19/7.56 divide_divide_nat: nat > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 13.19/7.56 divide_divide_rat: rat > rat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 13.19/7.56 divide_divide_real: real > real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 13.19/7.56 dvd_dvd_complex: complex > complex > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 13.19/7.56 dvd_dvd_int: int > int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 13.19/7.56 dvd_dvd_nat: nat > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
% 13.19/7.56 dvd_dvd_rat: rat > rat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
% 13.19/7.56 dvd_dvd_real: real > real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 13.19/7.56 modulo_modulo_int: int > int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 13.19/7.56 modulo_modulo_nat: nat > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 zero_n356916108424825756nteger: $o > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
% 13.19/7.56 zero_n1201886186963655149omplex: $o > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 13.19/7.56 zero_n2684676970156552555ol_int: $o > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 13.19/7.56 zero_n2687167440665602831ol_nat: $o > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
% 13.19/7.56 zero_n2052037380579107095ol_rat: $o > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
% 13.19/7.56 zero_n3304061248610475627l_real: $o > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
% 13.19/7.56 suminf_complex: ( nat > complex ) > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
% 13.19/7.56 suminf_int: ( nat > int ) > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
% 13.19/7.56 suminf_nat: ( nat > nat ) > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 13.19/7.56 suminf_real: ( nat > real ) > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
% 13.19/7.56 summable_complex: ( nat > complex ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osummable_001t__Int__Oint,type,
% 13.19/7.56 summable_int: ( nat > int ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
% 13.19/7.56 summable_nat: ( nat > nat ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 13.19/7.56 summable_real: ( nat > real ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
% 13.19/7.56 sums_complex: ( nat > complex ) > complex > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osums_001t__Int__Oint,type,
% 13.19/7.56 sums_int: ( nat > int ) > int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osums_001t__Nat__Onat,type,
% 13.19/7.56 sums_nat: ( nat > nat ) > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 13.19/7.56 sums_real: ( nat > real ) > real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001_Eo,type,
% 13.19/7.56 collect_o: ( $o > $o ) > set_o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
% 13.19/7.56 collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 13.19/7.56 collect_complex: ( complex > $o ) > set_complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 13.19/7.56 collect_int: ( int > $o ) > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
% 13.19/7.56 collect_list_o: ( list_o > $o ) > set_list_o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 13.19/7.56 collect_list_complex: ( list_complex > $o ) > set_list_complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
% 13.19/7.56 collect_list_int: ( list_int > $o ) > set_list_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
% 13.19/7.56 collect_list_nat: ( list_nat > $o ) > set_list_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.56 collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 13.19/7.56 collect_nat: ( nat > $o ) > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Num__Onum,type,
% 13.19/7.56 collect_num: ( num > $o ) > set_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
% 13.19/7.56 collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 13.19/7.56 collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_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,
% 13.19/7.56 collec7088162979684241874at_nat: ( produc859450856879609959at_nat > $o ) > set_Pr8693737435421807431at_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 13.19/7.56 collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
% 13.19/7.56 collect_rat: ( rat > $o ) > set_rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
% 13.19/7.56 collect_real: ( real > $o ) > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 13.19/7.56 collect_set_complex: ( set_complex > $o ) > set_set_complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
% 13.19/7.56 collect_set_int: ( set_int > $o ) > set_set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 13.19/7.56 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
% 13.19/7.56 image_int_int: ( int > int ) > set_int > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
% 13.19/7.56 image_nat_int: ( nat > int ) > set_nat > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.56 image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
% 13.19/7.56 image_nat_real: ( nat > real ) > set_nat > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
% 13.19/7.56 image_nat_char: ( nat > char ) > set_nat > set_char ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
% 13.19/7.56 image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
% 13.19/7.56 image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.56 image_real_real: ( real > real ) > set_real > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
% 13.19/7.56 image_char_nat: ( char > nat ) > set_char > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
% 13.19/7.56 insert_int: int > set_int > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
% 13.19/7.56 insert_real: real > set_real > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
% 13.19/7.56 vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
% 13.19/7.56 set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
% 13.19/7.56 set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
% 13.19/7.56 set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
% 13.19/7.56 set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
% 13.19/7.56 set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
% 13.19/7.56 set_or1266510415728281911st_int: int > int > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
% 13.19/7.56 set_or1269000886237332187st_nat: nat > nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
% 13.19/7.56 set_or7049704709247886629st_num: num > num > set_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
% 13.19/7.56 set_or633870826150836451st_rat: rat > rat > set_rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
% 13.19/7.56 set_or1222579329274155063t_real: real > real > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Int__Oint_J,type,
% 13.19/7.56 set_or370866239135849197et_int: set_int > set_int > set_set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
% 13.19/7.56 set_or4662586982721622107an_int: int > int > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
% 13.19/7.56 set_or4665077453230672383an_nat: nat > nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
% 13.19/7.56 set_ord_atLeast_nat: nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
% 13.19/7.56 set_ord_atLeast_real: real > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Int__Oint,type,
% 13.19/7.56 set_ord_atMost_int: int > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 13.19/7.56 set_ord_atMost_nat: nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 13.19/7.56 set_or6656581121297822940st_int: int > int > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 13.19/7.56 set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 13.19/7.56 set_or5832277885323065728an_int: int > int > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 13.19/7.56 set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 13.19/7.56 set_or1633881224788618240n_real: real > real > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 13.19/7.56 set_or1210151606488870762an_nat: nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 13.19/7.56 set_or5849166863359141190n_real: real > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 13.19/7.56 set_ord_lessThan_int: int > set_int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 13.19/7.56 set_ord_lessThan_nat: nat > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 13.19/7.56 set_ord_lessThan_num: num > set_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 13.19/7.56 set_ord_lessThan_rat: rat > set_rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 13.19/7.56 set_or5984915006950818249n_real: real > set_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_String_Oascii__of,type,
% 13.19/7.56 ascii_of: char > char ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_String_Ochar_OChar,type,
% 13.19/7.56 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 13.19/7.56 comm_s629917340098488124ar_nat: char > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_String_Ointeger__of__char,type,
% 13.19/7.56 integer_of_char: char > code_integer ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 13.19/7.56 unique3096191561947761185of_nat: nat > char ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.56 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 13.19/7.56 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 13.19/7.56 topolo4899668324122417113eq_int: ( nat > int ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 13.19/7.56 topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
% 13.19/7.56 topolo1459490580787246023eq_num: ( nat > num ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
% 13.19/7.56 topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 13.19/7.56 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
% 13.19/7.56 topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 13.19/7.56 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
% 13.19/7.56 topolo7531315842566124627t_real: ( nat > real ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 13.19/7.56 topolo2815343760600316023s_real: real > filter_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 13.19/7.56 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 13.19/7.56 topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 13.19/7.56 topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oarccos,type,
% 13.19/7.56 arccos: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 13.19/7.56 arcosh_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oarcsin,type,
% 13.19/7.56 arcsin: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oarctan,type,
% 13.19/7.56 arctan: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 13.19/7.56 arsinh_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 13.19/7.56 artanh_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 13.19/7.56 cos_complex: complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 13.19/7.56 cos_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Ocos__coeff,type,
% 13.19/7.56 cos_coeff: nat > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 13.19/7.56 cosh_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 13.19/7.56 cot_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 13.19/7.56 diffs_complex: ( nat > complex ) > nat > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
% 13.19/7.56 diffs_int: ( nat > int ) > nat > int ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
% 13.19/7.56 diffs_rat: ( nat > rat ) > nat > rat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 13.19/7.56 diffs_real: ( nat > real ) > nat > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 13.19/7.56 exp_complex: complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 13.19/7.56 exp_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 13.19/7.56 ln_ln_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Olog,type,
% 13.19/7.56 log: real > real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Opi,type,
% 13.19/7.56 pi: real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 13.19/7.56 powr_real: real > real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 13.19/7.56 sin_complex: complex > complex ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 13.19/7.56 sin_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Osin__coeff,type,
% 13.19/7.56 sin_coeff: nat > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 13.19/7.56 sinh_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 13.19/7.56 tan_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 13.19/7.56 tanh_real: real > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 13.19/7.56 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 13.19/7.56 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 13.19/7.56 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 13.19/7.56 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 13.19/7.56 vEBT_VEBT_high: nat > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 13.19/7.56 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 13.19/7.56 vEBT_VEBT_low: nat > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 13.19/7.56 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 13.19/7.56 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 13.19/7.56 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 13.19/7.56 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 13.19/7.56 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 13.19/7.56 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 13.19/7.56 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 13.19/7.56 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 13.19/7.56 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 13.19/7.56 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 13.19/7.56 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 13.19/7.56 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 13.19/7.56 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 13.19/7.56 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 13.19/7.56 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 13.19/7.56 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 13.19/7.56 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 13.19/7.56 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 13.19/7.56 vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 13.19/7.56 vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 13.19/7.56 vEBT_VEBT_less: option_nat > option_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 13.19/7.56 vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 13.19/7.56 vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 13.19/7.56 vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 13.19/7.56 vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 13.19/7.56 vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 13.19/7.56 vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.56 vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 13.19/7.56 vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 13.19/7.56 vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 13.19/7.56 vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 13.19/7.56 vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 13.19/7.56 vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Succ_Ois__succ__in__set,type,
% 13.19/7.56 vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Succ_Ovebt__succ,type,
% 13.19/7.56 vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
% 13.19/7.56 vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
% 13.19/7.56 accp_nat: ( nat > nat > $o ) > nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 13.19/7.56 accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 13.19/7.56 accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
% 13.19/7.56 accp_P3113834385874906142um_num: ( product_prod_num_num > product_prod_num_num > $o ) > product_prod_num_num > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 13.19/7.56 accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_Wellfounded_Oaccp_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_fChoice_001t__Real__Oreal,type,
% 13.19/7.56 fChoice_real: ( real > $o ) > real ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001_Eo,type,
% 13.19/7.56 member_o: $o > set_o > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 13.19/7.56 member_complex: complex > set_complex > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__Int__Oint,type,
% 13.19/7.56 member_int: int > set_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 13.19/7.56 member_list_o: list_o > set_list_o > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
% 13.19/7.56 member_list_int: list_int > set_list_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 13.19/7.56 member_list_nat: list_nat > set_list_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 13.19/7.56 member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__Nat__Onat,type,
% 13.19/7.56 member_nat: nat > set_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__Num__Onum,type,
% 13.19/7.56 member_num: num > set_num > $o ).
% 13.19/7.56
% 13.19/7.56 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,
% 13.19/7.56 member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__Rat__Orat,type,
% 13.19/7.56 member_rat: rat > set_rat > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__Real__Oreal,type,
% 13.19/7.56 member_real: real > set_real > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 13.19/7.56 member_set_int: set_int > set_set_int > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 13.19/7.56 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_deg____,type,
% 13.19/7.56 deg: nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_m____,type,
% 13.19/7.56 m: nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_ma____,type,
% 13.19/7.56 ma: nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_mi____,type,
% 13.19/7.56 mi: nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_na____,type,
% 13.19/7.56 na: nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_sc____,type,
% 13.19/7.56 sc: nat ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_summary____,type,
% 13.19/7.56 summary: vEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_treeList____,type,
% 13.19/7.56 treeList: list_VEBT_VEBT ).
% 13.19/7.56
% 13.19/7.56 thf(sy_v_xa____,type,
% 13.19/7.56 xa: nat ).
% 13.19/7.56
% 13.19/7.56 thf(fact_0__C4_Ohyps_C_I3_J,axiom,
% 13.19/7.56 ! [X: nat,Sx: nat] :
% 13.19/7.56 ( ( ( vEBT_vebt_succ @ summary @ X )
% 13.19/7.56 = ( some_nat @ Sx ) )
% 13.19/7.56 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ X @ Sx ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_1__092_060open_062vebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_092_060close_062,axiom,
% 13.19/7.56 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.19/7.56 = ( some_nat @ sc ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062sc_O_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 13.19/7.56 ~ ! [Sc: nat] :
% 13.19/7.56 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.19/7.56 != ( some_nat @ Sc ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_3_False,axiom,
% 13.19/7.56 ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.19/7.56 != none_nat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_4__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
% 13.19/7.56 ( ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = na ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_5_semiring__norm_I85_J,axiom,
% 13.19/7.56 ! [M: num] :
% 13.19/7.56 ( ( bit0 @ M )
% 13.19/7.56 != one ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_6_semiring__norm_I83_J,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( one
% 13.19/7.56 != ( bit0 @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_7__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
% 13.19/7.56 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 13.19/7.56
% 13.19/7.56 thf(fact_8_numeral__Bit0__div__2,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = ( numeral_numeral_nat @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_9_numeral__Bit0__div__2,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.19/7.56 = ( numeral_numeral_int @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_10_high__def,axiom,
% 13.19/7.56 ( vEBT_VEBT_high
% 13.19/7.56 = ( ^ [X2: nat,N2: nat] : ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_11_divide__numeral__1,axiom,
% 13.19/7.56 ! [A: complex] :
% 13.19/7.56 ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_12_divide__numeral__1,axiom,
% 13.19/7.56 ! [A: real] :
% 13.19/7.56 ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_13_divide__numeral__1,axiom,
% 13.19/7.56 ! [A: rat] :
% 13.19/7.56 ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_14_verit__eq__simplify_I8_J,axiom,
% 13.19/7.56 ! [X22: num,Y2: num] :
% 13.19/7.56 ( ( ( bit0 @ X22 )
% 13.19/7.56 = ( bit0 @ Y2 ) )
% 13.19/7.56 = ( X22 = Y2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_15_semiring__norm_I87_J,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ( bit0 @ M )
% 13.19/7.56 = ( bit0 @ N ) )
% 13.19/7.56 = ( M = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_16_numeral__eq__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ( numera6690914467698888265omplex @ M )
% 13.19/7.56 = ( numera6690914467698888265omplex @ N ) )
% 13.19/7.56 = ( M = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_17_numeral__eq__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ( numeral_numeral_real @ M )
% 13.19/7.56 = ( numeral_numeral_real @ N ) )
% 13.19/7.56 = ( M = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_18_numeral__eq__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ( numeral_numeral_rat @ M )
% 13.19/7.56 = ( numeral_numeral_rat @ N ) )
% 13.19/7.56 = ( M = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_19_numeral__eq__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ( numeral_numeral_nat @ M )
% 13.19/7.56 = ( numeral_numeral_nat @ N ) )
% 13.19/7.56 = ( M = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_20_numeral__eq__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ( numeral_numeral_int @ M )
% 13.19/7.56 = ( numeral_numeral_int @ N ) )
% 13.19/7.56 = ( M = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_21__C4_Ohyps_C_I5_J,axiom,
% 13.19/7.56 m = na ).
% 13.19/7.56
% 13.19/7.56 thf(fact_22_min__in__set__def,axiom,
% 13.19/7.56 ( vEBT_VEBT_min_in_set
% 13.19/7.56 = ( ^ [Xs: set_nat,X2: nat] :
% 13.19/7.56 ( ( member_nat @ X2 @ Xs )
% 13.19/7.56 & ! [Y: nat] :
% 13.19/7.56 ( ( member_nat @ Y @ Xs )
% 13.19/7.56 => ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_23_max__in__set__def,axiom,
% 13.19/7.56 ( vEBT_VEBT_max_in_set
% 13.19/7.56 = ( ^ [Xs: set_nat,X2: nat] :
% 13.19/7.56 ( ( member_nat @ X2 @ Xs )
% 13.19/7.56 & ! [Y: nat] :
% 13.19/7.56 ( ( member_nat @ Y @ Xs )
% 13.19/7.56 => ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_24_power__shift,axiom,
% 13.19/7.56 ! [X: nat,Y3: nat,Z: nat] :
% 13.19/7.56 ( ( ( power_power_nat @ X @ Y3 )
% 13.19/7.56 = Z )
% 13.19/7.56 = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y3 ) )
% 13.19/7.56 = ( some_nat @ Z ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_25_numeral__le__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 13.19/7.56 = ( ord_less_eq_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_26_numeral__le__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 13.19/7.56 = ( ord_less_eq_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_27_numeral__le__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.19/7.56 = ( ord_less_eq_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_28_numeral__le__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.19/7.56 = ( ord_less_eq_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_29_lesseq__shift,axiom,
% 13.19/7.56 ( ord_less_eq_nat
% 13.19/7.56 = ( ^ [X2: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_30_verit__comp__simplify1_I2_J,axiom,
% 13.19/7.56 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_31_verit__comp__simplify1_I2_J,axiom,
% 13.19/7.56 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_32_verit__comp__simplify1_I2_J,axiom,
% 13.19/7.56 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_33_verit__comp__simplify1_I2_J,axiom,
% 13.19/7.56 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_34_verit__comp__simplify1_I2_J,axiom,
% 13.19/7.56 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_35_verit__la__disequality,axiom,
% 13.19/7.56 ! [A: rat,B: rat] :
% 13.19/7.56 ( ( A = B )
% 13.19/7.56 | ~ ( ord_less_eq_rat @ A @ B )
% 13.19/7.56 | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_36_verit__la__disequality,axiom,
% 13.19/7.56 ! [A: num,B: num] :
% 13.19/7.56 ( ( A = B )
% 13.19/7.56 | ~ ( ord_less_eq_num @ A @ B )
% 13.19/7.56 | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_37_verit__la__disequality,axiom,
% 13.19/7.56 ! [A: nat,B: nat] :
% 13.19/7.56 ( ( A = B )
% 13.19/7.56 | ~ ( ord_less_eq_nat @ A @ B )
% 13.19/7.56 | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_38_verit__la__disequality,axiom,
% 13.19/7.56 ! [A: int,B: int] :
% 13.19/7.56 ( ( A = B )
% 13.19/7.56 | ~ ( ord_less_eq_int @ A @ B )
% 13.19/7.56 | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_39_div__le__dividend,axiom,
% 13.19/7.56 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_40_div__le__mono,axiom,
% 13.19/7.56 ! [M: nat,N: nat,K: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.56 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_41_verit__eq__simplify_I10_J,axiom,
% 13.19/7.56 ! [X22: num] :
% 13.19/7.56 ( one
% 13.19/7.56 != ( bit0 @ X22 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_42__092_060open_062mi_A_092_060le_062_Ax_092_060close_062,axiom,
% 13.19/7.56 ord_less_eq_nat @ mi @ xa ).
% 13.19/7.56
% 13.19/7.56 thf(fact_43_not__None__eq,axiom,
% 13.19/7.56 ! [X: option_nat] :
% 13.19/7.56 ( ( X != none_nat )
% 13.19/7.56 = ( ? [Y: nat] :
% 13.19/7.56 ( X
% 13.19/7.56 = ( some_nat @ Y ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_44_not__None__eq,axiom,
% 13.19/7.56 ! [X: option4927543243414619207at_nat] :
% 13.19/7.56 ( ( X != none_P5556105721700978146at_nat )
% 13.19/7.56 = ( ? [Y: product_prod_nat_nat] :
% 13.19/7.56 ( X
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_45_not__None__eq,axiom,
% 13.19/7.56 ! [X: option_num] :
% 13.19/7.56 ( ( X != none_num )
% 13.19/7.56 = ( ? [Y: num] :
% 13.19/7.56 ( X
% 13.19/7.56 = ( some_num @ Y ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_46_not__Some__eq,axiom,
% 13.19/7.56 ! [X: option_nat] :
% 13.19/7.56 ( ( ! [Y: nat] :
% 13.19/7.56 ( X
% 13.19/7.56 != ( some_nat @ Y ) ) )
% 13.19/7.56 = ( X = none_nat ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_47_not__Some__eq,axiom,
% 13.19/7.56 ! [X: option4927543243414619207at_nat] :
% 13.19/7.56 ( ( ! [Y: product_prod_nat_nat] :
% 13.19/7.56 ( X
% 13.19/7.56 != ( some_P7363390416028606310at_nat @ Y ) ) )
% 13.19/7.56 = ( X = none_P5556105721700978146at_nat ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_48_not__Some__eq,axiom,
% 13.19/7.56 ! [X: option_num] :
% 13.19/7.56 ( ( ! [Y: num] :
% 13.19/7.56 ( X
% 13.19/7.56 != ( some_num @ Y ) ) )
% 13.19/7.56 = ( X = none_num ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_49_power2__nat__le__imp__le,axiom,
% 13.19/7.56 ! [M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 13.19/7.56 => ( ord_less_eq_nat @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_50_power2__nat__le__eq__le,axiom,
% 13.19/7.56 ! [M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.56 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_51_self__le__ge2__pow,axiom,
% 13.19/7.56 ! [K: nat,M: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 13.19/7.56 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_52_power__minus__is__div,axiom,
% 13.19/7.56 ! [B: nat,A: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ B @ A )
% 13.19/7.56 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
% 13.19/7.56 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_53_pow__sum,axiom,
% 13.19/7.56 ! [A: nat,B: nat] :
% 13.19/7.56 ( ( 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 ) )
% 13.19/7.56 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_54__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
% 13.19/7.56 ord_less_eq_nat @ one_one_nat @ na ).
% 13.19/7.56
% 13.19/7.56 thf(fact_55__C4_Ohyps_C_I10_J,axiom,
% 13.19/7.56 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_56_option_Oinject,axiom,
% 13.19/7.56 ! [X22: nat,Y2: nat] :
% 13.19/7.56 ( ( ( some_nat @ X22 )
% 13.19/7.56 = ( some_nat @ Y2 ) )
% 13.19/7.56 = ( X22 = Y2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_57_option_Oinject,axiom,
% 13.19/7.56 ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 13.19/7.56 ( ( ( some_P7363390416028606310at_nat @ X22 )
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ Y2 ) )
% 13.19/7.56 = ( X22 = Y2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_58_option_Oinject,axiom,
% 13.19/7.56 ! [X22: num,Y2: num] :
% 13.19/7.56 ( ( ( some_num @ X22 )
% 13.19/7.56 = ( some_num @ Y2 ) )
% 13.19/7.56 = ( X22 = Y2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_59_zdiv__numeral__Bit0,axiom,
% 13.19/7.56 ! [V: num,W: num] :
% 13.19/7.56 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 13.19/7.56 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_60__C4_Ohyps_C_I9_J,axiom,
% 13.19/7.56 ord_less_eq_nat @ mi @ ma ).
% 13.19/7.56
% 13.19/7.56 thf(fact_61__C4_Ohyps_C_I6_J,axiom,
% 13.19/7.56 ( deg
% 13.19/7.56 = ( plus_plus_nat @ na @ m ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_62_high__bound__aux,axiom,
% 13.19/7.56 ! [Ma: nat,N: nat,M: nat] :
% 13.19/7.56 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 13.19/7.56 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_63_numeral__less__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 13.19/7.56 = ( ord_less_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_64_numeral__less__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 13.19/7.56 = ( ord_less_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_65_numeral__less__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.19/7.56 = ( ord_less_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_66_numeral__less__iff,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.19/7.56 = ( ord_less_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_67_add__numeral__left,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: complex] :
% 13.19/7.56 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 13.19/7.56 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_68_add__numeral__left,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: real] :
% 13.19/7.56 ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 13.19/7.56 = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_69_add__numeral__left,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: rat] :
% 13.19/7.56 ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 13.19/7.56 = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_70_add__numeral__left,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: nat] :
% 13.19/7.56 ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 13.19/7.56 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_71_add__numeral__left,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: int] :
% 13.19/7.56 ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 13.19/7.56 = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_72_numeral__plus__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 13.19/7.56 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_73_numeral__plus__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 13.19/7.56 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_74_numeral__plus__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 13.19/7.56 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_75_numeral__plus__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.19/7.56 = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_76_numeral__plus__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.19/7.56 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_77_mem__Collect__eq,axiom,
% 13.19/7.56 ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
% 13.19/7.56 ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
% 13.19/7.56 = ( P @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_78_mem__Collect__eq,axiom,
% 13.19/7.56 ! [A: complex,P: complex > $o] :
% 13.19/7.56 ( ( member_complex @ A @ ( collect_complex @ P ) )
% 13.19/7.56 = ( P @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_79_mem__Collect__eq,axiom,
% 13.19/7.56 ! [A: real,P: real > $o] :
% 13.19/7.56 ( ( member_real @ A @ ( collect_real @ P ) )
% 13.19/7.56 = ( P @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_80_mem__Collect__eq,axiom,
% 13.19/7.56 ! [A: list_nat,P: list_nat > $o] :
% 13.19/7.56 ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 13.19/7.56 = ( P @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_81_mem__Collect__eq,axiom,
% 13.19/7.56 ! [A: nat,P: nat > $o] :
% 13.19/7.56 ( ( member_nat @ A @ ( collect_nat @ P ) )
% 13.19/7.56 = ( P @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_82_mem__Collect__eq,axiom,
% 13.19/7.56 ! [A: int,P: int > $o] :
% 13.19/7.56 ( ( member_int @ A @ ( collect_int @ P ) )
% 13.19/7.56 = ( P @ A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_83_Collect__mem__eq,axiom,
% 13.19/7.56 ! [A2: set_Pr8693737435421807431at_nat] :
% 13.19/7.56 ( ( collec7088162979684241874at_nat
% 13.19/7.56 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A2 ) )
% 13.19/7.56 = A2 ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_84_Collect__mem__eq,axiom,
% 13.19/7.56 ! [A2: set_complex] :
% 13.19/7.56 ( ( collect_complex
% 13.19/7.56 @ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
% 13.19/7.56 = A2 ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_85_Collect__mem__eq,axiom,
% 13.19/7.56 ! [A2: set_real] :
% 13.19/7.56 ( ( collect_real
% 13.19/7.56 @ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
% 13.19/7.56 = A2 ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_86_Collect__mem__eq,axiom,
% 13.19/7.56 ! [A2: set_list_nat] :
% 13.19/7.56 ( ( collect_list_nat
% 13.19/7.56 @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A2 ) )
% 13.19/7.56 = A2 ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_87_Collect__mem__eq,axiom,
% 13.19/7.56 ! [A2: set_nat] :
% 13.19/7.56 ( ( collect_nat
% 13.19/7.56 @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
% 13.19/7.56 = A2 ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_88_Collect__mem__eq,axiom,
% 13.19/7.56 ! [A2: set_int] :
% 13.19/7.56 ( ( collect_int
% 13.19/7.56 @ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
% 13.19/7.56 = A2 ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_89_Collect__cong,axiom,
% 13.19/7.56 ! [P: complex > $o,Q: complex > $o] :
% 13.19/7.56 ( ! [X3: complex] :
% 13.19/7.56 ( ( P @ X3 )
% 13.19/7.56 = ( Q @ X3 ) )
% 13.19/7.56 => ( ( collect_complex @ P )
% 13.19/7.56 = ( collect_complex @ Q ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_90_Collect__cong,axiom,
% 13.19/7.56 ! [P: real > $o,Q: real > $o] :
% 13.19/7.56 ( ! [X3: real] :
% 13.19/7.56 ( ( P @ X3 )
% 13.19/7.56 = ( Q @ X3 ) )
% 13.19/7.56 => ( ( collect_real @ P )
% 13.19/7.56 = ( collect_real @ Q ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_91_Collect__cong,axiom,
% 13.19/7.56 ! [P: list_nat > $o,Q: list_nat > $o] :
% 13.19/7.56 ( ! [X3: list_nat] :
% 13.19/7.56 ( ( P @ X3 )
% 13.19/7.56 = ( Q @ X3 ) )
% 13.19/7.56 => ( ( collect_list_nat @ P )
% 13.19/7.56 = ( collect_list_nat @ Q ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_92_Collect__cong,axiom,
% 13.19/7.56 ! [P: nat > $o,Q: nat > $o] :
% 13.19/7.56 ( ! [X3: nat] :
% 13.19/7.56 ( ( P @ X3 )
% 13.19/7.56 = ( Q @ X3 ) )
% 13.19/7.56 => ( ( collect_nat @ P )
% 13.19/7.56 = ( collect_nat @ Q ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_93_Collect__cong,axiom,
% 13.19/7.56 ! [P: int > $o,Q: int > $o] :
% 13.19/7.56 ( ! [X3: int] :
% 13.19/7.56 ( ( P @ X3 )
% 13.19/7.56 = ( Q @ X3 ) )
% 13.19/7.56 => ( ( collect_int @ P )
% 13.19/7.56 = ( collect_int @ Q ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_94_power__one,axiom,
% 13.19/7.56 ! [N: nat] :
% 13.19/7.56 ( ( power_power_rat @ one_one_rat @ N )
% 13.19/7.56 = one_one_rat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_95_power__one,axiom,
% 13.19/7.56 ! [N: nat] :
% 13.19/7.56 ( ( power_power_nat @ one_one_nat @ N )
% 13.19/7.56 = one_one_nat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_96_power__one,axiom,
% 13.19/7.56 ! [N: nat] :
% 13.19/7.56 ( ( power_power_real @ one_one_real @ N )
% 13.19/7.56 = one_one_real ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_97_power__one,axiom,
% 13.19/7.56 ! [N: nat] :
% 13.19/7.56 ( ( power_power_int @ one_one_int @ N )
% 13.19/7.56 = one_one_int ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_98_power__one,axiom,
% 13.19/7.56 ! [N: nat] :
% 13.19/7.56 ( ( power_power_complex @ one_one_complex @ N )
% 13.19/7.56 = one_one_complex ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_99_power__one__right,axiom,
% 13.19/7.56 ! [A: nat] :
% 13.19/7.56 ( ( power_power_nat @ A @ one_one_nat )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_100_power__one__right,axiom,
% 13.19/7.56 ! [A: real] :
% 13.19/7.56 ( ( power_power_real @ A @ one_one_nat )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_101_power__one__right,axiom,
% 13.19/7.56 ! [A: int] :
% 13.19/7.56 ( ( power_power_int @ A @ one_one_nat )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_102_power__one__right,axiom,
% 13.19/7.56 ! [A: complex] :
% 13.19/7.56 ( ( power_power_complex @ A @ one_one_nat )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_103_semiring__norm_I71_J,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.19/7.56 = ( ord_less_eq_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_104_semiring__norm_I68_J,axiom,
% 13.19/7.56 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_105_numeral__eq__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ( numera6690914467698888265omplex @ N )
% 13.19/7.56 = one_one_complex )
% 13.19/7.56 = ( N = one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_106_numeral__eq__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ( numeral_numeral_real @ N )
% 13.19/7.56 = one_one_real )
% 13.19/7.56 = ( N = one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_107_numeral__eq__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ( numeral_numeral_rat @ N )
% 13.19/7.56 = one_one_rat )
% 13.19/7.56 = ( N = one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_108_numeral__eq__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ( numeral_numeral_nat @ N )
% 13.19/7.56 = one_one_nat )
% 13.19/7.56 = ( N = one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_109_numeral__eq__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ( numeral_numeral_int @ N )
% 13.19/7.56 = one_one_int )
% 13.19/7.56 = ( N = one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_110_one__eq__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( one_one_complex
% 13.19/7.56 = ( numera6690914467698888265omplex @ N ) )
% 13.19/7.56 = ( one = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_111_one__eq__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( one_one_real
% 13.19/7.56 = ( numeral_numeral_real @ N ) )
% 13.19/7.56 = ( one = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_112_one__eq__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( one_one_rat
% 13.19/7.56 = ( numeral_numeral_rat @ N ) )
% 13.19/7.56 = ( one = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_113_one__eq__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( one_one_nat
% 13.19/7.56 = ( numeral_numeral_nat @ N ) )
% 13.19/7.56 = ( one = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_114_one__eq__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( one_one_int
% 13.19/7.56 = ( numeral_numeral_int @ N ) )
% 13.19/7.56 = ( one = N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_115_power__inject__exp,axiom,
% 13.19/7.56 ! [A: real,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_real @ one_one_real @ A )
% 13.19/7.56 => ( ( ( power_power_real @ A @ M )
% 13.19/7.56 = ( power_power_real @ A @ N ) )
% 13.19/7.56 = ( M = N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_116_power__inject__exp,axiom,
% 13.19/7.56 ! [A: rat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_rat @ one_one_rat @ A )
% 13.19/7.56 => ( ( ( power_power_rat @ A @ M )
% 13.19/7.56 = ( power_power_rat @ A @ N ) )
% 13.19/7.56 = ( M = N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_117_power__inject__exp,axiom,
% 13.19/7.56 ! [A: nat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_nat @ one_one_nat @ A )
% 13.19/7.56 => ( ( ( power_power_nat @ A @ M )
% 13.19/7.56 = ( power_power_nat @ A @ N ) )
% 13.19/7.56 = ( M = N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_118_power__inject__exp,axiom,
% 13.19/7.56 ! [A: int,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_int @ one_one_int @ A )
% 13.19/7.56 => ( ( ( power_power_int @ A @ M )
% 13.19/7.56 = ( power_power_int @ A @ N ) )
% 13.19/7.56 = ( M = N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_119_semiring__norm_I69_J,axiom,
% 13.19/7.56 ! [M: num] :
% 13.19/7.56 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_120_one__less__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 13.19/7.56 = ( ord_less_num @ one @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_121_one__less__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 13.19/7.56 = ( ord_less_num @ one @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_122_one__less__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 13.19/7.56 = ( ord_less_num @ one @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_123_one__less__numeral__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 13.19/7.56 = ( ord_less_num @ one @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_124_numeral__plus__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ one_one_complex )
% 13.19/7.56 = ( numera6690914467698888265omplex @ ( plus_plus_num @ N @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_125_numeral__plus__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 13.19/7.56 = ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_126_numeral__plus__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 13.19/7.56 = ( numeral_numeral_rat @ ( plus_plus_num @ N @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_127_numeral__plus__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 13.19/7.56 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_128_numeral__plus__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 13.19/7.56 = ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_129_one__plus__numeral,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N ) )
% 13.19/7.56 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_130_one__plus__numeral,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 13.19/7.56 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_131_one__plus__numeral,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 13.19/7.56 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_132_one__plus__numeral,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 13.19/7.56 = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_133_one__plus__numeral,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 13.19/7.56 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_134_power__strict__increasing__iff,axiom,
% 13.19/7.56 ! [B: real,X: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_real @ one_one_real @ B )
% 13.19/7.56 => ( ( ord_less_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y3 ) )
% 13.19/7.56 = ( ord_less_nat @ X @ Y3 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_135_power__strict__increasing__iff,axiom,
% 13.19/7.56 ! [B: rat,X: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_rat @ one_one_rat @ B )
% 13.19/7.56 => ( ( ord_less_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y3 ) )
% 13.19/7.56 = ( ord_less_nat @ X @ Y3 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_136_power__strict__increasing__iff,axiom,
% 13.19/7.56 ! [B: nat,X: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_nat @ one_one_nat @ B )
% 13.19/7.56 => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y3 ) )
% 13.19/7.56 = ( ord_less_nat @ X @ Y3 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_137_power__strict__increasing__iff,axiom,
% 13.19/7.56 ! [B: int,X: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_int @ one_one_int @ B )
% 13.19/7.56 => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y3 ) )
% 13.19/7.56 = ( ord_less_nat @ X @ Y3 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_138_greater__shift,axiom,
% 13.19/7.56 ( ord_less_nat
% 13.19/7.56 = ( ^ [Y: nat,X2: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_139_one__add__one,axiom,
% 13.19/7.56 ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 13.19/7.56 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_140_one__add__one,axiom,
% 13.19/7.56 ( ( plus_plus_real @ one_one_real @ one_one_real )
% 13.19/7.56 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_141_one__add__one,axiom,
% 13.19/7.56 ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 13.19/7.56 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_142_one__add__one,axiom,
% 13.19/7.56 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 13.19/7.56 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_143_one__add__one,axiom,
% 13.19/7.56 ( ( plus_plus_int @ one_one_int @ one_one_int )
% 13.19/7.56 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_144_power__increasing__iff,axiom,
% 13.19/7.56 ! [B: real,X: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_real @ one_one_real @ B )
% 13.19/7.56 => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y3 ) )
% 13.19/7.56 = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_145_power__increasing__iff,axiom,
% 13.19/7.56 ! [B: rat,X: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_rat @ one_one_rat @ B )
% 13.19/7.56 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y3 ) )
% 13.19/7.56 = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_146_power__increasing__iff,axiom,
% 13.19/7.56 ! [B: nat,X: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_nat @ one_one_nat @ B )
% 13.19/7.56 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y3 ) )
% 13.19/7.56 = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_147_power__increasing__iff,axiom,
% 13.19/7.56 ! [B: int,X: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_int @ one_one_int @ B )
% 13.19/7.56 => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y3 ) )
% 13.19/7.56 = ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_148_add__self__div__2,axiom,
% 13.19/7.56 ! [M: nat] :
% 13.19/7.56 ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = M ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_149_numeral__le__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 13.19/7.56 = ( ord_less_eq_num @ N @ one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_150_numeral__le__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 13.19/7.56 = ( ord_less_eq_num @ N @ one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_151_numeral__le__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 13.19/7.56 = ( ord_less_eq_num @ N @ one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_152_numeral__le__one__iff,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 13.19/7.56 = ( ord_less_eq_num @ N @ one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_153__C4_Ohyps_C_I2_J,axiom,
% 13.19/7.56 vEBT_invar_vebt @ summary @ m ).
% 13.19/7.56
% 13.19/7.56 thf(fact_154_local_Opower__def,axiom,
% 13.19/7.56 ( vEBT_VEBT_power
% 13.19/7.56 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_155_verit__comp__simplify1_I1_J,axiom,
% 13.19/7.56 ! [A: real] :
% 13.19/7.56 ~ ( ord_less_real @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_156_verit__comp__simplify1_I1_J,axiom,
% 13.19/7.56 ! [A: rat] :
% 13.19/7.56 ~ ( ord_less_rat @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_157_verit__comp__simplify1_I1_J,axiom,
% 13.19/7.56 ! [A: num] :
% 13.19/7.56 ~ ( ord_less_num @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_158_verit__comp__simplify1_I1_J,axiom,
% 13.19/7.56 ! [A: nat] :
% 13.19/7.56 ~ ( ord_less_nat @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_159_verit__comp__simplify1_I1_J,axiom,
% 13.19/7.56 ! [A: int] :
% 13.19/7.56 ~ ( ord_less_int @ A @ A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_160_is__num__normalize_I1_J,axiom,
% 13.19/7.56 ! [A: real,B: real,C: real] :
% 13.19/7.56 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_161_is__num__normalize_I1_J,axiom,
% 13.19/7.56 ! [A: rat,B: rat,C: rat] :
% 13.19/7.56 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_162_is__num__normalize_I1_J,axiom,
% 13.19/7.56 ! [A: int,B: int,C: int] :
% 13.19/7.56 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_163_less__numeral__extra_I4_J,axiom,
% 13.19/7.56 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_164_less__numeral__extra_I4_J,axiom,
% 13.19/7.56 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_165_less__numeral__extra_I4_J,axiom,
% 13.19/7.56 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_166_less__numeral__extra_I4_J,axiom,
% 13.19/7.56 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_167_power__less__imp__less__exp,axiom,
% 13.19/7.56 ! [A: real,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_real @ one_one_real @ A )
% 13.19/7.56 => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 13.19/7.56 => ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_168_power__less__imp__less__exp,axiom,
% 13.19/7.56 ! [A: rat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_rat @ one_one_rat @ A )
% 13.19/7.56 => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 13.19/7.56 => ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_169_power__less__imp__less__exp,axiom,
% 13.19/7.56 ! [A: nat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_nat @ one_one_nat @ A )
% 13.19/7.56 => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 13.19/7.56 => ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_170_power__less__imp__less__exp,axiom,
% 13.19/7.56 ! [A: int,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_int @ one_one_int @ A )
% 13.19/7.56 => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 13.19/7.56 => ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_171_power__strict__increasing,axiom,
% 13.19/7.56 ! [N: nat,N3: nat,A: real] :
% 13.19/7.56 ( ( ord_less_nat @ N @ N3 )
% 13.19/7.56 => ( ( ord_less_real @ one_one_real @ A )
% 13.19/7.56 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_172_power__strict__increasing,axiom,
% 13.19/7.56 ! [N: nat,N3: nat,A: rat] :
% 13.19/7.56 ( ( ord_less_nat @ N @ N3 )
% 13.19/7.56 => ( ( ord_less_rat @ one_one_rat @ A )
% 13.19/7.56 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_173_power__strict__increasing,axiom,
% 13.19/7.56 ! [N: nat,N3: nat,A: nat] :
% 13.19/7.56 ( ( ord_less_nat @ N @ N3 )
% 13.19/7.56 => ( ( ord_less_nat @ one_one_nat @ A )
% 13.19/7.56 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_174_power__strict__increasing,axiom,
% 13.19/7.56 ! [N: nat,N3: nat,A: int] :
% 13.19/7.56 ( ( ord_less_nat @ N @ N3 )
% 13.19/7.56 => ( ( ord_less_int @ one_one_int @ A )
% 13.19/7.56 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_175_discrete,axiom,
% 13.19/7.56 ( ord_less_nat
% 13.19/7.56 = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_176_discrete,axiom,
% 13.19/7.56 ( ord_less_int
% 13.19/7.56 = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_177_one__plus__numeral__commute,axiom,
% 13.19/7.56 ! [X: num] :
% 13.19/7.56 ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
% 13.19/7.56 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_178_one__plus__numeral__commute,axiom,
% 13.19/7.56 ! [X: num] :
% 13.19/7.56 ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 13.19/7.56 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_179_one__plus__numeral__commute,axiom,
% 13.19/7.56 ! [X: num] :
% 13.19/7.56 ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 13.19/7.56 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_180_one__plus__numeral__commute,axiom,
% 13.19/7.56 ! [X: num] :
% 13.19/7.56 ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 13.19/7.56 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_181_one__plus__numeral__commute,axiom,
% 13.19/7.56 ! [X: num] :
% 13.19/7.56 ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 13.19/7.56 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_182_not__numeral__less__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_183_not__numeral__less__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_184_not__numeral__less__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_185_not__numeral__less__one,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_186_power__le__imp__le__exp,axiom,
% 13.19/7.56 ! [A: real,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_real @ one_one_real @ A )
% 13.19/7.56 => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) )
% 13.19/7.56 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_187_power__le__imp__le__exp,axiom,
% 13.19/7.56 ! [A: rat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_rat @ one_one_rat @ A )
% 13.19/7.56 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) )
% 13.19/7.56 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_188_power__le__imp__le__exp,axiom,
% 13.19/7.56 ! [A: nat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_nat @ one_one_nat @ A )
% 13.19/7.56 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 13.19/7.56 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_189_power__le__imp__le__exp,axiom,
% 13.19/7.56 ! [A: int,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_int @ one_one_int @ A )
% 13.19/7.56 => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 13.19/7.56 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_190_verit__comp__simplify1_I3_J,axiom,
% 13.19/7.56 ! [B2: real,A4: real] :
% 13.19/7.56 ( ( ~ ( ord_less_eq_real @ B2 @ A4 ) )
% 13.19/7.56 = ( ord_less_real @ A4 @ B2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_191_verit__comp__simplify1_I3_J,axiom,
% 13.19/7.56 ! [B2: rat,A4: rat] :
% 13.19/7.56 ( ( ~ ( ord_less_eq_rat @ B2 @ A4 ) )
% 13.19/7.56 = ( ord_less_rat @ A4 @ B2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_192_verit__comp__simplify1_I3_J,axiom,
% 13.19/7.56 ! [B2: num,A4: num] :
% 13.19/7.56 ( ( ~ ( ord_less_eq_num @ B2 @ A4 ) )
% 13.19/7.56 = ( ord_less_num @ A4 @ B2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_193_verit__comp__simplify1_I3_J,axiom,
% 13.19/7.56 ! [B2: nat,A4: nat] :
% 13.19/7.56 ( ( ~ ( ord_less_eq_nat @ B2 @ A4 ) )
% 13.19/7.56 = ( ord_less_nat @ A4 @ B2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_194_verit__comp__simplify1_I3_J,axiom,
% 13.19/7.56 ! [B2: int,A4: int] :
% 13.19/7.56 ( ( ~ ( ord_less_eq_int @ B2 @ A4 ) )
% 13.19/7.56 = ( ord_less_int @ A4 @ B2 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_195_le__num__One__iff,axiom,
% 13.19/7.56 ! [X: num] :
% 13.19/7.56 ( ( ord_less_eq_num @ X @ one )
% 13.19/7.56 = ( X = one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_196_le__numeral__extra_I4_J,axiom,
% 13.19/7.56 ord_less_eq_real @ one_one_real @ one_one_real ).
% 13.19/7.56
% 13.19/7.56 thf(fact_197_le__numeral__extra_I4_J,axiom,
% 13.19/7.56 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 13.19/7.56
% 13.19/7.56 thf(fact_198_le__numeral__extra_I4_J,axiom,
% 13.19/7.56 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 13.19/7.56
% 13.19/7.56 thf(fact_199_le__numeral__extra_I4_J,axiom,
% 13.19/7.56 ord_less_eq_int @ one_one_int @ one_one_int ).
% 13.19/7.56
% 13.19/7.56 thf(fact_200_ex__power__ivl1,axiom,
% 13.19/7.56 ! [B: nat,K: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 13.19/7.56 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 13.19/7.56 => ? [N4: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N4 ) @ K )
% 13.19/7.56 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_201_ex__power__ivl2,axiom,
% 13.19/7.56 ! [B: nat,K: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 13.19/7.56 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 13.19/7.56 => ? [N4: nat] :
% 13.19/7.56 ( ( ord_less_nat @ ( power_power_nat @ B @ N4 ) @ K )
% 13.19/7.56 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_202_one__le__power,axiom,
% 13.19/7.56 ! [A: real,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_real @ one_one_real @ A )
% 13.19/7.56 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_203_one__le__power,axiom,
% 13.19/7.56 ! [A: rat,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 13.19/7.56 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_204_one__le__power,axiom,
% 13.19/7.56 ! [A: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 13.19/7.56 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_205_one__le__power,axiom,
% 13.19/7.56 ! [A: int,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_int @ one_one_int @ A )
% 13.19/7.56 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_206_power__one__over,axiom,
% 13.19/7.56 ! [A: complex,N: nat] :
% 13.19/7.56 ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N )
% 13.19/7.56 = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_207_power__one__over,axiom,
% 13.19/7.56 ! [A: real,N: nat] :
% 13.19/7.56 ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N )
% 13.19/7.56 = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_208_power__one__over,axiom,
% 13.19/7.56 ! [A: rat,N: nat] :
% 13.19/7.56 ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N )
% 13.19/7.56 = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_209_nat__1__add__1,axiom,
% 13.19/7.56 ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 13.19/7.56 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_210_numeral__Bit0,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 13.19/7.56 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_211_numeral__Bit0,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 13.19/7.56 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_212_numeral__Bit0,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 13.19/7.56 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_213_numeral__Bit0,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 13.19/7.56 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_214_numeral__Bit0,axiom,
% 13.19/7.56 ! [N: num] :
% 13.19/7.56 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 13.19/7.56 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_215_power__increasing,axiom,
% 13.19/7.56 ! [N: nat,N3: nat,A: real] :
% 13.19/7.56 ( ( ord_less_eq_nat @ N @ N3 )
% 13.19/7.56 => ( ( ord_less_eq_real @ one_one_real @ A )
% 13.19/7.56 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_216_power__increasing,axiom,
% 13.19/7.56 ! [N: nat,N3: nat,A: rat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ N @ N3 )
% 13.19/7.56 => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 13.19/7.56 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_217_power__increasing,axiom,
% 13.19/7.56 ! [N: nat,N3: nat,A: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ N @ N3 )
% 13.19/7.56 => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 13.19/7.56 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_218_power__increasing,axiom,
% 13.19/7.56 ! [N: nat,N3: nat,A: int] :
% 13.19/7.56 ( ( ord_less_eq_nat @ N @ N3 )
% 13.19/7.56 => ( ( ord_less_eq_int @ one_one_int @ A )
% 13.19/7.56 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_219_one__le__numeral,axiom,
% 13.19/7.56 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_220_one__le__numeral,axiom,
% 13.19/7.56 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_221_one__le__numeral,axiom,
% 13.19/7.56 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_222_one__le__numeral,axiom,
% 13.19/7.56 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_223_numeral__One,axiom,
% 13.19/7.56 ( ( numera6690914467698888265omplex @ one )
% 13.19/7.56 = one_one_complex ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_224_numeral__One,axiom,
% 13.19/7.56 ( ( numeral_numeral_real @ one )
% 13.19/7.56 = one_one_real ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_225_numeral__One,axiom,
% 13.19/7.56 ( ( numeral_numeral_rat @ one )
% 13.19/7.56 = one_one_rat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_226_numeral__One,axiom,
% 13.19/7.56 ( ( numeral_numeral_nat @ one )
% 13.19/7.56 = one_one_nat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_227_numeral__One,axiom,
% 13.19/7.56 ( ( numeral_numeral_int @ one )
% 13.19/7.56 = one_one_int ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_228_numerals_I1_J,axiom,
% 13.19/7.56 ( ( numeral_numeral_nat @ one )
% 13.19/7.56 = one_one_nat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_229_is__succ__in__set__def,axiom,
% 13.19/7.56 ( vEBT_is_succ_in_set
% 13.19/7.56 = ( ^ [Xs: set_nat,X2: nat,Y: nat] :
% 13.19/7.56 ( ( member_nat @ Y @ Xs )
% 13.19/7.56 & ( ord_less_nat @ X2 @ Y )
% 13.19/7.56 & ! [Z2: nat] :
% 13.19/7.56 ( ( member_nat @ Z2 @ Xs )
% 13.19/7.56 => ( ( ord_less_nat @ X2 @ Z2 )
% 13.19/7.56 => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_230_power2__commute,axiom,
% 13.19/7.56 ! [X: complex,Y3: complex] :
% 13.19/7.56 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = ( power_power_complex @ ( minus_minus_complex @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_231_power2__commute,axiom,
% 13.19/7.56 ! [X: real,Y3: real] :
% 13.19/7.56 ( ( power_power_real @ ( minus_minus_real @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = ( power_power_real @ ( minus_minus_real @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_232_power2__commute,axiom,
% 13.19/7.56 ! [X: rat,Y3: rat] :
% 13.19/7.56 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = ( power_power_rat @ ( minus_minus_rat @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_233_power2__commute,axiom,
% 13.19/7.56 ! [X: int,Y3: int] :
% 13.19/7.56 ( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = ( power_power_int @ ( minus_minus_int @ Y3 @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_234_less__exp,axiom,
% 13.19/7.56 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_235_one__power2,axiom,
% 13.19/7.56 ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = one_one_rat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_236_one__power2,axiom,
% 13.19/7.56 ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = one_one_nat ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_237_one__power2,axiom,
% 13.19/7.56 ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = one_one_real ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_238_one__power2,axiom,
% 13.19/7.56 ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = one_one_int ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_239_one__power2,axiom,
% 13.19/7.56 ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.56 = one_one_complex ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_240_diff__le__diff__pow,axiom,
% 13.19/7.56 ! [K: nat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 13.19/7.56 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_241_power__divide,axiom,
% 13.19/7.56 ! [A: complex,B: complex,N: nat] :
% 13.19/7.56 ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
% 13.19/7.56 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_242_power__divide,axiom,
% 13.19/7.56 ! [A: real,B: real,N: nat] :
% 13.19/7.56 ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
% 13.19/7.56 = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_243_power__divide,axiom,
% 13.19/7.56 ! [A: rat,B: rat,N: nat] :
% 13.19/7.56 ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
% 13.19/7.56 = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_244_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 13.19/7.56 ( ( ( X = none_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: nat,B3: nat] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_nat @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_nat @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_245_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 13.19/7.56 ( ( ( X = none_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: nat,B3: product_prod_nat_nat] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_nat @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_246_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 13.19/7.56 ( ( ( X = none_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_num )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: nat,B3: num] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_nat @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_num @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_247_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 13.19/7.56 ( ( ( X = none_P5556105721700978146at_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: product_prod_nat_nat,B3: nat] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_nat @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_248_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 13.19/7.56 ( ( ( X = none_P5556105721700978146at_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_249_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 13.19/7.56 ( ( ( X = none_P5556105721700978146at_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_num )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: product_prod_nat_nat,B3: num] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_num @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_250_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 13.19/7.56 ( ( ( X = none_num )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: num,B3: nat] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_num @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_nat @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_251_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 13.19/7.56 ( ( ( X = none_num )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: num,B3: product_prod_nat_nat] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_num @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_252_combine__options__cases,axiom,
% 13.19/7.56 ! [X: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 13.19/7.56 ( ( ( X = none_num )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ( ( Y3 = none_num )
% 13.19/7.56 => ( P @ X @ Y3 ) )
% 13.19/7.56 => ( ! [A5: num,B3: num] :
% 13.19/7.56 ( ( X
% 13.19/7.56 = ( some_num @ A5 ) )
% 13.19/7.56 => ( ( Y3
% 13.19/7.56 = ( some_num @ B3 ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) )
% 13.19/7.56 => ( P @ X @ Y3 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_253_split__option__all,axiom,
% 13.19/7.56 ( ( ^ [P2: option_nat > $o] :
% 13.19/7.56 ! [X4: option_nat] : ( P2 @ X4 ) )
% 13.19/7.56 = ( ^ [P3: option_nat > $o] :
% 13.19/7.56 ( ( P3 @ none_nat )
% 13.19/7.56 & ! [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_254_split__option__all,axiom,
% 13.19/7.56 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 13.19/7.56 ! [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
% 13.19/7.56 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 13.19/7.56 ( ( P3 @ none_P5556105721700978146at_nat )
% 13.19/7.56 & ! [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_255_split__option__all,axiom,
% 13.19/7.56 ( ( ^ [P2: option_num > $o] :
% 13.19/7.56 ! [X4: option_num] : ( P2 @ X4 ) )
% 13.19/7.56 = ( ^ [P3: option_num > $o] :
% 13.19/7.56 ( ( P3 @ none_num )
% 13.19/7.56 & ! [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_256_split__option__ex,axiom,
% 13.19/7.56 ( ( ^ [P2: option_nat > $o] :
% 13.19/7.56 ? [X4: option_nat] : ( P2 @ X4 ) )
% 13.19/7.56 = ( ^ [P3: option_nat > $o] :
% 13.19/7.56 ( ( P3 @ none_nat )
% 13.19/7.56 | ? [X2: nat] : ( P3 @ ( some_nat @ X2 ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_257_split__option__ex,axiom,
% 13.19/7.56 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 13.19/7.56 ? [X4: option4927543243414619207at_nat] : ( P2 @ X4 ) )
% 13.19/7.56 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 13.19/7.56 ( ( P3 @ none_P5556105721700978146at_nat )
% 13.19/7.56 | ? [X2: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X2 ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_258_split__option__ex,axiom,
% 13.19/7.56 ( ( ^ [P2: option_num > $o] :
% 13.19/7.56 ? [X4: option_num] : ( P2 @ X4 ) )
% 13.19/7.56 = ( ^ [P3: option_num > $o] :
% 13.19/7.56 ( ( P3 @ none_num )
% 13.19/7.56 | ? [X2: num] : ( P3 @ ( some_num @ X2 ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_259_option_Oexhaust,axiom,
% 13.19/7.56 ! [Y3: option_nat] :
% 13.19/7.56 ( ( Y3 != none_nat )
% 13.19/7.56 => ~ ! [X23: nat] :
% 13.19/7.56 ( Y3
% 13.19/7.56 != ( some_nat @ X23 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_260_option_Oexhaust,axiom,
% 13.19/7.56 ! [Y3: option4927543243414619207at_nat] :
% 13.19/7.56 ( ( Y3 != none_P5556105721700978146at_nat )
% 13.19/7.56 => ~ ! [X23: product_prod_nat_nat] :
% 13.19/7.56 ( Y3
% 13.19/7.56 != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_261_option_Oexhaust,axiom,
% 13.19/7.56 ! [Y3: option_num] :
% 13.19/7.56 ( ( Y3 != none_num )
% 13.19/7.56 => ~ ! [X23: num] :
% 13.19/7.56 ( Y3
% 13.19/7.56 != ( some_num @ X23 ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_262_option_OdiscI,axiom,
% 13.19/7.56 ! [Option: option_nat,X22: nat] :
% 13.19/7.56 ( ( Option
% 13.19/7.56 = ( some_nat @ X22 ) )
% 13.19/7.56 => ( Option != none_nat ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_263_option_OdiscI,axiom,
% 13.19/7.56 ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 13.19/7.56 ( ( Option
% 13.19/7.56 = ( some_P7363390416028606310at_nat @ X22 ) )
% 13.19/7.56 => ( Option != none_P5556105721700978146at_nat ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_264_option_OdiscI,axiom,
% 13.19/7.56 ! [Option: option_num,X22: num] :
% 13.19/7.56 ( ( Option
% 13.19/7.56 = ( some_num @ X22 ) )
% 13.19/7.56 => ( Option != none_num ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_265_option_Odistinct_I1_J,axiom,
% 13.19/7.56 ! [X22: nat] :
% 13.19/7.56 ( none_nat
% 13.19/7.56 != ( some_nat @ X22 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_266_option_Odistinct_I1_J,axiom,
% 13.19/7.56 ! [X22: product_prod_nat_nat] :
% 13.19/7.56 ( none_P5556105721700978146at_nat
% 13.19/7.56 != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_267_option_Odistinct_I1_J,axiom,
% 13.19/7.56 ! [X22: num] :
% 13.19/7.56 ( none_num
% 13.19/7.56 != ( some_num @ X22 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_268_add__shift,axiom,
% 13.19/7.56 ! [X: nat,Y3: nat,Z: nat] :
% 13.19/7.56 ( ( ( plus_plus_nat @ X @ Y3 )
% 13.19/7.56 = Z )
% 13.19/7.56 = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y3 ) )
% 13.19/7.56 = ( some_nat @ Z ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_269_less__shift,axiom,
% 13.19/7.56 ( ord_less_nat
% 13.19/7.56 = ( ^ [X2: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X2 ) @ ( some_nat @ Y ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_270_Nat_Oadd__diff__assoc,axiom,
% 13.19/7.56 ! [K: nat,J: nat,I: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ K @ J )
% 13.19/7.56 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 13.19/7.56 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_271_Nat_Oadd__diff__assoc2,axiom,
% 13.19/7.56 ! [K: nat,J: nat,I: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ K @ J )
% 13.19/7.56 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 13.19/7.56 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_272_Nat_Odiff__diff__right,axiom,
% 13.19/7.56 ! [K: nat,J: nat,I: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ K @ J )
% 13.19/7.56 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 13.19/7.56 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_273_le__add__diff__inverse,axiom,
% 13.19/7.56 ! [B: real,A: real] :
% 13.19/7.56 ( ( ord_less_eq_real @ B @ A )
% 13.19/7.56 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 13.19/7.56 = A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_274_le__add__diff__inverse,axiom,
% 13.19/7.56 ! [B: rat,A: rat] :
% 13.19/7.56 ( ( ord_less_eq_rat @ B @ A )
% 13.19/7.56 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 13.19/7.56 = A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_275_le__add__diff__inverse,axiom,
% 13.19/7.56 ! [B: nat,A: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ B @ A )
% 13.19/7.56 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 13.19/7.56 = A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_276_le__add__diff__inverse,axiom,
% 13.19/7.56 ! [B: int,A: int] :
% 13.19/7.56 ( ( ord_less_eq_int @ B @ A )
% 13.19/7.56 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 13.19/7.56 = A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_277_le__add__diff__inverse2,axiom,
% 13.19/7.56 ! [B: real,A: real] :
% 13.19/7.56 ( ( ord_less_eq_real @ B @ A )
% 13.19/7.56 => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 13.19/7.56 = A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_278_le__add__diff__inverse2,axiom,
% 13.19/7.56 ! [B: rat,A: rat] :
% 13.19/7.56 ( ( ord_less_eq_rat @ B @ A )
% 13.19/7.56 => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 13.19/7.56 = A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_279_le__add__diff__inverse2,axiom,
% 13.19/7.56 ! [B: nat,A: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ B @ A )
% 13.19/7.56 => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 13.19/7.56 = A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_280_le__add__diff__inverse2,axiom,
% 13.19/7.56 ! [B: int,A: int] :
% 13.19/7.56 ( ( ord_less_eq_int @ B @ A )
% 13.19/7.56 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 13.19/7.56 = A ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_281__092_060open_062high_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
% 13.19/7.56 ( ( ord_less_nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) )
% 13.19/7.56 & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_282_div__exp__eq,axiom,
% 13.19/7.56 ! [A: nat,M: nat,N: nat] :
% 13.19/7.56 ( ( 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 ) )
% 13.19/7.56 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_283_div__exp__eq,axiom,
% 13.19/7.56 ! [A: int,M: nat,N: nat] :
% 13.19/7.56 ( ( 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 ) )
% 13.19/7.56 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_284_field__less__half__sum,axiom,
% 13.19/7.56 ! [X: real,Y3: real] :
% 13.19/7.56 ( ( ord_less_real @ X @ Y3 )
% 13.19/7.56 => ( ord_less_real @ X @ ( divide_divide_real @ ( plus_plus_real @ X @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_285_field__less__half__sum,axiom,
% 13.19/7.56 ! [X: rat,Y3: rat] :
% 13.19/7.56 ( ( ord_less_rat @ X @ Y3 )
% 13.19/7.56 => ( ord_less_rat @ X @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_286_succ__member,axiom,
% 13.19/7.56 ! [T: vEBT_VEBT,X: nat,Y3: nat] :
% 13.19/7.56 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y3 )
% 13.19/7.56 = ( ( vEBT_vebt_member @ T @ Y3 )
% 13.19/7.56 & ( ord_less_nat @ X @ Y3 )
% 13.19/7.56 & ! [Z2: nat] :
% 13.19/7.56 ( ( ( vEBT_vebt_member @ T @ Z2 )
% 13.19/7.56 & ( ord_less_nat @ X @ Z2 ) )
% 13.19/7.56 => ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_287_high__inv,axiom,
% 13.19/7.56 ! [X: nat,N: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.56 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 13.19/7.56 = Y3 ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_288_add__def,axiom,
% 13.19/7.56 ( vEBT_VEBT_add
% 13.19/7.56 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_289_bit__split__inv,axiom,
% 13.19/7.56 ! [X: nat,D: nat] :
% 13.19/7.56 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
% 13.19/7.56 = X ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_290_member__correct,axiom,
% 13.19/7.56 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 13.19/7.56 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.56 => ( ( vEBT_vebt_member @ T @ X )
% 13.19/7.56 = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_291_set__vebt__set__vebt_H__valid,axiom,
% 13.19/7.56 ! [T: vEBT_VEBT,N: nat] :
% 13.19/7.56 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.56 => ( ( vEBT_set_vebt @ T )
% 13.19/7.56 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_292_member__bound,axiom,
% 13.19/7.56 ! [Tree: vEBT_VEBT,X: nat,N: nat] :
% 13.19/7.56 ( ( vEBT_vebt_member @ Tree @ X )
% 13.19/7.56 => ( ( vEBT_invar_vebt @ Tree @ N )
% 13.19/7.56 => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_293_numeral__times__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) )
% 13.19/7.56 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_294_numeral__times__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 13.19/7.56 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_295_numeral__times__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 13.19/7.56 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_296_numeral__times__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.19/7.56 = ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_297_numeral__times__numeral,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.19/7.56 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_298_mult__numeral__left__semiring__numeral,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: complex] :
% 13.19/7.56 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
% 13.19/7.56 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_299_mult__numeral__left__semiring__numeral,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: real] :
% 13.19/7.56 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
% 13.19/7.56 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_300_mult__numeral__left__semiring__numeral,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: rat] :
% 13.19/7.56 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
% 13.19/7.56 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_301_mult__numeral__left__semiring__numeral,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: nat] :
% 13.19/7.56 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
% 13.19/7.56 = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_302_mult__numeral__left__semiring__numeral,axiom,
% 13.19/7.56 ! [V: num,W: num,Z: int] :
% 13.19/7.56 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
% 13.19/7.56 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_303_low__inv,axiom,
% 13.19/7.56 ! [X: nat,N: nat,Y3: nat] :
% 13.19/7.56 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.56 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
% 13.19/7.56 = X ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_304_bits__div__by__1,axiom,
% 13.19/7.56 ! [A: nat] :
% 13.19/7.56 ( ( divide_divide_nat @ A @ one_one_nat )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_305_bits__div__by__1,axiom,
% 13.19/7.56 ! [A: int] :
% 13.19/7.56 ( ( divide_divide_int @ A @ one_one_int )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_306_div__by__1,axiom,
% 13.19/7.56 ! [A: complex] :
% 13.19/7.56 ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_307_div__by__1,axiom,
% 13.19/7.56 ! [A: real] :
% 13.19/7.56 ( ( divide_divide_real @ A @ one_one_real )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_308_div__by__1,axiom,
% 13.19/7.56 ! [A: rat] :
% 13.19/7.56 ( ( divide_divide_rat @ A @ one_one_rat )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_309_div__by__1,axiom,
% 13.19/7.56 ! [A: nat] :
% 13.19/7.56 ( ( divide_divide_nat @ A @ one_one_nat )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_310_div__by__1,axiom,
% 13.19/7.56 ! [A: int] :
% 13.19/7.56 ( ( divide_divide_int @ A @ one_one_int )
% 13.19/7.56 = A ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_311_nat__add__left__cancel__less,axiom,
% 13.19/7.56 ! [K: nat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 13.19/7.56 = ( ord_less_nat @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_312_nat__add__left__cancel__le,axiom,
% 13.19/7.56 ! [K: nat,M: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 13.19/7.56 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_313_diff__diff__cancel,axiom,
% 13.19/7.56 ! [I: nat,N: nat] :
% 13.19/7.56 ( ( ord_less_eq_nat @ I @ N )
% 13.19/7.56 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 13.19/7.56 = I ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_314_diff__diff__left,axiom,
% 13.19/7.56 ! [I: nat,J: nat,K: nat] :
% 13.19/7.56 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 13.19/7.56 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_315_nat__1__eq__mult__iff,axiom,
% 13.19/7.56 ! [M: nat,N: nat] :
% 13.19/7.56 ( ( one_one_nat
% 13.19/7.56 = ( times_times_nat @ M @ N ) )
% 13.19/7.56 = ( ( M = one_one_nat )
% 13.19/7.56 & ( N = one_one_nat ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_316_nat__mult__eq__1__iff,axiom,
% 13.19/7.56 ! [M: nat,N: nat] :
% 13.19/7.56 ( ( ( times_times_nat @ M @ N )
% 13.19/7.56 = one_one_nat )
% 13.19/7.56 = ( ( M = one_one_nat )
% 13.19/7.56 & ( N = one_one_nat ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_317_semiring__norm_I6_J,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.19/7.56 = ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_318_semiring__norm_I78_J,axiom,
% 13.19/7.56 ! [M: num,N: num] :
% 13.19/7.56 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.19/7.56 = ( ord_less_num @ M @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_319_semiring__norm_I75_J,axiom,
% 13.19/7.56 ! [M: num] :
% 13.19/7.56 ~ ( ord_less_num @ M @ one ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_320_bit__concat__def,axiom,
% 13.19/7.56 ( vEBT_VEBT_bit_concat
% 13.19/7.56 = ( ^ [H: nat,L: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_321_distrib__left__numeral,axiom,
% 13.19/7.56 ! [V: num,B: complex,C: complex] :
% 13.19/7.56 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 13.19/7.56 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_322_distrib__left__numeral,axiom,
% 13.19/7.56 ! [V: num,B: real,C: real] :
% 13.19/7.56 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 13.19/7.56 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_323_distrib__left__numeral,axiom,
% 13.19/7.56 ! [V: num,B: rat,C: rat] :
% 13.19/7.56 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.56 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_324_distrib__left__numeral,axiom,
% 13.19/7.56 ! [V: num,B: nat,C: nat] :
% 13.19/7.56 ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 13.19/7.56 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_325_distrib__left__numeral,axiom,
% 13.19/7.56 ! [V: num,B: int,C: int] :
% 13.19/7.56 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 13.19/7.56 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_326_distrib__right__numeral,axiom,
% 13.19/7.56 ! [A: complex,B: complex,V: num] :
% 13.19/7.56 ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 13.19/7.56 = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_327_distrib__right__numeral,axiom,
% 13.19/7.56 ! [A: real,B: real,V: num] :
% 13.19/7.56 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 13.19/7.56 = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_328_distrib__right__numeral,axiom,
% 13.19/7.56 ! [A: rat,B: rat,V: num] :
% 13.19/7.56 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 13.19/7.56 = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_329_distrib__right__numeral,axiom,
% 13.19/7.56 ! [A: nat,B: nat,V: num] :
% 13.19/7.56 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 13.19/7.56 = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_330_distrib__right__numeral,axiom,
% 13.19/7.56 ! [A: int,B: int,V: num] :
% 13.19/7.56 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 13.19/7.56 = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_331_left__diff__distrib__numeral,axiom,
% 13.19/7.56 ! [A: complex,B: complex,V: num] :
% 13.19/7.56 ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 13.19/7.56 = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_332_left__diff__distrib__numeral,axiom,
% 13.19/7.56 ! [A: real,B: real,V: num] :
% 13.19/7.56 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 13.19/7.56 = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_333_left__diff__distrib__numeral,axiom,
% 13.19/7.56 ! [A: rat,B: rat,V: num] :
% 13.19/7.56 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 13.19/7.56 = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_334_left__diff__distrib__numeral,axiom,
% 13.19/7.56 ! [A: int,B: int,V: num] :
% 13.19/7.56 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 13.19/7.56 = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_335_right__diff__distrib__numeral,axiom,
% 13.19/7.56 ! [V: num,B: complex,C: complex] :
% 13.19/7.56 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
% 13.19/7.56 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_336_right__diff__distrib__numeral,axiom,
% 13.19/7.56 ! [V: num,B: real,C: real] :
% 13.19/7.56 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
% 13.19/7.56 = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_337_right__diff__distrib__numeral,axiom,
% 13.19/7.56 ! [V: num,B: rat,C: rat] :
% 13.19/7.56 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
% 13.19/7.56 = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_338_right__diff__distrib__numeral,axiom,
% 13.19/7.56 ! [V: num,B: int,C: int] :
% 13.19/7.56 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
% 13.19/7.56 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_339_semiring__norm_I2_J,axiom,
% 13.19/7.56 ( ( plus_plus_num @ one @ one )
% 13.19/7.56 = ( bit0 @ one ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_340_semiring__norm_I76_J,axiom,
% 13.19/7.56 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_341_le__divide__eq__numeral1_I1_J,axiom,
% 13.19/7.56 ! [A: real,B: real,W: num] :
% 13.19/7.56 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 13.19/7.56 = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_342_le__divide__eq__numeral1_I1_J,axiom,
% 13.19/7.56 ! [A: rat,B: rat,W: num] :
% 13.19/7.56 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 13.19/7.56 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_343_divide__le__eq__numeral1_I1_J,axiom,
% 13.19/7.56 ! [B: real,W: num,A: real] :
% 13.19/7.56 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 13.19/7.56 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_344_divide__le__eq__numeral1_I1_J,axiom,
% 13.19/7.56 ! [B: rat,W: num,A: rat] :
% 13.19/7.56 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 13.19/7.56 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_345_less__divide__eq__numeral1_I1_J,axiom,
% 13.19/7.56 ! [A: real,B: real,W: num] :
% 13.19/7.56 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 13.19/7.56 = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_346_less__divide__eq__numeral1_I1_J,axiom,
% 13.19/7.56 ! [A: rat,B: rat,W: num] :
% 13.19/7.56 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 13.19/7.56 = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_347_divide__less__eq__numeral1_I1_J,axiom,
% 13.19/7.56 ! [B: real,W: num,A: real] :
% 13.19/7.56 ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
% 13.19/7.56 = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_348_divide__less__eq__numeral1_I1_J,axiom,
% 13.19/7.56 ! [B: rat,W: num,A: rat] :
% 13.19/7.56 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
% 13.19/7.56 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_349_power__add__numeral,axiom,
% 13.19/7.56 ! [A: complex,M: num,N: num] :
% 13.19/7.56 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
% 13.19/7.56 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_350_power__add__numeral,axiom,
% 13.19/7.56 ! [A: real,M: num,N: num] :
% 13.19/7.56 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
% 13.19/7.56 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_351_power__add__numeral,axiom,
% 13.19/7.56 ! [A: rat,M: num,N: num] :
% 13.19/7.56 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
% 13.19/7.56 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_352_power__add__numeral,axiom,
% 13.19/7.56 ! [A: nat,M: num,N: num] :
% 13.19/7.56 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
% 13.19/7.56 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_353_power__add__numeral,axiom,
% 13.19/7.56 ! [A: int,M: num,N: num] :
% 13.19/7.56 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
% 13.19/7.56 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_354_power__add__numeral2,axiom,
% 13.19/7.56 ! [A: complex,M: num,N: num,B: complex] :
% 13.19/7.56 ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 13.19/7.56 = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_355_power__add__numeral2,axiom,
% 13.19/7.56 ! [A: real,M: num,N: num,B: real] :
% 13.19/7.56 ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 13.19/7.56 = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_356_power__add__numeral2,axiom,
% 13.19/7.56 ! [A: rat,M: num,N: num,B: rat] :
% 13.19/7.56 ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 13.19/7.56 = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_357_power__add__numeral2,axiom,
% 13.19/7.56 ! [A: nat,M: num,N: num,B: nat] :
% 13.19/7.56 ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 13.19/7.56 = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_358_power__add__numeral2,axiom,
% 13.19/7.56 ! [A: int,M: num,N: num,B: int] :
% 13.19/7.56 ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
% 13.19/7.56 = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) @ B ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_359_ring__class_Oring__distribs_I2_J,axiom,
% 13.19/7.56 ! [A: real,B: real,C: real] :
% 13.19/7.56 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_360_ring__class_Oring__distribs_I2_J,axiom,
% 13.19/7.56 ! [A: rat,B: rat,C: rat] :
% 13.19/7.56 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_361_ring__class_Oring__distribs_I2_J,axiom,
% 13.19/7.56 ! [A: int,B: int,C: int] :
% 13.19/7.56 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_362_ring__class_Oring__distribs_I1_J,axiom,
% 13.19/7.56 ! [A: real,B: real,C: real] :
% 13.19/7.56 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 13.19/7.56 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_363_ring__class_Oring__distribs_I1_J,axiom,
% 13.19/7.56 ! [A: rat,B: rat,C: rat] :
% 13.19/7.56 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.56 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_364_ring__class_Oring__distribs_I1_J,axiom,
% 13.19/7.56 ! [A: int,B: int,C: int] :
% 13.19/7.56 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 13.19/7.56 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_365_comm__semiring__class_Odistrib,axiom,
% 13.19/7.56 ! [A: real,B: real,C: real] :
% 13.19/7.56 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_366_comm__semiring__class_Odistrib,axiom,
% 13.19/7.56 ! [A: rat,B: rat,C: rat] :
% 13.19/7.56 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_367_comm__semiring__class_Odistrib,axiom,
% 13.19/7.56 ! [A: nat,B: nat,C: nat] :
% 13.19/7.56 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_368_comm__semiring__class_Odistrib,axiom,
% 13.19/7.56 ! [A: int,B: int,C: int] :
% 13.19/7.56 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_369_distrib__left,axiom,
% 13.19/7.56 ! [A: real,B: real,C: real] :
% 13.19/7.56 ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 13.19/7.56 = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_370_distrib__left,axiom,
% 13.19/7.56 ! [A: rat,B: rat,C: rat] :
% 13.19/7.56 ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.56 = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_371_distrib__left,axiom,
% 13.19/7.56 ! [A: nat,B: nat,C: nat] :
% 13.19/7.56 ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 13.19/7.56 = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_372_distrib__left,axiom,
% 13.19/7.56 ! [A: int,B: int,C: int] :
% 13.19/7.56 ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 13.19/7.56 = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_373_distrib__right,axiom,
% 13.19/7.56 ! [A: real,B: real,C: real] :
% 13.19/7.56 ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_374_distrib__right,axiom,
% 13.19/7.56 ! [A: rat,B: rat,C: rat] :
% 13.19/7.56 ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_375_distrib__right,axiom,
% 13.19/7.56 ! [A: nat,B: nat,C: nat] :
% 13.19/7.56 ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_376_distrib__right,axiom,
% 13.19/7.56 ! [A: int,B: int,C: int] :
% 13.19/7.56 ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.19/7.56 = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_377_combine__common__factor,axiom,
% 13.19/7.56 ! [A: real,E: real,B: real,C: real] :
% 13.19/7.56 ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
% 13.19/7.56 = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_378_combine__common__factor,axiom,
% 13.19/7.56 ! [A: rat,E: rat,B: rat,C: rat] :
% 13.19/7.56 ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
% 13.19/7.56 = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_379_combine__common__factor,axiom,
% 13.19/7.56 ! [A: nat,E: nat,B: nat,C: nat] :
% 13.19/7.56 ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
% 13.19/7.56 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_380_combine__common__factor,axiom,
% 13.19/7.56 ! [A: int,E: int,B: int,C: int] :
% 13.19/7.56 ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
% 13.19/7.56 = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% 13.19/7.56
% 13.19/7.56 thf(fact_381_left__diff__distrib,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_382_left__diff__distrib,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_383_left__diff__distrib,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_384_right__diff__distrib,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 13.19/7.57 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_385_right__diff__distrib,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 13.19/7.57 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_386_right__diff__distrib,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 13.19/7.57 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_387_left__diff__distrib_H,axiom,
% 13.19/7.57 ! [B: real,C: real,A: real] :
% 13.19/7.57 ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 13.19/7.57 = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_388_left__diff__distrib_H,axiom,
% 13.19/7.57 ! [B: rat,C: rat,A: rat] :
% 13.19/7.57 ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 13.19/7.57 = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_389_left__diff__distrib_H,axiom,
% 13.19/7.57 ! [B: nat,C: nat,A: nat] :
% 13.19/7.57 ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 13.19/7.57 = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_390_left__diff__distrib_H,axiom,
% 13.19/7.57 ! [B: int,C: int,A: int] :
% 13.19/7.57 ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 13.19/7.57 = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_391_right__diff__distrib_H,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 13.19/7.57 = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_392_right__diff__distrib_H,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 13.19/7.57 = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_393_right__diff__distrib_H,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 13.19/7.57 = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_394_right__diff__distrib_H,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 13.19/7.57 = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_395_le__cube,axiom,
% 13.19/7.57 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_396_le__square,axiom,
% 13.19/7.57 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_397_mult__le__mono,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ( ord_less_eq_nat @ K @ L2 )
% 13.19/7.57 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_398_mult__le__mono1,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_399_mult__le__mono2,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_400_add__mult__distrib,axiom,
% 13.19/7.57 ! [M: nat,N: nat,K: nat] :
% 13.19/7.57 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 13.19/7.57 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_401_add__mult__distrib2,axiom,
% 13.19/7.57 ! [K: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 13.19/7.57 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_402_diff__mult__distrib,axiom,
% 13.19/7.57 ! [M: nat,N: nat,K: nat] :
% 13.19/7.57 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 13.19/7.57 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_403_diff__mult__distrib2,axiom,
% 13.19/7.57 ! [K: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 13.19/7.57 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_404_nat__mult__1,axiom,
% 13.19/7.57 ! [N: nat] :
% 13.19/7.57 ( ( times_times_nat @ one_one_nat @ N )
% 13.19/7.57 = N ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_405_nat__mult__1__right,axiom,
% 13.19/7.57 ! [N: nat] :
% 13.19/7.57 ( ( times_times_nat @ N @ one_one_nat )
% 13.19/7.57 = N ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_406_two__realpow__ge__one,axiom,
% 13.19/7.57 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_407_add__One__commute,axiom,
% 13.19/7.57 ! [N: num] :
% 13.19/7.57 ( ( plus_plus_num @ one @ N )
% 13.19/7.57 = ( plus_plus_num @ N @ one ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_408_power__commutes,axiom,
% 13.19/7.57 ! [A: complex,N: nat] :
% 13.19/7.57 ( ( times_times_complex @ ( power_power_complex @ A @ N ) @ A )
% 13.19/7.57 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_409_power__commutes,axiom,
% 13.19/7.57 ! [A: real,N: nat] :
% 13.19/7.57 ( ( times_times_real @ ( power_power_real @ A @ N ) @ A )
% 13.19/7.57 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_410_power__commutes,axiom,
% 13.19/7.57 ! [A: rat,N: nat] :
% 13.19/7.57 ( ( times_times_rat @ ( power_power_rat @ A @ N ) @ A )
% 13.19/7.57 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_411_power__commutes,axiom,
% 13.19/7.57 ! [A: nat,N: nat] :
% 13.19/7.57 ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
% 13.19/7.57 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_412_power__commutes,axiom,
% 13.19/7.57 ! [A: int,N: nat] :
% 13.19/7.57 ( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
% 13.19/7.57 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_413_power__mult__distrib,axiom,
% 13.19/7.57 ! [A: complex,B: complex,N: nat] :
% 13.19/7.57 ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
% 13.19/7.57 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_414_power__mult__distrib,axiom,
% 13.19/7.57 ! [A: real,B: real,N: nat] :
% 13.19/7.57 ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
% 13.19/7.57 = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_415_power__mult__distrib,axiom,
% 13.19/7.57 ! [A: rat,B: rat,N: nat] :
% 13.19/7.57 ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
% 13.19/7.57 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_416_power__mult__distrib,axiom,
% 13.19/7.57 ! [A: nat,B: nat,N: nat] :
% 13.19/7.57 ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
% 13.19/7.57 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_417_power__mult__distrib,axiom,
% 13.19/7.57 ! [A: int,B: int,N: nat] :
% 13.19/7.57 ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
% 13.19/7.57 = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_418_power__commuting__commutes,axiom,
% 13.19/7.57 ! [X: complex,Y3: complex,N: nat] :
% 13.19/7.57 ( ( ( times_times_complex @ X @ Y3 )
% 13.19/7.57 = ( times_times_complex @ Y3 @ X ) )
% 13.19/7.57 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y3 )
% 13.19/7.57 = ( times_times_complex @ Y3 @ ( power_power_complex @ X @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_419_power__commuting__commutes,axiom,
% 13.19/7.57 ! [X: real,Y3: real,N: nat] :
% 13.19/7.57 ( ( ( times_times_real @ X @ Y3 )
% 13.19/7.57 = ( times_times_real @ Y3 @ X ) )
% 13.19/7.57 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y3 )
% 13.19/7.57 = ( times_times_real @ Y3 @ ( power_power_real @ X @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_420_power__commuting__commutes,axiom,
% 13.19/7.57 ! [X: rat,Y3: rat,N: nat] :
% 13.19/7.57 ( ( ( times_times_rat @ X @ Y3 )
% 13.19/7.57 = ( times_times_rat @ Y3 @ X ) )
% 13.19/7.57 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y3 )
% 13.19/7.57 = ( times_times_rat @ Y3 @ ( power_power_rat @ X @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_421_power__commuting__commutes,axiom,
% 13.19/7.57 ! [X: nat,Y3: nat,N: nat] :
% 13.19/7.57 ( ( ( times_times_nat @ X @ Y3 )
% 13.19/7.57 = ( times_times_nat @ Y3 @ X ) )
% 13.19/7.57 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y3 )
% 13.19/7.57 = ( times_times_nat @ Y3 @ ( power_power_nat @ X @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_422_power__commuting__commutes,axiom,
% 13.19/7.57 ! [X: int,Y3: int,N: nat] :
% 13.19/7.57 ( ( ( times_times_int @ X @ Y3 )
% 13.19/7.57 = ( times_times_int @ Y3 @ X ) )
% 13.19/7.57 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y3 )
% 13.19/7.57 = ( times_times_int @ Y3 @ ( power_power_int @ X @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_423_power__mult,axiom,
% 13.19/7.57 ! [A: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
% 13.19/7.57 = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_424_power__mult,axiom,
% 13.19/7.57 ! [A: real,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_real @ A @ ( times_times_nat @ M @ N ) )
% 13.19/7.57 = ( power_power_real @ ( power_power_real @ A @ M ) @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_425_power__mult,axiom,
% 13.19/7.57 ! [A: int,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
% 13.19/7.57 = ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_426_power__mult,axiom,
% 13.19/7.57 ! [A: complex,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_complex @ A @ ( times_times_nat @ M @ N ) )
% 13.19/7.57 = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_427_left__add__mult__distrib,axiom,
% 13.19/7.57 ! [I: nat,U: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 13.19/7.57 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_428_div__mult2__eq,axiom,
% 13.19/7.57 ! [M: nat,N: nat,Q2: nat] :
% 13.19/7.57 ( ( divide_divide_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 13.19/7.57 = ( divide_divide_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_429_less__1__mult,axiom,
% 13.19/7.57 ! [M: real,N: real] :
% 13.19/7.57 ( ( ord_less_real @ one_one_real @ M )
% 13.19/7.57 => ( ( ord_less_real @ one_one_real @ N )
% 13.19/7.57 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_430_less__1__mult,axiom,
% 13.19/7.57 ! [M: rat,N: rat] :
% 13.19/7.57 ( ( ord_less_rat @ one_one_rat @ M )
% 13.19/7.57 => ( ( ord_less_rat @ one_one_rat @ N )
% 13.19/7.57 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_431_less__1__mult,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_nat @ one_one_nat @ M )
% 13.19/7.57 => ( ( ord_less_nat @ one_one_nat @ N )
% 13.19/7.57 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_432_less__1__mult,axiom,
% 13.19/7.57 ! [M: int,N: int] :
% 13.19/7.57 ( ( ord_less_int @ one_one_int @ M )
% 13.19/7.57 => ( ( ord_less_int @ one_one_int @ N )
% 13.19/7.57 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_433_eq__add__iff1,axiom,
% 13.19/7.57 ! [A: real,E: real,C: real,B: real,D: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 13.19/7.57 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 13.19/7.57 = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
% 13.19/7.57 = D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_434_eq__add__iff1,axiom,
% 13.19/7.57 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 13.19/7.57 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 13.19/7.57 = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
% 13.19/7.57 = D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_435_eq__add__iff1,axiom,
% 13.19/7.57 ! [A: int,E: int,C: int,B: int,D: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 13.19/7.57 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 13.19/7.57 = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
% 13.19/7.57 = D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_436_eq__add__iff2,axiom,
% 13.19/7.57 ! [A: real,E: real,C: real,B: real,D: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
% 13.19/7.57 = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 13.19/7.57 = ( C
% 13.19/7.57 = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_437_eq__add__iff2,axiom,
% 13.19/7.57 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
% 13.19/7.57 = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 13.19/7.57 = ( C
% 13.19/7.57 = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_438_eq__add__iff2,axiom,
% 13.19/7.57 ! [A: int,E: int,C: int,B: int,D: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
% 13.19/7.57 = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 13.19/7.57 = ( C
% 13.19/7.57 = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_439_square__diff__square__factored,axiom,
% 13.19/7.57 ! [X: real,Y3: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) )
% 13.19/7.57 = ( times_times_real @ ( plus_plus_real @ X @ Y3 ) @ ( minus_minus_real @ X @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_440_square__diff__square__factored,axiom,
% 13.19/7.57 ! [X: rat,Y3: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 13.19/7.57 = ( times_times_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( minus_minus_rat @ X @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_441_square__diff__square__factored,axiom,
% 13.19/7.57 ! [X: int,Y3: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
% 13.19/7.57 = ( times_times_int @ ( plus_plus_int @ X @ Y3 ) @ ( minus_minus_int @ X @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_442_mult__diff__mult,axiom,
% 13.19/7.57 ! [X: real,Y3: real,A: real,B: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( times_times_real @ X @ Y3 ) @ ( times_times_real @ A @ B ) )
% 13.19/7.57 = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_443_mult__diff__mult,axiom,
% 13.19/7.57 ! [X: rat,Y3: rat,A: rat,B: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( times_times_rat @ X @ Y3 ) @ ( times_times_rat @ A @ B ) )
% 13.19/7.57 = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y3 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_444_mult__diff__mult,axiom,
% 13.19/7.57 ! [X: int,Y3: int,A: int,B: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( times_times_int @ X @ Y3 ) @ ( times_times_int @ A @ B ) )
% 13.19/7.57 = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y3 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_445_ordered__ring__class_Ole__add__iff2,axiom,
% 13.19/7.57 ! [A: real,E: real,C: real,B: real,D: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_446_ordered__ring__class_Ole__add__iff2,axiom,
% 13.19/7.57 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_447_ordered__ring__class_Ole__add__iff2,axiom,
% 13.19/7.57 ! [A: int,E: int,C: int,B: int,D: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_448_ordered__ring__class_Ole__add__iff1,axiom,
% 13.19/7.57 ! [A: real,E: real,C: real,B: real,D: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_449_ordered__ring__class_Ole__add__iff1,axiom,
% 13.19/7.57 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_450_ordered__ring__class_Ole__add__iff1,axiom,
% 13.19/7.57 ! [A: int,E: int,C: int,B: int,D: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_451_less__add__iff1,axiom,
% 13.19/7.57 ! [A: real,E: real,C: real,B: real,D: real] :
% 13.19/7.57 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_452_less__add__iff1,axiom,
% 13.19/7.57 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 13.19/7.57 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_453_less__add__iff1,axiom,
% 13.19/7.57 ! [A: int,E: int,C: int,B: int,D: int] :
% 13.19/7.57 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_454_less__add__iff2,axiom,
% 13.19/7.57 ! [A: real,E: real,C: real,B: real,D: real] :
% 13.19/7.57 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_455_less__add__iff2,axiom,
% 13.19/7.57 ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
% 13.19/7.57 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_456_less__add__iff2,axiom,
% 13.19/7.57 ! [A: int,E: int,C: int,B: int,D: int] :
% 13.19/7.57 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
% 13.19/7.57 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_457_square__diff__one__factored,axiom,
% 13.19/7.57 ! [X: complex] :
% 13.19/7.57 ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
% 13.19/7.57 = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_458_square__diff__one__factored,axiom,
% 13.19/7.57 ! [X: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
% 13.19/7.57 = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_459_square__diff__one__factored,axiom,
% 13.19/7.57 ! [X: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
% 13.19/7.57 = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_460_square__diff__one__factored,axiom,
% 13.19/7.57 ! [X: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
% 13.19/7.57 = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_461_mult__numeral__1__right,axiom,
% 13.19/7.57 ! [A: complex] :
% 13.19/7.57 ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_462_mult__numeral__1__right,axiom,
% 13.19/7.57 ! [A: real] :
% 13.19/7.57 ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_463_mult__numeral__1__right,axiom,
% 13.19/7.57 ! [A: rat] :
% 13.19/7.57 ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_464_mult__numeral__1__right,axiom,
% 13.19/7.57 ! [A: nat] :
% 13.19/7.57 ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_465_mult__numeral__1__right,axiom,
% 13.19/7.57 ! [A: int] :
% 13.19/7.57 ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_466_mult__numeral__1,axiom,
% 13.19/7.57 ! [A: complex] :
% 13.19/7.57 ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_467_mult__numeral__1,axiom,
% 13.19/7.57 ! [A: real] :
% 13.19/7.57 ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_468_mult__numeral__1,axiom,
% 13.19/7.57 ! [A: rat] :
% 13.19/7.57 ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_469_mult__numeral__1,axiom,
% 13.19/7.57 ! [A: nat] :
% 13.19/7.57 ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_470_mult__numeral__1,axiom,
% 13.19/7.57 ! [A: int] :
% 13.19/7.57 ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_471_left__right__inverse__power,axiom,
% 13.19/7.57 ! [X: complex,Y3: complex,N: nat] :
% 13.19/7.57 ( ( ( times_times_complex @ X @ Y3 )
% 13.19/7.57 = one_one_complex )
% 13.19/7.57 => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 13.19/7.57 = one_one_complex ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_472_left__right__inverse__power,axiom,
% 13.19/7.57 ! [X: real,Y3: real,N: nat] :
% 13.19/7.57 ( ( ( times_times_real @ X @ Y3 )
% 13.19/7.57 = one_one_real )
% 13.19/7.57 => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y3 @ N ) )
% 13.19/7.57 = one_one_real ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_473_left__right__inverse__power,axiom,
% 13.19/7.57 ! [X: rat,Y3: rat,N: nat] :
% 13.19/7.57 ( ( ( times_times_rat @ X @ Y3 )
% 13.19/7.57 = one_one_rat )
% 13.19/7.57 => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 13.19/7.57 = one_one_rat ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_474_left__right__inverse__power,axiom,
% 13.19/7.57 ! [X: nat,Y3: nat,N: nat] :
% 13.19/7.57 ( ( ( times_times_nat @ X @ Y3 )
% 13.19/7.57 = one_one_nat )
% 13.19/7.57 => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ N ) )
% 13.19/7.57 = one_one_nat ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_475_left__right__inverse__power,axiom,
% 13.19/7.57 ! [X: int,Y3: int,N: nat] :
% 13.19/7.57 ( ( ( times_times_int @ X @ Y3 )
% 13.19/7.57 = one_one_int )
% 13.19/7.57 => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) )
% 13.19/7.57 = one_one_int ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_476_power__add,axiom,
% 13.19/7.57 ! [A: complex,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N ) )
% 13.19/7.57 = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_477_power__add,axiom,
% 13.19/7.57 ! [A: real,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N ) )
% 13.19/7.57 = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_478_power__add,axiom,
% 13.19/7.57 ! [A: rat,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N ) )
% 13.19/7.57 = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_479_power__add,axiom,
% 13.19/7.57 ! [A: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
% 13.19/7.57 = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_480_power__add,axiom,
% 13.19/7.57 ! [A: int,M: nat,N: nat] :
% 13.19/7.57 ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
% 13.19/7.57 = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_481_less__mult__imp__div__less,axiom,
% 13.19/7.57 ! [M: nat,I: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 13.19/7.57 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_482_div__times__less__eq__dividend,axiom,
% 13.19/7.57 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_483_times__div__less__eq__dividend,axiom,
% 13.19/7.57 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_484_power__less__power__Suc,axiom,
% 13.19/7.57 ! [A: real,N: nat] :
% 13.19/7.57 ( ( ord_less_real @ one_one_real @ A )
% 13.19/7.57 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_485_power__less__power__Suc,axiom,
% 13.19/7.57 ! [A: rat,N: nat] :
% 13.19/7.57 ( ( ord_less_rat @ one_one_rat @ A )
% 13.19/7.57 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_486_power__less__power__Suc,axiom,
% 13.19/7.57 ! [A: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_nat @ one_one_nat @ A )
% 13.19/7.57 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_487_power__less__power__Suc,axiom,
% 13.19/7.57 ! [A: int,N: nat] :
% 13.19/7.57 ( ( ord_less_int @ one_one_int @ A )
% 13.19/7.57 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_488_power__gt1__lemma,axiom,
% 13.19/7.57 ! [A: real,N: nat] :
% 13.19/7.57 ( ( ord_less_real @ one_one_real @ A )
% 13.19/7.57 => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_489_power__gt1__lemma,axiom,
% 13.19/7.57 ! [A: rat,N: nat] :
% 13.19/7.57 ( ( ord_less_rat @ one_one_rat @ A )
% 13.19/7.57 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_490_power__gt1__lemma,axiom,
% 13.19/7.57 ! [A: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_nat @ one_one_nat @ A )
% 13.19/7.57 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_491_power__gt1__lemma,axiom,
% 13.19/7.57 ! [A: int,N: nat] :
% 13.19/7.57 ( ( ord_less_int @ one_one_int @ A )
% 13.19/7.57 => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_492_linorder__neqE__linordered__idom,axiom,
% 13.19/7.57 ! [X: real,Y3: real] :
% 13.19/7.57 ( ( X != Y3 )
% 13.19/7.57 => ( ~ ( ord_less_real @ X @ Y3 )
% 13.19/7.57 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_493_linorder__neqE__linordered__idom,axiom,
% 13.19/7.57 ! [X: rat,Y3: rat] :
% 13.19/7.57 ( ( X != Y3 )
% 13.19/7.57 => ( ~ ( ord_less_rat @ X @ Y3 )
% 13.19/7.57 => ( ord_less_rat @ Y3 @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_494_linorder__neqE__linordered__idom,axiom,
% 13.19/7.57 ! [X: int,Y3: int] :
% 13.19/7.57 ( ( X != Y3 )
% 13.19/7.57 => ( ~ ( ord_less_int @ X @ Y3 )
% 13.19/7.57 => ( ord_less_int @ Y3 @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_495_nat__diff__add__eq2,axiom,
% 13.19/7.57 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 13.19/7.57 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_496_nat__diff__add__eq1,axiom,
% 13.19/7.57 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ J @ I )
% 13.19/7.57 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 13.19/7.57 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_497_nat__le__add__iff2,axiom,
% 13.19/7.57 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 13.19/7.57 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_498_nat__le__add__iff1,axiom,
% 13.19/7.57 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ J @ I )
% 13.19/7.57 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 13.19/7.57 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_499_nat__eq__add__iff2,axiom,
% 13.19/7.57 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 13.19/7.57 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 13.19/7.57 = ( M
% 13.19/7.57 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_500_nat__eq__add__iff1,axiom,
% 13.19/7.57 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ J @ I )
% 13.19/7.57 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 13.19/7.57 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 13.19/7.57 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 13.19/7.57 = N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_501_nat__neq__iff,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( M != N )
% 13.19/7.57 = ( ( ord_less_nat @ M @ N )
% 13.19/7.57 | ( ord_less_nat @ N @ M ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_502_less__not__refl,axiom,
% 13.19/7.57 ! [N: nat] :
% 13.19/7.57 ~ ( ord_less_nat @ N @ N ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_503_less__not__refl2,axiom,
% 13.19/7.57 ! [N: nat,M: nat] :
% 13.19/7.57 ( ( ord_less_nat @ N @ M )
% 13.19/7.57 => ( M != N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_504_less__not__refl3,axiom,
% 13.19/7.57 ! [S: nat,T: nat] :
% 13.19/7.57 ( ( ord_less_nat @ S @ T )
% 13.19/7.57 => ( S != T ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_505_less__irrefl__nat,axiom,
% 13.19/7.57 ! [N: nat] :
% 13.19/7.57 ~ ( ord_less_nat @ N @ N ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_506_nat__less__induct,axiom,
% 13.19/7.57 ! [P: nat > $o,N: nat] :
% 13.19/7.57 ( ! [N4: nat] :
% 13.19/7.57 ( ! [M2: nat] :
% 13.19/7.57 ( ( ord_less_nat @ M2 @ N4 )
% 13.19/7.57 => ( P @ M2 ) )
% 13.19/7.57 => ( P @ N4 ) )
% 13.19/7.57 => ( P @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_507_infinite__descent,axiom,
% 13.19/7.57 ! [P: nat > $o,N: nat] :
% 13.19/7.57 ( ! [N4: nat] :
% 13.19/7.57 ( ~ ( P @ N4 )
% 13.19/7.57 => ? [M2: nat] :
% 13.19/7.57 ( ( ord_less_nat @ M2 @ N4 )
% 13.19/7.57 & ~ ( P @ M2 ) ) )
% 13.19/7.57 => ( P @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_508_linorder__neqE__nat,axiom,
% 13.19/7.57 ! [X: nat,Y3: nat] :
% 13.19/7.57 ( ( X != Y3 )
% 13.19/7.57 => ( ~ ( ord_less_nat @ X @ Y3 )
% 13.19/7.57 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_509_le__refl,axiom,
% 13.19/7.57 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_510_le__trans,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ( ord_less_eq_nat @ J @ K )
% 13.19/7.57 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_511_eq__imp__le,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( M = N )
% 13.19/7.57 => ( ord_less_eq_nat @ M @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_512_le__antisym,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.57 => ( ( ord_less_eq_nat @ N @ M )
% 13.19/7.57 => ( M = N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_513_nat__le__linear,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.57 | ( ord_less_eq_nat @ N @ M ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_514_Nat_Oex__has__greatest__nat,axiom,
% 13.19/7.57 ! [P: nat > $o,K: nat,B: nat] :
% 13.19/7.57 ( ( P @ K )
% 13.19/7.57 => ( ! [Y4: nat] :
% 13.19/7.57 ( ( P @ Y4 )
% 13.19/7.57 => ( ord_less_eq_nat @ Y4 @ B ) )
% 13.19/7.57 => ? [X3: nat] :
% 13.19/7.57 ( ( P @ X3 )
% 13.19/7.57 & ! [Y5: nat] :
% 13.19/7.57 ( ( P @ Y5 )
% 13.19/7.57 => ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_515_diff__commute,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 13.19/7.57 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_516_left__add__twice,axiom,
% 13.19/7.57 ! [A: complex,B: complex] :
% 13.19/7.57 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 13.19/7.57 = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_517_left__add__twice,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 13.19/7.57 = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_518_left__add__twice,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 13.19/7.57 = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_519_left__add__twice,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 13.19/7.57 = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_520_left__add__twice,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 13.19/7.57 = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_521_mult__2__right,axiom,
% 13.19/7.57 ! [Z: complex] :
% 13.19/7.57 ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_complex @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_522_mult__2__right,axiom,
% 13.19/7.57 ! [Z: real] :
% 13.19/7.57 ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_real @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_523_mult__2__right,axiom,
% 13.19/7.57 ! [Z: rat] :
% 13.19/7.57 ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_rat @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_524_mult__2__right,axiom,
% 13.19/7.57 ! [Z: nat] :
% 13.19/7.57 ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_nat @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_525_mult__2__right,axiom,
% 13.19/7.57 ! [Z: int] :
% 13.19/7.57 ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_int @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_526_mult__2,axiom,
% 13.19/7.57 ! [Z: complex] :
% 13.19/7.57 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
% 13.19/7.57 = ( plus_plus_complex @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_527_mult__2,axiom,
% 13.19/7.57 ! [Z: real] :
% 13.19/7.57 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
% 13.19/7.57 = ( plus_plus_real @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_528_mult__2,axiom,
% 13.19/7.57 ! [Z: rat] :
% 13.19/7.57 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
% 13.19/7.57 = ( plus_plus_rat @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_529_mult__2,axiom,
% 13.19/7.57 ! [Z: nat] :
% 13.19/7.57 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
% 13.19/7.57 = ( plus_plus_nat @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_530_mult__2,axiom,
% 13.19/7.57 ! [Z: int] :
% 13.19/7.57 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
% 13.19/7.57 = ( plus_plus_int @ Z @ Z ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_531_power2__eq__square,axiom,
% 13.19/7.57 ! [A: complex] :
% 13.19/7.57 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( times_times_complex @ A @ A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_532_power2__eq__square,axiom,
% 13.19/7.57 ! [A: real] :
% 13.19/7.57 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( times_times_real @ A @ A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_533_power2__eq__square,axiom,
% 13.19/7.57 ! [A: rat] :
% 13.19/7.57 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( times_times_rat @ A @ A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_534_power2__eq__square,axiom,
% 13.19/7.57 ! [A: nat] :
% 13.19/7.57 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( times_times_nat @ A @ A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_535_power2__eq__square,axiom,
% 13.19/7.57 ! [A: int] :
% 13.19/7.57 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( times_times_int @ A @ A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_536_power4__eq__xxxx,axiom,
% 13.19/7.57 ! [X: complex] :
% 13.19/7.57 ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.19/7.57 = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_537_power4__eq__xxxx,axiom,
% 13.19/7.57 ! [X: real] :
% 13.19/7.57 ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.19/7.57 = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_538_power4__eq__xxxx,axiom,
% 13.19/7.57 ! [X: rat] :
% 13.19/7.57 ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.19/7.57 = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_539_power4__eq__xxxx,axiom,
% 13.19/7.57 ! [X: nat] :
% 13.19/7.57 ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.19/7.57 = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_540_power4__eq__xxxx,axiom,
% 13.19/7.57 ! [X: int] :
% 13.19/7.57 ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.19/7.57 = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_541_power__even__eq,axiom,
% 13.19/7.57 ! [A: nat,N: nat] :
% 13.19/7.57 ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 = ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_542_power__even__eq,axiom,
% 13.19/7.57 ! [A: real,N: nat] :
% 13.19/7.57 ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 = ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_543_power__even__eq,axiom,
% 13.19/7.57 ! [A: int,N: nat] :
% 13.19/7.57 ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 = ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_544_power__even__eq,axiom,
% 13.19/7.57 ! [A: complex,N: nat] :
% 13.19/7.57 ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 = ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_545_nat__less__add__iff2,axiom,
% 13.19/7.57 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 13.19/7.57 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_546_nat__less__add__iff1,axiom,
% 13.19/7.57 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ J @ I )
% 13.19/7.57 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 13.19/7.57 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_547_power2__sum,axiom,
% 13.19/7.57 ! [X: complex,Y3: complex] :
% 13.19/7.57 ( ( power_power_complex @ ( plus_plus_complex @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_548_power2__sum,axiom,
% 13.19/7.57 ! [X: real,Y3: real] :
% 13.19/7.57 ( ( power_power_real @ ( plus_plus_real @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_549_power2__sum,axiom,
% 13.19/7.57 ! [X: rat,Y3: rat] :
% 13.19/7.57 ( ( power_power_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_550_power2__sum,axiom,
% 13.19/7.57 ! [X: nat,Y3: nat] :
% 13.19/7.57 ( ( power_power_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_551_power2__sum,axiom,
% 13.19/7.57 ! [X: int,Y3: int] :
% 13.19/7.57 ( ( power_power_int @ ( plus_plus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_552_add__diff__add,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real,D: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 13.19/7.57 = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_553_add__diff__add,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat,D: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 13.19/7.57 = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_554_add__diff__add,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int,D: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 13.19/7.57 = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_555_nat__less__le,axiom,
% 13.19/7.57 ( ord_less_nat
% 13.19/7.57 = ( ^ [M3: nat,N2: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.19/7.57 & ( M3 != N2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_556_less__imp__le__nat,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_nat @ M @ N )
% 13.19/7.57 => ( ord_less_eq_nat @ M @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_557_le__eq__less__or__eq,axiom,
% 13.19/7.57 ( ord_less_eq_nat
% 13.19/7.57 = ( ^ [M3: nat,N2: nat] :
% 13.19/7.57 ( ( ord_less_nat @ M3 @ N2 )
% 13.19/7.57 | ( M3 = N2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_558_less__or__eq__imp__le,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( ( ord_less_nat @ M @ N )
% 13.19/7.57 | ( M = N ) )
% 13.19/7.57 => ( ord_less_eq_nat @ M @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_559_le__neq__implies__less,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.57 => ( ( M != N )
% 13.19/7.57 => ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_560_less__mono__imp__le__mono,axiom,
% 13.19/7.57 ! [F: nat > nat,I: nat,J: nat] :
% 13.19/7.57 ( ! [I2: nat,J2: nat] :
% 13.19/7.57 ( ( ord_less_nat @ I2 @ J2 )
% 13.19/7.57 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 13.19/7.57 => ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_561_add__lessD1,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 13.19/7.57 => ( ord_less_nat @ I @ K ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_562_add__less__mono,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ord_less_nat @ I @ J )
% 13.19/7.57 => ( ( ord_less_nat @ K @ L2 )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_563_not__add__less1,axiom,
% 13.19/7.57 ! [I: nat,J: nat] :
% 13.19/7.57 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_564_not__add__less2,axiom,
% 13.19/7.57 ! [J: nat,I: nat] :
% 13.19/7.57 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_565_add__less__mono1,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( ord_less_nat @ I @ J )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_566_trans__less__add1,axiom,
% 13.19/7.57 ! [I: nat,J: nat,M: nat] :
% 13.19/7.57 ( ( ord_less_nat @ I @ J )
% 13.19/7.57 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_567_trans__less__add2,axiom,
% 13.19/7.57 ! [I: nat,J: nat,M: nat] :
% 13.19/7.57 ( ( ord_less_nat @ I @ J )
% 13.19/7.57 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_568_less__add__eq__less,axiom,
% 13.19/7.57 ! [K: nat,L2: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_nat @ K @ L2 )
% 13.19/7.57 => ( ( ( plus_plus_nat @ M @ L2 )
% 13.19/7.57 = ( plus_plus_nat @ K @ N ) )
% 13.19/7.57 => ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_569_add__leE,axiom,
% 13.19/7.57 ! [M: nat,K: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 13.19/7.57 => ~ ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.57 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_570_le__add1,axiom,
% 13.19/7.57 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_571_le__add2,axiom,
% 13.19/7.57 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_572_add__leD1,axiom,
% 13.19/7.57 ! [M: nat,K: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 13.19/7.57 => ( ord_less_eq_nat @ M @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_573_add__leD2,axiom,
% 13.19/7.57 ! [M: nat,K: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 13.19/7.57 => ( ord_less_eq_nat @ K @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_574_le__Suc__ex,axiom,
% 13.19/7.57 ! [K: nat,L2: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ L2 )
% 13.19/7.57 => ? [N4: nat] :
% 13.19/7.57 ( L2
% 13.19/7.57 = ( plus_plus_nat @ K @ N4 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_575_add__le__mono,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ( ord_less_eq_nat @ K @ L2 )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_576_add__le__mono1,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_577_trans__le__add1,axiom,
% 13.19/7.57 ! [I: nat,J: nat,M: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_578_trans__le__add2,axiom,
% 13.19/7.57 ! [I: nat,J: nat,M: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_579_nat__le__iff__add,axiom,
% 13.19/7.57 ( ord_less_eq_nat
% 13.19/7.57 = ( ^ [M3: nat,N2: nat] :
% 13.19/7.57 ? [K2: nat] :
% 13.19/7.57 ( N2
% 13.19/7.57 = ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_580_diff__less__mono2,axiom,
% 13.19/7.57 ! [M: nat,N: nat,L2: nat] :
% 13.19/7.57 ( ( ord_less_nat @ M @ N )
% 13.19/7.57 => ( ( ord_less_nat @ M @ L2 )
% 13.19/7.57 => ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_581_less__imp__diff__less,axiom,
% 13.19/7.57 ! [J: nat,K: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_nat @ J @ K )
% 13.19/7.57 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_582_eq__diff__iff,axiom,
% 13.19/7.57 ! [K: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ M )
% 13.19/7.57 => ( ( ord_less_eq_nat @ K @ N )
% 13.19/7.57 => ( ( ( minus_minus_nat @ M @ K )
% 13.19/7.57 = ( minus_minus_nat @ N @ K ) )
% 13.19/7.57 = ( M = N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_583_le__diff__iff,axiom,
% 13.19/7.57 ! [K: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ M )
% 13.19/7.57 => ( ( ord_less_eq_nat @ K @ N )
% 13.19/7.57 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 13.19/7.57 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_584_Nat_Odiff__diff__eq,axiom,
% 13.19/7.57 ! [K: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ M )
% 13.19/7.57 => ( ( ord_less_eq_nat @ K @ N )
% 13.19/7.57 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 13.19/7.57 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_585_diff__le__mono,axiom,
% 13.19/7.57 ! [M: nat,N: nat,L2: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.57 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_586_diff__le__self,axiom,
% 13.19/7.57 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_587_le__diff__iff_H,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ C )
% 13.19/7.57 => ( ( ord_less_eq_nat @ B @ C )
% 13.19/7.57 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 13.19/7.57 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_588_diff__le__mono2,axiom,
% 13.19/7.57 ! [M: nat,N: nat,L2: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.57 => ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_589_power2__diff,axiom,
% 13.19/7.57 ! [X: complex,Y3: complex] :
% 13.19/7.57 ( ( power_power_complex @ ( minus_minus_complex @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_590_power2__diff,axiom,
% 13.19/7.57 ! [X: real,Y3: real] :
% 13.19/7.57 ( ( power_power_real @ ( minus_minus_real @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_591_power2__diff,axiom,
% 13.19/7.57 ! [X: rat,Y3: rat] :
% 13.19/7.57 ( ( power_power_rat @ ( minus_minus_rat @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_592_power2__diff,axiom,
% 13.19/7.57 ! [X: int,Y3: int] :
% 13.19/7.57 ( ( power_power_int @ ( minus_minus_int @ X @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_593_Nat_Odiff__cancel,axiom,
% 13.19/7.57 ! [K: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 13.19/7.57 = ( minus_minus_nat @ M @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_594_diff__cancel2,axiom,
% 13.19/7.57 ! [M: nat,K: nat,N: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 13.19/7.57 = ( minus_minus_nat @ M @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_595_diff__add__inverse,axiom,
% 13.19/7.57 ! [N: nat,M: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 13.19/7.57 = M ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_596_diff__add__inverse2,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 13.19/7.57 = M ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_597_add__le__add__imp__diff__le,axiom,
% 13.19/7.57 ! [I: real,K: real,N: real,J: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 13.19/7.57 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 13.19/7.57 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 13.19/7.57 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 13.19/7.57 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_598_add__le__add__imp__diff__le,axiom,
% 13.19/7.57 ! [I: rat,K: rat,N: rat,J: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 13.19/7.57 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 13.19/7.57 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 13.19/7.57 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 13.19/7.57 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_599_add__le__add__imp__diff__le,axiom,
% 13.19/7.57 ! [I: nat,K: nat,N: nat,J: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 13.19/7.57 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 13.19/7.57 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 13.19/7.57 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 13.19/7.57 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_600_add__le__add__imp__diff__le,axiom,
% 13.19/7.57 ! [I: int,K: int,N: int,J: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 13.19/7.57 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 13.19/7.57 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 13.19/7.57 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 13.19/7.57 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_601_add__le__imp__le__diff,axiom,
% 13.19/7.57 ! [I: real,K: real,N: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 13.19/7.57 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_602_add__le__imp__le__diff,axiom,
% 13.19/7.57 ! [I: rat,K: rat,N: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 13.19/7.57 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_603_add__le__imp__le__diff,axiom,
% 13.19/7.57 ! [I: nat,K: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 13.19/7.57 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_604_add__le__imp__le__diff,axiom,
% 13.19/7.57 ! [I: int,K: int,N: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 13.19/7.57 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_605_less__add__one,axiom,
% 13.19/7.57 ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_606_less__add__one,axiom,
% 13.19/7.57 ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_607_less__add__one,axiom,
% 13.19/7.57 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_608_less__add__one,axiom,
% 13.19/7.57 ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_609_add__mono1,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_610_add__mono1,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_611_add__mono1,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_nat @ A @ B )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_612_add__mono1,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( ord_less_int @ A @ B )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_613_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ~ ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 13.19/7.57 = A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_614_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ~ ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 13.19/7.57 = A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_615_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ~ ( ord_less_nat @ A @ B )
% 13.19/7.57 => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 13.19/7.57 = A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_616_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ~ ( ord_less_int @ A @ B )
% 13.19/7.57 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 13.19/7.57 = A ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_617_mono__nat__linear__lb,axiom,
% 13.19/7.57 ! [F: nat > nat,M: nat,K: nat] :
% 13.19/7.57 ( ! [M4: nat,N4: nat] :
% 13.19/7.57 ( ( ord_less_nat @ M4 @ N4 )
% 13.19/7.57 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_618_diff__less__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_nat @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_nat @ C @ A )
% 13.19/7.57 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_619_less__diff__iff,axiom,
% 13.19/7.57 ! [K: nat,M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ M )
% 13.19/7.57 => ( ( ord_less_eq_nat @ K @ N )
% 13.19/7.57 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 13.19/7.57 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_620_less__diff__conv,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 13.19/7.57 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_621_add__diff__inverse__nat,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ~ ( ord_less_nat @ M @ N )
% 13.19/7.57 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 13.19/7.57 = M ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_622_Nat_Ole__imp__diff__is__add,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 => ( ( ( minus_minus_nat @ J @ I )
% 13.19/7.57 = K )
% 13.19/7.57 = ( J
% 13.19/7.57 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_623_Nat_Odiff__add__assoc2,axiom,
% 13.19/7.57 ! [K: nat,J: nat,I: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ J )
% 13.19/7.57 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 13.19/7.57 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_624_Nat_Odiff__add__assoc,axiom,
% 13.19/7.57 ! [K: nat,J: nat,I: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ J )
% 13.19/7.57 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 13.19/7.57 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_625_Nat_Ole__diff__conv2,axiom,
% 13.19/7.57 ! [K: nat,J: nat,I: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ J )
% 13.19/7.57 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 13.19/7.57 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_626_le__diff__conv,axiom,
% 13.19/7.57 ! [J: nat,K: nat,I: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 13.19/7.57 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_627_less__diff__conv2,axiom,
% 13.19/7.57 ! [K: nat,J: nat,I: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ K @ J )
% 13.19/7.57 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 13.19/7.57 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_628_field__sum__of__halves,axiom,
% 13.19/7.57 ! [X: real] :
% 13.19/7.57 ( ( plus_plus_real @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.19/7.57 = X ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_629_field__sum__of__halves,axiom,
% 13.19/7.57 ! [X: rat] :
% 13.19/7.57 ( ( plus_plus_rat @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 13.19/7.57 = X ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_630_post__member__pre__member,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat,Y3: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y3 )
% 13.19/7.57 => ( ( vEBT_vebt_member @ T @ Y3 )
% 13.19/7.57 | ( X = Y3 ) ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_631_mul__def,axiom,
% 13.19/7.57 ( vEBT_VEBT_mul
% 13.19/7.57 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_632_misiz,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,M: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( some_nat @ M )
% 13.19/7.57 = ( vEBT_vebt_mint @ T ) )
% 13.19/7.57 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_633_maxt__sound,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 13.19/7.57 => ( ( vEBT_vebt_maxt @ T )
% 13.19/7.57 = ( some_nat @ X ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_634_maxt__corr,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( vEBT_vebt_maxt @ T )
% 13.19/7.57 = ( some_nat @ X ) )
% 13.19/7.57 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_635_mul__shift,axiom,
% 13.19/7.57 ! [X: nat,Y3: nat,Z: nat] :
% 13.19/7.57 ( ( ( times_times_nat @ X @ Y3 )
% 13.19/7.57 = Z )
% 13.19/7.57 = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y3 ) )
% 13.19/7.57 = ( some_nat @ Z ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_636_mint__sound,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
% 13.19/7.57 => ( ( vEBT_vebt_mint @ T )
% 13.19/7.57 = ( some_nat @ X ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_637_mint__corr,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( vEBT_vebt_mint @ T )
% 13.19/7.57 = ( some_nat @ X ) )
% 13.19/7.57 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_638_maxt__corr__help,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( vEBT_vebt_maxt @ T )
% 13.19/7.57 = ( some_nat @ Maxi ) )
% 13.19/7.57 => ( ( vEBT_vebt_member @ T @ X )
% 13.19/7.57 => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_639_sum__squares__bound,axiom,
% 13.19/7.57 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_640_sum__squares__bound,axiom,
% 13.19/7.57 ! [X: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_641_mint__corr__help,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( vEBT_vebt_mint @ T )
% 13.19/7.57 = ( some_nat @ Mini ) )
% 13.19/7.57 => ( ( vEBT_vebt_member @ T @ X )
% 13.19/7.57 => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_642_maxt__member,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( vEBT_vebt_maxt @ T )
% 13.19/7.57 = ( some_nat @ Maxi ) )
% 13.19/7.57 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_643_mint__member,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( vEBT_vebt_mint @ T )
% 13.19/7.57 = ( some_nat @ Maxi ) )
% 13.19/7.57 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_644_semiring__norm_I13_J,axiom,
% 13.19/7.57 ! [M: num,N: num] :
% 13.19/7.57 ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.19/7.57 = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_645_semiring__norm_I12_J,axiom,
% 13.19/7.57 ! [N: num] :
% 13.19/7.57 ( ( times_times_num @ one @ N )
% 13.19/7.57 = N ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_646_semiring__norm_I11_J,axiom,
% 13.19/7.57 ! [M: num] :
% 13.19/7.57 ( ( times_times_num @ M @ one )
% 13.19/7.57 = M ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_647_num__double,axiom,
% 13.19/7.57 ! [N: num] :
% 13.19/7.57 ( ( times_times_num @ ( bit0 @ one ) @ N )
% 13.19/7.57 = ( bit0 @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_648_power__mult__numeral,axiom,
% 13.19/7.57 ! [A: nat,M: num,N: num] :
% 13.19/7.57 ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 13.19/7.57 = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_649_power__mult__numeral,axiom,
% 13.19/7.57 ! [A: real,M: num,N: num] :
% 13.19/7.57 ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 13.19/7.57 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_650_power__mult__numeral,axiom,
% 13.19/7.57 ! [A: int,M: num,N: num] :
% 13.19/7.57 ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 13.19/7.57 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_651_power__mult__numeral,axiom,
% 13.19/7.57 ! [A: complex,M: num,N: num] :
% 13.19/7.57 ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) )
% 13.19/7.57 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_652_complete__real,axiom,
% 13.19/7.57 ! [S2: set_real] :
% 13.19/7.57 ( ? [X5: real] : ( member_real @ X5 @ S2 )
% 13.19/7.57 => ( ? [Z3: real] :
% 13.19/7.57 ! [X3: real] :
% 13.19/7.57 ( ( member_real @ X3 @ S2 )
% 13.19/7.57 => ( ord_less_eq_real @ X3 @ Z3 ) )
% 13.19/7.57 => ? [Y4: real] :
% 13.19/7.57 ( ! [X5: real] :
% 13.19/7.57 ( ( member_real @ X5 @ S2 )
% 13.19/7.57 => ( ord_less_eq_real @ X5 @ Y4 ) )
% 13.19/7.57 & ! [Z3: real] :
% 13.19/7.57 ( ! [X3: real] :
% 13.19/7.57 ( ( member_real @ X3 @ S2 )
% 13.19/7.57 => ( ord_less_eq_real @ X3 @ Z3 ) )
% 13.19/7.57 => ( ord_less_eq_real @ Y4 @ Z3 ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_653_real__arch__pow,axiom,
% 13.19/7.57 ! [X: real,Y3: real] :
% 13.19/7.57 ( ( ord_less_real @ one_one_real @ X )
% 13.19/7.57 => ? [N4: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X @ N4 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_654_less__eq__real__def,axiom,
% 13.19/7.57 ( ord_less_eq_real
% 13.19/7.57 = ( ^ [X2: real,Y: real] :
% 13.19/7.57 ( ( ord_less_real @ X2 @ Y )
% 13.19/7.57 | ( X2 = Y ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_655_four__x__squared,axiom,
% 13.19/7.57 ! [X: real] :
% 13.19/7.57 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.57 = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_656_L2__set__mult__ineq__lemma,axiom,
% 13.19/7.57 ! [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 ) ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_657_div__mult2__numeral__eq,axiom,
% 13.19/7.57 ! [A: nat,K: num,L2: num] :
% 13.19/7.57 ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L2 ) )
% 13.19/7.57 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_658_div__mult2__numeral__eq,axiom,
% 13.19/7.57 ! [A: int,K: num,L2: num] :
% 13.19/7.57 ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L2 ) )
% 13.19/7.57 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_659_real__average__minus__second,axiom,
% 13.19/7.57 ! [B: real,A: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 13.19/7.57 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_660_real__average__minus__first,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
% 13.19/7.57 = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_661_minminNull,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT] :
% 13.19/7.57 ( ( ( vEBT_vebt_mint @ T )
% 13.19/7.57 = none_nat )
% 13.19/7.57 => ( vEBT_VEBT_minNull @ T ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_662_minNullmin,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT] :
% 13.19/7.57 ( ( vEBT_VEBT_minNull @ T )
% 13.19/7.57 => ( ( vEBT_vebt_mint @ T )
% 13.19/7.57 = none_nat ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_663_enat__ord__number_I1_J,axiom,
% 13.19/7.57 ! [M: num,N: num] :
% 13.19/7.57 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 13.19/7.57 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_664_enat__ord__number_I2_J,axiom,
% 13.19/7.57 ! [M: num,N: num] :
% 13.19/7.57 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 13.19/7.57 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_665_add__diff__cancel__right_H,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_666_add__diff__cancel__right_H,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_667_add__diff__cancel__right_H,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_668_add__diff__cancel__right_H,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_669_add__diff__cancel__right,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 13.19/7.57 = ( minus_minus_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_670_add__diff__cancel__right,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.57 = ( minus_minus_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_671_add__diff__cancel__right,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 13.19/7.57 = ( minus_minus_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_672_add__diff__cancel__right,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 13.19/7.57 = ( minus_minus_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_673_add__diff__cancel__left_H,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 13.19/7.57 = B ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_674_add__diff__cancel__left_H,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 13.19/7.57 = B ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_675_add__diff__cancel__left_H,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 13.19/7.57 = B ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_676_add__diff__cancel__left_H,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 13.19/7.57 = B ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_677_add__diff__cancel__left,axiom,
% 13.19/7.57 ! [C: real,A: real,B: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 13.19/7.57 = ( minus_minus_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_678_add__diff__cancel__left,axiom,
% 13.19/7.57 ! [C: rat,A: rat,B: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 13.19/7.57 = ( minus_minus_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_679_add__diff__cancel__left,axiom,
% 13.19/7.57 ! [C: nat,A: nat,B: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 13.19/7.57 = ( minus_minus_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_680_add__diff__cancel__left,axiom,
% 13.19/7.57 ! [C: int,A: int,B: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 13.19/7.57 = ( minus_minus_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_681_diff__add__cancel,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_682_diff__add__cancel,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_683_diff__add__cancel,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_684_min__Null__member,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,X: nat] :
% 13.19/7.57 ( ( vEBT_VEBT_minNull @ T )
% 13.19/7.57 => ~ ( vEBT_vebt_member @ T @ X ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_685_add__left__cancel,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ A @ B )
% 13.19/7.57 = ( plus_plus_real @ A @ C ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_686_add__left__cancel,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ A @ B )
% 13.19/7.57 = ( plus_plus_rat @ A @ C ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_687_add__left__cancel,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ( plus_plus_nat @ A @ B )
% 13.19/7.57 = ( plus_plus_nat @ A @ C ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_688_add__left__cancel,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ A @ B )
% 13.19/7.57 = ( plus_plus_int @ A @ C ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_689_add__right__cancel,axiom,
% 13.19/7.57 ! [B: real,A: real,C: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ B @ A )
% 13.19/7.57 = ( plus_plus_real @ C @ A ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_690_add__right__cancel,axiom,
% 13.19/7.57 ! [B: rat,A: rat,C: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ B @ A )
% 13.19/7.57 = ( plus_plus_rat @ C @ A ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_691_add__right__cancel,axiom,
% 13.19/7.57 ! [B: nat,A: nat,C: nat] :
% 13.19/7.57 ( ( ( plus_plus_nat @ B @ A )
% 13.19/7.57 = ( plus_plus_nat @ C @ A ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_692_add__right__cancel,axiom,
% 13.19/7.57 ! [B: int,A: int,C: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ B @ A )
% 13.19/7.57 = ( plus_plus_int @ C @ A ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_693_add__le__cancel__left,axiom,
% 13.19/7.57 ! [C: real,A: real,B: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 13.19/7.57 = ( ord_less_eq_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_694_add__le__cancel__left,axiom,
% 13.19/7.57 ! [C: rat,A: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 13.19/7.57 = ( ord_less_eq_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_695_add__le__cancel__left,axiom,
% 13.19/7.57 ! [C: nat,A: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 13.19/7.57 = ( ord_less_eq_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_696_add__le__cancel__left,axiom,
% 13.19/7.57 ! [C: int,A: int,B: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 13.19/7.57 = ( ord_less_eq_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_697_add__le__cancel__right,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 13.19/7.57 = ( ord_less_eq_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_698_add__le__cancel__right,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.57 = ( ord_less_eq_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_699_add__le__cancel__right,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 13.19/7.57 = ( ord_less_eq_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_700_add__le__cancel__right,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 13.19/7.57 = ( ord_less_eq_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_701_add__less__cancel__left,axiom,
% 13.19/7.57 ! [C: real,A: real,B: real] :
% 13.19/7.57 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 13.19/7.57 = ( ord_less_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_702_add__less__cancel__left,axiom,
% 13.19/7.57 ! [C: rat,A: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 13.19/7.57 = ( ord_less_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_703_add__less__cancel__left,axiom,
% 13.19/7.57 ! [C: nat,A: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 13.19/7.57 = ( ord_less_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_704_add__less__cancel__left,axiom,
% 13.19/7.57 ! [C: int,A: int,B: int] :
% 13.19/7.57 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 13.19/7.57 = ( ord_less_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_705_add__less__cancel__right,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 13.19/7.57 = ( ord_less_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_706_add__less__cancel__right,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.57 = ( ord_less_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_707_add__less__cancel__right,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 13.19/7.57 = ( ord_less_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_708_add__less__cancel__right,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 13.19/7.57 = ( ord_less_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_709_mult_Oright__neutral,axiom,
% 13.19/7.57 ! [A: complex] :
% 13.19/7.57 ( ( times_times_complex @ A @ one_one_complex )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_710_mult_Oright__neutral,axiom,
% 13.19/7.57 ! [A: real] :
% 13.19/7.57 ( ( times_times_real @ A @ one_one_real )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_711_mult_Oright__neutral,axiom,
% 13.19/7.57 ! [A: rat] :
% 13.19/7.57 ( ( times_times_rat @ A @ one_one_rat )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_712_mult_Oright__neutral,axiom,
% 13.19/7.57 ! [A: nat] :
% 13.19/7.57 ( ( times_times_nat @ A @ one_one_nat )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_713_mult_Oright__neutral,axiom,
% 13.19/7.57 ! [A: int] :
% 13.19/7.57 ( ( times_times_int @ A @ one_one_int )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_714_mult__1,axiom,
% 13.19/7.57 ! [A: complex] :
% 13.19/7.57 ( ( times_times_complex @ one_one_complex @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_715_mult__1,axiom,
% 13.19/7.57 ! [A: real] :
% 13.19/7.57 ( ( times_times_real @ one_one_real @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_716_mult__1,axiom,
% 13.19/7.57 ! [A: rat] :
% 13.19/7.57 ( ( times_times_rat @ one_one_rat @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_717_mult__1,axiom,
% 13.19/7.57 ! [A: nat] :
% 13.19/7.57 ( ( times_times_nat @ one_one_nat @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_718_mult__1,axiom,
% 13.19/7.57 ! [A: int] :
% 13.19/7.57 ( ( times_times_int @ one_one_int @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_719_add__diff__cancel,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_720_add__diff__cancel,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_721_add__diff__cancel,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_722_real__divide__square__eq,axiom,
% 13.19/7.57 ! [R: real,A: real] :
% 13.19/7.57 ( ( divide_divide_real @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ R ) )
% 13.19/7.57 = ( divide_divide_real @ A @ R ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_723_add__diff__assoc__enat,axiom,
% 13.19/7.57 ! [Z: extended_enat,Y3: extended_enat,X: extended_enat] :
% 13.19/7.57 ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 13.19/7.57 => ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 13.19/7.57 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y3 ) @ Z ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_724_enat__less__induct,axiom,
% 13.19/7.57 ! [P: extended_enat > $o,N: extended_enat] :
% 13.19/7.57 ( ! [N4: extended_enat] :
% 13.19/7.57 ( ! [M2: extended_enat] :
% 13.19/7.57 ( ( ord_le72135733267957522d_enat @ M2 @ N4 )
% 13.19/7.57 => ( P @ M2 ) )
% 13.19/7.57 => ( P @ N4 ) )
% 13.19/7.57 => ( P @ N ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_725_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 13.19/7.57 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_726_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 13.19/7.57 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_727_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 13.19/7.57 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_728_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 13.19/7.57 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_729_mult_Oassoc,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 13.19/7.57 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_730_mult_Oassoc,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 13.19/7.57 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_731_mult_Oassoc,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 13.19/7.57 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_732_mult_Oassoc,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 13.19/7.57 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_733_mult_Ocommute,axiom,
% 13.19/7.57 ( times_times_real
% 13.19/7.57 = ( ^ [A3: real,B4: real] : ( times_times_real @ B4 @ A3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_734_mult_Ocommute,axiom,
% 13.19/7.57 ( times_times_rat
% 13.19/7.57 = ( ^ [A3: rat,B4: rat] : ( times_times_rat @ B4 @ A3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_735_mult_Ocommute,axiom,
% 13.19/7.57 ( times_times_nat
% 13.19/7.57 = ( ^ [A3: nat,B4: nat] : ( times_times_nat @ B4 @ A3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_736_mult_Ocommute,axiom,
% 13.19/7.57 ( times_times_int
% 13.19/7.57 = ( ^ [A3: int,B4: int] : ( times_times_int @ B4 @ A3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_737_mult_Oleft__commute,axiom,
% 13.19/7.57 ! [B: real,A: real,C: real] :
% 13.19/7.57 ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 13.19/7.57 = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_738_mult_Oleft__commute,axiom,
% 13.19/7.57 ! [B: rat,A: rat,C: rat] :
% 13.19/7.57 ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 13.19/7.57 = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_739_mult_Oleft__commute,axiom,
% 13.19/7.57 ! [B: nat,A: nat,C: nat] :
% 13.19/7.57 ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 13.19/7.57 = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_740_mult_Oleft__commute,axiom,
% 13.19/7.57 ! [B: int,A: int,C: int] :
% 13.19/7.57 ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 13.19/7.57 = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_741_one__reorient,axiom,
% 13.19/7.57 ! [X: complex] :
% 13.19/7.57 ( ( one_one_complex = X )
% 13.19/7.57 = ( X = one_one_complex ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_742_one__reorient,axiom,
% 13.19/7.57 ! [X: real] :
% 13.19/7.57 ( ( one_one_real = X )
% 13.19/7.57 = ( X = one_one_real ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_743_one__reorient,axiom,
% 13.19/7.57 ! [X: rat] :
% 13.19/7.57 ( ( one_one_rat = X )
% 13.19/7.57 = ( X = one_one_rat ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_744_one__reorient,axiom,
% 13.19/7.57 ! [X: nat] :
% 13.19/7.57 ( ( one_one_nat = X )
% 13.19/7.57 = ( X = one_one_nat ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_745_one__reorient,axiom,
% 13.19/7.57 ! [X: int] :
% 13.19/7.57 ( ( one_one_int = X )
% 13.19/7.57 = ( X = one_one_int ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_746_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 13.19/7.57 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_747_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 13.19/7.57 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_748_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.19/7.57 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_749_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.19/7.57 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_750_add__mono__thms__linordered__semiring_I4_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ( plus_plus_real @ I @ K )
% 13.19/7.57 = ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_751_add__mono__thms__linordered__semiring_I4_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ( plus_plus_rat @ I @ K )
% 13.19/7.57 = ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_752_add__mono__thms__linordered__semiring_I4_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ( plus_plus_nat @ I @ K )
% 13.19/7.57 = ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_753_add__mono__thms__linordered__semiring_I4_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ( plus_plus_int @ I @ K )
% 13.19/7.57 = ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_754_group__cancel_Oadd1,axiom,
% 13.19/7.57 ! [A2: real,K: real,A: real,B: real] :
% 13.19/7.57 ( ( A2
% 13.19/7.57 = ( plus_plus_real @ K @ A ) )
% 13.19/7.57 => ( ( plus_plus_real @ A2 @ B )
% 13.19/7.57 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_755_group__cancel_Oadd1,axiom,
% 13.19/7.57 ! [A2: rat,K: rat,A: rat,B: rat] :
% 13.19/7.57 ( ( A2
% 13.19/7.57 = ( plus_plus_rat @ K @ A ) )
% 13.19/7.57 => ( ( plus_plus_rat @ A2 @ B )
% 13.19/7.57 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_756_group__cancel_Oadd1,axiom,
% 13.19/7.57 ! [A2: nat,K: nat,A: nat,B: nat] :
% 13.19/7.57 ( ( A2
% 13.19/7.57 = ( plus_plus_nat @ K @ A ) )
% 13.19/7.57 => ( ( plus_plus_nat @ A2 @ B )
% 13.19/7.57 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_757_group__cancel_Oadd1,axiom,
% 13.19/7.57 ! [A2: int,K: int,A: int,B: int] :
% 13.19/7.57 ( ( A2
% 13.19/7.57 = ( plus_plus_int @ K @ A ) )
% 13.19/7.57 => ( ( plus_plus_int @ A2 @ B )
% 13.19/7.57 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_758_group__cancel_Oadd2,axiom,
% 13.19/7.57 ! [B5: real,K: real,B: real,A: real] :
% 13.19/7.57 ( ( B5
% 13.19/7.57 = ( plus_plus_real @ K @ B ) )
% 13.19/7.57 => ( ( plus_plus_real @ A @ B5 )
% 13.19/7.57 = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_759_group__cancel_Oadd2,axiom,
% 13.19/7.57 ! [B5: rat,K: rat,B: rat,A: rat] :
% 13.19/7.57 ( ( B5
% 13.19/7.57 = ( plus_plus_rat @ K @ B ) )
% 13.19/7.57 => ( ( plus_plus_rat @ A @ B5 )
% 13.19/7.57 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_760_group__cancel_Oadd2,axiom,
% 13.19/7.57 ! [B5: nat,K: nat,B: nat,A: nat] :
% 13.19/7.57 ( ( B5
% 13.19/7.57 = ( plus_plus_nat @ K @ B ) )
% 13.19/7.57 => ( ( plus_plus_nat @ A @ B5 )
% 13.19/7.57 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_761_group__cancel_Oadd2,axiom,
% 13.19/7.57 ! [B5: int,K: int,B: int,A: int] :
% 13.19/7.57 ( ( B5
% 13.19/7.57 = ( plus_plus_int @ K @ B ) )
% 13.19/7.57 => ( ( plus_plus_int @ A @ B5 )
% 13.19/7.57 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_762_add_Oassoc,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 13.19/7.57 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_763_add_Oassoc,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 13.19/7.57 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_764_add_Oassoc,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.19/7.57 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_765_add_Oassoc,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.19/7.57 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_766_add_Oleft__cancel,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ A @ B )
% 13.19/7.57 = ( plus_plus_real @ A @ C ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_767_add_Oleft__cancel,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ A @ B )
% 13.19/7.57 = ( plus_plus_rat @ A @ C ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_768_add_Oleft__cancel,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ A @ B )
% 13.19/7.57 = ( plus_plus_int @ A @ C ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_769_add_Oright__cancel,axiom,
% 13.19/7.57 ! [B: real,A: real,C: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ B @ A )
% 13.19/7.57 = ( plus_plus_real @ C @ A ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_770_add_Oright__cancel,axiom,
% 13.19/7.57 ! [B: rat,A: rat,C: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ B @ A )
% 13.19/7.57 = ( plus_plus_rat @ C @ A ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_771_add_Oright__cancel,axiom,
% 13.19/7.57 ! [B: int,A: int,C: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ B @ A )
% 13.19/7.57 = ( plus_plus_int @ C @ A ) )
% 13.19/7.57 = ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_772_add_Ocommute,axiom,
% 13.19/7.57 ( plus_plus_real
% 13.19/7.57 = ( ^ [A3: real,B4: real] : ( plus_plus_real @ B4 @ A3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_773_add_Ocommute,axiom,
% 13.19/7.57 ( plus_plus_rat
% 13.19/7.57 = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ B4 @ A3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_774_add_Ocommute,axiom,
% 13.19/7.57 ( plus_plus_nat
% 13.19/7.57 = ( ^ [A3: nat,B4: nat] : ( plus_plus_nat @ B4 @ A3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_775_add_Ocommute,axiom,
% 13.19/7.57 ( plus_plus_int
% 13.19/7.57 = ( ^ [A3: int,B4: int] : ( plus_plus_int @ B4 @ A3 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_776_add_Oleft__commute,axiom,
% 13.19/7.57 ! [B: real,A: real,C: real] :
% 13.19/7.57 ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 13.19/7.57 = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_777_add_Oleft__commute,axiom,
% 13.19/7.57 ! [B: rat,A: rat,C: rat] :
% 13.19/7.57 ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 13.19/7.57 = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_778_add_Oleft__commute,axiom,
% 13.19/7.57 ! [B: nat,A: nat,C: nat] :
% 13.19/7.57 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 13.19/7.57 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_779_add_Oleft__commute,axiom,
% 13.19/7.57 ! [B: int,A: int,C: int] :
% 13.19/7.57 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 13.19/7.57 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_780_add__left__imp__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ A @ B )
% 13.19/7.57 = ( plus_plus_real @ A @ C ) )
% 13.19/7.57 => ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_781_add__left__imp__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ A @ B )
% 13.19/7.57 = ( plus_plus_rat @ A @ C ) )
% 13.19/7.57 => ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_782_add__left__imp__eq,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ( plus_plus_nat @ A @ B )
% 13.19/7.57 = ( plus_plus_nat @ A @ C ) )
% 13.19/7.57 => ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_783_add__left__imp__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ A @ B )
% 13.19/7.57 = ( plus_plus_int @ A @ C ) )
% 13.19/7.57 => ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_784_add__right__imp__eq,axiom,
% 13.19/7.57 ! [B: real,A: real,C: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ B @ A )
% 13.19/7.57 = ( plus_plus_real @ C @ A ) )
% 13.19/7.57 => ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_785_add__right__imp__eq,axiom,
% 13.19/7.57 ! [B: rat,A: rat,C: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ B @ A )
% 13.19/7.57 = ( plus_plus_rat @ C @ A ) )
% 13.19/7.57 => ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_786_add__right__imp__eq,axiom,
% 13.19/7.57 ! [B: nat,A: nat,C: nat] :
% 13.19/7.57 ( ( ( plus_plus_nat @ B @ A )
% 13.19/7.57 = ( plus_plus_nat @ C @ A ) )
% 13.19/7.57 => ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_787_add__right__imp__eq,axiom,
% 13.19/7.57 ! [B: int,A: int,C: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ B @ A )
% 13.19/7.57 = ( plus_plus_int @ C @ A ) )
% 13.19/7.57 => ( B = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_788_diff__eq__diff__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.57 ( ( ( minus_minus_real @ A @ B )
% 13.19/7.57 = ( minus_minus_real @ C @ D ) )
% 13.19/7.57 => ( ( A = B )
% 13.19/7.57 = ( C = D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_789_diff__eq__diff__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.57 ( ( ( minus_minus_rat @ A @ B )
% 13.19/7.57 = ( minus_minus_rat @ C @ D ) )
% 13.19/7.57 => ( ( A = B )
% 13.19/7.57 = ( C = D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_790_diff__eq__diff__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.57 ( ( ( minus_minus_int @ A @ B )
% 13.19/7.57 = ( minus_minus_int @ C @ D ) )
% 13.19/7.57 => ( ( A = B )
% 13.19/7.57 = ( C = D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_791_diff__right__commute,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 13.19/7.57 = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_792_diff__right__commute,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 13.19/7.57 = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_793_diff__right__commute,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 13.19/7.57 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_794_diff__right__commute,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 13.19/7.57 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_795_add__mono__thms__linordered__semiring_I3_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( ord_less_eq_real @ I @ J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_796_add__mono__thms__linordered__semiring_I3_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( ord_less_eq_rat @ I @ J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_797_add__mono__thms__linordered__semiring_I3_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_798_add__mono__thms__linordered__semiring_I3_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( ord_less_eq_int @ I @ J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_799_add__mono__thms__linordered__semiring_I2_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( ord_less_eq_real @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_800_add__mono__thms__linordered__semiring_I2_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( ord_less_eq_rat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_801_add__mono__thms__linordered__semiring_I2_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( ord_less_eq_nat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_802_add__mono__thms__linordered__semiring_I2_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( ord_less_eq_int @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_803_add__mono__thms__linordered__semiring_I1_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( ord_less_eq_real @ I @ J )
% 13.19/7.57 & ( ord_less_eq_real @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_804_add__mono__thms__linordered__semiring_I1_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( ord_less_eq_rat @ I @ J )
% 13.19/7.57 & ( ord_less_eq_rat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_805_add__mono__thms__linordered__semiring_I1_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 & ( ord_less_eq_nat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_806_add__mono__thms__linordered__semiring_I1_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( ord_less_eq_int @ I @ J )
% 13.19/7.57 & ( ord_less_eq_int @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_807_add__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_real @ C @ D )
% 13.19/7.57 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_808_add__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_rat @ C @ D )
% 13.19/7.57 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_809_add__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_nat @ C @ D )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_810_add__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_int @ C @ D )
% 13.19/7.57 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_811_add__left__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.57 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_812_add__left__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.57 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_813_add__left__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_814_add__left__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.57 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_815_less__eqE,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ~ ! [C2: nat] :
% 13.19/7.57 ( B
% 13.19/7.57 != ( plus_plus_nat @ A @ C2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_816_add__right__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.57 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_817_add__right__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.57 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_818_add__right__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_819_add__right__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.57 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_820_le__iff__add,axiom,
% 13.19/7.57 ( ord_less_eq_nat
% 13.19/7.57 = ( ^ [A3: nat,B4: nat] :
% 13.19/7.57 ? [C3: nat] :
% 13.19/7.57 ( B4
% 13.19/7.57 = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_821_add__le__imp__le__left,axiom,
% 13.19/7.57 ! [C: real,A: real,B: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 13.19/7.57 => ( ord_less_eq_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_822_add__le__imp__le__left,axiom,
% 13.19/7.57 ! [C: rat,A: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 13.19/7.57 => ( ord_less_eq_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_823_add__le__imp__le__left,axiom,
% 13.19/7.57 ! [C: nat,A: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 13.19/7.57 => ( ord_less_eq_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_824_add__le__imp__le__left,axiom,
% 13.19/7.57 ! [C: int,A: int,B: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 13.19/7.57 => ( ord_less_eq_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_825_add__le__imp__le__right,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 13.19/7.57 => ( ord_less_eq_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_826_add__le__imp__le__right,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.57 => ( ord_less_eq_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_827_add__le__imp__le__right,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 13.19/7.57 => ( ord_less_eq_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_828_add__le__imp__le__right,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 13.19/7.57 => ( ord_less_eq_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_829_diff__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,D: real,C: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_real @ D @ C )
% 13.19/7.57 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_830_diff__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,D: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_rat @ D @ C )
% 13.19/7.57 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_831_diff__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,D: int,C: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_int @ D @ C )
% 13.19/7.57 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_832_diff__left__mono,axiom,
% 13.19/7.57 ! [B: real,A: real,C: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ B @ A )
% 13.19/7.57 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_833_diff__left__mono,axiom,
% 13.19/7.57 ! [B: rat,A: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ B @ A )
% 13.19/7.57 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_834_diff__left__mono,axiom,
% 13.19/7.57 ! [B: int,A: int,C: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ B @ A )
% 13.19/7.57 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_835_diff__right__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.57 => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_836_diff__right__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.57 => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_837_diff__right__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.57 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_838_diff__eq__diff__less__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.57 ( ( ( minus_minus_real @ A @ B )
% 13.19/7.57 = ( minus_minus_real @ C @ D ) )
% 13.19/7.57 => ( ( ord_less_eq_real @ A @ B )
% 13.19/7.57 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_839_diff__eq__diff__less__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.57 ( ( ( minus_minus_rat @ A @ B )
% 13.19/7.57 = ( minus_minus_rat @ C @ D ) )
% 13.19/7.57 => ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.57 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_840_diff__eq__diff__less__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.57 ( ( ( minus_minus_int @ A @ B )
% 13.19/7.57 = ( minus_minus_int @ C @ D ) )
% 13.19/7.57 => ( ( ord_less_eq_int @ A @ B )
% 13.19/7.57 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_841_add__mono__thms__linordered__field_I5_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( ord_less_real @ I @ J )
% 13.19/7.57 & ( ord_less_real @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_842_add__mono__thms__linordered__field_I5_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( ord_less_rat @ I @ J )
% 13.19/7.57 & ( ord_less_rat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_843_add__mono__thms__linordered__field_I5_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( ord_less_nat @ I @ J )
% 13.19/7.57 & ( ord_less_nat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_844_add__mono__thms__linordered__field_I5_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( ord_less_int @ I @ J )
% 13.19/7.57 & ( ord_less_int @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_845_add__mono__thms__linordered__field_I2_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( ord_less_real @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_846_add__mono__thms__linordered__field_I2_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( ord_less_rat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_847_add__mono__thms__linordered__field_I2_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( ord_less_nat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_848_add__mono__thms__linordered__field_I2_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( I = J )
% 13.19/7.57 & ( ord_less_int @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_849_add__mono__thms__linordered__field_I1_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( ord_less_real @ I @ J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_850_add__mono__thms__linordered__field_I1_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( ord_less_rat @ I @ J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_851_add__mono__thms__linordered__field_I1_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( ord_less_nat @ I @ J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_852_add__mono__thms__linordered__field_I1_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( ord_less_int @ I @ J )
% 13.19/7.57 & ( K = L2 ) )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_853_add__strict__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ( ord_less_real @ C @ D )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_854_add__strict__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ( ord_less_rat @ C @ D )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_855_add__strict__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.57 ( ( ord_less_nat @ A @ B )
% 13.19/7.57 => ( ( ord_less_nat @ C @ D )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_856_add__strict__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.57 ( ( ord_less_int @ A @ B )
% 13.19/7.57 => ( ( ord_less_int @ C @ D )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_857_add__strict__left__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_858_add__strict__left__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_859_add__strict__left__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_nat @ A @ B )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_860_add__strict__left__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ord_less_int @ A @ B )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_861_add__strict__right__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_862_add__strict__right__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_863_add__strict__right__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_nat @ A @ B )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_864_add__strict__right__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ord_less_int @ A @ B )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_865_add__less__imp__less__left,axiom,
% 13.19/7.57 ! [C: real,A: real,B: real] :
% 13.19/7.57 ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 13.19/7.57 => ( ord_less_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_866_add__less__imp__less__left,axiom,
% 13.19/7.57 ! [C: rat,A: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 13.19/7.57 => ( ord_less_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_867_add__less__imp__less__left,axiom,
% 13.19/7.57 ! [C: nat,A: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 13.19/7.57 => ( ord_less_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_868_add__less__imp__less__left,axiom,
% 13.19/7.57 ! [C: int,A: int,B: int] :
% 13.19/7.57 ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 13.19/7.57 => ( ord_less_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_869_add__less__imp__less__right,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 13.19/7.57 => ( ord_less_real @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_870_add__less__imp__less__right,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.57 => ( ord_less_rat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_871_add__less__imp__less__right,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 13.19/7.57 => ( ord_less_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_872_add__less__imp__less__right,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 13.19/7.57 => ( ord_less_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_873_diff__strict__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,D: real,C: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ( ord_less_real @ D @ C )
% 13.19/7.57 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_874_diff__strict__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,D: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ( ord_less_rat @ D @ C )
% 13.19/7.57 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_875_diff__strict__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,D: int,C: int] :
% 13.19/7.57 ( ( ord_less_int @ A @ B )
% 13.19/7.57 => ( ( ord_less_int @ D @ C )
% 13.19/7.57 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_876_diff__eq__diff__less,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.57 ( ( ( minus_minus_real @ A @ B )
% 13.19/7.57 = ( minus_minus_real @ C @ D ) )
% 13.19/7.57 => ( ( ord_less_real @ A @ B )
% 13.19/7.57 = ( ord_less_real @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_877_diff__eq__diff__less,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.57 ( ( ( minus_minus_rat @ A @ B )
% 13.19/7.57 = ( minus_minus_rat @ C @ D ) )
% 13.19/7.57 => ( ( ord_less_rat @ A @ B )
% 13.19/7.57 = ( ord_less_rat @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_878_diff__eq__diff__less,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.57 ( ( ( minus_minus_int @ A @ B )
% 13.19/7.57 = ( minus_minus_int @ C @ D ) )
% 13.19/7.57 => ( ( ord_less_int @ A @ B )
% 13.19/7.57 = ( ord_less_int @ C @ D ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_879_diff__strict__left__mono,axiom,
% 13.19/7.57 ! [B: real,A: real,C: real] :
% 13.19/7.57 ( ( ord_less_real @ B @ A )
% 13.19/7.57 => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_880_diff__strict__left__mono,axiom,
% 13.19/7.57 ! [B: rat,A: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_rat @ B @ A )
% 13.19/7.57 => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_881_diff__strict__left__mono,axiom,
% 13.19/7.57 ! [B: int,A: int,C: int] :
% 13.19/7.57 ( ( ord_less_int @ B @ A )
% 13.19/7.57 => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_882_diff__strict__right__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_883_diff__strict__right__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_884_diff__strict__right__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ord_less_int @ A @ B )
% 13.19/7.57 => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_885_comm__monoid__mult__class_Omult__1,axiom,
% 13.19/7.57 ! [A: complex] :
% 13.19/7.57 ( ( times_times_complex @ one_one_complex @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_886_comm__monoid__mult__class_Omult__1,axiom,
% 13.19/7.57 ! [A: real] :
% 13.19/7.57 ( ( times_times_real @ one_one_real @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_887_comm__monoid__mult__class_Omult__1,axiom,
% 13.19/7.57 ! [A: rat] :
% 13.19/7.57 ( ( times_times_rat @ one_one_rat @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_888_comm__monoid__mult__class_Omult__1,axiom,
% 13.19/7.57 ! [A: nat] :
% 13.19/7.57 ( ( times_times_nat @ one_one_nat @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_889_comm__monoid__mult__class_Omult__1,axiom,
% 13.19/7.57 ! [A: int] :
% 13.19/7.57 ( ( times_times_int @ one_one_int @ A )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_890_mult_Ocomm__neutral,axiom,
% 13.19/7.57 ! [A: complex] :
% 13.19/7.57 ( ( times_times_complex @ A @ one_one_complex )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_891_mult_Ocomm__neutral,axiom,
% 13.19/7.57 ! [A: real] :
% 13.19/7.57 ( ( times_times_real @ A @ one_one_real )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_892_mult_Ocomm__neutral,axiom,
% 13.19/7.57 ! [A: rat] :
% 13.19/7.57 ( ( times_times_rat @ A @ one_one_rat )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_893_mult_Ocomm__neutral,axiom,
% 13.19/7.57 ! [A: nat] :
% 13.19/7.57 ( ( times_times_nat @ A @ one_one_nat )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_894_mult_Ocomm__neutral,axiom,
% 13.19/7.57 ! [A: int] :
% 13.19/7.57 ( ( times_times_int @ A @ one_one_int )
% 13.19/7.57 = A ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_895_group__cancel_Osub1,axiom,
% 13.19/7.57 ! [A2: real,K: real,A: real,B: real] :
% 13.19/7.57 ( ( A2
% 13.19/7.57 = ( plus_plus_real @ K @ A ) )
% 13.19/7.57 => ( ( minus_minus_real @ A2 @ B )
% 13.19/7.57 = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_896_group__cancel_Osub1,axiom,
% 13.19/7.57 ! [A2: rat,K: rat,A: rat,B: rat] :
% 13.19/7.57 ( ( A2
% 13.19/7.57 = ( plus_plus_rat @ K @ A ) )
% 13.19/7.57 => ( ( minus_minus_rat @ A2 @ B )
% 13.19/7.57 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_897_group__cancel_Osub1,axiom,
% 13.19/7.57 ! [A2: int,K: int,A: int,B: int] :
% 13.19/7.57 ( ( A2
% 13.19/7.57 = ( plus_plus_int @ K @ A ) )
% 13.19/7.57 => ( ( minus_minus_int @ A2 @ B )
% 13.19/7.57 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_898_diff__eq__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ( minus_minus_real @ A @ B )
% 13.19/7.57 = C )
% 13.19/7.57 = ( A
% 13.19/7.57 = ( plus_plus_real @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_899_diff__eq__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ( minus_minus_rat @ A @ B )
% 13.19/7.57 = C )
% 13.19/7.57 = ( A
% 13.19/7.57 = ( plus_plus_rat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_900_diff__eq__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ( minus_minus_int @ A @ B )
% 13.19/7.57 = C )
% 13.19/7.57 = ( A
% 13.19/7.57 = ( plus_plus_int @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_901_eq__diff__eq,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( A
% 13.19/7.57 = ( minus_minus_real @ C @ B ) )
% 13.19/7.57 = ( ( plus_plus_real @ A @ B )
% 13.19/7.57 = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_902_eq__diff__eq,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( A
% 13.19/7.57 = ( minus_minus_rat @ C @ B ) )
% 13.19/7.57 = ( ( plus_plus_rat @ A @ B )
% 13.19/7.57 = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_903_eq__diff__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( A
% 13.19/7.57 = ( minus_minus_int @ C @ B ) )
% 13.19/7.57 = ( ( plus_plus_int @ A @ B )
% 13.19/7.57 = C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_904_add__diff__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 13.19/7.57 = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_905_add__diff__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 13.19/7.57 = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_906_add__diff__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 13.19/7.57 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_907_diff__diff__eq2,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 13.19/7.57 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_908_diff__diff__eq2,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 13.19/7.57 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_909_diff__diff__eq2,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 13.19/7.57 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_910_diff__add__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_911_diff__add__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_912_diff__add__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_913_diff__add__eq__diff__diff__swap,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 13.19/7.57 = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_914_diff__add__eq__diff__diff__swap,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 13.19/7.57 = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_915_diff__add__eq__diff__diff__swap,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 13.19/7.57 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_916_add__implies__diff,axiom,
% 13.19/7.57 ! [C: real,B: real,A: real] :
% 13.19/7.57 ( ( ( plus_plus_real @ C @ B )
% 13.19/7.57 = A )
% 13.19/7.57 => ( C
% 13.19/7.57 = ( minus_minus_real @ A @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_917_add__implies__diff,axiom,
% 13.19/7.57 ! [C: rat,B: rat,A: rat] :
% 13.19/7.57 ( ( ( plus_plus_rat @ C @ B )
% 13.19/7.57 = A )
% 13.19/7.57 => ( C
% 13.19/7.57 = ( minus_minus_rat @ A @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_918_add__implies__diff,axiom,
% 13.19/7.57 ! [C: nat,B: nat,A: nat] :
% 13.19/7.57 ( ( ( plus_plus_nat @ C @ B )
% 13.19/7.57 = A )
% 13.19/7.57 => ( C
% 13.19/7.57 = ( minus_minus_nat @ A @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_919_add__implies__diff,axiom,
% 13.19/7.57 ! [C: int,B: int,A: int] :
% 13.19/7.57 ( ( ( plus_plus_int @ C @ B )
% 13.19/7.57 = A )
% 13.19/7.57 => ( C
% 13.19/7.57 = ( minus_minus_int @ A @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_920_diff__diff__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_921_diff__diff__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_922_diff__diff__eq,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_923_diff__diff__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 13.19/7.57 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_924_add__less__le__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_real @ C @ D )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_925_add__less__le__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_rat @ C @ D )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_926_add__less__le__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.57 ( ( ord_less_nat @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_nat @ C @ D )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_927_add__less__le__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.57 ( ( ord_less_int @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_int @ C @ D )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_928_add__le__less__mono,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.57 => ( ( ord_less_real @ C @ D )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_929_add__le__less__mono,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.57 => ( ( ord_less_rat @ C @ D )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_930_add__le__less__mono,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( ord_less_nat @ C @ D )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_931_add__le__less__mono,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.57 => ( ( ord_less_int @ C @ D )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_932_add__mono__thms__linordered__field_I3_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( ord_less_real @ I @ J )
% 13.19/7.57 & ( ord_less_eq_real @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_933_add__mono__thms__linordered__field_I3_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( ord_less_rat @ I @ J )
% 13.19/7.57 & ( ord_less_eq_rat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_934_add__mono__thms__linordered__field_I3_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( ord_less_nat @ I @ J )
% 13.19/7.57 & ( ord_less_eq_nat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_935_add__mono__thms__linordered__field_I3_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( ord_less_int @ I @ J )
% 13.19/7.57 & ( ord_less_eq_int @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_936_add__mono__thms__linordered__field_I4_J,axiom,
% 13.19/7.57 ! [I: real,J: real,K: real,L2: real] :
% 13.19/7.57 ( ( ( ord_less_eq_real @ I @ J )
% 13.19/7.57 & ( ord_less_real @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_937_add__mono__thms__linordered__field_I4_J,axiom,
% 13.19/7.57 ! [I: rat,J: rat,K: rat,L2: rat] :
% 13.19/7.57 ( ( ( ord_less_eq_rat @ I @ J )
% 13.19/7.57 & ( ord_less_rat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_938_add__mono__thms__linordered__field_I4_J,axiom,
% 13.19/7.57 ! [I: nat,J: nat,K: nat,L2: nat] :
% 13.19/7.57 ( ( ( ord_less_eq_nat @ I @ J )
% 13.19/7.57 & ( ord_less_nat @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_939_add__mono__thms__linordered__field_I4_J,axiom,
% 13.19/7.57 ! [I: int,J: int,K: int,L2: int] :
% 13.19/7.57 ( ( ( ord_less_eq_int @ I @ J )
% 13.19/7.57 & ( ord_less_int @ K @ L2 ) )
% 13.19/7.57 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L2 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_940_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( ( minus_minus_nat @ B @ A )
% 13.19/7.57 = C )
% 13.19/7.57 = ( B
% 13.19/7.57 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_941_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 13.19/7.57 = B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_942_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 13.19/7.57 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_943_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 13.19/7.57 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_944_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 13.19/7.57 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_945_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 13.19/7.57 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_946_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 13.19/7.57 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_947_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 13.19/7.57 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_948_le__add__diff,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_949_diff__add,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.57 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 13.19/7.57 = B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_950_le__diff__eq,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 13.19/7.57 = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_951_le__diff__eq,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 13.19/7.57 = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_952_le__diff__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 13.19/7.57 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_953_diff__le__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 13.19/7.57 = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_954_diff__le__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 13.19/7.57 = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_955_diff__le__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 13.19/7.57 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_956_diff__less__eq,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 13.19/7.57 = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_957_diff__less__eq,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 13.19/7.57 = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_958_diff__less__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 13.19/7.57 = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_959_less__diff__eq,axiom,
% 13.19/7.57 ! [A: real,C: real,B: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 13.19/7.57 = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_960_less__diff__eq,axiom,
% 13.19/7.57 ! [A: rat,C: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 13.19/7.57 = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_961_less__diff__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 13.19/7.57 = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_962_times__divide__eq__right,axiom,
% 13.19/7.57 ! [A: complex,B: complex,C: complex] :
% 13.19/7.57 ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 13.19/7.57 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_963_times__divide__eq__right,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 13.19/7.57 = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_964_times__divide__eq__right,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.57 = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_965_divide__divide__eq__right,axiom,
% 13.19/7.57 ! [A: complex,B: complex,C: complex] :
% 13.19/7.57 ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 13.19/7.57 = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_966_divide__divide__eq__right,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 13.19/7.57 = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_967_divide__divide__eq__right,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.57 = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_968_divide__divide__eq__left,axiom,
% 13.19/7.57 ! [A: complex,B: complex,C: complex] :
% 13.19/7.57 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 13.19/7.57 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_969_divide__divide__eq__left,axiom,
% 13.19/7.57 ! [A: real,B: real,C: real] :
% 13.19/7.57 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 13.19/7.57 = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_970_divide__divide__eq__left,axiom,
% 13.19/7.57 ! [A: rat,B: rat,C: rat] :
% 13.19/7.57 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 13.19/7.57 = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_971_times__divide__eq__left,axiom,
% 13.19/7.57 ! [B: complex,C: complex,A: complex] :
% 13.19/7.57 ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 13.19/7.57 = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_972_times__divide__eq__left,axiom,
% 13.19/7.57 ! [B: real,C: real,A: real] :
% 13.19/7.57 ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 13.19/7.57 = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_973_times__divide__eq__left,axiom,
% 13.19/7.57 ! [B: rat,C: rat,A: rat] :
% 13.19/7.57 ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 13.19/7.57 = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_974_valid__insert__both__member__options__pres,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat,Y3: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 13.19/7.57 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_975_valid__insert__both__member__options__add,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.57 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_976_maxt__corr__help__empty,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( vEBT_vebt_maxt @ T )
% 13.19/7.57 = none_nat )
% 13.19/7.57 => ( ( vEBT_VEBT_set_vebt @ T )
% 13.19/7.57 = bot_bot_set_nat ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_977_mint__corr__help__empty,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( ( vEBT_vebt_mint @ T )
% 13.19/7.57 = none_nat )
% 13.19/7.57 => ( ( vEBT_VEBT_set_vebt @ T )
% 13.19/7.57 = bot_bot_set_nat ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_978_less__half__sum,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_979_less__half__sum,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_980_gt__half__sum,axiom,
% 13.19/7.57 ! [A: real,B: real] :
% 13.19/7.57 ( ( ord_less_real @ A @ B )
% 13.19/7.57 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_981_gt__half__sum,axiom,
% 13.19/7.57 ! [A: rat,B: rat] :
% 13.19/7.57 ( ( ord_less_rat @ A @ B )
% 13.19/7.57 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_982_low__def,axiom,
% 13.19/7.57 ( vEBT_VEBT_low
% 13.19/7.57 = ( ^ [X2: nat,N2: nat] : ( modulo_modulo_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_983_not__min__Null__member,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT] :
% 13.19/7.57 ( ~ ( vEBT_VEBT_minNull @ T )
% 13.19/7.57 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_984_valid__member__both__member__options,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 13.19/7.57 => ( vEBT_vebt_member @ T @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_985_both__member__options__equiv__member,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,N: nat,X: nat] :
% 13.19/7.57 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.57 => ( ( vEBT_V8194947554948674370ptions @ T @ X )
% 13.19/7.57 = ( vEBT_vebt_member @ T @ X ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_986_maxbmo,axiom,
% 13.19/7.57 ! [T: vEBT_VEBT,X: nat] :
% 13.19/7.57 ( ( ( vEBT_vebt_maxt @ T )
% 13.19/7.57 = ( some_nat @ X ) )
% 13.19/7.57 => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_987_mod__mod__trivial,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 13.19/7.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_988_mod__mod__trivial,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_989_mod__mod__trivial,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_990_mod__add__self1,axiom,
% 13.19/7.57 ! [B: nat,A: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 13.19/7.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_991_mod__add__self1,axiom,
% 13.19/7.57 ! [B: int,A: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_992_mod__add__self1,axiom,
% 13.19/7.57 ! [B: code_integer,A: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_993_mod__add__self2,axiom,
% 13.19/7.57 ! [A: nat,B: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 13.19/7.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_994_mod__add__self2,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_995_mod__add__self2,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_996_minus__mod__self2,axiom,
% 13.19/7.57 ! [A: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_997_minus__mod__self2,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_998_mod__less,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ( ord_less_nat @ M @ N )
% 13.19/7.57 => ( ( modulo_modulo_nat @ M @ N )
% 13.19/7.57 = M ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_999_mod__mult__self1,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 13.19/7.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1000_mod__mult__self1,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1001_mod__mult__self1,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1002_mod__mult__self2,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 13.19/7.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1003_mod__mult__self2,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1004_mod__mult__self2,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1005_mod__mult__self3,axiom,
% 13.19/7.57 ! [C: nat,B: nat,A: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 13.19/7.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1006_mod__mult__self3,axiom,
% 13.19/7.57 ! [C: int,B: int,A: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1007_mod__mult__self3,axiom,
% 13.19/7.57 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1008_mod__mult__self4,axiom,
% 13.19/7.57 ! [B: nat,C: nat,A: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 13.19/7.57 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1009_mod__mult__self4,axiom,
% 13.19/7.57 ! [B: int,C: int,A: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1010_mod__mult__self4,axiom,
% 13.19/7.57 ! [B: code_integer,C: code_integer,A: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1011_one__mod__two__eq__one,axiom,
% 13.19/7.57 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = one_one_nat ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1012_one__mod__two__eq__one,axiom,
% 13.19/7.57 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.19/7.57 = one_one_int ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1013_one__mod__two__eq__one,axiom,
% 13.19/7.57 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.19/7.57 = one_one_Code_integer ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1014_bits__one__mod__two__eq__one,axiom,
% 13.19/7.57 ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.57 = one_one_nat ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1015_bits__one__mod__two__eq__one,axiom,
% 13.19/7.57 ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.19/7.57 = one_one_int ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1016_bits__one__mod__two__eq__one,axiom,
% 13.19/7.57 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.19/7.57 = one_one_Code_integer ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1017_mod__mult__eq,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1018_mod__mult__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1019_mod__mult__eq,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1020_mod__mult__cong,axiom,
% 13.19/7.57 ! [A: nat,C: nat,A4: nat,B: nat,B2: nat] :
% 13.19/7.57 ( ( ( modulo_modulo_nat @ A @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ A4 @ C ) )
% 13.19/7.57 => ( ( ( modulo_modulo_nat @ B @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ B2 @ C ) )
% 13.19/7.57 => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B2 ) @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1021_mod__mult__cong,axiom,
% 13.19/7.57 ! [A: int,C: int,A4: int,B: int,B2: int] :
% 13.19/7.57 ( ( ( modulo_modulo_int @ A @ C )
% 13.19/7.57 = ( modulo_modulo_int @ A4 @ C ) )
% 13.19/7.57 => ( ( ( modulo_modulo_int @ B @ C )
% 13.19/7.57 = ( modulo_modulo_int @ B2 @ C ) )
% 13.19/7.57 => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( times_times_int @ A4 @ B2 ) @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1022_mod__mult__cong,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B2: code_integer] :
% 13.19/7.57 ( ( ( modulo364778990260209775nteger @ A @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A4 @ C ) )
% 13.19/7.57 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 13.19/7.57 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B2 ) @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1023_mod__mult__mult2,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 13.19/7.57 = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1024_mod__mult__mult2,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 13.19/7.57 = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1025_mod__mult__mult2,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 13.19/7.57 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1026_mult__mod__right,axiom,
% 13.19/7.57 ! [C: nat,A: nat,B: nat] :
% 13.19/7.57 ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 13.19/7.57 = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1027_mult__mod__right,axiom,
% 13.19/7.57 ! [C: int,A: int,B: int] :
% 13.19/7.57 ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 13.19/7.57 = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1028_mult__mod__right,axiom,
% 13.19/7.57 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.19/7.57 ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1029_mod__mult__left__eq,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1030_mod__mult__left__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1031_mod__mult__left__eq,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1032_mod__mult__right__eq,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1033_mod__mult__right__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1034_mod__mult__right__eq,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1035_mod__add__eq,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1036_mod__add__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1037_mod__add__eq,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1038_mod__add__cong,axiom,
% 13.19/7.57 ! [A: nat,C: nat,A4: nat,B: nat,B2: nat] :
% 13.19/7.57 ( ( ( modulo_modulo_nat @ A @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ A4 @ C ) )
% 13.19/7.57 => ( ( ( modulo_modulo_nat @ B @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ B2 @ C ) )
% 13.19/7.57 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B2 ) @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1039_mod__add__cong,axiom,
% 13.19/7.57 ! [A: int,C: int,A4: int,B: int,B2: int] :
% 13.19/7.57 ( ( ( modulo_modulo_int @ A @ C )
% 13.19/7.57 = ( modulo_modulo_int @ A4 @ C ) )
% 13.19/7.57 => ( ( ( modulo_modulo_int @ B @ C )
% 13.19/7.57 = ( modulo_modulo_int @ B2 @ C ) )
% 13.19/7.57 => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B2 ) @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1040_mod__add__cong,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B2: code_integer] :
% 13.19/7.57 ( ( ( modulo364778990260209775nteger @ A @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A4 @ C ) )
% 13.19/7.57 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 13.19/7.57 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B2 ) @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1041_mod__add__left__eq,axiom,
% 13.19/7.57 ! [A: nat,C: nat,B: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1042_mod__add__left__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1043_mod__add__left__eq,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1044_mod__add__right__eq,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1045_mod__add__right__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1046_mod__add__right__eq,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1047_mod__diff__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1048_mod__diff__eq,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1049_mod__diff__cong,axiom,
% 13.19/7.57 ! [A: int,C: int,A4: int,B: int,B2: int] :
% 13.19/7.57 ( ( ( modulo_modulo_int @ A @ C )
% 13.19/7.57 = ( modulo_modulo_int @ A4 @ C ) )
% 13.19/7.57 => ( ( ( modulo_modulo_int @ B @ C )
% 13.19/7.57 = ( modulo_modulo_int @ B2 @ C ) )
% 13.19/7.57 => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B2 ) @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1050_mod__diff__cong,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B2: code_integer] :
% 13.19/7.57 ( ( ( modulo364778990260209775nteger @ A @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ A4 @ C ) )
% 13.19/7.57 => ( ( ( modulo364778990260209775nteger @ B @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ B2 @ C ) )
% 13.19/7.57 => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B2 ) @ C ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1051_mod__diff__left__eq,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1052_mod__diff__left__eq,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1053_mod__diff__right__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1054_mod__diff__right__eq,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1055_power__mod,axiom,
% 13.19/7.57 ! [A: nat,B: nat,N: nat] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
% 13.19/7.57 = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1056_power__mod,axiom,
% 13.19/7.57 ! [A: int,B: int,N: nat] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
% 13.19/7.57 = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1057_power__mod,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer,N: nat] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1058_mod__less__eq__dividend,axiom,
% 13.19/7.57 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1059_cong__exp__iff__simps_I9_J,axiom,
% 13.19/7.57 ! [M: num,Q2: num,N: num] :
% 13.19/7.57 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.19/7.57 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 13.19/7.57 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 13.19/7.57 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1060_cong__exp__iff__simps_I9_J,axiom,
% 13.19/7.57 ! [M: num,Q2: num,N: num] :
% 13.19/7.57 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.19/7.57 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 13.19/7.57 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 13.19/7.57 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1061_cong__exp__iff__simps_I9_J,axiom,
% 13.19/7.57 ! [M: num,Q2: num,N: num] :
% 13.19/7.57 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 13.19/7.57 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1062_cong__exp__iff__simps_I4_J,axiom,
% 13.19/7.57 ! [M: num,N: num] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
% 13.19/7.57 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1063_cong__exp__iff__simps_I4_J,axiom,
% 13.19/7.57 ! [M: num,N: num] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
% 13.19/7.57 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1064_cong__exp__iff__simps_I4_J,axiom,
% 13.19/7.57 ! [M: num,N: num] :
% 13.19/7.57 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
% 13.19/7.57 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1065_mod__eqE,axiom,
% 13.19/7.57 ! [A: int,C: int,B: int] :
% 13.19/7.57 ( ( ( modulo_modulo_int @ A @ C )
% 13.19/7.57 = ( modulo_modulo_int @ B @ C ) )
% 13.19/7.57 => ~ ! [D3: int] :
% 13.19/7.57 ( B
% 13.19/7.57 != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1066_mod__eqE,axiom,
% 13.19/7.57 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.19/7.57 ( ( ( modulo364778990260209775nteger @ A @ C )
% 13.19/7.57 = ( modulo364778990260209775nteger @ B @ C ) )
% 13.19/7.57 => ~ ! [D3: code_integer] :
% 13.19/7.57 ( B
% 13.19/7.57 != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1067_div__add1__eq,axiom,
% 13.19/7.57 ! [A: nat,B: nat,C: nat] :
% 13.19/7.57 ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.19/7.57 = ( 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 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1068_div__add1__eq,axiom,
% 13.19/7.57 ! [A: int,B: int,C: int] :
% 13.19/7.57 ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.19/7.57 = ( 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 ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1069_div__add1__eq,axiom,
% 13.19/7.57 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.19/7.57 ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 13.19/7.57 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1070_mod__if,axiom,
% 13.19/7.57 ( modulo_modulo_nat
% 13.19/7.57 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1071_mod__geq,axiom,
% 13.19/7.57 ! [M: nat,N: nat] :
% 13.19/7.57 ( ~ ( ord_less_nat @ M @ N )
% 13.19/7.57 => ( ( modulo_modulo_nat @ M @ N )
% 13.19/7.57 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1072_le__mod__geq,axiom,
% 13.19/7.57 ! [N: nat,M: nat] :
% 13.19/7.57 ( ( ord_less_eq_nat @ N @ M )
% 13.19/7.57 => ( ( modulo_modulo_nat @ M @ N )
% 13.19/7.57 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1073_nat__mod__eq__iff,axiom,
% 13.19/7.57 ! [X: nat,N: nat,Y3: nat] :
% 13.19/7.57 ( ( ( modulo_modulo_nat @ X @ N )
% 13.19/7.57 = ( modulo_modulo_nat @ Y3 @ N ) )
% 13.19/7.57 = ( ? [Q1: nat,Q22: nat] :
% 13.19/7.57 ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
% 13.19/7.57 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1074_cong__exp__iff__simps_I8_J,axiom,
% 13.19/7.57 ! [M: num,Q2: num] :
% 13.19/7.57 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.19/7.57 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 13.19/7.57
% 13.19/7.57 thf(fact_1075_cong__exp__iff__simps_I8_J,axiom,
% 13.19/7.57 ! [M: num,Q2: num] :
% 13.19/7.57 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.19/7.57 != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1076_cong__exp__iff__simps_I8_J,axiom,
% 13.19/7.58 ! [M: num,Q2: num] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.19/7.58 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1077_cong__exp__iff__simps_I6_J,axiom,
% 13.19/7.58 ! [Q2: num,N: num] :
% 13.19/7.58 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.19/7.58 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1078_cong__exp__iff__simps_I6_J,axiom,
% 13.19/7.58 ! [Q2: num,N: num] :
% 13.19/7.58 ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.19/7.58 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1079_cong__exp__iff__simps_I6_J,axiom,
% 13.19/7.58 ! [Q2: num,N: num] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.19/7.58 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1080_div__mult1__eq,axiom,
% 13.19/7.58 ! [A: nat,B: nat,C: nat] :
% 13.19/7.58 ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 13.19/7.58 = ( 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 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1081_div__mult1__eq,axiom,
% 13.19/7.58 ! [A: int,B: int,C: int] :
% 13.19/7.58 ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 13.19/7.58 = ( 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 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1082_div__mult1__eq,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.19/7.58 ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 13.19/7.58 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1083_cancel__div__mod__rules_I2_J,axiom,
% 13.19/7.58 ! [B: nat,A: nat,C: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 13.19/7.58 = ( plus_plus_nat @ A @ C ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1084_cancel__div__mod__rules_I2_J,axiom,
% 13.19/7.58 ! [B: int,A: int,C: int] :
% 13.19/7.58 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 13.19/7.58 = ( plus_plus_int @ A @ C ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1085_cancel__div__mod__rules_I2_J,axiom,
% 13.19/7.58 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.19/7.58 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 13.19/7.58 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1086_cancel__div__mod__rules_I1_J,axiom,
% 13.19/7.58 ! [A: nat,B: nat,C: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 13.19/7.58 = ( plus_plus_nat @ A @ C ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1087_cancel__div__mod__rules_I1_J,axiom,
% 13.19/7.58 ! [A: int,B: int,C: int] :
% 13.19/7.58 ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 13.19/7.58 = ( plus_plus_int @ A @ C ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1088_cancel__div__mod__rules_I1_J,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.19/7.58 ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 13.19/7.58 = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1089_mod__div__decomp,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( A
% 13.19/7.58 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1090_mod__div__decomp,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( A
% 13.19/7.58 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1091_mod__div__decomp,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( A
% 13.19/7.58 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1092_div__mult__mod__eq,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1093_div__mult__mod__eq,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1094_div__mult__mod__eq,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1095_mod__div__mult__eq,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1096_mod__div__mult__eq,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1097_mod__div__mult__eq,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1098_mod__mult__div__eq,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1099_mod__mult__div__eq,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1100_mod__mult__div__eq,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1101_mult__div__mod__eq,axiom,
% 13.19/7.58 ! [B: nat,A: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1102_mult__div__mod__eq,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1103_mult__div__mod__eq,axiom,
% 13.19/7.58 ! [B: code_integer,A: code_integer] :
% 13.19/7.58 ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1104_minus__div__mult__eq__mod,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 13.19/7.58 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1105_minus__div__mult__eq__mod,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 13.19/7.58 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1106_minus__div__mult__eq__mod,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 13.19/7.58 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1107_minus__mod__eq__div__mult,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 13.19/7.58 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1108_minus__mod__eq__div__mult,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 13.19/7.58 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1109_minus__mod__eq__div__mult,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 13.19/7.58 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1110_minus__mod__eq__mult__div,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 13.19/7.58 = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1111_minus__mod__eq__mult__div,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 13.19/7.58 = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1112_minus__mod__eq__mult__div,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 13.19/7.58 = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1113_minus__mult__div__eq__mod,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 13.19/7.58 = ( modulo_modulo_nat @ A @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1114_minus__mult__div__eq__mod,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 13.19/7.58 = ( modulo_modulo_int @ A @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1115_minus__mult__div__eq__mod,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 13.19/7.58 = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1116_mod__eq__nat1E,axiom,
% 13.19/7.58 ! [M: nat,Q2: nat,N: nat] :
% 13.19/7.58 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 13.19/7.58 = ( modulo_modulo_nat @ N @ Q2 ) )
% 13.19/7.58 => ( ( ord_less_eq_nat @ N @ M )
% 13.19/7.58 => ~ ! [S3: nat] :
% 13.19/7.58 ( M
% 13.19/7.58 != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1117_mod__eq__nat2E,axiom,
% 13.19/7.58 ! [M: nat,Q2: nat,N: nat] :
% 13.19/7.58 ( ( ( modulo_modulo_nat @ M @ Q2 )
% 13.19/7.58 = ( modulo_modulo_nat @ N @ Q2 ) )
% 13.19/7.58 => ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.58 => ~ ! [S3: nat] :
% 13.19/7.58 ( N
% 13.19/7.58 != ( plus_plus_nat @ M @ ( times_times_nat @ Q2 @ S3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1118_nat__mod__eq__lemma,axiom,
% 13.19/7.58 ! [X: nat,N: nat,Y3: nat] :
% 13.19/7.58 ( ( ( modulo_modulo_nat @ X @ N )
% 13.19/7.58 = ( modulo_modulo_nat @ Y3 @ N ) )
% 13.19/7.58 => ( ( ord_less_eq_nat @ Y3 @ X )
% 13.19/7.58 => ? [Q3: nat] :
% 13.19/7.58 ( X
% 13.19/7.58 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1119_div__mod__decomp,axiom,
% 13.19/7.58 ! [A2: nat,N: nat] :
% 13.19/7.58 ( A2
% 13.19/7.58 = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N ) @ N ) @ ( modulo_modulo_nat @ A2 @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1120_mod__mult2__eq,axiom,
% 13.19/7.58 ! [M: nat,N: nat,Q2: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N @ Q2 ) )
% 13.19/7.58 = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1121_modulo__nat__def,axiom,
% 13.19/7.58 ( modulo_modulo_nat
% 13.19/7.58 = ( ^ [M3: nat,N2: nat] : ( minus_minus_nat @ M3 @ ( times_times_nat @ ( divide_divide_nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1122_linordered__field__no__ub,axiom,
% 13.19/7.58 ! [X5: real] :
% 13.19/7.58 ? [X_1: real] : ( ord_less_real @ X5 @ X_1 ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1123_linordered__field__no__ub,axiom,
% 13.19/7.58 ! [X5: rat] :
% 13.19/7.58 ? [X_1: rat] : ( ord_less_rat @ X5 @ X_1 ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1124_linordered__field__no__lb,axiom,
% 13.19/7.58 ! [X5: real] :
% 13.19/7.58 ? [Y4: real] : ( ord_less_real @ Y4 @ X5 ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1125_linordered__field__no__lb,axiom,
% 13.19/7.58 ! [X5: rat] :
% 13.19/7.58 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X5 ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1126_div__exp__mod__exp__eq,axiom,
% 13.19/7.58 ! [A: nat,N: nat,M: nat] :
% 13.19/7.58 ( ( 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 ) )
% 13.19/7.58 = ( 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 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1127_div__exp__mod__exp__eq,axiom,
% 13.19/7.58 ! [A: int,N: nat,M: nat] :
% 13.19/7.58 ( ( 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 ) )
% 13.19/7.58 = ( 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 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1128_div__exp__mod__exp__eq,axiom,
% 13.19/7.58 ! [A: code_integer,N: nat,M: nat] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 13.19/7.58 = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1129_mult__exp__mod__exp__eq,axiom,
% 13.19/7.58 ! [M: nat,N: nat,A: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.58 => ( ( 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 ) )
% 13.19/7.58 = ( 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 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1130_mult__exp__mod__exp__eq,axiom,
% 13.19/7.58 ! [M: nat,N: nat,A: int] :
% 13.19/7.58 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.58 => ( ( 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 ) )
% 13.19/7.58 = ( 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 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1131_mult__exp__mod__exp__eq,axiom,
% 13.19/7.58 ! [M: nat,N: nat,A: code_integer] :
% 13.19/7.58 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.58 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.19/7.58 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1132_times__divide__times__eq,axiom,
% 13.19/7.58 ! [X: complex,Y3: complex,Z: complex,W: complex] :
% 13.19/7.58 ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ Y3 @ W ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1133_times__divide__times__eq,axiom,
% 13.19/7.58 ! [X: real,Y3: real,Z: real,W: real] :
% 13.19/7.58 ( ( times_times_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 13.19/7.58 = ( divide_divide_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y3 @ W ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1134_times__divide__times__eq,axiom,
% 13.19/7.58 ! [X: rat,Y3: rat,Z: rat,W: rat] :
% 13.19/7.58 ( ( times_times_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 13.19/7.58 = ( divide_divide_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y3 @ W ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1135_divide__divide__times__eq,axiom,
% 13.19/7.58 ! [X: complex,Y3: complex,Z: complex,W: complex] :
% 13.19/7.58 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ ( divide1717551699836669952omplex @ Z @ W ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W ) @ ( times_times_complex @ Y3 @ Z ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1136_divide__divide__times__eq,axiom,
% 13.19/7.58 ! [X: real,Y3: real,Z: real,W: real] :
% 13.19/7.58 ( ( divide_divide_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ Z @ W ) )
% 13.19/7.58 = ( divide_divide_real @ ( times_times_real @ X @ W ) @ ( times_times_real @ Y3 @ Z ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1137_divide__divide__times__eq,axiom,
% 13.19/7.58 ! [X: rat,Y3: rat,Z: rat,W: rat] :
% 13.19/7.58 ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ Z @ W ) )
% 13.19/7.58 = ( divide_divide_rat @ ( times_times_rat @ X @ W ) @ ( times_times_rat @ Y3 @ Z ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1138_divide__divide__eq__left_H,axiom,
% 13.19/7.58 ! [A: complex,B: complex,C: complex] :
% 13.19/7.58 ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 13.19/7.58 = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1139_divide__divide__eq__left_H,axiom,
% 13.19/7.58 ! [A: real,B: real,C: real] :
% 13.19/7.58 ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 13.19/7.58 = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1140_divide__divide__eq__left_H,axiom,
% 13.19/7.58 ! [A: rat,B: rat,C: rat] :
% 13.19/7.58 ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 13.19/7.58 = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1141_add__divide__distrib,axiom,
% 13.19/7.58 ! [A: complex,B: complex,C: complex] :
% 13.19/7.58 ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
% 13.19/7.58 = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1142_add__divide__distrib,axiom,
% 13.19/7.58 ! [A: real,B: real,C: real] :
% 13.19/7.58 ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 13.19/7.58 = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1143_add__divide__distrib,axiom,
% 13.19/7.58 ! [A: rat,B: rat,C: rat] :
% 13.19/7.58 ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 13.19/7.58 = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1144_diff__divide__distrib,axiom,
% 13.19/7.58 ! [A: complex,B: complex,C: complex] :
% 13.19/7.58 ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
% 13.19/7.58 = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1145_diff__divide__distrib,axiom,
% 13.19/7.58 ! [A: real,B: real,C: real] :
% 13.19/7.58 ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 13.19/7.58 = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1146_diff__divide__distrib,axiom,
% 13.19/7.58 ! [A: rat,B: rat,C: rat] :
% 13.19/7.58 ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 13.19/7.58 = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1147_buildup__gives__empty,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 13.19/7.58 = bot_bot_set_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1148__C4_Ohyps_C_I11_J,axiom,
% 13.19/7.58 ( ( mi != ma )
% 13.19/7.58 => ! [I3: nat] :
% 13.19/7.58 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 13.19/7.58 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 13.19/7.58 = I3 )
% 13.19/7.58 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 13.19/7.58 & ! [X5: nat] :
% 13.19/7.58 ( ( ( ( vEBT_VEBT_high @ X5 @ na )
% 13.19/7.58 = I3 )
% 13.19/7.58 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
% 13.19/7.58 => ( ( ord_less_nat @ mi @ X5 )
% 13.19/7.58 & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1149_VEBT__internal_Ooption__shift_Oelims,axiom,
% 13.19/7.58 ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 13.19/7.58 ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa @ Xb )
% 13.19/7.58 = Y3 )
% 13.19/7.58 => ( ( ( Xa = none_P5556105721700978146at_nat )
% 13.19/7.58 => ( Y3 != none_P5556105721700978146at_nat ) )
% 13.19/7.58 => ( ( ? [V2: product_prod_nat_nat] :
% 13.19/7.58 ( Xa
% 13.19/7.58 = ( some_P7363390416028606310at_nat @ V2 ) )
% 13.19/7.58 => ( ( Xb = none_P5556105721700978146at_nat )
% 13.19/7.58 => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 13.19/7.58 => ~ ! [A5: product_prod_nat_nat] :
% 13.19/7.58 ( ( Xa
% 13.19/7.58 = ( some_P7363390416028606310at_nat @ A5 ) )
% 13.19/7.58 => ! [B3: product_prod_nat_nat] :
% 13.19/7.58 ( ( Xb
% 13.19/7.58 = ( some_P7363390416028606310at_nat @ B3 ) )
% 13.19/7.58 => ( Y3
% 13.19/7.58 != ( some_P7363390416028606310at_nat @ ( X @ A5 @ B3 ) ) ) ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1150_VEBT__internal_Ooption__shift_Oelims,axiom,
% 13.19/7.58 ! [X: num > num > num,Xa: option_num,Xb: option_num,Y3: option_num] :
% 13.19/7.58 ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa @ Xb )
% 13.19/7.58 = Y3 )
% 13.19/7.58 => ( ( ( Xa = none_num )
% 13.19/7.58 => ( Y3 != none_num ) )
% 13.19/7.58 => ( ( ? [V2: num] :
% 13.19/7.58 ( Xa
% 13.19/7.58 = ( some_num @ V2 ) )
% 13.19/7.58 => ( ( Xb = none_num )
% 13.19/7.58 => ( Y3 != none_num ) ) )
% 13.19/7.58 => ~ ! [A5: num] :
% 13.19/7.58 ( ( Xa
% 13.19/7.58 = ( some_num @ A5 ) )
% 13.19/7.58 => ! [B3: num] :
% 13.19/7.58 ( ( Xb
% 13.19/7.58 = ( some_num @ B3 ) )
% 13.19/7.58 => ( Y3
% 13.19/7.58 != ( some_num @ ( X @ A5 @ B3 ) ) ) ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1151_VEBT__internal_Ooption__shift_Oelims,axiom,
% 13.19/7.58 ! [X: nat > nat > nat,Xa: option_nat,Xb: option_nat,Y3: option_nat] :
% 13.19/7.58 ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa @ Xb )
% 13.19/7.58 = Y3 )
% 13.19/7.58 => ( ( ( Xa = none_nat )
% 13.19/7.58 => ( Y3 != none_nat ) )
% 13.19/7.58 => ( ( ? [V2: nat] :
% 13.19/7.58 ( Xa
% 13.19/7.58 = ( some_nat @ V2 ) )
% 13.19/7.58 => ( ( Xb = none_nat )
% 13.19/7.58 => ( Y3 != none_nat ) ) )
% 13.19/7.58 => ~ ! [A5: nat] :
% 13.19/7.58 ( ( Xa
% 13.19/7.58 = ( some_nat @ A5 ) )
% 13.19/7.58 => ! [B3: nat] :
% 13.19/7.58 ( ( Xb
% 13.19/7.58 = ( some_nat @ B3 ) )
% 13.19/7.58 => ( Y3
% 13.19/7.58 != ( some_nat @ ( X @ A5 @ B3 ) ) ) ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1152_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 13.19/7.58 ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 13.19/7.58 ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 13.19/7.58 = none_P5556105721700978146at_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1153_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 13.19/7.58 ! [Uw: num > num > num,V: num] :
% 13.19/7.58 ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 13.19/7.58 = none_num ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1154_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 13.19/7.58 ! [Uw: nat > nat > nat,V: nat] :
% 13.19/7.58 ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 13.19/7.58 = none_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1155_obtain__set__succ,axiom,
% 13.19/7.58 ! [X: nat,Z: nat,A2: set_nat,B5: set_nat] :
% 13.19/7.58 ( ( ord_less_nat @ X @ Z )
% 13.19/7.58 => ( ( vEBT_VEBT_max_in_set @ A2 @ Z )
% 13.19/7.58 => ( ( finite_finite_nat @ B5 )
% 13.19/7.58 => ( ( A2 = B5 )
% 13.19/7.58 => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A2 @ X @ X_1 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1156_True,axiom,
% 13.19/7.58 ord_less_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ treeList ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1157_dbl__simps_I3_J,axiom,
% 13.19/7.58 ( ( neg_nu7009210354673126013omplex @ one_one_complex )
% 13.19/7.58 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1158_dbl__simps_I3_J,axiom,
% 13.19/7.58 ( ( neg_numeral_dbl_real @ one_one_real )
% 13.19/7.58 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1159_dbl__simps_I3_J,axiom,
% 13.19/7.58 ( ( neg_numeral_dbl_rat @ one_one_rat )
% 13.19/7.58 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1160_dbl__simps_I3_J,axiom,
% 13.19/7.58 ( ( neg_numeral_dbl_int @ one_one_int )
% 13.19/7.58 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1161_divmod__digit__1_I1_J,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 13.19/7.58 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 13.19/7.58 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 13.19/7.58 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 13.19/7.58 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1162_divmod__digit__1_I1_J,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.58 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.58 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 13.19/7.58 => ( ( 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 )
% 13.19/7.58 = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1163_divmod__digit__1_I1_J,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.58 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 13.19/7.58 => ( ( 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 )
% 13.19/7.58 = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1164_diff__shunt__var,axiom,
% 13.19/7.58 ! [X: set_real,Y3: set_real] :
% 13.19/7.58 ( ( ( minus_minus_set_real @ X @ Y3 )
% 13.19/7.58 = bot_bot_set_real )
% 13.19/7.58 = ( ord_less_eq_set_real @ X @ Y3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1165_diff__shunt__var,axiom,
% 13.19/7.58 ! [X: set_nat,Y3: set_nat] :
% 13.19/7.58 ( ( ( minus_minus_set_nat @ X @ Y3 )
% 13.19/7.58 = bot_bot_set_nat )
% 13.19/7.58 = ( ord_less_eq_set_nat @ X @ Y3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1166_diff__shunt__var,axiom,
% 13.19/7.58 ! [X: set_int,Y3: set_int] :
% 13.19/7.58 ( ( ( minus_minus_set_int @ X @ Y3 )
% 13.19/7.58 = bot_bot_set_int )
% 13.19/7.58 = ( ord_less_eq_set_int @ X @ Y3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1167_valid__0__not,axiom,
% 13.19/7.58 ! [T: vEBT_VEBT] :
% 13.19/7.58 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1168_valid__tree__deg__neq__0,axiom,
% 13.19/7.58 ! [T: vEBT_VEBT] :
% 13.19/7.58 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1169_deg__not__0,axiom,
% 13.19/7.58 ! [T: vEBT_VEBT,N: nat] :
% 13.19/7.58 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.58 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1170_buildup__gives__valid,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1171_set__vebt__finite,axiom,
% 13.19/7.58 ! [T: vEBT_VEBT,N: nat] :
% 13.19/7.58 ( ( vEBT_invar_vebt @ T @ N )
% 13.19/7.58 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1172_succ__none__empty,axiom,
% 13.19/7.58 ! [Xs2: set_nat,A: nat] :
% 13.19/7.58 ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_1 )
% 13.19/7.58 => ( ( finite_finite_nat @ Xs2 )
% 13.19/7.58 => ~ ? [X5: nat] :
% 13.19/7.58 ( ( member_nat @ X5 @ Xs2 )
% 13.19/7.58 & ( ord_less_nat @ A @ X5 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1173_bot__nat__0_Onot__eq__extremum,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( A != zero_zero_nat )
% 13.19/7.58 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1174_neq0__conv,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( N != zero_zero_nat )
% 13.19/7.58 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1175_less__nat__zero__code,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1176_bot__nat__0_Oextremum,axiom,
% 13.19/7.58 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1177_le0,axiom,
% 13.19/7.58 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1178__C4_Ohyps_C_I4_J,axiom,
% 13.19/7.58 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 13.19/7.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1179_add__is__0,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ( plus_plus_nat @ M @ N )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 = ( ( M = zero_zero_nat )
% 13.19/7.58 & ( N = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1180_Nat_Oadd__0__right,axiom,
% 13.19/7.58 ! [M: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 13.19/7.58 = M ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1181_diff__0__eq__0,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1182_diff__self__eq__0,axiom,
% 13.19/7.58 ! [M: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ M @ M )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1183_mult__is__0,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ M @ N )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 = ( ( M = zero_zero_nat )
% 13.19/7.58 | ( N = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1184_mult__0__right,axiom,
% 13.19/7.58 ! [M: nat] :
% 13.19/7.58 ( ( times_times_nat @ M @ zero_zero_nat )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1185_mult__cancel1,axiom,
% 13.19/7.58 ! [K: nat,M: nat,N: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ K @ M )
% 13.19/7.58 = ( times_times_nat @ K @ N ) )
% 13.19/7.58 = ( ( M = N )
% 13.19/7.58 | ( K = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1186_mult__cancel2,axiom,
% 13.19/7.58 ! [M: nat,K: nat,N: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ M @ K )
% 13.19/7.58 = ( times_times_nat @ N @ K ) )
% 13.19/7.58 = ( ( M = N )
% 13.19/7.58 | ( K = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1187_div__pos__pos__trivial,axiom,
% 13.19/7.58 ! [K: int,L2: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.19/7.58 => ( ( ord_less_int @ K @ L2 )
% 13.19/7.58 => ( ( divide_divide_int @ K @ L2 )
% 13.19/7.58 = zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1188_div__neg__neg__trivial,axiom,
% 13.19/7.58 ! [K: int,L2: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 13.19/7.58 => ( ( ord_less_int @ L2 @ K )
% 13.19/7.58 => ( ( divide_divide_int @ K @ L2 )
% 13.19/7.58 = zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1189_i0__less,axiom,
% 13.19/7.58 ! [N: extended_enat] :
% 13.19/7.58 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 13.19/7.58 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1190_idiff__0__right,axiom,
% 13.19/7.58 ! [N: extended_enat] :
% 13.19/7.58 ( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
% 13.19/7.58 = N ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1191_idiff__0,axiom,
% 13.19/7.58 ! [N: extended_enat] :
% 13.19/7.58 ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
% 13.19/7.58 = zero_z5237406670263579293d_enat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1192_le__zero__eq,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 13.19/7.58 = ( N = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1193_not__gr__zero,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 13.19/7.58 = ( N = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1194_mult__zero__left,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( times_times_complex @ zero_zero_complex @ A )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1195_mult__zero__left,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( times_times_real @ zero_zero_real @ A )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1196_mult__zero__left,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( times_times_rat @ zero_zero_rat @ A )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1197_mult__zero__left,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( times_times_nat @ zero_zero_nat @ A )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1198_mult__zero__left,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( times_times_int @ zero_zero_int @ A )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1199_mult__zero__right,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( times_times_complex @ A @ zero_zero_complex )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1200_mult__zero__right,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( times_times_real @ A @ zero_zero_real )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1201_mult__zero__right,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( times_times_rat @ A @ zero_zero_rat )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1202_mult__zero__right,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( times_times_nat @ A @ zero_zero_nat )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1203_mult__zero__right,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( times_times_int @ A @ zero_zero_int )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1204_mult__eq__0__iff,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( ( times_times_complex @ A @ B )
% 13.19/7.58 = zero_zero_complex )
% 13.19/7.58 = ( ( A = zero_zero_complex )
% 13.19/7.58 | ( B = zero_zero_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1205_mult__eq__0__iff,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ( times_times_real @ A @ B )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 = ( ( A = zero_zero_real )
% 13.19/7.58 | ( B = zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1206_mult__eq__0__iff,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ( times_times_rat @ A @ B )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 = ( ( A = zero_zero_rat )
% 13.19/7.58 | ( B = zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1207_mult__eq__0__iff,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ A @ B )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 = ( ( A = zero_zero_nat )
% 13.19/7.58 | ( B = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1208_mult__eq__0__iff,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ( times_times_int @ A @ B )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 = ( ( A = zero_zero_int )
% 13.19/7.58 | ( B = zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1209_mult__cancel__left,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( ( times_times_complex @ C @ A )
% 13.19/7.58 = ( times_times_complex @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_complex )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1210_mult__cancel__left,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( ( times_times_real @ C @ A )
% 13.19/7.58 = ( times_times_real @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_real )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1211_mult__cancel__left,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( ( times_times_rat @ C @ A )
% 13.19/7.58 = ( times_times_rat @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_rat )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1212_mult__cancel__left,axiom,
% 13.19/7.58 ! [C: nat,A: nat,B: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ C @ A )
% 13.19/7.58 = ( times_times_nat @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_nat )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1213_mult__cancel__left,axiom,
% 13.19/7.58 ! [C: int,A: int,B: int] :
% 13.19/7.58 ( ( ( times_times_int @ C @ A )
% 13.19/7.58 = ( times_times_int @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_int )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1214_mult__cancel__right,axiom,
% 13.19/7.58 ! [A: complex,C: complex,B: complex] :
% 13.19/7.58 ( ( ( times_times_complex @ A @ C )
% 13.19/7.58 = ( times_times_complex @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_complex )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1215_mult__cancel__right,axiom,
% 13.19/7.58 ! [A: real,C: real,B: real] :
% 13.19/7.58 ( ( ( times_times_real @ A @ C )
% 13.19/7.58 = ( times_times_real @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_real )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1216_mult__cancel__right,axiom,
% 13.19/7.58 ! [A: rat,C: rat,B: rat] :
% 13.19/7.58 ( ( ( times_times_rat @ A @ C )
% 13.19/7.58 = ( times_times_rat @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_rat )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1217_mult__cancel__right,axiom,
% 13.19/7.58 ! [A: nat,C: nat,B: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ A @ C )
% 13.19/7.58 = ( times_times_nat @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_nat )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1218_mult__cancel__right,axiom,
% 13.19/7.58 ! [A: int,C: int,B: int] :
% 13.19/7.58 ( ( ( times_times_int @ A @ C )
% 13.19/7.58 = ( times_times_int @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_int )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1219_add_Oright__neutral,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1220_add_Oright__neutral,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( plus_plus_real @ A @ zero_zero_real )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1221_add_Oright__neutral,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1222_add_Oright__neutral,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1223_add_Oright__neutral,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( plus_plus_int @ A @ zero_zero_int )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1224_double__zero__sym,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( zero_zero_real
% 13.19/7.58 = ( plus_plus_real @ A @ A ) )
% 13.19/7.58 = ( A = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1225_double__zero__sym,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( zero_zero_rat
% 13.19/7.58 = ( plus_plus_rat @ A @ A ) )
% 13.19/7.58 = ( A = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1226_double__zero__sym,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( zero_zero_int
% 13.19/7.58 = ( plus_plus_int @ A @ A ) )
% 13.19/7.58 = ( A = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1227_add__cancel__left__left,axiom,
% 13.19/7.58 ! [B: complex,A: complex] :
% 13.19/7.58 ( ( ( plus_plus_complex @ B @ A )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1228_add__cancel__left__left,axiom,
% 13.19/7.58 ! [B: real,A: real] :
% 13.19/7.58 ( ( ( plus_plus_real @ B @ A )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1229_add__cancel__left__left,axiom,
% 13.19/7.58 ! [B: rat,A: rat] :
% 13.19/7.58 ( ( ( plus_plus_rat @ B @ A )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1230_add__cancel__left__left,axiom,
% 13.19/7.58 ! [B: nat,A: nat] :
% 13.19/7.58 ( ( ( plus_plus_nat @ B @ A )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1231_add__cancel__left__left,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( ( plus_plus_int @ B @ A )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1232_add__cancel__left__right,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( ( plus_plus_complex @ A @ B )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1233_add__cancel__left__right,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ( plus_plus_real @ A @ B )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1234_add__cancel__left__right,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ( plus_plus_rat @ A @ B )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1235_add__cancel__left__right,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ( plus_plus_nat @ A @ B )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1236_add__cancel__left__right,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ( plus_plus_int @ A @ B )
% 13.19/7.58 = A )
% 13.19/7.58 = ( B = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1237_add__cancel__right__left,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_complex @ B @ A ) )
% 13.19/7.58 = ( B = zero_zero_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1238_add__cancel__right__left,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_real @ B @ A ) )
% 13.19/7.58 = ( B = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1239_add__cancel__right__left,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_rat @ B @ A ) )
% 13.19/7.58 = ( B = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1240_add__cancel__right__left,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_nat @ B @ A ) )
% 13.19/7.58 = ( B = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1241_add__cancel__right__left,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_int @ B @ A ) )
% 13.19/7.58 = ( B = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1242_add__cancel__right__right,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_complex @ A @ B ) )
% 13.19/7.58 = ( B = zero_zero_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1243_add__cancel__right__right,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_real @ A @ B ) )
% 13.19/7.58 = ( B = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1244_add__cancel__right__right,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_rat @ A @ B ) )
% 13.19/7.58 = ( B = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1245_add__cancel__right__right,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_nat @ A @ B ) )
% 13.19/7.58 = ( B = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1246_add__cancel__right__right,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( plus_plus_int @ A @ B ) )
% 13.19/7.58 = ( B = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1247_add__eq__0__iff__both__eq__0,axiom,
% 13.19/7.58 ! [X: nat,Y3: nat] :
% 13.19/7.58 ( ( ( plus_plus_nat @ X @ Y3 )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 = ( ( X = zero_zero_nat )
% 13.19/7.58 & ( Y3 = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1248_zero__eq__add__iff__both__eq__0,axiom,
% 13.19/7.58 ! [X: nat,Y3: nat] :
% 13.19/7.58 ( ( zero_zero_nat
% 13.19/7.58 = ( plus_plus_nat @ X @ Y3 ) )
% 13.19/7.58 = ( ( X = zero_zero_nat )
% 13.19/7.58 & ( Y3 = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1249_add__0,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1250_add__0,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( plus_plus_real @ zero_zero_real @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1251_add__0,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1252_add__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1253_add__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( plus_plus_int @ zero_zero_int @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1254_diff__self,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( minus_minus_complex @ A @ A )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1255_diff__self,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( minus_minus_real @ A @ A )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1256_diff__self,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( minus_minus_rat @ A @ A )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1257_diff__self,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( minus_minus_int @ A @ A )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1258_diff__0__right,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1259_diff__0__right,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( minus_minus_real @ A @ zero_zero_real )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1260_diff__0__right,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1261_diff__0__right,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( minus_minus_int @ A @ zero_zero_int )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1262_zero__diff,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ zero_zero_nat @ A )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1263_diff__zero,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( minus_minus_complex @ A @ zero_zero_complex )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1264_diff__zero,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( minus_minus_real @ A @ zero_zero_real )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1265_diff__zero,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( minus_minus_rat @ A @ zero_zero_rat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1266_diff__zero,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ A @ zero_zero_nat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1267_diff__zero,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( minus_minus_int @ A @ zero_zero_int )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1268_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( minus_minus_complex @ A @ A )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1269_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( minus_minus_real @ A @ A )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1270_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( minus_minus_rat @ A @ A )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1271_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ A @ A )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1272_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( minus_minus_int @ A @ A )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1273_bits__div__by__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1274_bits__div__by__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( divide_divide_int @ A @ zero_zero_int )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1275_bits__div__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1276_bits__div__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( divide_divide_int @ zero_zero_int @ A )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1277_div__by__0,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1278_div__by__0,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( divide_divide_real @ A @ zero_zero_real )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1279_div__by__0,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1280_div__by__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( divide_divide_nat @ A @ zero_zero_nat )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1281_div__by__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( divide_divide_int @ A @ zero_zero_int )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1282_div__0,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1283_div__0,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( divide_divide_real @ zero_zero_real @ A )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1284_div__0,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( divide_divide_rat @ zero_zero_rat @ A )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1285_div__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( divide_divide_nat @ zero_zero_nat @ A )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1286_div__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( divide_divide_int @ zero_zero_int @ A )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1287_division__ring__divide__zero,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1288_division__ring__divide__zero,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( divide_divide_real @ A @ zero_zero_real )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1289_division__ring__divide__zero,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( divide_divide_rat @ A @ zero_zero_rat )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1290_divide__cancel__right,axiom,
% 13.19/7.58 ! [A: complex,C: complex,B: complex] :
% 13.19/7.58 ( ( ( divide1717551699836669952omplex @ A @ C )
% 13.19/7.58 = ( divide1717551699836669952omplex @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_complex )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1291_divide__cancel__right,axiom,
% 13.19/7.58 ! [A: real,C: real,B: real] :
% 13.19/7.58 ( ( ( divide_divide_real @ A @ C )
% 13.19/7.58 = ( divide_divide_real @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_real )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1292_divide__cancel__right,axiom,
% 13.19/7.58 ! [A: rat,C: rat,B: rat] :
% 13.19/7.58 ( ( ( divide_divide_rat @ A @ C )
% 13.19/7.58 = ( divide_divide_rat @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_rat )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1293_divide__cancel__left,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( ( divide1717551699836669952omplex @ C @ A )
% 13.19/7.58 = ( divide1717551699836669952omplex @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_complex )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1294_divide__cancel__left,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( ( divide_divide_real @ C @ A )
% 13.19/7.58 = ( divide_divide_real @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_real )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1295_divide__cancel__left,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( ( divide_divide_rat @ C @ A )
% 13.19/7.58 = ( divide_divide_rat @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_rat )
% 13.19/7.58 | ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1296_divide__eq__0__iff,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( ( divide1717551699836669952omplex @ A @ B )
% 13.19/7.58 = zero_zero_complex )
% 13.19/7.58 = ( ( A = zero_zero_complex )
% 13.19/7.58 | ( B = zero_zero_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1297_divide__eq__0__iff,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ( divide_divide_real @ A @ B )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 = ( ( A = zero_zero_real )
% 13.19/7.58 | ( B = zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1298_divide__eq__0__iff,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ( divide_divide_rat @ A @ B )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 = ( ( A = zero_zero_rat )
% 13.19/7.58 | ( B = zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1299_bits__mod__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1300_bits__mod__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1301_bits__mod__0,axiom,
% 13.19/7.58 ! [A: code_integer] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1302_mod__self,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ A @ A )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1303_mod__self,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( modulo_modulo_int @ A @ A )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1304_mod__self,axiom,
% 13.19/7.58 ! [A: code_integer] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ A @ A )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1305_mod__by__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1306_mod__by__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( modulo_modulo_int @ A @ zero_zero_int )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1307_mod__by__0,axiom,
% 13.19/7.58 ! [A: code_integer] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1308_mod__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1309_mod__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( modulo_modulo_int @ zero_zero_int @ A )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1310_mod__0,axiom,
% 13.19/7.58 ! [A: code_integer] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1311_add__gr__0,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 13.19/7.58 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.19/7.58 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1312_zero__less__diff,axiom,
% 13.19/7.58 ! [N: nat,M: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 13.19/7.58 = ( ord_less_nat @ M @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1313_nat__0__less__mult__iff,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 13.19/7.58 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.19/7.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1314_mult__less__cancel2,axiom,
% 13.19/7.58 ! [M: nat,K: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 13.19/7.58 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.58 & ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1315_nat__mult__less__cancel__disj,axiom,
% 13.19/7.58 ! [K: nat,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.19/7.58 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.58 & ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1316_not__real__square__gt__zero,axiom,
% 13.19/7.58 ! [X: real] :
% 13.19/7.58 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
% 13.19/7.58 = ( X = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1317_diff__is__0__eq_H,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.58 => ( ( minus_minus_nat @ M @ N )
% 13.19/7.58 = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1318_diff__is__0__eq,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ( minus_minus_nat @ M @ N )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1319_div__less,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ M @ N )
% 13.19/7.58 => ( ( divide_divide_nat @ M @ N )
% 13.19/7.58 = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1320_less__one,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ N @ one_one_nat )
% 13.19/7.58 = ( N = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1321_nat__zero__less__power__iff,axiom,
% 13.19/7.58 ! [X: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
% 13.19/7.58 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 13.19/7.58 | ( N = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1322_nat__mult__div__cancel__disj,axiom,
% 13.19/7.58 ! [K: nat,M: nat,N: nat] :
% 13.19/7.58 ( ( ( K = zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.19/7.58 = zero_zero_nat ) )
% 13.19/7.58 & ( ( K != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.19/7.58 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1323__092_060open_062length_AtreeList_A_061_A2_A_094_An_092_060close_062,axiom,
% 13.19/7.58 ( ( size_s6755466524823107622T_VEBT @ treeList )
% 13.19/7.58 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ na ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1324_dbl__simps_I2_J,axiom,
% 13.19/7.58 ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1325_dbl__simps_I2_J,axiom,
% 13.19/7.58 ( ( neg_numeral_dbl_real @ zero_zero_real )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1326_dbl__simps_I2_J,axiom,
% 13.19/7.58 ( ( neg_numeral_dbl_rat @ zero_zero_rat )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1327_dbl__simps_I2_J,axiom,
% 13.19/7.58 ( ( neg_numeral_dbl_int @ zero_zero_int )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1328_zero__le__double__add__iff__zero__le__single__add,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 13.19/7.58 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1329_zero__le__double__add__iff__zero__le__single__add,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 13.19/7.58 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1330_zero__le__double__add__iff__zero__le__single__add,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 13.19/7.58 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1331_double__add__le__zero__iff__single__add__le__zero,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 13.19/7.58 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1332_double__add__le__zero__iff__single__add__le__zero,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 13.19/7.58 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1333_double__add__le__zero__iff__single__add__le__zero,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 13.19/7.58 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1334_le__add__same__cancel2,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 13.19/7.58 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1335_le__add__same__cancel2,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 13.19/7.58 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1336_le__add__same__cancel2,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 13.19/7.58 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1337_le__add__same__cancel2,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 13.19/7.58 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1338_le__add__same__cancel1,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1339_le__add__same__cancel1,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1340_le__add__same__cancel1,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1341_le__add__same__cancel1,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1342_add__le__same__cancel2,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 13.19/7.58 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1343_add__le__same__cancel2,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 13.19/7.58 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1344_add__le__same__cancel2,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 13.19/7.58 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1345_add__le__same__cancel2,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 13.19/7.58 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1346_add__le__same__cancel1,axiom,
% 13.19/7.58 ! [B: real,A: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 13.19/7.58 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1347_add__le__same__cancel1,axiom,
% 13.19/7.58 ! [B: rat,A: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 13.19/7.58 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1348_add__le__same__cancel1,axiom,
% 13.19/7.58 ! [B: nat,A: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 13.19/7.58 = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1349_add__le__same__cancel1,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 13.19/7.58 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1350_diff__ge__0__iff__ge,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_real @ B @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1351_diff__ge__0__iff__ge,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_rat @ B @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1352_diff__ge__0__iff__ge,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_int @ B @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1353_add__less__same__cancel1,axiom,
% 13.19/7.58 ! [B: real,A: real] :
% 13.19/7.58 ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 13.19/7.58 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1354_add__less__same__cancel1,axiom,
% 13.19/7.58 ! [B: rat,A: rat] :
% 13.19/7.58 ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 13.19/7.58 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1355_add__less__same__cancel1,axiom,
% 13.19/7.58 ! [B: nat,A: nat] :
% 13.19/7.58 ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 13.19/7.58 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1356_add__less__same__cancel1,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 13.19/7.58 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1357_add__less__same__cancel2,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 13.19/7.58 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1358_add__less__same__cancel2,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 13.19/7.58 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1359_add__less__same__cancel2,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 13.19/7.58 = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1360_add__less__same__cancel2,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 13.19/7.58 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1361_less__add__same__cancel1,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 13.19/7.58 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1362_less__add__same__cancel1,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 13.19/7.58 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1363_less__add__same__cancel1,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 13.19/7.58 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1364_less__add__same__cancel1,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 13.19/7.58 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1365_less__add__same__cancel2,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 13.19/7.58 = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1366_less__add__same__cancel2,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 13.19/7.58 = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1367_less__add__same__cancel2,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 13.19/7.58 = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1368_less__add__same__cancel2,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 13.19/7.58 = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1369_double__add__less__zero__iff__single__add__less__zero,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 13.19/7.58 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1370_double__add__less__zero__iff__single__add__less__zero,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 13.19/7.58 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1371_double__add__less__zero__iff__single__add__less__zero,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 13.19/7.58 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1372_zero__less__double__add__iff__zero__less__single__add,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 13.19/7.58 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1373_zero__less__double__add__iff__zero__less__single__add,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 13.19/7.58 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1374_zero__less__double__add__iff__zero__less__single__add,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 13.19/7.58 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1375_diff__gt__0__iff__gt,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 13.19/7.58 = ( ord_less_real @ B @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1376_diff__gt__0__iff__gt,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 13.19/7.58 = ( ord_less_rat @ B @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1377_diff__gt__0__iff__gt,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 13.19/7.58 = ( ord_less_int @ B @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1378_mult__cancel__right2,axiom,
% 13.19/7.58 ! [A: complex,C: complex] :
% 13.19/7.58 ( ( ( times_times_complex @ A @ C )
% 13.19/7.58 = C )
% 13.19/7.58 = ( ( C = zero_zero_complex )
% 13.19/7.58 | ( A = one_one_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1379_mult__cancel__right2,axiom,
% 13.19/7.58 ! [A: real,C: real] :
% 13.19/7.58 ( ( ( times_times_real @ A @ C )
% 13.19/7.58 = C )
% 13.19/7.58 = ( ( C = zero_zero_real )
% 13.19/7.58 | ( A = one_one_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1380_mult__cancel__right2,axiom,
% 13.19/7.58 ! [A: rat,C: rat] :
% 13.19/7.58 ( ( ( times_times_rat @ A @ C )
% 13.19/7.58 = C )
% 13.19/7.58 = ( ( C = zero_zero_rat )
% 13.19/7.58 | ( A = one_one_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1381_mult__cancel__right2,axiom,
% 13.19/7.58 ! [A: int,C: int] :
% 13.19/7.58 ( ( ( times_times_int @ A @ C )
% 13.19/7.58 = C )
% 13.19/7.58 = ( ( C = zero_zero_int )
% 13.19/7.58 | ( A = one_one_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1382_mult__cancel__right1,axiom,
% 13.19/7.58 ! [C: complex,B: complex] :
% 13.19/7.58 ( ( C
% 13.19/7.58 = ( times_times_complex @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_complex )
% 13.19/7.58 | ( B = one_one_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1383_mult__cancel__right1,axiom,
% 13.19/7.58 ! [C: real,B: real] :
% 13.19/7.58 ( ( C
% 13.19/7.58 = ( times_times_real @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_real )
% 13.19/7.58 | ( B = one_one_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1384_mult__cancel__right1,axiom,
% 13.19/7.58 ! [C: rat,B: rat] :
% 13.19/7.58 ( ( C
% 13.19/7.58 = ( times_times_rat @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_rat )
% 13.19/7.58 | ( B = one_one_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1385_mult__cancel__right1,axiom,
% 13.19/7.58 ! [C: int,B: int] :
% 13.19/7.58 ( ( C
% 13.19/7.58 = ( times_times_int @ B @ C ) )
% 13.19/7.58 = ( ( C = zero_zero_int )
% 13.19/7.58 | ( B = one_one_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1386_mult__cancel__left2,axiom,
% 13.19/7.58 ! [C: complex,A: complex] :
% 13.19/7.58 ( ( ( times_times_complex @ C @ A )
% 13.19/7.58 = C )
% 13.19/7.58 = ( ( C = zero_zero_complex )
% 13.19/7.58 | ( A = one_one_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1387_mult__cancel__left2,axiom,
% 13.19/7.58 ! [C: real,A: real] :
% 13.19/7.58 ( ( ( times_times_real @ C @ A )
% 13.19/7.58 = C )
% 13.19/7.58 = ( ( C = zero_zero_real )
% 13.19/7.58 | ( A = one_one_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1388_mult__cancel__left2,axiom,
% 13.19/7.58 ! [C: rat,A: rat] :
% 13.19/7.58 ( ( ( times_times_rat @ C @ A )
% 13.19/7.58 = C )
% 13.19/7.58 = ( ( C = zero_zero_rat )
% 13.19/7.58 | ( A = one_one_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1389_mult__cancel__left2,axiom,
% 13.19/7.58 ! [C: int,A: int] :
% 13.19/7.58 ( ( ( times_times_int @ C @ A )
% 13.19/7.58 = C )
% 13.19/7.58 = ( ( C = zero_zero_int )
% 13.19/7.58 | ( A = one_one_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1390_mult__cancel__left1,axiom,
% 13.19/7.58 ! [C: complex,B: complex] :
% 13.19/7.58 ( ( C
% 13.19/7.58 = ( times_times_complex @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_complex )
% 13.19/7.58 | ( B = one_one_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1391_mult__cancel__left1,axiom,
% 13.19/7.58 ! [C: real,B: real] :
% 13.19/7.58 ( ( C
% 13.19/7.58 = ( times_times_real @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_real )
% 13.19/7.58 | ( B = one_one_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1392_mult__cancel__left1,axiom,
% 13.19/7.58 ! [C: rat,B: rat] :
% 13.19/7.58 ( ( C
% 13.19/7.58 = ( times_times_rat @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_rat )
% 13.19/7.58 | ( B = one_one_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1393_mult__cancel__left1,axiom,
% 13.19/7.58 ! [C: int,B: int] :
% 13.19/7.58 ( ( C
% 13.19/7.58 = ( times_times_int @ C @ B ) )
% 13.19/7.58 = ( ( C = zero_zero_int )
% 13.19/7.58 | ( B = one_one_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1394_sum__squares__eq__zero__iff,axiom,
% 13.19/7.58 ! [X: real,Y3: real] :
% 13.19/7.58 ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 = ( ( X = zero_zero_real )
% 13.19/7.58 & ( Y3 = zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1395_sum__squares__eq__zero__iff,axiom,
% 13.19/7.58 ! [X: rat,Y3: rat] :
% 13.19/7.58 ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 = ( ( X = zero_zero_rat )
% 13.19/7.58 & ( Y3 = zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1396_sum__squares__eq__zero__iff,axiom,
% 13.19/7.58 ! [X: int,Y3: int] :
% 13.19/7.58 ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 = ( ( X = zero_zero_int )
% 13.19/7.58 & ( Y3 = zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1397_nonzero__mult__div__cancel__left,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( A != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 13.19/7.58 = B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1398_nonzero__mult__div__cancel__left,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( A != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 13.19/7.58 = B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1399_nonzero__mult__div__cancel__left,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( A != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 13.19/7.58 = B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1400_nonzero__mult__div__cancel__left,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( A != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 13.19/7.58 = B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1401_nonzero__mult__div__cancel__left,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( A != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 13.19/7.58 = B ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1402_nonzero__mult__div__cancel__right,axiom,
% 13.19/7.58 ! [B: complex,A: complex] :
% 13.19/7.58 ( ( B != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 13.19/7.58 = A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1403_nonzero__mult__div__cancel__right,axiom,
% 13.19/7.58 ! [B: real,A: real] :
% 13.19/7.58 ( ( B != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 13.19/7.58 = A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1404_nonzero__mult__div__cancel__right,axiom,
% 13.19/7.58 ! [B: rat,A: rat] :
% 13.19/7.58 ( ( B != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 13.19/7.58 = A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1405_nonzero__mult__div__cancel__right,axiom,
% 13.19/7.58 ! [B: nat,A: nat] :
% 13.19/7.58 ( ( B != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 13.19/7.58 = A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1406_nonzero__mult__div__cancel__right,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( B != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 13.19/7.58 = A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1407_div__mult__mult1,axiom,
% 13.19/7.58 ! [C: nat,A: nat,B: nat] :
% 13.19/7.58 ( ( C != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 13.19/7.58 = ( divide_divide_nat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1408_div__mult__mult1,axiom,
% 13.19/7.58 ! [C: int,A: int,B: int] :
% 13.19/7.58 ( ( C != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.58 = ( divide_divide_int @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1409_div__mult__mult2,axiom,
% 13.19/7.58 ! [C: nat,A: nat,B: nat] :
% 13.19/7.58 ( ( C != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 13.19/7.58 = ( divide_divide_nat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1410_div__mult__mult2,axiom,
% 13.19/7.58 ! [C: int,A: int,B: int] :
% 13.19/7.58 ( ( C != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 13.19/7.58 = ( divide_divide_int @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1411_div__mult__mult1__if,axiom,
% 13.19/7.58 ! [C: nat,A: nat,B: nat] :
% 13.19/7.58 ( ( ( C = zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 13.19/7.58 = zero_zero_nat ) )
% 13.19/7.58 & ( ( C != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 13.19/7.58 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1412_div__mult__mult1__if,axiom,
% 13.19/7.58 ! [C: int,A: int,B: int] :
% 13.19/7.58 ( ( ( C = zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.58 = zero_zero_int ) )
% 13.19/7.58 & ( ( C != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.58 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1413_mult__divide__mult__cancel__left__if,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( ( C = zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 13.19/7.58 = zero_zero_complex ) )
% 13.19/7.58 & ( ( C != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1414_mult__divide__mult__cancel__left__if,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( ( C = zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.58 = zero_zero_real ) )
% 13.19/7.58 & ( ( C != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.58 = ( divide_divide_real @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1415_mult__divide__mult__cancel__left__if,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( ( C = zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.58 = zero_zero_rat ) )
% 13.19/7.58 & ( ( C != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.58 = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1416_nonzero__mult__divide__mult__cancel__left,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( C != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1417_nonzero__mult__divide__mult__cancel__left,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( C != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.58 = ( divide_divide_real @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1418_nonzero__mult__divide__mult__cancel__left,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( C != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.58 = ( divide_divide_rat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1419_nonzero__mult__divide__mult__cancel__left2,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( C != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1420_nonzero__mult__divide__mult__cancel__left2,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( C != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 13.19/7.58 = ( divide_divide_real @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1421_nonzero__mult__divide__mult__cancel__left2,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( C != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 13.19/7.58 = ( divide_divide_rat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1422_nonzero__mult__divide__mult__cancel__right,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( C != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1423_nonzero__mult__divide__mult__cancel__right,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( C != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 13.19/7.58 = ( divide_divide_real @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1424_nonzero__mult__divide__mult__cancel__right,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( C != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 13.19/7.58 = ( divide_divide_rat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1425_nonzero__mult__divide__mult__cancel__right2,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( C != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1426_nonzero__mult__divide__mult__cancel__right2,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( C != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 13.19/7.58 = ( divide_divide_real @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1427_nonzero__mult__divide__mult__cancel__right2,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( C != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.58 = ( divide_divide_rat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1428_diff__numeral__special_I9_J,axiom,
% 13.19/7.58 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1429_diff__numeral__special_I9_J,axiom,
% 13.19/7.58 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1430_diff__numeral__special_I9_J,axiom,
% 13.19/7.58 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1431_diff__numeral__special_I9_J,axiom,
% 13.19/7.58 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1432_diff__add__zero,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1433_div__self,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( A != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ A @ A )
% 13.19/7.58 = one_one_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1434_div__self,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( A != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ A @ A )
% 13.19/7.58 = one_one_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1435_div__self,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( A != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ A @ A )
% 13.19/7.58 = one_one_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1436_div__self,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( A != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ A @ A )
% 13.19/7.58 = one_one_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1437_div__self,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( A != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ A @ A )
% 13.19/7.58 = one_one_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1438_zero__eq__1__divide__iff,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( zero_zero_real
% 13.19/7.58 = ( divide_divide_real @ one_one_real @ A ) )
% 13.19/7.58 = ( A = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1439_zero__eq__1__divide__iff,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( zero_zero_rat
% 13.19/7.58 = ( divide_divide_rat @ one_one_rat @ A ) )
% 13.19/7.58 = ( A = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1440_one__divide__eq__0__iff,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ( divide_divide_real @ one_one_real @ A )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 = ( A = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1441_one__divide__eq__0__iff,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ( divide_divide_rat @ one_one_rat @ A )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 = ( A = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1442_eq__divide__eq__1,axiom,
% 13.19/7.58 ! [B: real,A: real] :
% 13.19/7.58 ( ( one_one_real
% 13.19/7.58 = ( divide_divide_real @ B @ A ) )
% 13.19/7.58 = ( ( A != zero_zero_real )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1443_eq__divide__eq__1,axiom,
% 13.19/7.58 ! [B: rat,A: rat] :
% 13.19/7.58 ( ( one_one_rat
% 13.19/7.58 = ( divide_divide_rat @ B @ A ) )
% 13.19/7.58 = ( ( A != zero_zero_rat )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1444_divide__eq__eq__1,axiom,
% 13.19/7.58 ! [B: real,A: real] :
% 13.19/7.58 ( ( ( divide_divide_real @ B @ A )
% 13.19/7.58 = one_one_real )
% 13.19/7.58 = ( ( A != zero_zero_real )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1445_divide__eq__eq__1,axiom,
% 13.19/7.58 ! [B: rat,A: rat] :
% 13.19/7.58 ( ( ( divide_divide_rat @ B @ A )
% 13.19/7.58 = one_one_rat )
% 13.19/7.58 = ( ( A != zero_zero_rat )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1446_divide__self__if,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( ( A = zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ A @ A )
% 13.19/7.58 = zero_zero_complex ) )
% 13.19/7.58 & ( ( A != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ A @ A )
% 13.19/7.58 = one_one_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1447_divide__self__if,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ( A = zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ A @ A )
% 13.19/7.58 = zero_zero_real ) )
% 13.19/7.58 & ( ( A != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ A @ A )
% 13.19/7.58 = one_one_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1448_divide__self__if,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ( A = zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ A @ A )
% 13.19/7.58 = zero_zero_rat ) )
% 13.19/7.58 & ( ( A != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ A @ A )
% 13.19/7.58 = one_one_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1449_divide__self,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( A != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ A @ A )
% 13.19/7.58 = one_one_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1450_divide__self,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( A != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ A @ A )
% 13.19/7.58 = one_one_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1451_divide__self,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( A != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ A @ A )
% 13.19/7.58 = one_one_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1452_one__eq__divide__iff,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( one_one_complex
% 13.19/7.58 = ( divide1717551699836669952omplex @ A @ B ) )
% 13.19/7.58 = ( ( B != zero_zero_complex )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1453_one__eq__divide__iff,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( one_one_real
% 13.19/7.58 = ( divide_divide_real @ A @ B ) )
% 13.19/7.58 = ( ( B != zero_zero_real )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1454_one__eq__divide__iff,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( one_one_rat
% 13.19/7.58 = ( divide_divide_rat @ A @ B ) )
% 13.19/7.58 = ( ( B != zero_zero_rat )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1455_divide__eq__1__iff,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( ( divide1717551699836669952omplex @ A @ B )
% 13.19/7.58 = one_one_complex )
% 13.19/7.58 = ( ( B != zero_zero_complex )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1456_divide__eq__1__iff,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ( divide_divide_real @ A @ B )
% 13.19/7.58 = one_one_real )
% 13.19/7.58 = ( ( B != zero_zero_real )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1457_divide__eq__1__iff,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ( divide_divide_rat @ A @ B )
% 13.19/7.58 = one_one_rat )
% 13.19/7.58 = ( ( B != zero_zero_rat )
% 13.19/7.58 & ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1458_power__zero__numeral,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 13.19/7.58 = zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1459_power__zero__numeral,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1460_power__zero__numeral,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 13.19/7.58 = zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1461_power__zero__numeral,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1462_power__zero__numeral,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 13.19/7.58 = zero_zero_complex ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1463_power__eq__0__iff,axiom,
% 13.19/7.58 ! [A: rat,N: nat] :
% 13.19/7.58 ( ( ( power_power_rat @ A @ N )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 = ( ( A = zero_zero_rat )
% 13.19/7.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1464_power__eq__0__iff,axiom,
% 13.19/7.58 ! [A: nat,N: nat] :
% 13.19/7.58 ( ( ( power_power_nat @ A @ N )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 = ( ( A = zero_zero_nat )
% 13.19/7.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1465_power__eq__0__iff,axiom,
% 13.19/7.58 ! [A: real,N: nat] :
% 13.19/7.58 ( ( ( power_power_real @ A @ N )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 = ( ( A = zero_zero_real )
% 13.19/7.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1466_power__eq__0__iff,axiom,
% 13.19/7.58 ! [A: int,N: nat] :
% 13.19/7.58 ( ( ( power_power_int @ A @ N )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 = ( ( A = zero_zero_int )
% 13.19/7.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1467_power__eq__0__iff,axiom,
% 13.19/7.58 ! [A: complex,N: nat] :
% 13.19/7.58 ( ( ( power_power_complex @ A @ N )
% 13.19/7.58 = zero_zero_complex )
% 13.19/7.58 = ( ( A = zero_zero_complex )
% 13.19/7.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1468_mod__mult__self2__is__0,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1469_mod__mult__self2__is__0,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1470_mod__mult__self2__is__0,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1471_mod__mult__self1__is__0,axiom,
% 13.19/7.58 ! [B: nat,A: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1472_mod__mult__self1__is__0,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1473_mod__mult__self1__is__0,axiom,
% 13.19/7.58 ! [B: code_integer,A: code_integer] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1474_bits__mod__by__1,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1475_bits__mod__by__1,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( modulo_modulo_int @ A @ one_one_int )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1476_bits__mod__by__1,axiom,
% 13.19/7.58 ! [A: code_integer] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1477_mod__by__1,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ A @ one_one_nat )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1478_mod__by__1,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( modulo_modulo_int @ A @ one_one_int )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1479_mod__by__1,axiom,
% 13.19/7.58 ! [A: code_integer] :
% 13.19/7.58 ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1480_bits__mod__div__trivial,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1481_bits__mod__div__trivial,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1482_bits__mod__div__trivial,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1483_mod__div__trivial,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1484_mod__div__trivial,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1485_mod__div__trivial,axiom,
% 13.19/7.58 ! [A: code_integer,B: code_integer] :
% 13.19/7.58 ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 13.19/7.58 = zero_z3403309356797280102nteger ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1486_zmod__numeral__Bit0,axiom,
% 13.19/7.58 ! [V: num,W: num] :
% 13.19/7.58 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 13.19/7.58 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1487_mult__le__cancel2,axiom,
% 13.19/7.58 ! [M: nat,K: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 13.19/7.58 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.58 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1488_nat__mult__le__cancel__disj,axiom,
% 13.19/7.58 ! [K: nat,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.19/7.58 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.58 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1489_div__mult__self1__is__m,axiom,
% 13.19/7.58 ! [N: nat,M: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 13.19/7.58 = M ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1490_div__mult__self__is__m,axiom,
% 13.19/7.58 ! [N: nat,M: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 13.19/7.58 = M ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1491_dbl__simps_I5_J,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
% 13.19/7.58 = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1492_dbl__simps_I5_J,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
% 13.19/7.58 = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1493_dbl__simps_I5_J,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) )
% 13.19/7.58 = ( numeral_numeral_rat @ ( bit0 @ K ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1494_dbl__simps_I5_J,axiom,
% 13.19/7.58 ! [K: num] :
% 13.19/7.58 ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
% 13.19/7.58 = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1495__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
% 13.19/7.58 vEBT_invar_vebt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ na ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1496__C33_C,axiom,
% 13.19/7.58 ~ ? [U2: nat] :
% 13.19/7.58 ( ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U2 )
% 13.19/7.58 & ( ord_less_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ U2 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1497_divide__le__0__1__iff,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 13.19/7.58 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1498_divide__le__0__1__iff,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 13.19/7.58 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1499_zero__le__divide__1__iff,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 13.19/7.58 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1500_zero__le__divide__1__iff,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 13.19/7.58 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1501_zero__less__divide__1__iff,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 13.19/7.58 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1502_zero__less__divide__1__iff,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 13.19/7.58 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1503_less__divide__eq__1__pos,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.58 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 13.19/7.58 = ( ord_less_real @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1504_less__divide__eq__1__pos,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.58 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 13.19/7.58 = ( ord_less_rat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1505_less__divide__eq__1__neg,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.58 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 13.19/7.58 = ( ord_less_real @ B @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1506_less__divide__eq__1__neg,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.58 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 13.19/7.58 = ( ord_less_rat @ B @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1507_divide__less__eq__1__pos,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.58 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 13.19/7.58 = ( ord_less_real @ B @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1508_divide__less__eq__1__pos,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.58 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 13.19/7.58 = ( ord_less_rat @ B @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1509_divide__less__eq__1__neg,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.58 => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 13.19/7.58 = ( ord_less_real @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1510_divide__less__eq__1__neg,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.58 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 13.19/7.58 = ( ord_less_rat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1511_divide__less__0__1__iff,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 13.19/7.58 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1512_divide__less__0__1__iff,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 13.19/7.58 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1513_eq__divide__eq__numeral1_I1_J,axiom,
% 13.19/7.58 ! [A: complex,B: complex,W: num] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
% 13.19/7.58 = ( ( ( ( numera6690914467698888265omplex @ W )
% 13.19/7.58 != zero_zero_complex )
% 13.19/7.58 => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
% 13.19/7.58 = B ) )
% 13.19/7.58 & ( ( ( numera6690914467698888265omplex @ W )
% 13.19/7.58 = zero_zero_complex )
% 13.19/7.58 => ( A = zero_zero_complex ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1514_eq__divide__eq__numeral1_I1_J,axiom,
% 13.19/7.58 ! [A: real,B: real,W: num] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
% 13.19/7.58 = ( ( ( ( numeral_numeral_real @ W )
% 13.19/7.58 != zero_zero_real )
% 13.19/7.58 => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
% 13.19/7.58 = B ) )
% 13.19/7.58 & ( ( ( numeral_numeral_real @ W )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 => ( A = zero_zero_real ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1515_eq__divide__eq__numeral1_I1_J,axiom,
% 13.19/7.58 ! [A: rat,B: rat,W: num] :
% 13.19/7.58 ( ( A
% 13.19/7.58 = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
% 13.19/7.58 = ( ( ( ( numeral_numeral_rat @ W )
% 13.19/7.58 != zero_zero_rat )
% 13.19/7.58 => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
% 13.19/7.58 = B ) )
% 13.19/7.58 & ( ( ( numeral_numeral_rat @ W )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 => ( A = zero_zero_rat ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1516_divide__eq__eq__numeral1_I1_J,axiom,
% 13.19/7.58 ! [B: complex,W: num,A: complex] :
% 13.19/7.58 ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
% 13.19/7.58 = A )
% 13.19/7.58 = ( ( ( ( numera6690914467698888265omplex @ W )
% 13.19/7.58 != zero_zero_complex )
% 13.19/7.58 => ( B
% 13.19/7.58 = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
% 13.19/7.58 & ( ( ( numera6690914467698888265omplex @ W )
% 13.19/7.58 = zero_zero_complex )
% 13.19/7.58 => ( A = zero_zero_complex ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1517_divide__eq__eq__numeral1_I1_J,axiom,
% 13.19/7.58 ! [B: real,W: num,A: real] :
% 13.19/7.58 ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
% 13.19/7.58 = A )
% 13.19/7.58 = ( ( ( ( numeral_numeral_real @ W )
% 13.19/7.58 != zero_zero_real )
% 13.19/7.58 => ( B
% 13.19/7.58 = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
% 13.19/7.58 & ( ( ( numeral_numeral_real @ W )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 => ( A = zero_zero_real ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1518_divide__eq__eq__numeral1_I1_J,axiom,
% 13.19/7.58 ! [B: rat,W: num,A: rat] :
% 13.19/7.58 ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
% 13.19/7.58 = A )
% 13.19/7.58 = ( ( ( ( numeral_numeral_rat @ W )
% 13.19/7.58 != zero_zero_rat )
% 13.19/7.58 => ( B
% 13.19/7.58 = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
% 13.19/7.58 & ( ( ( numeral_numeral_rat @ W )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 => ( A = zero_zero_rat ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1519_nonzero__divide__mult__cancel__right,axiom,
% 13.19/7.58 ! [B: complex,A: complex] :
% 13.19/7.58 ( ( B != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1520_nonzero__divide__mult__cancel__right,axiom,
% 13.19/7.58 ! [B: real,A: real] :
% 13.19/7.58 ( ( B != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 13.19/7.58 = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1521_nonzero__divide__mult__cancel__right,axiom,
% 13.19/7.58 ! [B: rat,A: rat] :
% 13.19/7.58 ( ( B != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 13.19/7.58 = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1522_nonzero__divide__mult__cancel__left,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( A != zero_zero_complex )
% 13.19/7.58 => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 13.19/7.58 = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1523_nonzero__divide__mult__cancel__left,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( A != zero_zero_real )
% 13.19/7.58 => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 13.19/7.58 = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1524_nonzero__divide__mult__cancel__left,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( A != zero_zero_rat )
% 13.19/7.58 => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 13.19/7.58 = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1525_div__mult__self4,axiom,
% 13.19/7.58 ! [B: nat,C: nat,A: nat] :
% 13.19/7.58 ( ( B != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 13.19/7.58 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1526_div__mult__self4,axiom,
% 13.19/7.58 ! [B: int,C: int,A: int] :
% 13.19/7.58 ( ( B != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 13.19/7.58 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1527_div__mult__self3,axiom,
% 13.19/7.58 ! [B: nat,C: nat,A: nat] :
% 13.19/7.58 ( ( B != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 13.19/7.58 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1528_div__mult__self3,axiom,
% 13.19/7.58 ! [B: int,C: int,A: int] :
% 13.19/7.58 ( ( B != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 13.19/7.58 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1529_div__mult__self2,axiom,
% 13.19/7.58 ! [B: nat,A: nat,C: nat] :
% 13.19/7.58 ( ( B != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 13.19/7.58 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1530_div__mult__self2,axiom,
% 13.19/7.58 ! [B: int,A: int,C: int] :
% 13.19/7.58 ( ( B != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 13.19/7.58 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1531_div__mult__self1,axiom,
% 13.19/7.58 ! [B: nat,A: nat,C: nat] :
% 13.19/7.58 ( ( B != zero_zero_nat )
% 13.19/7.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 13.19/7.58 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1532_div__mult__self1,axiom,
% 13.19/7.58 ! [B: int,A: int,C: int] :
% 13.19/7.58 ( ( B != zero_zero_int )
% 13.19/7.58 => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 13.19/7.58 = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1533_power__mono__iff,axiom,
% 13.19/7.58 ! [A: real,B: real,N: nat] :
% 13.19/7.58 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.58 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 13.19/7.58 = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1534_power__mono__iff,axiom,
% 13.19/7.58 ! [A: rat,B: rat,N: nat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 13.19/7.58 = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1535_power__mono__iff,axiom,
% 13.19/7.58 ! [A: nat,B: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 13.19/7.58 = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1536_power__mono__iff,axiom,
% 13.19/7.58 ! [A: int,B: int,N: nat] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.58 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.58 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 13.19/7.58 = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1537_half__negative__int__iff,axiom,
% 13.19/7.58 ! [K: int] :
% 13.19/7.58 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 13.19/7.58 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1538_half__nonnegative__int__iff,axiom,
% 13.19/7.58 ! [K: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1539_i1,axiom,
% 13.19/7.58 ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.19/7.58 = none_nat )
% 13.19/7.58 | ~ ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1540_divide__le__eq__1__neg,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.58 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 13.19/7.58 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1541_divide__le__eq__1__neg,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.58 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 13.19/7.58 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1542_divide__le__eq__1__pos,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.58 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 13.19/7.58 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1543_divide__le__eq__1__pos,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.58 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 13.19/7.58 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1544_le__divide__eq__1__neg,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.58 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 13.19/7.58 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1545_le__divide__eq__1__neg,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.58 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 13.19/7.58 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1546_le__divide__eq__1__pos,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.58 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 13.19/7.58 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1547_le__divide__eq__1__pos,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.58 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 13.19/7.58 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1548_power__strict__decreasing__iff,axiom,
% 13.19/7.58 ! [B: real,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ B )
% 13.19/7.58 => ( ( ord_less_real @ B @ one_one_real )
% 13.19/7.58 => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 13.19/7.58 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1549_power__strict__decreasing__iff,axiom,
% 13.19/7.58 ! [B: rat,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ B )
% 13.19/7.58 => ( ( ord_less_rat @ B @ one_one_rat )
% 13.19/7.58 => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 13.19/7.58 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1550_power__strict__decreasing__iff,axiom,
% 13.19/7.58 ! [B: nat,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.58 => ( ( ord_less_nat @ B @ one_one_nat )
% 13.19/7.58 => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 13.19/7.58 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1551_power__strict__decreasing__iff,axiom,
% 13.19/7.58 ! [B: int,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.58 => ( ( ord_less_int @ B @ one_one_int )
% 13.19/7.58 => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 13.19/7.58 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1552_zero__eq__power2,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 = ( A = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1553_zero__eq__power2,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 = ( A = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1554_zero__eq__power2,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 = ( A = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1555_zero__eq__power2,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 = ( A = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1556_zero__eq__power2,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_complex )
% 13.19/7.58 = ( A = zero_zero_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1557_add__self__mod__2,axiom,
% 13.19/7.58 ! [M: nat] :
% 13.19/7.58 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1558_bits__1__div__2,axiom,
% 13.19/7.58 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1559_bits__1__div__2,axiom,
% 13.19/7.58 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1560_one__div__two__eq__zero,axiom,
% 13.19/7.58 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1561_one__div__two__eq__zero,axiom,
% 13.19/7.58 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.19/7.58 = zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1562_power__decreasing__iff,axiom,
% 13.19/7.58 ! [B: real,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ B )
% 13.19/7.58 => ( ( ord_less_real @ B @ one_one_real )
% 13.19/7.58 => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N ) )
% 13.19/7.58 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1563_power__decreasing__iff,axiom,
% 13.19/7.58 ! [B: rat,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ B )
% 13.19/7.58 => ( ( ord_less_rat @ B @ one_one_rat )
% 13.19/7.58 => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N ) )
% 13.19/7.58 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1564_power__decreasing__iff,axiom,
% 13.19/7.58 ! [B: nat,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.58 => ( ( ord_less_nat @ B @ one_one_nat )
% 13.19/7.58 => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
% 13.19/7.58 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1565_power__decreasing__iff,axiom,
% 13.19/7.58 ! [B: int,M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.58 => ( ( ord_less_int @ B @ one_one_int )
% 13.19/7.58 => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
% 13.19/7.58 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1566_power2__eq__iff__nonneg,axiom,
% 13.19/7.58 ! [X: real,Y3: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.58 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.19/7.58 => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( X = Y3 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1567_power2__eq__iff__nonneg,axiom,
% 13.19/7.58 ! [X: rat,Y3: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.19/7.58 => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( X = Y3 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1568_power2__eq__iff__nonneg,axiom,
% 13.19/7.58 ! [X: nat,Y3: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 13.19/7.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 13.19/7.58 => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( X = Y3 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1569_power2__eq__iff__nonneg,axiom,
% 13.19/7.58 ! [X: int,Y3: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.19/7.58 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.19/7.58 => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( X = Y3 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1570_power2__less__eq__zero__iff,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 13.19/7.58 = ( A = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1571_power2__less__eq__zero__iff,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 13.19/7.58 = ( A = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1572_power2__less__eq__zero__iff,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 13.19/7.58 = ( A = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1573_zero__less__power2,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( A != zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1574_zero__less__power2,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( A != zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1575_zero__less__power2,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( A != zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1576_sum__power2__eq__zero__iff,axiom,
% 13.19/7.58 ! [X: rat,Y3: rat] :
% 13.19/7.58 ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 = ( ( X = zero_zero_rat )
% 13.19/7.58 & ( Y3 = zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1577_sum__power2__eq__zero__iff,axiom,
% 13.19/7.58 ! [X: real,Y3: real] :
% 13.19/7.58 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 = ( ( X = zero_zero_real )
% 13.19/7.58 & ( Y3 = zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1578_sum__power2__eq__zero__iff,axiom,
% 13.19/7.58 ! [X: int,Y3: int] :
% 13.19/7.58 ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 = ( ( X = zero_zero_int )
% 13.19/7.58 & ( Y3 = zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1579_mod2__gr__0,axiom,
% 13.19/7.58 ! [M: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.19/7.58 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.19/7.58 = one_one_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1580__C4_Ohyps_C_I1_J,axiom,
% 13.19/7.58 ! [X5: vEBT_VEBT] :
% 13.19/7.58 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 13.19/7.58 => ( ( vEBT_invar_vebt @ X5 @ na )
% 13.19/7.58 & ! [Xa2: nat,Xb2: nat] :
% 13.19/7.58 ( ( ( vEBT_vebt_succ @ X5 @ Xa2 )
% 13.19/7.58 = ( some_nat @ Xb2 ) )
% 13.19/7.58 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ Xa2 @ Xb2 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1581__C4_Ohyps_C_I7_J,axiom,
% 13.19/7.58 ! [I3: nat] :
% 13.19/7.58 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 13.19/7.58 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I3 ) @ X6 ) )
% 13.19/7.58 = ( vEBT_V8194947554948674370ptions @ summary @ I3 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1582_zero__reorient,axiom,
% 13.19/7.58 ! [X: complex] :
% 13.19/7.58 ( ( zero_zero_complex = X )
% 13.19/7.58 = ( X = zero_zero_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1583_zero__reorient,axiom,
% 13.19/7.58 ! [X: real] :
% 13.19/7.58 ( ( zero_zero_real = X )
% 13.19/7.58 = ( X = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1584_zero__reorient,axiom,
% 13.19/7.58 ! [X: rat] :
% 13.19/7.58 ( ( zero_zero_rat = X )
% 13.19/7.58 = ( X = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1585_zero__reorient,axiom,
% 13.19/7.58 ! [X: nat] :
% 13.19/7.58 ( ( zero_zero_nat = X )
% 13.19/7.58 = ( X = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1586_zero__reorient,axiom,
% 13.19/7.58 ! [X: int] :
% 13.19/7.58 ( ( zero_zero_int = X )
% 13.19/7.58 = ( X = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1587_zdiv__mono__strict,axiom,
% 13.19/7.58 ! [A2: int,B5: int,N: int] :
% 13.19/7.58 ( ( ord_less_int @ A2 @ B5 )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ N )
% 13.19/7.58 => ( ( ( modulo_modulo_int @ A2 @ N )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 => ( ( ( modulo_modulo_int @ B5 @ N )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 => ( ord_less_int @ ( divide_divide_int @ A2 @ N ) @ ( divide_divide_int @ B5 @ N ) ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1588_verit__le__mono__div__int,axiom,
% 13.19/7.58 ! [A2: int,B5: int,N: int] :
% 13.19/7.58 ( ( ord_less_int @ A2 @ B5 )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ N )
% 13.19/7.58 => ( ord_less_eq_int
% 13.19/7.58 @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N )
% 13.19/7.58 @ ( if_int
% 13.19/7.58 @ ( ( modulo_modulo_int @ B5 @ N )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 @ one_one_int
% 13.19/7.58 @ zero_zero_int ) )
% 13.19/7.58 @ ( divide_divide_int @ B5 @ N ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1589_zmod__zmult2__eq,axiom,
% 13.19/7.58 ! [C: int,A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.58 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 13.19/7.58 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1590_split__pos__lemma,axiom,
% 13.19/7.58 ! [K: int,P: int > int > $o,N: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ K )
% 13.19/7.58 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 13.19/7.58 = ( ! [I4: int,J3: int] :
% 13.19/7.58 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 13.19/7.58 & ( ord_less_int @ J3 @ K )
% 13.19/7.58 & ( N
% 13.19/7.58 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 13.19/7.58 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1591_split__neg__lemma,axiom,
% 13.19/7.58 ! [K: int,P: int > int > $o,N: int] :
% 13.19/7.58 ( ( ord_less_int @ K @ zero_zero_int )
% 13.19/7.58 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 13.19/7.58 = ( ! [I4: int,J3: int] :
% 13.19/7.58 ( ( ( ord_less_int @ K @ J3 )
% 13.19/7.58 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 13.19/7.58 & ( N
% 13.19/7.58 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 13.19/7.58 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1592_bounded__nat__set__is__finite,axiom,
% 13.19/7.58 ! [N3: set_nat,N: nat] :
% 13.19/7.58 ( ! [X3: nat] :
% 13.19/7.58 ( ( member_nat @ X3 @ N3 )
% 13.19/7.58 => ( ord_less_nat @ X3 @ N ) )
% 13.19/7.58 => ( finite_finite_nat @ N3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1593_finite__nat__set__iff__bounded,axiom,
% 13.19/7.58 ( finite_finite_nat
% 13.19/7.58 = ( ^ [N5: set_nat] :
% 13.19/7.58 ? [M3: nat] :
% 13.19/7.58 ! [X2: nat] :
% 13.19/7.58 ( ( member_nat @ X2 @ N5 )
% 13.19/7.58 => ( ord_less_nat @ X2 @ M3 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1594_finite__nat__set__iff__bounded__le,axiom,
% 13.19/7.58 ( finite_finite_nat
% 13.19/7.58 = ( ^ [N5: set_nat] :
% 13.19/7.58 ? [M3: nat] :
% 13.19/7.58 ! [X2: nat] :
% 13.19/7.58 ( ( member_nat @ X2 @ N5 )
% 13.19/7.58 => ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1595_size__neq__size__imp__neq,axiom,
% 13.19/7.58 ! [X: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 13.19/7.58 ( ( ( size_s6755466524823107622T_VEBT @ X )
% 13.19/7.58 != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 13.19/7.58 => ( X != Y3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1596_size__neq__size__imp__neq,axiom,
% 13.19/7.58 ! [X: list_o,Y3: list_o] :
% 13.19/7.58 ( ( ( size_size_list_o @ X )
% 13.19/7.58 != ( size_size_list_o @ Y3 ) )
% 13.19/7.58 => ( X != Y3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1597_size__neq__size__imp__neq,axiom,
% 13.19/7.58 ! [X: list_nat,Y3: list_nat] :
% 13.19/7.58 ( ( ( size_size_list_nat @ X )
% 13.19/7.58 != ( size_size_list_nat @ Y3 ) )
% 13.19/7.58 => ( X != Y3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1598_size__neq__size__imp__neq,axiom,
% 13.19/7.58 ! [X: list_int,Y3: list_int] :
% 13.19/7.58 ( ( ( size_size_list_int @ X )
% 13.19/7.58 != ( size_size_list_int @ Y3 ) )
% 13.19/7.58 => ( X != Y3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1599_size__neq__size__imp__neq,axiom,
% 13.19/7.58 ! [X: num,Y3: num] :
% 13.19/7.58 ( ( ( size_size_num @ X )
% 13.19/7.58 != ( size_size_num @ Y3 ) )
% 13.19/7.58 => ( X != Y3 ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1600_power__0__left,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ( N = zero_zero_nat )
% 13.19/7.58 => ( ( power_power_rat @ zero_zero_rat @ N )
% 13.19/7.58 = one_one_rat ) )
% 13.19/7.58 & ( ( N != zero_zero_nat )
% 13.19/7.58 => ( ( power_power_rat @ zero_zero_rat @ N )
% 13.19/7.58 = zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1601_power__0__left,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ( N = zero_zero_nat )
% 13.19/7.58 => ( ( power_power_nat @ zero_zero_nat @ N )
% 13.19/7.58 = one_one_nat ) )
% 13.19/7.58 & ( ( N != zero_zero_nat )
% 13.19/7.58 => ( ( power_power_nat @ zero_zero_nat @ N )
% 13.19/7.58 = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1602_power__0__left,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ( N = zero_zero_nat )
% 13.19/7.58 => ( ( power_power_real @ zero_zero_real @ N )
% 13.19/7.58 = one_one_real ) )
% 13.19/7.58 & ( ( N != zero_zero_nat )
% 13.19/7.58 => ( ( power_power_real @ zero_zero_real @ N )
% 13.19/7.58 = zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1603_power__0__left,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ( N = zero_zero_nat )
% 13.19/7.58 => ( ( power_power_int @ zero_zero_int @ N )
% 13.19/7.58 = one_one_int ) )
% 13.19/7.58 & ( ( N != zero_zero_nat )
% 13.19/7.58 => ( ( power_power_int @ zero_zero_int @ N )
% 13.19/7.58 = zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1604_power__0__left,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ( N = zero_zero_nat )
% 13.19/7.58 => ( ( power_power_complex @ zero_zero_complex @ N )
% 13.19/7.58 = one_one_complex ) )
% 13.19/7.58 & ( ( N != zero_zero_nat )
% 13.19/7.58 => ( ( power_power_complex @ zero_zero_complex @ N )
% 13.19/7.58 = zero_zero_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1605_zero__power,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( power_power_rat @ zero_zero_rat @ N )
% 13.19/7.58 = zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1606_zero__power,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( power_power_nat @ zero_zero_nat @ N )
% 13.19/7.58 = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1607_zero__power,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( power_power_real @ zero_zero_real @ N )
% 13.19/7.58 = zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1608_zero__power,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( power_power_int @ zero_zero_int @ N )
% 13.19/7.58 = zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1609_zero__power,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 => ( ( power_power_complex @ zero_zero_complex @ N )
% 13.19/7.58 = zero_zero_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1610_div__mod__decomp__int,axiom,
% 13.19/7.58 ! [A2: int,N: int] :
% 13.19/7.58 ( A2
% 13.19/7.58 = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N ) @ N ) @ ( modulo_modulo_int @ A2 @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1611_zero__le,axiom,
% 13.19/7.58 ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1612_le__numeral__extra_I3_J,axiom,
% 13.19/7.58 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1613_le__numeral__extra_I3_J,axiom,
% 13.19/7.58 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1614_le__numeral__extra_I3_J,axiom,
% 13.19/7.58 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1615_le__numeral__extra_I3_J,axiom,
% 13.19/7.58 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1616_gr__zeroI,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( N != zero_zero_nat )
% 13.19/7.58 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1617_not__less__zero,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1618_gr__implies__not__zero,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ M @ N )
% 13.19/7.58 => ( N != zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1619_zero__less__iff__neq__zero,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.58 = ( N != zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1620_field__lbound__gt__zero,axiom,
% 13.19/7.58 ! [D1: real,D22: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ D1 )
% 13.19/7.58 => ( ( ord_less_real @ zero_zero_real @ D22 )
% 13.19/7.58 => ? [E2: real] :
% 13.19/7.58 ( ( ord_less_real @ zero_zero_real @ E2 )
% 13.19/7.58 & ( ord_less_real @ E2 @ D1 )
% 13.19/7.58 & ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1621_field__lbound__gt__zero,axiom,
% 13.19/7.58 ! [D1: rat,D22: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 13.19/7.58 => ( ( ord_less_rat @ zero_zero_rat @ D22 )
% 13.19/7.58 => ? [E2: rat] :
% 13.19/7.58 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 13.19/7.58 & ( ord_less_rat @ E2 @ D1 )
% 13.19/7.58 & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1622_less__numeral__extra_I3_J,axiom,
% 13.19/7.58 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1623_less__numeral__extra_I3_J,axiom,
% 13.19/7.58 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1624_less__numeral__extra_I3_J,axiom,
% 13.19/7.58 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1625_less__numeral__extra_I3_J,axiom,
% 13.19/7.58 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1626_zero__neq__numeral,axiom,
% 13.19/7.58 ! [N: num] :
% 13.19/7.58 ( zero_zero_complex
% 13.19/7.58 != ( numera6690914467698888265omplex @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1627_zero__neq__numeral,axiom,
% 13.19/7.58 ! [N: num] :
% 13.19/7.58 ( zero_zero_real
% 13.19/7.58 != ( numeral_numeral_real @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1628_zero__neq__numeral,axiom,
% 13.19/7.58 ! [N: num] :
% 13.19/7.58 ( zero_zero_rat
% 13.19/7.58 != ( numeral_numeral_rat @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1629_zero__neq__numeral,axiom,
% 13.19/7.58 ! [N: num] :
% 13.19/7.58 ( zero_zero_nat
% 13.19/7.58 != ( numeral_numeral_nat @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1630_zero__neq__numeral,axiom,
% 13.19/7.58 ! [N: num] :
% 13.19/7.58 ( zero_zero_int
% 13.19/7.58 != ( numeral_numeral_int @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1631_mult__not__zero,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( ( times_times_complex @ A @ B )
% 13.19/7.58 != zero_zero_complex )
% 13.19/7.58 => ( ( A != zero_zero_complex )
% 13.19/7.58 & ( B != zero_zero_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1632_mult__not__zero,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ( times_times_real @ A @ B )
% 13.19/7.58 != zero_zero_real )
% 13.19/7.58 => ( ( A != zero_zero_real )
% 13.19/7.58 & ( B != zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1633_mult__not__zero,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ( times_times_rat @ A @ B )
% 13.19/7.58 != zero_zero_rat )
% 13.19/7.58 => ( ( A != zero_zero_rat )
% 13.19/7.58 & ( B != zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1634_mult__not__zero,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ A @ B )
% 13.19/7.58 != zero_zero_nat )
% 13.19/7.58 => ( ( A != zero_zero_nat )
% 13.19/7.58 & ( B != zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1635_mult__not__zero,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ( times_times_int @ A @ B )
% 13.19/7.58 != zero_zero_int )
% 13.19/7.58 => ( ( A != zero_zero_int )
% 13.19/7.58 & ( B != zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1636_divisors__zero,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( ( times_times_complex @ A @ B )
% 13.19/7.58 = zero_zero_complex )
% 13.19/7.58 => ( ( A = zero_zero_complex )
% 13.19/7.58 | ( B = zero_zero_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1637_divisors__zero,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( ( times_times_real @ A @ B )
% 13.19/7.58 = zero_zero_real )
% 13.19/7.58 => ( ( A = zero_zero_real )
% 13.19/7.58 | ( B = zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1638_divisors__zero,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( ( times_times_rat @ A @ B )
% 13.19/7.58 = zero_zero_rat )
% 13.19/7.58 => ( ( A = zero_zero_rat )
% 13.19/7.58 | ( B = zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1639_divisors__zero,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ A @ B )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 => ( ( A = zero_zero_nat )
% 13.19/7.58 | ( B = zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1640_divisors__zero,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ( times_times_int @ A @ B )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 => ( ( A = zero_zero_int )
% 13.19/7.58 | ( B = zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1641_no__zero__divisors,axiom,
% 13.19/7.58 ! [A: complex,B: complex] :
% 13.19/7.58 ( ( A != zero_zero_complex )
% 13.19/7.58 => ( ( B != zero_zero_complex )
% 13.19/7.58 => ( ( times_times_complex @ A @ B )
% 13.19/7.58 != zero_zero_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1642_no__zero__divisors,axiom,
% 13.19/7.58 ! [A: real,B: real] :
% 13.19/7.58 ( ( A != zero_zero_real )
% 13.19/7.58 => ( ( B != zero_zero_real )
% 13.19/7.58 => ( ( times_times_real @ A @ B )
% 13.19/7.58 != zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1643_no__zero__divisors,axiom,
% 13.19/7.58 ! [A: rat,B: rat] :
% 13.19/7.58 ( ( A != zero_zero_rat )
% 13.19/7.58 => ( ( B != zero_zero_rat )
% 13.19/7.58 => ( ( times_times_rat @ A @ B )
% 13.19/7.58 != zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1644_no__zero__divisors,axiom,
% 13.19/7.58 ! [A: nat,B: nat] :
% 13.19/7.58 ( ( A != zero_zero_nat )
% 13.19/7.58 => ( ( B != zero_zero_nat )
% 13.19/7.58 => ( ( times_times_nat @ A @ B )
% 13.19/7.58 != zero_zero_nat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1645_no__zero__divisors,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( A != zero_zero_int )
% 13.19/7.58 => ( ( B != zero_zero_int )
% 13.19/7.58 => ( ( times_times_int @ A @ B )
% 13.19/7.58 != zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1646_mult__left__cancel,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( C != zero_zero_complex )
% 13.19/7.58 => ( ( ( times_times_complex @ C @ A )
% 13.19/7.58 = ( times_times_complex @ C @ B ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1647_mult__left__cancel,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( C != zero_zero_real )
% 13.19/7.58 => ( ( ( times_times_real @ C @ A )
% 13.19/7.58 = ( times_times_real @ C @ B ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1648_mult__left__cancel,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( C != zero_zero_rat )
% 13.19/7.58 => ( ( ( times_times_rat @ C @ A )
% 13.19/7.58 = ( times_times_rat @ C @ B ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1649_mult__left__cancel,axiom,
% 13.19/7.58 ! [C: nat,A: nat,B: nat] :
% 13.19/7.58 ( ( C != zero_zero_nat )
% 13.19/7.58 => ( ( ( times_times_nat @ C @ A )
% 13.19/7.58 = ( times_times_nat @ C @ B ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1650_mult__left__cancel,axiom,
% 13.19/7.58 ! [C: int,A: int,B: int] :
% 13.19/7.58 ( ( C != zero_zero_int )
% 13.19/7.58 => ( ( ( times_times_int @ C @ A )
% 13.19/7.58 = ( times_times_int @ C @ B ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1651_mult__right__cancel,axiom,
% 13.19/7.58 ! [C: complex,A: complex,B: complex] :
% 13.19/7.58 ( ( C != zero_zero_complex )
% 13.19/7.58 => ( ( ( times_times_complex @ A @ C )
% 13.19/7.58 = ( times_times_complex @ B @ C ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1652_mult__right__cancel,axiom,
% 13.19/7.58 ! [C: real,A: real,B: real] :
% 13.19/7.58 ( ( C != zero_zero_real )
% 13.19/7.58 => ( ( ( times_times_real @ A @ C )
% 13.19/7.58 = ( times_times_real @ B @ C ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1653_mult__right__cancel,axiom,
% 13.19/7.58 ! [C: rat,A: rat,B: rat] :
% 13.19/7.58 ( ( C != zero_zero_rat )
% 13.19/7.58 => ( ( ( times_times_rat @ A @ C )
% 13.19/7.58 = ( times_times_rat @ B @ C ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1654_mult__right__cancel,axiom,
% 13.19/7.58 ! [C: nat,A: nat,B: nat] :
% 13.19/7.58 ( ( C != zero_zero_nat )
% 13.19/7.58 => ( ( ( times_times_nat @ A @ C )
% 13.19/7.58 = ( times_times_nat @ B @ C ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1655_mult__right__cancel,axiom,
% 13.19/7.58 ! [C: int,A: int,B: int] :
% 13.19/7.58 ( ( C != zero_zero_int )
% 13.19/7.58 => ( ( ( times_times_int @ A @ C )
% 13.19/7.58 = ( times_times_int @ B @ C ) )
% 13.19/7.58 = ( A = B ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1656_zero__neq__one,axiom,
% 13.19/7.58 zero_zero_complex != one_one_complex ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1657_zero__neq__one,axiom,
% 13.19/7.58 zero_zero_real != one_one_real ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1658_zero__neq__one,axiom,
% 13.19/7.58 zero_zero_rat != one_one_rat ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1659_zero__neq__one,axiom,
% 13.19/7.58 zero_zero_nat != one_one_nat ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1660_zero__neq__one,axiom,
% 13.19/7.58 zero_zero_int != one_one_int ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1661_comm__monoid__add__class_Oadd__0,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1662_comm__monoid__add__class_Oadd__0,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( plus_plus_real @ zero_zero_real @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1663_comm__monoid__add__class_Oadd__0,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1664_comm__monoid__add__class_Oadd__0,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ zero_zero_nat @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1665_comm__monoid__add__class_Oadd__0,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( plus_plus_int @ zero_zero_int @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1666_add_Ocomm__neutral,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1667_add_Ocomm__neutral,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( plus_plus_real @ A @ zero_zero_real )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1668_add_Ocomm__neutral,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1669_add_Ocomm__neutral,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1670_add_Ocomm__neutral,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( plus_plus_int @ A @ zero_zero_int )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1671_add_Ogroup__left__neutral,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( plus_plus_complex @ zero_zero_complex @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1672_add_Ogroup__left__neutral,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( plus_plus_real @ zero_zero_real @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1673_add_Ogroup__left__neutral,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( plus_plus_rat @ zero_zero_rat @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1674_add_Ogroup__left__neutral,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( plus_plus_int @ zero_zero_int @ A )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1675_verit__sum__simplify,axiom,
% 13.19/7.58 ! [A: complex] :
% 13.19/7.58 ( ( plus_plus_complex @ A @ zero_zero_complex )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1676_verit__sum__simplify,axiom,
% 13.19/7.58 ! [A: real] :
% 13.19/7.58 ( ( plus_plus_real @ A @ zero_zero_real )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1677_verit__sum__simplify,axiom,
% 13.19/7.58 ! [A: rat] :
% 13.19/7.58 ( ( plus_plus_rat @ A @ zero_zero_rat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1678_verit__sum__simplify,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ A @ zero_zero_nat )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1679_verit__sum__simplify,axiom,
% 13.19/7.58 ! [A: int] :
% 13.19/7.58 ( ( plus_plus_int @ A @ zero_zero_int )
% 13.19/7.58 = A ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1680_eq__iff__diff__eq__0,axiom,
% 13.19/7.58 ( ( ^ [Y6: complex,Z4: complex] : ( Y6 = Z4 ) )
% 13.19/7.58 = ( ^ [A3: complex,B4: complex] :
% 13.19/7.58 ( ( minus_minus_complex @ A3 @ B4 )
% 13.19/7.58 = zero_zero_complex ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1681_eq__iff__diff__eq__0,axiom,
% 13.19/7.58 ( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
% 13.19/7.58 = ( ^ [A3: real,B4: real] :
% 13.19/7.58 ( ( minus_minus_real @ A3 @ B4 )
% 13.19/7.58 = zero_zero_real ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1682_eq__iff__diff__eq__0,axiom,
% 13.19/7.58 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 13.19/7.58 = ( ^ [A3: rat,B4: rat] :
% 13.19/7.58 ( ( minus_minus_rat @ A3 @ B4 )
% 13.19/7.58 = zero_zero_rat ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1683_eq__iff__diff__eq__0,axiom,
% 13.19/7.58 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.19/7.58 = ( ^ [A3: int,B4: int] :
% 13.19/7.58 ( ( minus_minus_int @ A3 @ B4 )
% 13.19/7.58 = zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1684_power__not__zero,axiom,
% 13.19/7.58 ! [A: rat,N: nat] :
% 13.19/7.58 ( ( A != zero_zero_rat )
% 13.19/7.58 => ( ( power_power_rat @ A @ N )
% 13.19/7.58 != zero_zero_rat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1685_power__not__zero,axiom,
% 13.19/7.58 ! [A: nat,N: nat] :
% 13.19/7.58 ( ( A != zero_zero_nat )
% 13.19/7.58 => ( ( power_power_nat @ A @ N )
% 13.19/7.58 != zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1686_power__not__zero,axiom,
% 13.19/7.58 ! [A: real,N: nat] :
% 13.19/7.58 ( ( A != zero_zero_real )
% 13.19/7.58 => ( ( power_power_real @ A @ N )
% 13.19/7.58 != zero_zero_real ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1687_power__not__zero,axiom,
% 13.19/7.58 ! [A: int,N: nat] :
% 13.19/7.58 ( ( A != zero_zero_int )
% 13.19/7.58 => ( ( power_power_int @ A @ N )
% 13.19/7.58 != zero_zero_int ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1688_power__not__zero,axiom,
% 13.19/7.58 ! [A: complex,N: nat] :
% 13.19/7.58 ( ( A != zero_zero_complex )
% 13.19/7.58 => ( ( power_power_complex @ A @ N )
% 13.19/7.58 != zero_zero_complex ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1689_bot__nat__0_Oextremum__strict,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1690_gr0I,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( N != zero_zero_nat )
% 13.19/7.58 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1691_not__gr0,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 13.19/7.58 = ( N = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1692_not__less0,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1693_less__zeroE,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1694_gr__implies__not0,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ord_less_nat @ M @ N )
% 13.19/7.58 => ( N != zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1695_infinite__descent0,axiom,
% 13.19/7.58 ! [P: nat > $o,N: nat] :
% 13.19/7.58 ( ( P @ zero_zero_nat )
% 13.19/7.58 => ( ! [N4: nat] :
% 13.19/7.58 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 13.19/7.58 => ( ~ ( P @ N4 )
% 13.19/7.58 => ? [M2: nat] :
% 13.19/7.58 ( ( ord_less_nat @ M2 @ N4 )
% 13.19/7.58 & ~ ( P @ M2 ) ) ) )
% 13.19/7.58 => ( P @ N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1696_less__eq__nat_Osimps_I1_J,axiom,
% 13.19/7.58 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1697_bot__nat__0_Oextremum__unique,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 13.19/7.58 = ( A = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1698_bot__nat__0_Oextremum__uniqueI,axiom,
% 13.19/7.58 ! [A: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 13.19/7.58 => ( A = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1699_le__0__eq,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 13.19/7.58 = ( N = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1700_plus__nat_Oadd__0,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 13.19/7.58 = N ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1701_add__eq__self__zero,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ( plus_plus_nat @ M @ N )
% 13.19/7.58 = M )
% 13.19/7.58 => ( N = zero_zero_nat ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1702_ex__min__if__finite,axiom,
% 13.19/7.58 ! [S2: set_real] :
% 13.19/7.58 ( ( finite_finite_real @ S2 )
% 13.19/7.58 => ( ( S2 != bot_bot_set_real )
% 13.19/7.58 => ? [X3: real] :
% 13.19/7.58 ( ( member_real @ X3 @ S2 )
% 13.19/7.58 & ~ ? [Xa2: real] :
% 13.19/7.58 ( ( member_real @ Xa2 @ S2 )
% 13.19/7.58 & ( ord_less_real @ Xa2 @ X3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1703_ex__min__if__finite,axiom,
% 13.19/7.58 ! [S2: set_rat] :
% 13.19/7.58 ( ( finite_finite_rat @ S2 )
% 13.19/7.58 => ( ( S2 != bot_bot_set_rat )
% 13.19/7.58 => ? [X3: rat] :
% 13.19/7.58 ( ( member_rat @ X3 @ S2 )
% 13.19/7.58 & ~ ? [Xa2: rat] :
% 13.19/7.58 ( ( member_rat @ Xa2 @ S2 )
% 13.19/7.58 & ( ord_less_rat @ Xa2 @ X3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1704_ex__min__if__finite,axiom,
% 13.19/7.58 ! [S2: set_num] :
% 13.19/7.58 ( ( finite_finite_num @ S2 )
% 13.19/7.58 => ( ( S2 != bot_bot_set_num )
% 13.19/7.58 => ? [X3: num] :
% 13.19/7.58 ( ( member_num @ X3 @ S2 )
% 13.19/7.58 & ~ ? [Xa2: num] :
% 13.19/7.58 ( ( member_num @ Xa2 @ S2 )
% 13.19/7.58 & ( ord_less_num @ Xa2 @ X3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1705_ex__min__if__finite,axiom,
% 13.19/7.58 ! [S2: set_nat] :
% 13.19/7.58 ( ( finite_finite_nat @ S2 )
% 13.19/7.58 => ( ( S2 != bot_bot_set_nat )
% 13.19/7.58 => ? [X3: nat] :
% 13.19/7.58 ( ( member_nat @ X3 @ S2 )
% 13.19/7.58 & ~ ? [Xa2: nat] :
% 13.19/7.58 ( ( member_nat @ Xa2 @ S2 )
% 13.19/7.58 & ( ord_less_nat @ Xa2 @ X3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1706_ex__min__if__finite,axiom,
% 13.19/7.58 ! [S2: set_int] :
% 13.19/7.58 ( ( finite_finite_int @ S2 )
% 13.19/7.58 => ( ( S2 != bot_bot_set_int )
% 13.19/7.58 => ? [X3: int] :
% 13.19/7.58 ( ( member_int @ X3 @ S2 )
% 13.19/7.58 & ~ ? [Xa2: int] :
% 13.19/7.58 ( ( member_int @ Xa2 @ S2 )
% 13.19/7.58 & ( ord_less_int @ Xa2 @ X3 ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1707_infinite__growing,axiom,
% 13.19/7.58 ! [X7: set_real] :
% 13.19/7.58 ( ( X7 != bot_bot_set_real )
% 13.19/7.58 => ( ! [X3: real] :
% 13.19/7.58 ( ( member_real @ X3 @ X7 )
% 13.19/7.58 => ? [Xa2: real] :
% 13.19/7.58 ( ( member_real @ Xa2 @ X7 )
% 13.19/7.58 & ( ord_less_real @ X3 @ Xa2 ) ) )
% 13.19/7.58 => ~ ( finite_finite_real @ X7 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1708_infinite__growing,axiom,
% 13.19/7.58 ! [X7: set_rat] :
% 13.19/7.58 ( ( X7 != bot_bot_set_rat )
% 13.19/7.58 => ( ! [X3: rat] :
% 13.19/7.58 ( ( member_rat @ X3 @ X7 )
% 13.19/7.58 => ? [Xa2: rat] :
% 13.19/7.58 ( ( member_rat @ Xa2 @ X7 )
% 13.19/7.58 & ( ord_less_rat @ X3 @ Xa2 ) ) )
% 13.19/7.58 => ~ ( finite_finite_rat @ X7 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1709_infinite__growing,axiom,
% 13.19/7.58 ! [X7: set_num] :
% 13.19/7.58 ( ( X7 != bot_bot_set_num )
% 13.19/7.58 => ( ! [X3: num] :
% 13.19/7.58 ( ( member_num @ X3 @ X7 )
% 13.19/7.58 => ? [Xa2: num] :
% 13.19/7.58 ( ( member_num @ Xa2 @ X7 )
% 13.19/7.58 & ( ord_less_num @ X3 @ Xa2 ) ) )
% 13.19/7.58 => ~ ( finite_finite_num @ X7 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1710_infinite__growing,axiom,
% 13.19/7.58 ! [X7: set_nat] :
% 13.19/7.58 ( ( X7 != bot_bot_set_nat )
% 13.19/7.58 => ( ! [X3: nat] :
% 13.19/7.58 ( ( member_nat @ X3 @ X7 )
% 13.19/7.58 => ? [Xa2: nat] :
% 13.19/7.58 ( ( member_nat @ Xa2 @ X7 )
% 13.19/7.58 & ( ord_less_nat @ X3 @ Xa2 ) ) )
% 13.19/7.58 => ~ ( finite_finite_nat @ X7 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1711_infinite__growing,axiom,
% 13.19/7.58 ! [X7: set_int] :
% 13.19/7.58 ( ( X7 != bot_bot_set_int )
% 13.19/7.58 => ( ! [X3: int] :
% 13.19/7.58 ( ( member_int @ X3 @ X7 )
% 13.19/7.58 => ? [Xa2: int] :
% 13.19/7.58 ( ( member_int @ Xa2 @ X7 )
% 13.19/7.58 & ( ord_less_int @ X3 @ Xa2 ) ) )
% 13.19/7.58 => ~ ( finite_finite_int @ X7 ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1712_minus__nat_Odiff__0,axiom,
% 13.19/7.58 ! [M: nat] :
% 13.19/7.58 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 13.19/7.58 = M ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1713_diffs0__imp__equal,axiom,
% 13.19/7.58 ! [M: nat,N: nat] :
% 13.19/7.58 ( ( ( minus_minus_nat @ M @ N )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 => ( ( ( minus_minus_nat @ N @ M )
% 13.19/7.58 = zero_zero_nat )
% 13.19/7.58 => ( M = N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1714_mult__0,axiom,
% 13.19/7.58 ! [N: nat] :
% 13.19/7.58 ( ( times_times_nat @ zero_zero_nat @ N )
% 13.19/7.58 = zero_zero_nat ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1715_nat__mult__eq__cancel__disj,axiom,
% 13.19/7.58 ! [K: nat,M: nat,N: nat] :
% 13.19/7.58 ( ( ( times_times_nat @ K @ M )
% 13.19/7.58 = ( times_times_nat @ K @ N ) )
% 13.19/7.58 = ( ( K = zero_zero_nat )
% 13.19/7.58 | ( M = N ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1716_div__pos__geq,axiom,
% 13.19/7.58 ! [L2: int,K: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ L2 )
% 13.19/7.58 => ( ( ord_less_eq_int @ L2 @ K )
% 13.19/7.58 => ( ( divide_divide_int @ K @ L2 )
% 13.19/7.58 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) @ one_one_int ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1717_int__div__less__self,axiom,
% 13.19/7.58 ! [X: int,K: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ X )
% 13.19/7.58 => ( ( ord_less_int @ one_one_int @ K )
% 13.19/7.58 => ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1718_nonneg1__imp__zdiv__pos__iff,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 13.19/7.58 = ( ( ord_less_eq_int @ B @ A )
% 13.19/7.58 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1719_pos__imp__zdiv__nonneg__iff,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.58 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1720_neg__imp__zdiv__nonneg__iff,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.58 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 13.19/7.58 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1721_pos__imp__zdiv__pos__iff,axiom,
% 13.19/7.58 ! [K: int,I: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ K )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 13.19/7.58 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1722_pos__imp__zdiv__neg__iff,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.58 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 13.19/7.58 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1723_neg__imp__zdiv__neg__iff,axiom,
% 13.19/7.58 ! [B: int,A: int] :
% 13.19/7.58 ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.58 => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 13.19/7.58 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1724_div__nonpos__pos__le0,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.58 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1725_div__nonneg__neg__le0,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.58 => ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.58 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1726_div__neg__pos__less0,axiom,
% 13.19/7.58 ! [A: int,B: int] :
% 13.19/7.58 ( ( ord_less_int @ A @ zero_zero_int )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.58 => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1727_div__positive__int,axiom,
% 13.19/7.58 ! [L2: int,K: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ L2 @ K )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ L2 )
% 13.19/7.58 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1728_div__int__pos__iff,axiom,
% 13.19/7.58 ! [K: int,L2: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L2 ) )
% 13.19/7.58 = ( ( K = zero_zero_int )
% 13.19/7.58 | ( L2 = zero_zero_int )
% 13.19/7.58 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.19/7.58 & ( ord_less_eq_int @ zero_zero_int @ L2 ) )
% 13.19/7.58 | ( ( ord_less_int @ K @ zero_zero_int )
% 13.19/7.58 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1729_zdiv__mono2__neg,axiom,
% 13.19/7.58 ! [A: int,B2: int,B: int] :
% 13.19/7.58 ( ( ord_less_int @ A @ zero_zero_int )
% 13.19/7.58 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 13.19/7.58 => ( ( ord_less_eq_int @ B2 @ B )
% 13.19/7.58 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1730_zdiv__mono1__neg,axiom,
% 13.19/7.58 ! [A: int,A4: int,B: int] :
% 13.19/7.58 ( ( ord_less_eq_int @ A @ A4 )
% 13.19/7.58 => ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.58 => ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.19/7.58
% 13.19/7.58 thf(fact_1731_zdiv__eq__0__iff,axiom,
% 13.19/7.58 ! [I: int,K: int] :
% 13.19/7.58 ( ( ( divide_divide_int @ I @ K )
% 13.19/7.58 = zero_zero_int )
% 13.19/7.58 = ( ( K = zero_zero_int )
% 13.19/7.58 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 13.19/7.59 & ( ord_less_int @ I @ K ) )
% 13.19/7.59 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 13.19/7.59 & ( ord_less_int @ K @ I ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1732_zdiv__mono2,axiom,
% 13.19/7.59 ! [A: int,B2: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 13.19/7.59 => ( ( ord_less_eq_int @ B2 @ B )
% 13.19/7.59 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1733_zdiv__mono1,axiom,
% 13.19/7.59 ! [A: int,A4: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ A4 )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1734_split__zdiv,axiom,
% 13.19/7.59 ! [P: int > $o,N: int,K: int] :
% 13.19/7.59 ( ( P @ ( divide_divide_int @ N @ K ) )
% 13.19/7.59 = ( ( ( K = zero_zero_int )
% 13.19/7.59 => ( P @ zero_zero_int ) )
% 13.19/7.59 & ( ( ord_less_int @ zero_zero_int @ K )
% 13.19/7.59 => ! [I4: int,J3: int] :
% 13.19/7.59 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 13.19/7.59 & ( ord_less_int @ J3 @ K )
% 13.19/7.59 & ( N
% 13.19/7.59 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 13.19/7.59 => ( P @ I4 ) ) )
% 13.19/7.59 & ( ( ord_less_int @ K @ zero_zero_int )
% 13.19/7.59 => ! [I4: int,J3: int] :
% 13.19/7.59 ( ( ( ord_less_int @ K @ J3 )
% 13.19/7.59 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 13.19/7.59 & ( N
% 13.19/7.59 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 13.19/7.59 => ( P @ I4 ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1735_int__div__neg__eq,axiom,
% 13.19/7.59 ! [A: int,B: int,Q2: int,R: int] :
% 13.19/7.59 ( ( A
% 13.19/7.59 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 13.19/7.59 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_int @ B @ R )
% 13.19/7.59 => ( ( divide_divide_int @ A @ B )
% 13.19/7.59 = Q2 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1736_int__div__pos__eq,axiom,
% 13.19/7.59 ! [A: int,B: int,Q2: int,R: int] :
% 13.19/7.59 ( ( A
% 13.19/7.59 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 13.19/7.59 => ( ( ord_less_int @ R @ B )
% 13.19/7.59 => ( ( divide_divide_int @ A @ B )
% 13.19/7.59 = Q2 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1737_zdiv__zmult2__eq,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 13.19/7.59 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1738_neg__zmod__mult__2,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( 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 ) )
% 13.19/7.59 = ( 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 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1739_pos__zmod__mult__2,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( 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 ) )
% 13.19/7.59 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1740_enat__0__less__mult__iff,axiom,
% 13.19/7.59 ! [M: extended_enat,N: extended_enat] :
% 13.19/7.59 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 13.19/7.59 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 13.19/7.59 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1741_not__iless0,axiom,
% 13.19/7.59 ! [N: extended_enat] :
% 13.19/7.59 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1742_iadd__is__0,axiom,
% 13.19/7.59 ! [M: extended_enat,N: extended_enat] :
% 13.19/7.59 ( ( ( plus_p3455044024723400733d_enat @ M @ N )
% 13.19/7.59 = zero_z5237406670263579293d_enat )
% 13.19/7.59 = ( ( M = zero_z5237406670263579293d_enat )
% 13.19/7.59 & ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1743_ile0__eq,axiom,
% 13.19/7.59 ! [N: extended_enat] :
% 13.19/7.59 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 13.19/7.59 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1744_i0__lb,axiom,
% 13.19/7.59 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1745_power__eq__iff__eq__base,axiom,
% 13.19/7.59 ! [N: nat,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ( ( power_power_real @ A @ N )
% 13.19/7.59 = ( power_power_real @ B @ N ) )
% 13.19/7.59 = ( A = B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1746_power__eq__iff__eq__base,axiom,
% 13.19/7.59 ! [N: nat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ( ( power_power_rat @ A @ N )
% 13.19/7.59 = ( power_power_rat @ B @ N ) )
% 13.19/7.59 = ( A = B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1747_power__eq__iff__eq__base,axiom,
% 13.19/7.59 ! [N: nat,A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ( ( power_power_nat @ A @ N )
% 13.19/7.59 = ( power_power_nat @ B @ N ) )
% 13.19/7.59 = ( A = B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1748_power__eq__iff__eq__base,axiom,
% 13.19/7.59 ! [N: nat,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ( ( power_power_int @ A @ N )
% 13.19/7.59 = ( power_power_int @ B @ N ) )
% 13.19/7.59 = ( A = B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1749_power__eq__imp__eq__base,axiom,
% 13.19/7.59 ! [A: real,N: nat,B: real] :
% 13.19/7.59 ( ( ( power_power_real @ A @ N )
% 13.19/7.59 = ( power_power_real @ B @ N ) )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( A = B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1750_power__eq__imp__eq__base,axiom,
% 13.19/7.59 ! [A: rat,N: nat,B: rat] :
% 13.19/7.59 ( ( ( power_power_rat @ A @ N )
% 13.19/7.59 = ( power_power_rat @ B @ N ) )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( A = B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1751_power__eq__imp__eq__base,axiom,
% 13.19/7.59 ! [A: nat,N: nat,B: nat] :
% 13.19/7.59 ( ( ( power_power_nat @ A @ N )
% 13.19/7.59 = ( power_power_nat @ B @ N ) )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( A = B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1752_power__eq__imp__eq__base,axiom,
% 13.19/7.59 ! [A: int,N: nat,B: int] :
% 13.19/7.59 ( ( ( power_power_int @ A @ N )
% 13.19/7.59 = ( power_power_int @ B @ N ) )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( A = B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1753_realpow__pos__nth__unique,axiom,
% 13.19/7.59 ! [N: nat,A: real] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ? [X3: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ X3 )
% 13.19/7.59 & ( ( power_power_real @ X3 @ N )
% 13.19/7.59 = A )
% 13.19/7.59 & ! [Y5: real] :
% 13.19/7.59 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 13.19/7.59 & ( ( power_power_real @ Y5 @ N )
% 13.19/7.59 = A ) )
% 13.19/7.59 => ( Y5 = X3 ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1754_realpow__pos__nth,axiom,
% 13.19/7.59 ! [N: nat,A: real] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ? [R2: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ R2 )
% 13.19/7.59 & ( ( power_power_real @ R2 @ N )
% 13.19/7.59 = A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1755_power__strict__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,N: nat] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1756_power__strict__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,N: nat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1757_power__strict__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1758_power__strict__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,N: nat] :
% 13.19/7.59 ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1759_dbl__def,axiom,
% 13.19/7.59 ( neg_numeral_dbl_real
% 13.19/7.59 = ( ^ [X2: real] : ( plus_plus_real @ X2 @ X2 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1760_dbl__def,axiom,
% 13.19/7.59 ( neg_numeral_dbl_rat
% 13.19/7.59 = ( ^ [X2: rat] : ( plus_plus_rat @ X2 @ X2 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1761_dbl__def,axiom,
% 13.19/7.59 ( neg_numeral_dbl_int
% 13.19/7.59 = ( ^ [X2: int] : ( plus_plus_int @ X2 @ X2 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1762_zero__le__numeral,axiom,
% 13.19/7.59 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1763_zero__le__numeral,axiom,
% 13.19/7.59 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1764_zero__le__numeral,axiom,
% 13.19/7.59 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1765_zero__le__numeral,axiom,
% 13.19/7.59 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1766_not__numeral__le__zero,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1767_not__numeral__le__zero,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1768_not__numeral__le__zero,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1769_not__numeral__le__zero,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1770_mult__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1771_mult__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1772_mult__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1773_mult__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1774_mult__mono_H,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1775_mult__mono_H,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1776_mult__mono_H,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1777_mult__mono_H,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1778_zero__le__square,axiom,
% 13.19/7.59 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1779_zero__le__square,axiom,
% 13.19/7.59 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1780_zero__le__square,axiom,
% 13.19/7.59 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1781_split__mult__pos__le,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 13.19/7.59 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1782_split__mult__pos__le,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 13.19/7.59 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1783_split__mult__pos__le,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 13.19/7.59 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 13.19/7.59 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1784_mult__left__mono__neg,axiom,
% 13.19/7.59 ! [B: real,A: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ B @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1785_mult__left__mono__neg,axiom,
% 13.19/7.59 ! [B: rat,A: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ B @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1786_mult__left__mono__neg,axiom,
% 13.19/7.59 ! [B: int,A: int,C: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ B @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1787_mult__nonpos__nonpos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1788_mult__nonpos__nonpos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1789_mult__nonpos__nonpos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1790_mult__left__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1791_mult__left__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1792_mult__left__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1793_mult__left__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1794_mult__right__mono__neg,axiom,
% 13.19/7.59 ! [B: real,A: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ B @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1795_mult__right__mono__neg,axiom,
% 13.19/7.59 ! [B: rat,A: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ B @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1796_mult__right__mono__neg,axiom,
% 13.19/7.59 ! [B: int,A: int,C: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ B @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1797_mult__right__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1798_mult__right__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1799_mult__right__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1800_mult__right__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1801_mult__le__0__iff,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 13.19/7.59 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1802_mult__le__0__iff,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 13.19/7.59 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1803_mult__le__0__iff,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 13.19/7.59 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 13.19/7.59 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1804_split__mult__neg__le,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 13.19/7.59 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1805_split__mult__neg__le,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 13.19/7.59 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1806_split__mult__neg__le,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 13.19/7.59 | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 13.19/7.59 & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1807_split__mult__neg__le,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 13.19/7.59 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1808_mult__nonneg__nonneg,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1809_mult__nonneg__nonneg,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1810_mult__nonneg__nonneg,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1811_mult__nonneg__nonneg,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1812_mult__nonneg__nonpos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1813_mult__nonneg__nonpos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1814_mult__nonneg__nonpos,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1815_mult__nonneg__nonpos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1816_mult__nonpos__nonneg,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1817_mult__nonpos__nonneg,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1818_mult__nonpos__nonneg,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1819_mult__nonpos__nonneg,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1820_mult__nonneg__nonpos2,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1821_mult__nonneg__nonpos2,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1822_mult__nonneg__nonpos2,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1823_mult__nonneg__nonpos2,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1824_zero__le__mult__iff,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 13.19/7.59 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1825_zero__le__mult__iff,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 13.19/7.59 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1826_zero__le__mult__iff,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 13.19/7.59 | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1827_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1828_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1829_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1830_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1831_zero__less__numeral,axiom,
% 13.19/7.59 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1832_zero__less__numeral,axiom,
% 13.19/7.59 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1833_zero__less__numeral,axiom,
% 13.19/7.59 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1834_zero__less__numeral,axiom,
% 13.19/7.59 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1835_not__numeral__less__zero,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1836_not__numeral__less__zero,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1837_not__numeral__less__zero,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1838_not__numeral__less__zero,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1839_not__one__le__zero,axiom,
% 13.19/7.59 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1840_not__one__le__zero,axiom,
% 13.19/7.59 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1841_not__one__le__zero,axiom,
% 13.19/7.59 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1842_not__one__le__zero,axiom,
% 13.19/7.59 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1843_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 13.19/7.59 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1844_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 13.19/7.59 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1845_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 13.19/7.59 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1846_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 13.19/7.59 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1847_zero__less__one__class_Ozero__le__one,axiom,
% 13.19/7.59 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1848_zero__less__one__class_Ozero__le__one,axiom,
% 13.19/7.59 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1849_zero__less__one__class_Ozero__le__one,axiom,
% 13.19/7.59 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1850_zero__less__one__class_Ozero__le__one,axiom,
% 13.19/7.59 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1851_add__nonpos__eq__0__iff,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 13.19/7.59 => ( ( ( plus_plus_real @ X @ Y3 )
% 13.19/7.59 = zero_zero_real )
% 13.19/7.59 = ( ( X = zero_zero_real )
% 13.19/7.59 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1852_add__nonpos__eq__0__iff,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 13.19/7.59 => ( ( ( plus_plus_rat @ X @ Y3 )
% 13.19/7.59 = zero_zero_rat )
% 13.19/7.59 = ( ( X = zero_zero_rat )
% 13.19/7.59 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1853_add__nonpos__eq__0__iff,axiom,
% 13.19/7.59 ! [X: nat,Y3: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ X @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 13.19/7.59 => ( ( ( plus_plus_nat @ X @ Y3 )
% 13.19/7.59 = zero_zero_nat )
% 13.19/7.59 = ( ( X = zero_zero_nat )
% 13.19/7.59 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1854_add__nonpos__eq__0__iff,axiom,
% 13.19/7.59 ! [X: int,Y3: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ X @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 13.19/7.59 => ( ( ( plus_plus_int @ X @ Y3 )
% 13.19/7.59 = zero_zero_int )
% 13.19/7.59 = ( ( X = zero_zero_int )
% 13.19/7.59 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1855_add__nonneg__eq__0__iff,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ( ( plus_plus_real @ X @ Y3 )
% 13.19/7.59 = zero_zero_real )
% 13.19/7.59 = ( ( X = zero_zero_real )
% 13.19/7.59 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1856_add__nonneg__eq__0__iff,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ( ( plus_plus_rat @ X @ Y3 )
% 13.19/7.59 = zero_zero_rat )
% 13.19/7.59 = ( ( X = zero_zero_rat )
% 13.19/7.59 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1857_add__nonneg__eq__0__iff,axiom,
% 13.19/7.59 ! [X: nat,Y3: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 13.19/7.59 => ( ( ( plus_plus_nat @ X @ Y3 )
% 13.19/7.59 = zero_zero_nat )
% 13.19/7.59 = ( ( X = zero_zero_nat )
% 13.19/7.59 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1858_add__nonneg__eq__0__iff,axiom,
% 13.19/7.59 ! [X: int,Y3: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.19/7.59 => ( ( ( plus_plus_int @ X @ Y3 )
% 13.19/7.59 = zero_zero_int )
% 13.19/7.59 = ( ( X = zero_zero_int )
% 13.19/7.59 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1859_add__nonpos__nonpos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1860_add__nonpos__nonpos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1861_add__nonpos__nonpos,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 13.19/7.59 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1862_add__nonpos__nonpos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1863_add__nonneg__nonneg,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1864_add__nonneg__nonneg,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1865_add__nonneg__nonneg,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1866_add__nonneg__nonneg,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1867_add__increasing2,axiom,
% 13.19/7.59 ! [C: real,B: real,A: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ A )
% 13.19/7.59 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1868_add__increasing2,axiom,
% 13.19/7.59 ! [C: rat,B: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ A )
% 13.19/7.59 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1869_add__increasing2,axiom,
% 13.19/7.59 ! [C: nat,B: nat,A: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ A )
% 13.19/7.59 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1870_add__increasing2,axiom,
% 13.19/7.59 ! [C: int,B: int,A: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ A )
% 13.19/7.59 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1871_add__decreasing2,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1872_add__decreasing2,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1873_add__decreasing2,axiom,
% 13.19/7.59 ! [C: nat,A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.59 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1874_add__decreasing2,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ A @ B )
% 13.19/7.59 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1875_add__increasing,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ C )
% 13.19/7.59 => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1876_add__increasing,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1877_add__increasing,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ C )
% 13.19/7.59 => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1878_add__increasing,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ C )
% 13.19/7.59 => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1879_add__decreasing,axiom,
% 13.19/7.59 ! [A: real,C: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ C @ B )
% 13.19/7.59 => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1880_add__decreasing,axiom,
% 13.19/7.59 ! [A: rat,C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ C @ B )
% 13.19/7.59 => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1881_add__decreasing,axiom,
% 13.19/7.59 ! [A: nat,C: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_eq_nat @ C @ B )
% 13.19/7.59 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1882_add__decreasing,axiom,
% 13.19/7.59 ! [A: int,C: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ C @ B )
% 13.19/7.59 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1883_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1884_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1885_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1886_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1887_mult__less__cancel__right__disj,axiom,
% 13.19/7.59 ! [A: real,C: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 & ( ord_less_real @ A @ B ) )
% 13.19/7.59 | ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 & ( ord_less_real @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1888_mult__less__cancel__right__disj,axiom,
% 13.19/7.59 ! [A: rat,C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 & ( ord_less_rat @ A @ B ) )
% 13.19/7.59 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1889_mult__less__cancel__right__disj,axiom,
% 13.19/7.59 ! [A: int,C: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 & ( ord_less_int @ A @ B ) )
% 13.19/7.59 | ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.59 & ( ord_less_int @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1890_mult__strict__right__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1891_mult__strict__right__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1892_mult__strict__right__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1893_mult__strict__right__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1894_mult__strict__right__mono__neg,axiom,
% 13.19/7.59 ! [B: real,A: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ B @ A )
% 13.19/7.59 => ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1895_mult__strict__right__mono__neg,axiom,
% 13.19/7.59 ! [B: rat,A: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ B @ A )
% 13.19/7.59 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1896_mult__strict__right__mono__neg,axiom,
% 13.19/7.59 ! [B: int,A: int,C: int] :
% 13.19/7.59 ( ( ord_less_int @ B @ A )
% 13.19/7.59 => ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1897_mult__less__cancel__left__disj,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 & ( ord_less_real @ A @ B ) )
% 13.19/7.59 | ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 & ( ord_less_real @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1898_mult__less__cancel__left__disj,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 & ( ord_less_rat @ A @ B ) )
% 13.19/7.59 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1899_mult__less__cancel__left__disj,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 & ( ord_less_int @ A @ B ) )
% 13.19/7.59 | ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.59 & ( ord_less_int @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1900_mult__strict__left__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1901_mult__strict__left__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1902_mult__strict__left__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1903_mult__strict__left__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1904_mult__strict__left__mono__neg,axiom,
% 13.19/7.59 ! [B: real,A: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ B @ A )
% 13.19/7.59 => ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1905_mult__strict__left__mono__neg,axiom,
% 13.19/7.59 ! [B: rat,A: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ B @ A )
% 13.19/7.59 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1906_mult__strict__left__mono__neg,axiom,
% 13.19/7.59 ! [B: int,A: int,C: int] :
% 13.19/7.59 ( ( ord_less_int @ B @ A )
% 13.19/7.59 => ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1907_mult__less__cancel__left__pos,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 = ( ord_less_real @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1908_mult__less__cancel__left__pos,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 = ( ord_less_rat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1909_mult__less__cancel__left__pos,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 = ( ord_less_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1910_mult__less__cancel__left__neg,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 = ( ord_less_real @ B @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1911_mult__less__cancel__left__neg,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 = ( ord_less_rat @ B @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1912_mult__less__cancel__left__neg,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 = ( ord_less_int @ B @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1913_zero__less__mult__pos2,axiom,
% 13.19/7.59 ! [B: real,A: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1914_zero__less__mult__pos2,axiom,
% 13.19/7.59 ! [B: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1915_zero__less__mult__pos2,axiom,
% 13.19/7.59 ! [B: nat,A: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1916_zero__less__mult__pos2,axiom,
% 13.19/7.59 ! [B: int,A: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1917_zero__less__mult__pos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1918_zero__less__mult__pos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1919_zero__less__mult__pos,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1920_zero__less__mult__pos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1921_zero__less__mult__iff,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_real @ zero_zero_real @ B ) )
% 13.19/7.59 | ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1922_zero__less__mult__iff,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 13.19/7.59 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1923_zero__less__mult__iff,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 & ( ord_less_int @ zero_zero_int @ B ) )
% 13.19/7.59 | ( ( ord_less_int @ A @ zero_zero_int )
% 13.19/7.59 & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1924_mult__pos__neg2,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1925_mult__pos__neg2,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1926_mult__pos__neg2,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1927_mult__pos__neg2,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1928_mult__pos__pos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1929_mult__pos__pos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1930_mult__pos__pos,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1931_mult__pos__pos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1932_mult__pos__neg,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1933_mult__pos__neg,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1934_mult__pos__neg,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1935_mult__pos__neg,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1936_mult__neg__pos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1937_mult__neg__pos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1938_mult__neg__pos,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1939_mult__neg__pos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1940_mult__less__0__iff,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_real @ B @ zero_zero_real ) )
% 13.19/7.59 | ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1941_mult__less__0__iff,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 13.19/7.59 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1942_mult__less__0__iff,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 13.19/7.59 = ( ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 & ( ord_less_int @ B @ zero_zero_int ) )
% 13.19/7.59 | ( ( ord_less_int @ A @ zero_zero_int )
% 13.19/7.59 & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1943_not__square__less__zero,axiom,
% 13.19/7.59 ! [A: real] :
% 13.19/7.59 ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1944_not__square__less__zero,axiom,
% 13.19/7.59 ! [A: rat] :
% 13.19/7.59 ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1945_not__square__less__zero,axiom,
% 13.19/7.59 ! [A: int] :
% 13.19/7.59 ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1946_mult__neg__neg,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1947_mult__neg__neg,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1948_mult__neg__neg,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1949_le__iff__diff__le__0,axiom,
% 13.19/7.59 ( ord_less_eq_real
% 13.19/7.59 = ( ^ [A3: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1950_le__iff__diff__le__0,axiom,
% 13.19/7.59 ( ord_less_eq_rat
% 13.19/7.59 = ( ^ [A3: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B4 ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1951_le__iff__diff__le__0,axiom,
% 13.19/7.59 ( ord_less_eq_int
% 13.19/7.59 = ( ^ [A3: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1952_zero__less__one,axiom,
% 13.19/7.59 ord_less_real @ zero_zero_real @ one_one_real ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1953_zero__less__one,axiom,
% 13.19/7.59 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1954_zero__less__one,axiom,
% 13.19/7.59 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1955_zero__less__one,axiom,
% 13.19/7.59 ord_less_int @ zero_zero_int @ one_one_int ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1956_not__one__less__zero,axiom,
% 13.19/7.59 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1957_not__one__less__zero,axiom,
% 13.19/7.59 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1958_not__one__less__zero,axiom,
% 13.19/7.59 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1959_not__one__less__zero,axiom,
% 13.19/7.59 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1960_less__numeral__extra_I1_J,axiom,
% 13.19/7.59 ord_less_real @ zero_zero_real @ one_one_real ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1961_less__numeral__extra_I1_J,axiom,
% 13.19/7.59 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1962_less__numeral__extra_I1_J,axiom,
% 13.19/7.59 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1963_less__numeral__extra_I1_J,axiom,
% 13.19/7.59 ord_less_int @ zero_zero_int @ one_one_int ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1964_add__neg__neg,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1965_add__neg__neg,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1966_add__neg__neg,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 13.19/7.59 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1967_add__neg__neg,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1968_add__pos__pos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1969_add__pos__pos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1970_add__pos__pos,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1971_add__pos__pos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1972_canonically__ordered__monoid__add__class_OlessE,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ~ ! [C2: nat] :
% 13.19/7.59 ( ( B
% 13.19/7.59 = ( plus_plus_nat @ A @ C2 ) )
% 13.19/7.59 => ( C2 = zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1973_pos__add__strict,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_real @ B @ C )
% 13.19/7.59 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1974_pos__add__strict,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_rat @ B @ C )
% 13.19/7.59 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1975_pos__add__strict,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ B @ C )
% 13.19/7.59 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1976_pos__add__strict,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ B @ C )
% 13.19/7.59 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1977_add__less__zeroD,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ X @ zero_zero_real )
% 13.19/7.59 | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1978_add__less__zeroD,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y3 ) @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ X @ zero_zero_rat )
% 13.19/7.59 | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1979_add__less__zeroD,axiom,
% 13.19/7.59 ! [X: int,Y3: int] :
% 13.19/7.59 ( ( ord_less_int @ ( plus_plus_int @ X @ Y3 ) @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_int @ X @ zero_zero_int )
% 13.19/7.59 | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1980_divide__le__0__iff,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 13.19/7.59 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1981_divide__le__0__iff,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 13.19/7.59 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1982_divide__right__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1983_divide__right__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1984_zero__le__divide__iff,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 13.19/7.59 | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1985_zero__le__divide__iff,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 13.19/7.59 | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1986_divide__nonneg__nonneg,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1987_divide__nonneg__nonneg,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1988_divide__nonneg__nonpos,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1989_divide__nonneg__nonpos,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1990_divide__nonpos__nonneg,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1991_divide__nonpos__nonneg,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1992_divide__nonpos__nonpos,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1993_divide__nonpos__nonpos,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1994_divide__right__mono__neg,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1995_divide__right__mono__neg,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1996_less__iff__diff__less__0,axiom,
% 13.19/7.59 ( ord_less_real
% 13.19/7.59 = ( ^ [A3: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B4 ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1997_less__iff__diff__less__0,axiom,
% 13.19/7.59 ( ord_less_rat
% 13.19/7.59 = ( ^ [A3: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B4 ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1998_less__iff__diff__less__0,axiom,
% 13.19/7.59 ( ord_less_int
% 13.19/7.59 = ( ^ [A3: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B4 ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_1999_divide__strict__right__mono__neg,axiom,
% 13.19/7.59 ! [B: real,A: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ B @ A )
% 13.19/7.59 => ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2000_divide__strict__right__mono__neg,axiom,
% 13.19/7.59 ! [B: rat,A: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ B @ A )
% 13.19/7.59 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2001_divide__strict__right__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2002_divide__strict__right__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2003_zero__less__divide__iff,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_real @ zero_zero_real @ B ) )
% 13.19/7.59 | ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2004_zero__less__divide__iff,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_rat @ zero_zero_rat @ B ) )
% 13.19/7.59 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2005_divide__less__cancel,axiom,
% 13.19/7.59 ! [A: real,C: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ B @ A ) )
% 13.19/7.59 & ( C != zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2006_divide__less__cancel,axiom,
% 13.19/7.59 ! [A: rat,C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ B @ A ) )
% 13.19/7.59 & ( C != zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2007_divide__less__0__iff,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_real @ B @ zero_zero_real ) )
% 13.19/7.59 | ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2008_divide__less__0__iff,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_rat @ B @ zero_zero_rat ) )
% 13.19/7.59 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2009_divide__pos__pos,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2010_divide__pos__pos,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2011_divide__pos__neg,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2012_divide__pos__neg,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2013_divide__neg__pos,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_real @ X @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2014_divide__neg__pos,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_rat @ X @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2015_divide__neg__neg,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_real @ X @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2016_divide__neg__neg,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_rat @ X @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2017_power__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2018_power__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2019_power__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2020_power__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2021_zero__le__power,axiom,
% 13.19/7.59 ! [A: real,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2022_zero__le__power,axiom,
% 13.19/7.59 ! [A: rat,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2023_zero__le__power,axiom,
% 13.19/7.59 ! [A: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2024_zero__le__power,axiom,
% 13.19/7.59 ! [A: int,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2025_zero__less__power,axiom,
% 13.19/7.59 ! [A: real,N: nat] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2026_zero__less__power,axiom,
% 13.19/7.59 ! [A: rat,N: nat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2027_zero__less__power,axiom,
% 13.19/7.59 ! [A: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2028_zero__less__power,axiom,
% 13.19/7.59 ! [A: int,N: nat] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2029_frac__eq__eq,axiom,
% 13.19/7.59 ! [Y3: complex,Z: complex,X: complex,W: complex] :
% 13.19/7.59 ( ( Y3 != zero_zero_complex )
% 13.19/7.59 => ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( ( divide1717551699836669952omplex @ X @ Y3 )
% 13.19/7.59 = ( divide1717551699836669952omplex @ W @ Z ) )
% 13.19/7.59 = ( ( times_times_complex @ X @ Z )
% 13.19/7.59 = ( times_times_complex @ W @ Y3 ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2030_frac__eq__eq,axiom,
% 13.19/7.59 ! [Y3: real,Z: real,X: real,W: real] :
% 13.19/7.59 ( ( Y3 != zero_zero_real )
% 13.19/7.59 => ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( ( divide_divide_real @ X @ Y3 )
% 13.19/7.59 = ( divide_divide_real @ W @ Z ) )
% 13.19/7.59 = ( ( times_times_real @ X @ Z )
% 13.19/7.59 = ( times_times_real @ W @ Y3 ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2031_frac__eq__eq,axiom,
% 13.19/7.59 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 13.19/7.59 ( ( Y3 != zero_zero_rat )
% 13.19/7.59 => ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( ( divide_divide_rat @ X @ Y3 )
% 13.19/7.59 = ( divide_divide_rat @ W @ Z ) )
% 13.19/7.59 = ( ( times_times_rat @ X @ Z )
% 13.19/7.59 = ( times_times_rat @ W @ Y3 ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2032_divide__eq__eq,axiom,
% 13.19/7.59 ! [B: complex,C: complex,A: complex] :
% 13.19/7.59 ( ( ( divide1717551699836669952omplex @ B @ C )
% 13.19/7.59 = A )
% 13.19/7.59 = ( ( ( C != zero_zero_complex )
% 13.19/7.59 => ( B
% 13.19/7.59 = ( times_times_complex @ A @ C ) ) )
% 13.19/7.59 & ( ( C = zero_zero_complex )
% 13.19/7.59 => ( A = zero_zero_complex ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2033_divide__eq__eq,axiom,
% 13.19/7.59 ! [B: real,C: real,A: real] :
% 13.19/7.59 ( ( ( divide_divide_real @ B @ C )
% 13.19/7.59 = A )
% 13.19/7.59 = ( ( ( C != zero_zero_real )
% 13.19/7.59 => ( B
% 13.19/7.59 = ( times_times_real @ A @ C ) ) )
% 13.19/7.59 & ( ( C = zero_zero_real )
% 13.19/7.59 => ( A = zero_zero_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2034_divide__eq__eq,axiom,
% 13.19/7.59 ! [B: rat,C: rat,A: rat] :
% 13.19/7.59 ( ( ( divide_divide_rat @ B @ C )
% 13.19/7.59 = A )
% 13.19/7.59 = ( ( ( C != zero_zero_rat )
% 13.19/7.59 => ( B
% 13.19/7.59 = ( times_times_rat @ A @ C ) ) )
% 13.19/7.59 & ( ( C = zero_zero_rat )
% 13.19/7.59 => ( A = zero_zero_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2035_eq__divide__eq,axiom,
% 13.19/7.59 ! [A: complex,B: complex,C: complex] :
% 13.19/7.59 ( ( A
% 13.19/7.59 = ( divide1717551699836669952omplex @ B @ C ) )
% 13.19/7.59 = ( ( ( C != zero_zero_complex )
% 13.19/7.59 => ( ( times_times_complex @ A @ C )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( C = zero_zero_complex )
% 13.19/7.59 => ( A = zero_zero_complex ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2036_eq__divide__eq,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( A
% 13.19/7.59 = ( divide_divide_real @ B @ C ) )
% 13.19/7.59 = ( ( ( C != zero_zero_real )
% 13.19/7.59 => ( ( times_times_real @ A @ C )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( C = zero_zero_real )
% 13.19/7.59 => ( A = zero_zero_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2037_eq__divide__eq,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( A
% 13.19/7.59 = ( divide_divide_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( C != zero_zero_rat )
% 13.19/7.59 => ( ( times_times_rat @ A @ C )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( C = zero_zero_rat )
% 13.19/7.59 => ( A = zero_zero_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2038_divide__eq__imp,axiom,
% 13.19/7.59 ! [C: complex,B: complex,A: complex] :
% 13.19/7.59 ( ( C != zero_zero_complex )
% 13.19/7.59 => ( ( B
% 13.19/7.59 = ( times_times_complex @ A @ C ) )
% 13.19/7.59 => ( ( divide1717551699836669952omplex @ B @ C )
% 13.19/7.59 = A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2039_divide__eq__imp,axiom,
% 13.19/7.59 ! [C: real,B: real,A: real] :
% 13.19/7.59 ( ( C != zero_zero_real )
% 13.19/7.59 => ( ( B
% 13.19/7.59 = ( times_times_real @ A @ C ) )
% 13.19/7.59 => ( ( divide_divide_real @ B @ C )
% 13.19/7.59 = A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2040_divide__eq__imp,axiom,
% 13.19/7.59 ! [C: rat,B: rat,A: rat] :
% 13.19/7.59 ( ( C != zero_zero_rat )
% 13.19/7.59 => ( ( B
% 13.19/7.59 = ( times_times_rat @ A @ C ) )
% 13.19/7.59 => ( ( divide_divide_rat @ B @ C )
% 13.19/7.59 = A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2041_eq__divide__imp,axiom,
% 13.19/7.59 ! [C: complex,A: complex,B: complex] :
% 13.19/7.59 ( ( C != zero_zero_complex )
% 13.19/7.59 => ( ( ( times_times_complex @ A @ C )
% 13.19/7.59 = B )
% 13.19/7.59 => ( A
% 13.19/7.59 = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2042_eq__divide__imp,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( C != zero_zero_real )
% 13.19/7.59 => ( ( ( times_times_real @ A @ C )
% 13.19/7.59 = B )
% 13.19/7.59 => ( A
% 13.19/7.59 = ( divide_divide_real @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2043_eq__divide__imp,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( C != zero_zero_rat )
% 13.19/7.59 => ( ( ( times_times_rat @ A @ C )
% 13.19/7.59 = B )
% 13.19/7.59 => ( A
% 13.19/7.59 = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2044_nonzero__divide__eq__eq,axiom,
% 13.19/7.59 ! [C: complex,B: complex,A: complex] :
% 13.19/7.59 ( ( C != zero_zero_complex )
% 13.19/7.59 => ( ( ( divide1717551699836669952omplex @ B @ C )
% 13.19/7.59 = A )
% 13.19/7.59 = ( B
% 13.19/7.59 = ( times_times_complex @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2045_nonzero__divide__eq__eq,axiom,
% 13.19/7.59 ! [C: real,B: real,A: real] :
% 13.19/7.59 ( ( C != zero_zero_real )
% 13.19/7.59 => ( ( ( divide_divide_real @ B @ C )
% 13.19/7.59 = A )
% 13.19/7.59 = ( B
% 13.19/7.59 = ( times_times_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2046_nonzero__divide__eq__eq,axiom,
% 13.19/7.59 ! [C: rat,B: rat,A: rat] :
% 13.19/7.59 ( ( C != zero_zero_rat )
% 13.19/7.59 => ( ( ( divide_divide_rat @ B @ C )
% 13.19/7.59 = A )
% 13.19/7.59 = ( B
% 13.19/7.59 = ( times_times_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2047_nonzero__eq__divide__eq,axiom,
% 13.19/7.59 ! [C: complex,A: complex,B: complex] :
% 13.19/7.59 ( ( C != zero_zero_complex )
% 13.19/7.59 => ( ( A
% 13.19/7.59 = ( divide1717551699836669952omplex @ B @ C ) )
% 13.19/7.59 = ( ( times_times_complex @ A @ C )
% 13.19/7.59 = B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2048_nonzero__eq__divide__eq,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( C != zero_zero_real )
% 13.19/7.59 => ( ( A
% 13.19/7.59 = ( divide_divide_real @ B @ C ) )
% 13.19/7.59 = ( ( times_times_real @ A @ C )
% 13.19/7.59 = B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2049_nonzero__eq__divide__eq,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( C != zero_zero_rat )
% 13.19/7.59 => ( ( A
% 13.19/7.59 = ( divide_divide_rat @ B @ C ) )
% 13.19/7.59 = ( ( times_times_rat @ A @ C )
% 13.19/7.59 = B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2050_right__inverse__eq,axiom,
% 13.19/7.59 ! [B: complex,A: complex] :
% 13.19/7.59 ( ( B != zero_zero_complex )
% 13.19/7.59 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 13.19/7.59 = one_one_complex )
% 13.19/7.59 = ( A = B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2051_right__inverse__eq,axiom,
% 13.19/7.59 ! [B: real,A: real] :
% 13.19/7.59 ( ( B != zero_zero_real )
% 13.19/7.59 => ( ( ( divide_divide_real @ A @ B )
% 13.19/7.59 = one_one_real )
% 13.19/7.59 = ( A = B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2052_right__inverse__eq,axiom,
% 13.19/7.59 ! [B: rat,A: rat] :
% 13.19/7.59 ( ( B != zero_zero_rat )
% 13.19/7.59 => ( ( ( divide_divide_rat @ A @ B )
% 13.19/7.59 = one_one_rat )
% 13.19/7.59 = ( A = B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2053_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 13.19/7.59 ! [A: code_integer,B: code_integer] :
% 13.19/7.59 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 13.19/7.59 => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2054_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2055_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2056_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 13.19/7.59 ! [B: nat,A: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2057_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 13.19/7.59 ! [B: int,A: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2058_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 13.19/7.59 ! [B: code_integer,A: code_integer] :
% 13.19/7.59 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 13.19/7.59 => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2059_power__0,axiom,
% 13.19/7.59 ! [A: rat] :
% 13.19/7.59 ( ( power_power_rat @ A @ zero_zero_nat )
% 13.19/7.59 = one_one_rat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2060_power__0,axiom,
% 13.19/7.59 ! [A: nat] :
% 13.19/7.59 ( ( power_power_nat @ A @ zero_zero_nat )
% 13.19/7.59 = one_one_nat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2061_power__0,axiom,
% 13.19/7.59 ! [A: real] :
% 13.19/7.59 ( ( power_power_real @ A @ zero_zero_nat )
% 13.19/7.59 = one_one_real ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2062_power__0,axiom,
% 13.19/7.59 ! [A: int] :
% 13.19/7.59 ( ( power_power_int @ A @ zero_zero_nat )
% 13.19/7.59 = one_one_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2063_power__0,axiom,
% 13.19/7.59 ! [A: complex] :
% 13.19/7.59 ( ( power_power_complex @ A @ zero_zero_nat )
% 13.19/7.59 = one_one_complex ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2064_mod__eq__self__iff__div__eq__0,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ( modulo_modulo_nat @ A @ B )
% 13.19/7.59 = A )
% 13.19/7.59 = ( ( divide_divide_nat @ A @ B )
% 13.19/7.59 = zero_zero_nat ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2065_mod__eq__self__iff__div__eq__0,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ( modulo_modulo_int @ A @ B )
% 13.19/7.59 = A )
% 13.19/7.59 = ( ( divide_divide_int @ A @ B )
% 13.19/7.59 = zero_zero_int ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2066_mod__eq__self__iff__div__eq__0,axiom,
% 13.19/7.59 ! [A: code_integer,B: code_integer] :
% 13.19/7.59 ( ( ( modulo364778990260209775nteger @ A @ B )
% 13.19/7.59 = A )
% 13.19/7.59 = ( ( divide6298287555418463151nteger @ A @ B )
% 13.19/7.59 = zero_z3403309356797280102nteger ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2067_ex__least__nat__le,axiom,
% 13.19/7.59 ! [P: nat > $o,N: nat] :
% 13.19/7.59 ( ( P @ N )
% 13.19/7.59 => ( ~ ( P @ zero_zero_nat )
% 13.19/7.59 => ? [K3: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ K3 @ N )
% 13.19/7.59 & ! [I3: nat] :
% 13.19/7.59 ( ( ord_less_nat @ I3 @ K3 )
% 13.19/7.59 => ~ ( P @ I3 ) )
% 13.19/7.59 & ( P @ K3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2068_less__imp__add__positive,axiom,
% 13.19/7.59 ! [I: nat,J: nat] :
% 13.19/7.59 ( ( ord_less_nat @ I @ J )
% 13.19/7.59 => ? [K3: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ K3 )
% 13.19/7.59 & ( ( plus_plus_nat @ I @ K3 )
% 13.19/7.59 = J ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2069_diff__less,axiom,
% 13.19/7.59 ! [N: nat,M: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.19/7.59 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2070_mult__less__mono2,axiom,
% 13.19/7.59 ! [I: nat,J: nat,K: nat] :
% 13.19/7.59 ( ( ord_less_nat @ I @ J )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2071_mult__less__mono1,axiom,
% 13.19/7.59 ! [I: nat,J: nat,K: nat] :
% 13.19/7.59 ( ( ord_less_nat @ I @ J )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2072_nat__mult__less__cancel1,axiom,
% 13.19/7.59 ! [K: nat,M: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.59 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.19/7.59 = ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2073_nat__mult__eq__cancel1,axiom,
% 13.19/7.59 ! [K: nat,M: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.59 => ( ( ( times_times_nat @ K @ M )
% 13.19/7.59 = ( times_times_nat @ K @ N ) )
% 13.19/7.59 = ( M = N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2074_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 13.19/7.59 ! [M: nat,N: nat] :
% 13.19/7.59 ( ( ( divide_divide_nat @ M @ N )
% 13.19/7.59 = zero_zero_nat )
% 13.19/7.59 = ( ( ord_less_nat @ M @ N )
% 13.19/7.59 | ( N = zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2075_diff__add__0,axiom,
% 13.19/7.59 ! [N: nat,M: nat] :
% 13.19/7.59 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 13.19/7.59 = zero_zero_nat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2076_nat__power__less__imp__less,axiom,
% 13.19/7.59 ! [I: nat,M: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ I )
% 13.19/7.59 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 13.19/7.59 => ( ord_less_nat @ M @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2077_mult__eq__self__implies__10,axiom,
% 13.19/7.59 ! [M: nat,N: nat] :
% 13.19/7.59 ( ( M
% 13.19/7.59 = ( times_times_nat @ M @ N ) )
% 13.19/7.59 => ( ( N = one_one_nat )
% 13.19/7.59 | ( M = zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2078_mod__less__divisor,axiom,
% 13.19/7.59 ! [N: nat,M: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2079_mod__eq__0D,axiom,
% 13.19/7.59 ! [M: nat,D: nat] :
% 13.19/7.59 ( ( ( modulo_modulo_nat @ M @ D )
% 13.19/7.59 = zero_zero_nat )
% 13.19/7.59 => ? [Q3: nat] :
% 13.19/7.59 ( M
% 13.19/7.59 = ( times_times_nat @ D @ Q3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2080_mult__less__le__imp__less,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2081_mult__less__le__imp__less,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2082_mult__less__le__imp__less,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2083_mult__less__le__imp__less,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2084_mult__le__less__imp__less,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_real @ C @ D )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2085_mult__le__less__imp__less,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_rat @ C @ D )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2086_mult__le__less__imp__less,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_nat @ C @ D )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2087_mult__le__less__imp__less,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_int @ C @ D )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2088_mult__right__le__imp__le,axiom,
% 13.19/7.59 ! [A: real,C: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2089_mult__right__le__imp__le,axiom,
% 13.19/7.59 ! [A: rat,C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2090_mult__right__le__imp__le,axiom,
% 13.19/7.59 ! [A: nat,C: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2091_mult__right__le__imp__le,axiom,
% 13.19/7.59 ! [A: int,C: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2092_mult__left__le__imp__le,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2093_mult__left__le__imp__le,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2094_mult__left__le__imp__le,axiom,
% 13.19/7.59 ! [C: nat,A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2095_mult__left__le__imp__le,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2096_mult__le__cancel__left__pos,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2097_mult__le__cancel__left__pos,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2098_mult__le__cancel__left__pos,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2099_mult__le__cancel__left__neg,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 = ( ord_less_eq_real @ B @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2100_mult__le__cancel__left__neg,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2101_mult__le__cancel__left__neg,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2102_mult__less__cancel__right,axiom,
% 13.19/7.59 ! [A: real,C: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2103_mult__less__cancel__right,axiom,
% 13.19/7.59 ! [A: rat,C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2104_mult__less__cancel__right,axiom,
% 13.19/7.59 ! [A: int,C: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2105_mult__strict__mono_H,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_real @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2106_mult__strict__mono_H,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_rat @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2107_mult__strict__mono_H,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_nat @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2108_mult__strict__mono_H,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_int @ C @ D )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2109_mult__right__less__imp__less,axiom,
% 13.19/7.59 ! [A: real,C: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2110_mult__right__less__imp__less,axiom,
% 13.19/7.59 ! [A: rat,C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2111_mult__right__less__imp__less,axiom,
% 13.19/7.59 ! [A: nat,C: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2112_mult__right__less__imp__less,axiom,
% 13.19/7.59 ! [A: int,C: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2113_mult__less__cancel__left,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2114_mult__less__cancel__left,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2115_mult__less__cancel__left,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2116_mult__strict__mono,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real,D: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_real @ C @ D )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2117_mult__strict__mono,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_rat @ C @ D )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2118_mult__strict__mono,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( ord_less_nat @ C @ D )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2119_mult__strict__mono,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int,D: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( ord_less_int @ C @ D )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2120_mult__left__less__imp__less,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2121_mult__left__less__imp__less,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2122_mult__left__less__imp__less,axiom,
% 13.19/7.59 ! [C: nat,A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ord_less_nat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2123_mult__left__less__imp__less,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2124_mult__le__cancel__right,axiom,
% 13.19/7.59 ! [A: real,C: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2125_mult__le__cancel__right,axiom,
% 13.19/7.59 ! [A: rat,C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2126_mult__le__cancel__right,axiom,
% 13.19/7.59 ! [A: int,C: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2127_mult__le__cancel__left,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2128_mult__le__cancel__left,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2129_mult__le__cancel__left,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.19/7.59 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_eq_int @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2130_add__strict__increasing2,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_real @ B @ C )
% 13.19/7.59 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2131_add__strict__increasing2,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_rat @ B @ C )
% 13.19/7.59 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2132_add__strict__increasing2,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ B @ C )
% 13.19/7.59 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2133_add__strict__increasing2,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ B @ C )
% 13.19/7.59 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2134_add__strict__increasing,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ C )
% 13.19/7.59 => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2135_add__strict__increasing,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ C )
% 13.19/7.59 => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2136_add__strict__increasing,axiom,
% 13.19/7.59 ! [A: nat,B: nat,C: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ C )
% 13.19/7.59 => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2137_add__strict__increasing,axiom,
% 13.19/7.59 ! [A: int,B: int,C: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ C )
% 13.19/7.59 => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2138_add__pos__nonneg,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2139_add__pos__nonneg,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2140_add__pos__nonneg,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2141_add__pos__nonneg,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2142_add__nonpos__neg,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2143_add__nonpos__neg,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2144_add__nonpos__neg,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_nat @ B @ zero_zero_nat )
% 13.19/7.59 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2145_add__nonpos__neg,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2146_add__nonneg__pos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2147_add__nonneg__pos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2148_add__nonneg__pos,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2149_add__nonneg__pos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2150_add__neg__nonpos,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2151_add__neg__nonpos,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2152_add__neg__nonpos,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A @ zero_zero_nat )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 13.19/7.59 => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2153_add__neg__nonpos,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ A @ zero_zero_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2154_field__le__epsilon,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ! [E2: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ E2 )
% 13.19/7.59 => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y3 @ E2 ) ) )
% 13.19/7.59 => ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2155_field__le__epsilon,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ! [E2: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 13.19/7.59 => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y3 @ E2 ) ) )
% 13.19/7.59 => ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2156_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( divide_divide_nat @ A @ B )
% 13.19/7.59 = zero_zero_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2157_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( divide_divide_int @ A @ B )
% 13.19/7.59 = zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2158_div__positive,axiom,
% 13.19/7.59 ! [B: nat,A: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ A )
% 13.19/7.59 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2159_div__positive,axiom,
% 13.19/7.59 ! [B: int,A: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ A )
% 13.19/7.59 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2160_frac__le,axiom,
% 13.19/7.59 ! [Y3: real,X: real,W: real,Z: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ W )
% 13.19/7.59 => ( ( ord_less_eq_real @ W @ Z )
% 13.19/7.59 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2161_frac__le,axiom,
% 13.19/7.59 ! [Y3: rat,X: rat,W: rat,Z: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ( ord_less_eq_rat @ X @ Y3 )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 13.19/7.59 => ( ( ord_less_eq_rat @ W @ Z )
% 13.19/7.59 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2162_frac__less,axiom,
% 13.19/7.59 ! [X: real,Y3: real,W: real,Z: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_real @ X @ Y3 )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ W )
% 13.19/7.59 => ( ( ord_less_eq_real @ W @ Z )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2163_frac__less,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat,W: rat,Z: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_rat @ X @ Y3 )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 13.19/7.59 => ( ( ord_less_eq_rat @ W @ Z )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2164_frac__less2,axiom,
% 13.19/7.59 ! [X: real,Y3: real,W: real,Z: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ W )
% 13.19/7.59 => ( ( ord_less_real @ W @ Z )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ X @ Z ) @ ( divide_divide_real @ Y3 @ W ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2165_frac__less2,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat,W: rat,Z: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_eq_rat @ X @ Y3 )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ W )
% 13.19/7.59 => ( ( ord_less_rat @ W @ Z )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ X @ Z ) @ ( divide_divide_rat @ Y3 @ W ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2166_divide__le__cancel,axiom,
% 13.19/7.59 ! [A: real,C: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_eq_real @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2167_divide__le__cancel,axiom,
% 13.19/7.59 ! [A: rat,C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_eq_rat @ A @ B ) )
% 13.19/7.59 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2168_divide__nonneg__neg,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2169_divide__nonneg__neg,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2170_divide__nonneg__pos,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2171_divide__nonneg__pos,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2172_divide__nonpos__neg,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 13.19/7.59 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2173_divide__nonpos__neg,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2174_divide__nonpos__pos,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2175_divide__nonpos__pos,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ X @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2176_mult__left__le,axiom,
% 13.19/7.59 ! [C: real,A: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ C @ one_one_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2177_mult__left__le,axiom,
% 13.19/7.59 ! [C: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2178_mult__left__le,axiom,
% 13.19/7.59 ! [C: nat,A: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2179_mult__left__le,axiom,
% 13.19/7.59 ! [C: int,A: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ C @ one_one_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2180_mult__le__one,axiom,
% 13.19/7.59 ! [A: real,B: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ A @ one_one_real )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ( ord_less_eq_real @ B @ one_one_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2181_mult__le__one,axiom,
% 13.19/7.59 ! [A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ A @ one_one_rat )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2182_mult__le__one,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ A @ one_one_nat )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 13.19/7.59 => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2183_mult__le__one,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ A @ one_one_int )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ( ord_less_eq_int @ B @ one_one_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2184_mult__right__le__one__le,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ X @ Y3 ) @ X ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2185_mult__right__le__one__le,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y3 ) @ X ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2186_mult__right__le__one__le,axiom,
% 13.19/7.59 ! [X: int,Y3: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.19/7.59 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ X @ Y3 ) @ X ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2187_mult__left__le__one__le,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X ) @ X ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2188_mult__left__le__one__le,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X ) @ X ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2189_mult__left__le__one__le,axiom,
% 13.19/7.59 ! [X: int,Y3: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.19/7.59 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X ) @ X ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2190_sum__squares__ge__zero,axiom,
% 13.19/7.59 ! [X: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2191_sum__squares__ge__zero,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2192_sum__squares__ge__zero,axiom,
% 13.19/7.59 ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2193_sum__squares__le__zero__iff,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 13.19/7.59 = ( ( X = zero_zero_real )
% 13.19/7.59 & ( Y3 = zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2194_sum__squares__le__zero__iff,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 13.19/7.59 = ( ( X = zero_zero_rat )
% 13.19/7.59 & ( Y3 = zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2195_sum__squares__le__zero__iff,axiom,
% 13.19/7.59 ! [X: int,Y3: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 13.19/7.59 = ( ( X = zero_zero_int )
% 13.19/7.59 & ( Y3 = zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2196_power__less__imp__less__base,axiom,
% 13.19/7.59 ! [A: real,N: nat,B: real] :
% 13.19/7.59 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
% 13.19/7.59 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.19/7.59 => ( ord_less_real @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2197_power__less__imp__less__base,axiom,
% 13.19/7.59 ! [A: rat,N: nat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
% 13.19/7.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.19/7.59 => ( ord_less_rat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2198_power__less__imp__less__base,axiom,
% 13.19/7.59 ! [A: nat,N: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 13.19/7.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_nat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2199_power__less__imp__less__base,axiom,
% 13.19/7.59 ! [A: int,N: nat,B: int] :
% 13.19/7.59 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 13.19/7.59 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2200_not__sum__squares__lt__zero,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2201_not__sum__squares__lt__zero,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2202_not__sum__squares__lt__zero,axiom,
% 13.19/7.59 ! [X: int,Y3: int] :
% 13.19/7.59 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2203_sum__squares__gt__zero__iff,axiom,
% 13.19/7.59 ! [X: real,Y3: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 13.19/7.59 = ( ( X != zero_zero_real )
% 13.19/7.59 | ( Y3 != zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2204_sum__squares__gt__zero__iff,axiom,
% 13.19/7.59 ! [X: rat,Y3: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 13.19/7.59 = ( ( X != zero_zero_rat )
% 13.19/7.59 | ( Y3 != zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2205_sum__squares__gt__zero__iff,axiom,
% 13.19/7.59 ! [X: int,Y3: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 13.19/7.59 = ( ( X != zero_zero_int )
% 13.19/7.59 | ( Y3 != zero_zero_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2206_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 13.19/7.59 ! [C: nat,A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.19/7.59 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 13.19/7.59 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2207_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 13.19/7.59 ! [C: int,A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 13.19/7.59 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2208_zero__less__two,axiom,
% 13.19/7.59 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2209_zero__less__two,axiom,
% 13.19/7.59 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2210_zero__less__two,axiom,
% 13.19/7.59 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2211_zero__less__two,axiom,
% 13.19/7.59 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2212_divide__strict__left__mono__neg,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ B )
% 13.19/7.59 => ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2213_divide__strict__left__mono__neg,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ B )
% 13.19/7.59 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2214_divide__strict__left__mono,axiom,
% 13.19/7.59 ! [B: real,A: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ B @ A )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2215_divide__strict__left__mono,axiom,
% 13.19/7.59 ! [B: rat,A: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ B @ A )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2216_mult__imp__less__div__pos,axiom,
% 13.19/7.59 ! [Y3: real,Z: real,X: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X )
% 13.19/7.59 => ( ord_less_real @ Z @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2217_mult__imp__less__div__pos,axiom,
% 13.19/7.59 ! [Y3: rat,Z: rat,X: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X )
% 13.19/7.59 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2218_mult__imp__div__pos__less,axiom,
% 13.19/7.59 ! [Y3: real,X: real,Z: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ( ord_less_real @ X @ ( times_times_real @ Z @ Y3 ) )
% 13.19/7.59 => ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2219_mult__imp__div__pos__less,axiom,
% 13.19/7.59 ! [Y3: rat,X: rat,Z: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.19/7.59 => ( ( ord_less_rat @ X @ ( times_times_rat @ Z @ Y3 ) )
% 13.19/7.59 => ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2220_pos__less__divide__eq,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 13.19/7.59 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2221_pos__less__divide__eq,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.59 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2222_pos__divide__less__eq,axiom,
% 13.19/7.59 ! [C: real,B: real,A: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 13.19/7.59 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2223_pos__divide__less__eq,axiom,
% 13.19/7.59 ! [C: rat,B: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 13.19/7.59 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2224_neg__less__divide__eq,axiom,
% 13.19/7.59 ! [C: real,A: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 13.19/7.59 = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2225_neg__less__divide__eq,axiom,
% 13.19/7.59 ! [C: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.59 = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2226_neg__divide__less__eq,axiom,
% 13.19/7.59 ! [C: real,B: real,A: real] :
% 13.19/7.59 ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 13.19/7.59 = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2227_neg__divide__less__eq,axiom,
% 13.19/7.59 ! [C: rat,B: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 13.19/7.59 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2228_less__divide__eq,axiom,
% 13.19/7.59 ! [A: real,B: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 13.19/7.59 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 13.19/7.59 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2229_less__divide__eq,axiom,
% 13.19/7.59 ! [A: rat,B: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 13.19/7.59 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 13.19/7.59 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2230_divide__less__eq,axiom,
% 13.19/7.59 ! [B: real,C: real,A: real] :
% 13.19/7.59 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 13.19/7.59 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 13.19/7.59 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2231_divide__less__eq,axiom,
% 13.19/7.59 ! [B: rat,C: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 13.19/7.59 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 13.19/7.59 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2232_less__divide__eq__1,axiom,
% 13.19/7.59 ! [B: real,A: real] :
% 13.19/7.59 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_real @ A @ B ) )
% 13.19/7.59 | ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_real @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2233_less__divide__eq__1,axiom,
% 13.19/7.59 ! [B: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_rat @ A @ B ) )
% 13.19/7.59 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_rat @ B @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2234_divide__less__eq__1,axiom,
% 13.19/7.59 ! [B: real,A: real] :
% 13.19/7.59 ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 13.19/7.59 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.59 & ( ord_less_real @ B @ A ) )
% 13.19/7.59 | ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.59 & ( ord_less_real @ A @ B ) )
% 13.19/7.59 | ( A = zero_zero_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2235_divide__less__eq__1,axiom,
% 13.19/7.59 ! [B: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 13.19/7.59 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.59 & ( ord_less_rat @ B @ A ) )
% 13.19/7.59 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.59 & ( ord_less_rat @ A @ B ) )
% 13.19/7.59 | ( A = zero_zero_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2236_power__le__one,axiom,
% 13.19/7.59 ! [A: real,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.19/7.59 => ( ( ord_less_eq_real @ A @ one_one_real )
% 13.19/7.59 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ one_one_real ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2237_power__le__one,axiom,
% 13.19/7.59 ! [A: rat,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.19/7.59 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 13.19/7.59 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ one_one_rat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2238_power__le__one,axiom,
% 13.19/7.59 ! [A: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 13.19/7.59 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2239_power__le__one,axiom,
% 13.19/7.59 ! [A: int,N: nat] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_eq_int @ A @ one_one_int )
% 13.19/7.59 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2240_eq__divide__eq__numeral_I1_J,axiom,
% 13.19/7.59 ! [W: num,B: complex,C: complex] :
% 13.19/7.59 ( ( ( numera6690914467698888265omplex @ W )
% 13.19/7.59 = ( divide1717551699836669952omplex @ B @ C ) )
% 13.19/7.59 = ( ( ( C != zero_zero_complex )
% 13.19/7.59 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( C = zero_zero_complex )
% 13.19/7.59 => ( ( numera6690914467698888265omplex @ W )
% 13.19/7.59 = zero_zero_complex ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2241_eq__divide__eq__numeral_I1_J,axiom,
% 13.19/7.59 ! [W: num,B: real,C: real] :
% 13.19/7.59 ( ( ( numeral_numeral_real @ W )
% 13.19/7.59 = ( divide_divide_real @ B @ C ) )
% 13.19/7.59 = ( ( ( C != zero_zero_real )
% 13.19/7.59 => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( C = zero_zero_real )
% 13.19/7.59 => ( ( numeral_numeral_real @ W )
% 13.19/7.59 = zero_zero_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2242_eq__divide__eq__numeral_I1_J,axiom,
% 13.19/7.59 ! [W: num,B: rat,C: rat] :
% 13.19/7.59 ( ( ( numeral_numeral_rat @ W )
% 13.19/7.59 = ( divide_divide_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( C != zero_zero_rat )
% 13.19/7.59 => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( C = zero_zero_rat )
% 13.19/7.59 => ( ( numeral_numeral_rat @ W )
% 13.19/7.59 = zero_zero_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2243_divide__eq__eq__numeral_I1_J,axiom,
% 13.19/7.59 ! [B: complex,C: complex,W: num] :
% 13.19/7.59 ( ( ( divide1717551699836669952omplex @ B @ C )
% 13.19/7.59 = ( numera6690914467698888265omplex @ W ) )
% 13.19/7.59 = ( ( ( C != zero_zero_complex )
% 13.19/7.59 => ( B
% 13.19/7.59 = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
% 13.19/7.59 & ( ( C = zero_zero_complex )
% 13.19/7.59 => ( ( numera6690914467698888265omplex @ W )
% 13.19/7.59 = zero_zero_complex ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2244_divide__eq__eq__numeral_I1_J,axiom,
% 13.19/7.59 ! [B: real,C: real,W: num] :
% 13.19/7.59 ( ( ( divide_divide_real @ B @ C )
% 13.19/7.59 = ( numeral_numeral_real @ W ) )
% 13.19/7.59 = ( ( ( C != zero_zero_real )
% 13.19/7.59 => ( B
% 13.19/7.59 = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 13.19/7.59 & ( ( C = zero_zero_real )
% 13.19/7.59 => ( ( numeral_numeral_real @ W )
% 13.19/7.59 = zero_zero_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2245_divide__eq__eq__numeral_I1_J,axiom,
% 13.19/7.59 ! [B: rat,C: rat,W: num] :
% 13.19/7.59 ( ( ( divide_divide_rat @ B @ C )
% 13.19/7.59 = ( numeral_numeral_rat @ W ) )
% 13.19/7.59 = ( ( ( C != zero_zero_rat )
% 13.19/7.59 => ( B
% 13.19/7.59 = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 13.19/7.59 & ( ( C = zero_zero_rat )
% 13.19/7.59 => ( ( numeral_numeral_rat @ W )
% 13.19/7.59 = zero_zero_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2246_divide__add__eq__iff,axiom,
% 13.19/7.59 ! [Z: complex,X: complex,Y3: complex] :
% 13.19/7.59 ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y3 )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2247_divide__add__eq__iff,axiom,
% 13.19/7.59 ! [Z: real,X: real,Y3: real] :
% 13.19/7.59 ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z ) @ Y3 )
% 13.19/7.59 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2248_divide__add__eq__iff,axiom,
% 13.19/7.59 ! [Z: rat,X: rat,Y3: rat] :
% 13.19/7.59 ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z ) @ Y3 )
% 13.19/7.59 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2249_add__divide__eq__iff,axiom,
% 13.19/7.59 ! [Z: complex,X: complex,Y3: complex] :
% 13.19/7.59 ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2250_add__divide__eq__iff,axiom,
% 13.19/7.59 ! [Z: real,X: real,Y3: real] :
% 13.19/7.59 ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y3 @ Z ) )
% 13.19/7.59 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2251_add__divide__eq__iff,axiom,
% 13.19/7.59 ! [Z: rat,X: rat,Y3: rat] :
% 13.19/7.59 ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y3 @ Z ) )
% 13.19/7.59 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2252_add__num__frac,axiom,
% 13.19/7.59 ! [Y3: complex,Z: complex,X: complex] :
% 13.19/7.59 ( ( Y3 != zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X @ Y3 ) )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2253_add__num__frac,axiom,
% 13.19/7.59 ! [Y3: real,Z: real,X: real] :
% 13.19/7.59 ( ( Y3 != zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X @ Y3 ) )
% 13.19/7.59 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2254_add__num__frac,axiom,
% 13.19/7.59 ! [Y3: rat,Z: rat,X: rat] :
% 13.19/7.59 ( ( Y3 != zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X @ Y3 ) )
% 13.19/7.59 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2255_add__frac__num,axiom,
% 13.19/7.59 ! [Y3: complex,X: complex,Z: complex] :
% 13.19/7.59 ( ( Y3 != zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ Z )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2256_add__frac__num,axiom,
% 13.19/7.59 ! [Y3: real,X: real,Z: real] :
% 13.19/7.59 ( ( Y3 != zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y3 ) @ Z )
% 13.19/7.59 = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2257_add__frac__num,axiom,
% 13.19/7.59 ! [Y3: rat,X: rat,Z: rat] :
% 13.19/7.59 ( ( Y3 != zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z )
% 13.19/7.59 = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2258_add__frac__eq,axiom,
% 13.19/7.59 ! [Y3: complex,Z: complex,X: complex,W: complex] :
% 13.19/7.59 ( ( Y3 != zero_zero_complex )
% 13.19/7.59 => ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2259_add__frac__eq,axiom,
% 13.19/7.59 ! [Y3: real,Z: real,X: real,W: real] :
% 13.19/7.59 ( ( Y3 != zero_zero_real )
% 13.19/7.59 => ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 13.19/7.59 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2260_add__frac__eq,axiom,
% 13.19/7.59 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 13.19/7.59 ( ( Y3 != zero_zero_rat )
% 13.19/7.59 => ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 13.19/7.59 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2261_add__divide__eq__if__simps_I1_J,axiom,
% 13.19/7.59 ! [Z: complex,A: complex,B: complex] :
% 13.19/7.59 ( ( ( Z = zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 13.19/7.59 = A ) )
% 13.19/7.59 & ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2262_add__divide__eq__if__simps_I1_J,axiom,
% 13.19/7.59 ! [Z: real,A: real,B: real] :
% 13.19/7.59 ( ( ( Z = zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 13.19/7.59 = A ) )
% 13.19/7.59 & ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 13.19/7.59 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2263_add__divide__eq__if__simps_I1_J,axiom,
% 13.19/7.59 ! [Z: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ( Z = zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 13.19/7.59 = A ) )
% 13.19/7.59 & ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 13.19/7.59 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2264_add__divide__eq__if__simps_I2_J,axiom,
% 13.19/7.59 ! [Z: complex,A: complex,B: complex] :
% 13.19/7.59 ( ( ( Z = zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2265_add__divide__eq__if__simps_I2_J,axiom,
% 13.19/7.59 ! [Z: real,A: real,B: real] :
% 13.19/7.59 ( ( ( Z = zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 13.19/7.59 = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2266_add__divide__eq__if__simps_I2_J,axiom,
% 13.19/7.59 ! [Z: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ( Z = zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 13.19/7.59 = B ) )
% 13.19/7.59 & ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 13.19/7.59 = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2267_div__add__self2,axiom,
% 13.19/7.59 ! [B: nat,A: nat] :
% 13.19/7.59 ( ( B != zero_zero_nat )
% 13.19/7.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 13.19/7.59 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2268_div__add__self2,axiom,
% 13.19/7.59 ! [B: int,A: int] :
% 13.19/7.59 ( ( B != zero_zero_int )
% 13.19/7.59 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 13.19/7.59 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2269_div__add__self1,axiom,
% 13.19/7.59 ! [B: nat,A: nat] :
% 13.19/7.59 ( ( B != zero_zero_nat )
% 13.19/7.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 13.19/7.59 = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2270_div__add__self1,axiom,
% 13.19/7.59 ! [B: int,A: int] :
% 13.19/7.59 ( ( B != zero_zero_int )
% 13.19/7.59 => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 13.19/7.59 = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2271_divide__diff__eq__iff,axiom,
% 13.19/7.59 ! [Z: complex,X: complex,Y3: complex] :
% 13.19/7.59 ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z ) @ Y3 )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2272_divide__diff__eq__iff,axiom,
% 13.19/7.59 ! [Z: real,X: real,Y3: real] :
% 13.19/7.59 ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z ) @ Y3 )
% 13.19/7.59 = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2273_divide__diff__eq__iff,axiom,
% 13.19/7.59 ! [Z: rat,X: rat,Y3: rat] :
% 13.19/7.59 ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z ) @ Y3 )
% 13.19/7.59 = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2274_diff__divide__eq__iff,axiom,
% 13.19/7.59 ! [Z: complex,X: complex,Y3: complex] :
% 13.19/7.59 ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2275_diff__divide__eq__iff,axiom,
% 13.19/7.59 ! [Z: real,X: real,Y3: real] :
% 13.19/7.59 ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y3 @ Z ) )
% 13.19/7.59 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2276_diff__divide__eq__iff,axiom,
% 13.19/7.59 ! [Z: rat,X: rat,Y3: rat] :
% 13.19/7.59 ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y3 @ Z ) )
% 13.19/7.59 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ Y3 ) @ Z ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2277_diff__frac__eq,axiom,
% 13.19/7.59 ! [Y3: complex,Z: complex,X: complex,W: complex] :
% 13.19/7.59 ( ( Y3 != zero_zero_complex )
% 13.19/7.59 => ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y3 ) @ ( divide1717551699836669952omplex @ W @ Z ) )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z ) @ ( times_times_complex @ W @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2278_diff__frac__eq,axiom,
% 13.19/7.59 ! [Y3: real,Z: real,X: real,W: real] :
% 13.19/7.59 ( ( Y3 != zero_zero_real )
% 13.19/7.59 => ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 13.19/7.59 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2279_diff__frac__eq,axiom,
% 13.19/7.59 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 13.19/7.59 ( ( Y3 != zero_zero_rat )
% 13.19/7.59 => ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 13.19/7.59 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2280_add__divide__eq__if__simps_I4_J,axiom,
% 13.19/7.59 ! [Z: complex,A: complex,B: complex] :
% 13.19/7.59 ( ( ( Z = zero_zero_complex )
% 13.19/7.59 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 13.19/7.59 = A ) )
% 13.19/7.59 & ( ( Z != zero_zero_complex )
% 13.19/7.59 => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
% 13.19/7.59 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2281_add__divide__eq__if__simps_I4_J,axiom,
% 13.19/7.59 ! [Z: real,A: real,B: real] :
% 13.19/7.59 ( ( ( Z = zero_zero_real )
% 13.19/7.59 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 13.19/7.59 = A ) )
% 13.19/7.59 & ( ( Z != zero_zero_real )
% 13.19/7.59 => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
% 13.19/7.59 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2282_add__divide__eq__if__simps_I4_J,axiom,
% 13.19/7.59 ! [Z: rat,A: rat,B: rat] :
% 13.19/7.59 ( ( ( Z = zero_zero_rat )
% 13.19/7.59 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 13.19/7.59 = A ) )
% 13.19/7.59 & ( ( Z != zero_zero_rat )
% 13.19/7.59 => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
% 13.19/7.59 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2283_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 13.19/7.59 ! [B: code_integer,A: code_integer] :
% 13.19/7.59 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 13.19/7.59 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2284_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 13.19/7.59 ! [B: nat,A: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.19/7.59 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2285_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 13.19/7.59 ! [B: int,A: int] :
% 13.19/7.59 ( ( ord_less_int @ zero_zero_int @ B )
% 13.19/7.59 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2286_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 13.19/7.59 ! [A: code_integer,B: code_integer] :
% 13.19/7.59 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 13.19/7.59 => ( ( ord_le6747313008572928689nteger @ A @ B )
% 13.19/7.59 => ( ( modulo364778990260209775nteger @ A @ B )
% 13.19/7.59 = A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2287_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 13.19/7.59 ! [A: nat,B: nat] :
% 13.19/7.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.19/7.59 => ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( ( modulo_modulo_nat @ A @ B )
% 13.19/7.59 = A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2288_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 13.19/7.59 ! [A: int,B: int] :
% 13.19/7.59 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.19/7.59 => ( ( ord_less_int @ A @ B )
% 13.19/7.59 => ( ( modulo_modulo_int @ A @ B )
% 13.19/7.59 = A ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2289_cong__exp__iff__simps_I2_J,axiom,
% 13.19/7.59 ! [N: num,Q2: num] :
% 13.19/7.59 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.19/7.59 = zero_zero_nat )
% 13.19/7.59 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 13.19/7.59 = zero_zero_nat ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2290_cong__exp__iff__simps_I2_J,axiom,
% 13.19/7.59 ! [N: num,Q2: num] :
% 13.19/7.59 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.19/7.59 = zero_zero_int )
% 13.19/7.59 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 13.19/7.59 = zero_zero_int ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2291_cong__exp__iff__simps_I2_J,axiom,
% 13.19/7.59 ! [N: num,Q2: num] :
% 13.19/7.59 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.19/7.59 = zero_z3403309356797280102nteger )
% 13.19/7.59 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 13.19/7.59 = zero_z3403309356797280102nteger ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2292_cong__exp__iff__simps_I1_J,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 13.19/7.59 = zero_zero_nat ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2293_cong__exp__iff__simps_I1_J,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 13.19/7.59 = zero_zero_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2294_cong__exp__iff__simps_I1_J,axiom,
% 13.19/7.59 ! [N: num] :
% 13.19/7.59 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 13.19/7.59 = zero_z3403309356797280102nteger ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2295_bounded__Max__nat,axiom,
% 13.19/7.59 ! [P: nat > $o,X: nat,M5: nat] :
% 13.19/7.59 ( ( P @ X )
% 13.19/7.59 => ( ! [X3: nat] :
% 13.19/7.59 ( ( P @ X3 )
% 13.19/7.59 => ( ord_less_eq_nat @ X3 @ M5 ) )
% 13.19/7.59 => ~ ! [M4: nat] :
% 13.19/7.59 ( ( P @ M4 )
% 13.19/7.59 => ~ ! [X5: nat] :
% 13.19/7.59 ( ( P @ X5 )
% 13.19/7.59 => ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2296_not__exp__less__eq__0__int,axiom,
% 13.19/7.59 ! [N: nat] :
% 13.19/7.59 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2297_nat__mult__le__cancel1,axiom,
% 13.19/7.59 ! [K: nat,M: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.59 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.19/7.59 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2298_nat__diff__split,axiom,
% 13.19/7.59 ! [P: nat > $o,A: nat,B: nat] :
% 13.19/7.59 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 13.19/7.59 = ( ( ( ord_less_nat @ A @ B )
% 13.19/7.59 => ( P @ zero_zero_nat ) )
% 13.19/7.59 & ! [D2: nat] :
% 13.19/7.59 ( ( A
% 13.19/7.59 = ( plus_plus_nat @ B @ D2 ) )
% 13.19/7.59 => ( P @ D2 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2299_nat__diff__split__asm,axiom,
% 13.19/7.59 ! [P: nat > $o,A: nat,B: nat] :
% 13.19/7.59 ( ( P @ ( minus_minus_nat @ A @ B ) )
% 13.19/7.59 = ( ~ ( ( ( ord_less_nat @ A @ B )
% 13.19/7.59 & ~ ( P @ zero_zero_nat ) )
% 13.19/7.59 | ? [D2: nat] :
% 13.19/7.59 ( ( A
% 13.19/7.59 = ( plus_plus_nat @ B @ D2 ) )
% 13.19/7.59 & ~ ( P @ D2 ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2300_div__le__mono2,axiom,
% 13.19/7.59 ! [M: nat,N: nat,K: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.19/7.59 => ( ( ord_less_eq_nat @ M @ N )
% 13.19/7.59 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2301_div__greater__zero__iff,axiom,
% 13.19/7.59 ! [M: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 13.19/7.59 = ( ( ord_less_eq_nat @ N @ M )
% 13.19/7.59 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2302_real__arch__pow__inv,axiom,
% 13.19/7.59 ! [Y3: real,X: real] :
% 13.19/7.59 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.59 => ( ( ord_less_real @ X @ one_one_real )
% 13.19/7.59 => ? [N4: nat] : ( ord_less_real @ ( power_power_real @ X @ N4 ) @ Y3 ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2303_div__less__iff__less__mult,axiom,
% 13.19/7.59 ! [Q2: nat,M: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 13.19/7.59 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q2 ) @ N )
% 13.19/7.59 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2304_nat__mult__div__cancel1,axiom,
% 13.19/7.59 ! [K: nat,M: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.19/7.59 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.19/7.59 = ( divide_divide_nat @ M @ N ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2305_div__eq__dividend__iff,axiom,
% 13.19/7.59 ! [M: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.19/7.59 => ( ( ( divide_divide_nat @ M @ N )
% 13.19/7.59 = M )
% 13.19/7.59 = ( N = one_one_nat ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2306_div__less__dividend,axiom,
% 13.19/7.59 ! [N: nat,M: nat] :
% 13.19/7.59 ( ( ord_less_nat @ one_one_nat @ N )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.19/7.59 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2307_mod__le__divisor,axiom,
% 13.19/7.59 ! [N: nat,M: nat] :
% 13.19/7.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2308_div__less__mono,axiom,
% 13.19/7.59 ! [A2: nat,B5: nat,N: nat] :
% 13.19/7.59 ( ( ord_less_nat @ A2 @ B5 )
% 13.19/7.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.19/7.59 => ( ( ( modulo_modulo_nat @ A2 @ N )
% 13.19/7.59 = zero_zero_nat )
% 13.19/7.59 => ( ( ( modulo_modulo_nat @ B5 @ N )
% 13.19/7.59 = zero_zero_nat )
% 13.19/7.59 => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2309_mult__less__cancel__right2,axiom,
% 13.19/7.59 ! [A: real,C: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ A @ one_one_real ) )
% 13.19/7.59 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2310_mult__less__cancel__right2,axiom,
% 13.19/7.59 ! [A: rat,C: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ A @ one_one_rat ) )
% 13.19/7.59 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2311_mult__less__cancel__right2,axiom,
% 13.19/7.59 ! [A: int,C: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 13.19/7.59 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ A @ one_one_int ) )
% 13.19/7.59 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2312_mult__less__cancel__right1,axiom,
% 13.19/7.59 ! [C: real,B: real] :
% 13.19/7.59 ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ one_one_real @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2313_mult__less__cancel__right1,axiom,
% 13.19/7.59 ! [C: rat,B: rat] :
% 13.19/7.59 ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ one_one_rat @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2314_mult__less__cancel__right1,axiom,
% 13.19/7.59 ! [C: int,B: int] :
% 13.19/7.59 ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 13.19/7.59 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.59 => ( ord_less_int @ one_one_int @ B ) )
% 13.19/7.59 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.59 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2315_mult__less__cancel__left2,axiom,
% 13.19/7.59 ! [C: real,A: real] :
% 13.19/7.59 ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 13.19/7.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.59 => ( ord_less_real @ A @ one_one_real ) )
% 13.19/7.59 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.59 => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2316_mult__less__cancel__left2,axiom,
% 13.19/7.59 ! [C: rat,A: rat] :
% 13.19/7.59 ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 13.19/7.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.59 => ( ord_less_rat @ A @ one_one_rat ) )
% 13.19/7.59 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.59 => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 13.19/7.59
% 13.19/7.59 thf(fact_2317_mult__less__cancel__left2,axiom,
% 13.19/7.59 ! [C: int,A: int] :
% 13.19/7.59 ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 13.19/7.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.60 => ( ord_less_int @ A @ one_one_int ) )
% 13.19/7.60 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.60 => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2318_mult__less__cancel__left1,axiom,
% 13.19/7.60 ! [C: real,B: real] :
% 13.19/7.60 ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 13.19/7.60 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_real @ one_one_real @ B ) )
% 13.19/7.60 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2319_mult__less__cancel__left1,axiom,
% 13.19/7.60 ! [C: rat,B: rat] :
% 13.19/7.60 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 13.19/7.60 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_rat @ one_one_rat @ B ) )
% 13.19/7.60 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2320_mult__less__cancel__left1,axiom,
% 13.19/7.60 ! [C: int,B: int] :
% 13.19/7.60 ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 13.19/7.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.19/7.60 => ( ord_less_int @ one_one_int @ B ) )
% 13.19/7.60 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 13.19/7.60 => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2321_mult__le__cancel__right2,axiom,
% 13.19/7.60 ! [A: real,C: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_eq_real @ A @ one_one_real ) )
% 13.19/7.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2322_mult__le__cancel__right2,axiom,
% 13.19/7.60 ! [A: rat,C: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 13.19/7.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2323_mult__le__cancel__right2,axiom,
% 13.19/7.60 ! [A: int,C: int] :
% 13.19/7.60 ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 13.19/7.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.60 => ( ord_less_eq_int @ A @ one_one_int ) )
% 13.19/7.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.60 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2324_mult__le__cancel__right1,axiom,
% 13.19/7.60 ! [C: real,B: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_eq_real @ one_one_real @ B ) )
% 13.19/7.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2325_mult__le__cancel__right1,axiom,
% 13.19/7.60 ! [C: rat,B: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 13.19/7.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2326_mult__le__cancel__right1,axiom,
% 13.19/7.60 ! [C: int,B: int] :
% 13.19/7.60 ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 13.19/7.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.60 => ( ord_less_eq_int @ one_one_int @ B ) )
% 13.19/7.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.60 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2327_mult__le__cancel__left2,axiom,
% 13.19/7.60 ! [C: real,A: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_eq_real @ A @ one_one_real ) )
% 13.19/7.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2328_mult__le__cancel__left2,axiom,
% 13.19/7.60 ! [C: rat,A: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 13.19/7.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2329_mult__le__cancel__left2,axiom,
% 13.19/7.60 ! [C: int,A: int] :
% 13.19/7.60 ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 13.19/7.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.60 => ( ord_less_eq_int @ A @ one_one_int ) )
% 13.19/7.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.60 => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2330_mult__le__cancel__left1,axiom,
% 13.19/7.60 ! [C: real,B: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_eq_real @ one_one_real @ B ) )
% 13.19/7.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2331_mult__le__cancel__left1,axiom,
% 13.19/7.60 ! [C: rat,B: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 13.19/7.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2332_mult__le__cancel__left1,axiom,
% 13.19/7.60 ! [C: int,B: int] :
% 13.19/7.60 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 13.19/7.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 13.19/7.60 => ( ord_less_eq_int @ one_one_int @ B ) )
% 13.19/7.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 13.19/7.60 => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2333_field__le__mult__one__interval,axiom,
% 13.19/7.60 ! [X: real,Y3: real] :
% 13.19/7.60 ( ! [Z5: real] :
% 13.19/7.60 ( ( ord_less_real @ zero_zero_real @ Z5 )
% 13.19/7.60 => ( ( ord_less_real @ Z5 @ one_one_real )
% 13.19/7.60 => ( ord_less_eq_real @ ( times_times_real @ Z5 @ X ) @ Y3 ) ) )
% 13.19/7.60 => ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2334_field__le__mult__one__interval,axiom,
% 13.19/7.60 ! [X: rat,Y3: rat] :
% 13.19/7.60 ( ! [Z5: rat] :
% 13.19/7.60 ( ( ord_less_rat @ zero_zero_rat @ Z5 )
% 13.19/7.60 => ( ( ord_less_rat @ Z5 @ one_one_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ ( times_times_rat @ Z5 @ X ) @ Y3 ) ) )
% 13.19/7.60 => ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2335_divide__le__eq,axiom,
% 13.19/7.60 ! [B: real,C: real,A: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 13.19/7.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 13.19/7.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2336_divide__le__eq,axiom,
% 13.19/7.60 ! [B: rat,C: rat,A: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 13.19/7.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 13.19/7.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2337_le__divide__eq,axiom,
% 13.19/7.60 ! [A: real,B: real,C: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 13.19/7.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 13.19/7.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2338_le__divide__eq,axiom,
% 13.19/7.60 ! [A: rat,B: rat,C: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 13.19/7.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 13.19/7.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2339_divide__left__mono,axiom,
% 13.19/7.60 ! [B: real,A: real,C: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ B @ A )
% 13.19/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 13.19/7.60 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2340_divide__left__mono,axiom,
% 13.19/7.60 ! [B: rat,A: rat,C: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ B @ A )
% 13.19/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 13.19/7.60 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2341_neg__divide__le__eq,axiom,
% 13.19/7.60 ! [C: real,B: real,A: real] :
% 13.19/7.60 ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 13.19/7.60 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2342_neg__divide__le__eq,axiom,
% 13.19/7.60 ! [C: rat,B: rat,A: rat] :
% 13.19/7.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 13.19/7.60 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2343_neg__le__divide__eq,axiom,
% 13.19/7.60 ! [C: real,A: real,B: real] :
% 13.19/7.60 ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 13.19/7.60 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2344_neg__le__divide__eq,axiom,
% 13.19/7.60 ! [C: rat,A: rat,B: rat] :
% 13.19/7.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.60 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2345_pos__divide__le__eq,axiom,
% 13.19/7.60 ! [C: real,B: real,A: real] :
% 13.19/7.60 ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 13.19/7.60 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2346_pos__divide__le__eq,axiom,
% 13.19/7.60 ! [C: rat,B: rat,A: rat] :
% 13.19/7.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 13.19/7.60 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2347_pos__le__divide__eq,axiom,
% 13.19/7.60 ! [C: real,A: real,B: real] :
% 13.19/7.60 ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 13.19/7.60 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2348_pos__le__divide__eq,axiom,
% 13.19/7.60 ! [C: rat,A: rat,B: rat] :
% 13.19/7.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.60 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2349_mult__imp__div__pos__le,axiom,
% 13.19/7.60 ! [Y3: real,X: real,Z: real] :
% 13.19/7.60 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.60 => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z @ Y3 ) )
% 13.19/7.60 => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ Z ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2350_mult__imp__div__pos__le,axiom,
% 13.19/7.60 ! [Y3: rat,X: rat,Z: rat] :
% 13.19/7.60 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.19/7.60 => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z @ Y3 ) )
% 13.19/7.60 => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ Z ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2351_mult__imp__le__div__pos,axiom,
% 13.19/7.60 ! [Y3: real,Z: real,X: real] :
% 13.19/7.60 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.19/7.60 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X )
% 13.19/7.60 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2352_mult__imp__le__div__pos,axiom,
% 13.19/7.60 ! [Y3: rat,Z: rat,X: rat] :
% 13.19/7.60 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.19/7.60 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X )
% 13.19/7.60 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2353_divide__left__mono__neg,axiom,
% 13.19/7.60 ! [A: real,B: real,C: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ A @ B )
% 13.19/7.60 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 13.19/7.60 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2354_divide__left__mono__neg,axiom,
% 13.19/7.60 ! [A: rat,B: rat,C: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ A @ B )
% 13.19/7.60 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 13.19/7.60 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2355_divide__le__eq__1,axiom,
% 13.19/7.60 ! [B: real,A: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.60 & ( ord_less_eq_real @ B @ A ) )
% 13.19/7.60 | ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.60 & ( ord_less_eq_real @ A @ B ) )
% 13.19/7.60 | ( A = zero_zero_real ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2356_divide__le__eq__1,axiom,
% 13.19/7.60 ! [B: rat,A: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.60 & ( ord_less_eq_rat @ B @ A ) )
% 13.19/7.60 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.60 & ( ord_less_eq_rat @ A @ B ) )
% 13.19/7.60 | ( A = zero_zero_rat ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2357_le__divide__eq__1,axiom,
% 13.19/7.60 ! [B: real,A: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.60 & ( ord_less_eq_real @ A @ B ) )
% 13.19/7.60 | ( ( ord_less_real @ A @ zero_zero_real )
% 13.19/7.60 & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2358_le__divide__eq__1,axiom,
% 13.19/7.60 ! [B: rat,A: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.60 & ( ord_less_eq_rat @ A @ B ) )
% 13.19/7.60 | ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.19/7.60 & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2359_convex__bound__le,axiom,
% 13.19/7.60 ! [X: real,A: real,Y3: real,U: real,V: real] :
% 13.19/7.60 ( ( ord_less_eq_real @ X @ A )
% 13.19/7.60 => ( ( ord_less_eq_real @ Y3 @ A )
% 13.19/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 13.19/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 13.19/7.60 => ( ( ( plus_plus_real @ U @ V )
% 13.19/7.60 = one_one_real )
% 13.19/7.60 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2360_convex__bound__le,axiom,
% 13.19/7.60 ! [X: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 13.19/7.60 ( ( ord_less_eq_rat @ X @ A )
% 13.19/7.60 => ( ( ord_less_eq_rat @ Y3 @ A )
% 13.19/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 13.19/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 13.19/7.60 => ( ( ( plus_plus_rat @ U @ V )
% 13.19/7.60 = one_one_rat )
% 13.19/7.60 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2361_convex__bound__le,axiom,
% 13.19/7.60 ! [X: int,A: int,Y3: int,U: int,V: int] :
% 13.19/7.60 ( ( ord_less_eq_int @ X @ A )
% 13.19/7.60 => ( ( ord_less_eq_int @ Y3 @ A )
% 13.19/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 13.19/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 13.19/7.60 => ( ( ( plus_plus_int @ U @ V )
% 13.19/7.60 = one_one_int )
% 13.19/7.60 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2362_less__divide__eq__numeral_I1_J,axiom,
% 13.19/7.60 ! [W: num,B: real,C: real] :
% 13.19/7.60 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 13.19/7.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 13.19/7.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2363_less__divide__eq__numeral_I1_J,axiom,
% 13.19/7.60 ! [W: num,B: rat,C: rat] :
% 13.19/7.60 ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 13.19/7.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 13.19/7.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2364_divide__less__eq__numeral_I1_J,axiom,
% 13.19/7.60 ! [B: real,C: real,W: num] :
% 13.19/7.60 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 13.19/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 13.19/7.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.19/7.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 13.19/7.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.19/7.60 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2365_divide__less__eq__numeral_I1_J,axiom,
% 13.19/7.60 ! [B: rat,C: rat,W: num] :
% 13.19/7.60 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 13.19/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 13.19/7.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.19/7.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 13.19/7.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.19/7.60 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2366_frac__le__eq,axiom,
% 13.19/7.60 ! [Y3: real,Z: real,X: real,W: real] :
% 13.19/7.60 ( ( Y3 != zero_zero_real )
% 13.19/7.60 => ( ( Z != zero_zero_real )
% 13.19/7.60 => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 13.19/7.60 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2367_frac__le__eq,axiom,
% 13.19/7.60 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 13.19/7.60 ( ( Y3 != zero_zero_rat )
% 13.19/7.60 => ( ( Z != zero_zero_rat )
% 13.19/7.60 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 13.19/7.60 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2368_frac__less__eq,axiom,
% 13.19/7.60 ! [Y3: real,Z: real,X: real,W: real] :
% 13.19/7.60 ( ( Y3 != zero_zero_real )
% 13.19/7.60 => ( ( Z != zero_zero_real )
% 13.19/7.60 => ( ( ord_less_real @ ( divide_divide_real @ X @ Y3 ) @ ( divide_divide_real @ W @ Z ) )
% 13.19/7.60 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ W @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2369_frac__less__eq,axiom,
% 13.19/7.60 ! [Y3: rat,Z: rat,X: rat,W: rat] :
% 13.19/7.60 ( ( Y3 != zero_zero_rat )
% 13.19/7.60 => ( ( Z != zero_zero_rat )
% 13.19/7.60 => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y3 ) @ ( divide_divide_rat @ W @ Z ) )
% 13.19/7.60 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ W @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2370_power__Suc__less,axiom,
% 13.19/7.60 ! [A: real,N: nat] :
% 13.19/7.60 ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.60 => ( ( ord_less_real @ A @ one_one_real )
% 13.19/7.60 => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N ) ) @ ( power_power_real @ A @ N ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2371_power__Suc__less,axiom,
% 13.19/7.60 ! [A: rat,N: nat] :
% 13.19/7.60 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.19/7.60 => ( ( ord_less_rat @ A @ one_one_rat )
% 13.19/7.60 => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) @ ( power_power_rat @ A @ N ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2372_power__Suc__less,axiom,
% 13.19/7.60 ! [A: nat,N: nat] :
% 13.19/7.60 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.19/7.60 => ( ( ord_less_nat @ A @ one_one_nat )
% 13.19/7.60 => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2373_power__Suc__less,axiom,
% 13.19/7.60 ! [A: int,N: nat] :
% 13.19/7.60 ( ( ord_less_int @ zero_zero_int @ A )
% 13.19/7.60 => ( ( ord_less_int @ A @ one_one_int )
% 13.19/7.60 => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2374_power__strict__decreasing,axiom,
% 13.19/7.60 ! [N: nat,N3: nat,A: real] :
% 13.19/7.60 ( ( ord_less_nat @ N @ N3 )
% 13.19/7.60 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.19/7.60 => ( ( ord_less_real @ A @ one_one_real )
% 13.19/7.60 => ( ord_less_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 13.19/7.60
% 13.19/7.60 thf(fact_2375_power__strict__decreasing,axiom,
% 13.25/7.60 ! [N: nat,N3: nat,A: rat] :
% 13.25/7.60 ( ( ord_less_nat @ N @ N3 )
% 13.25/7.60 => ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.25/7.60 => ( ( ord_less_rat @ A @ one_one_rat )
% 13.25/7.60 => ( ord_less_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2376_power__strict__decreasing,axiom,
% 13.25/7.60 ! [N: nat,N3: nat,A: nat] :
% 13.25/7.60 ( ( ord_less_nat @ N @ N3 )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.25/7.60 => ( ( ord_less_nat @ A @ one_one_nat )
% 13.25/7.60 => ( ord_less_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2377_power__strict__decreasing,axiom,
% 13.25/7.60 ! [N: nat,N3: nat,A: int] :
% 13.25/7.60 ( ( ord_less_nat @ N @ N3 )
% 13.25/7.60 => ( ( ord_less_int @ zero_zero_int @ A )
% 13.25/7.60 => ( ( ord_less_int @ A @ one_one_int )
% 13.25/7.60 => ( ord_less_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2378_power__decreasing,axiom,
% 13.25/7.60 ! [N: nat,N3: nat,A: real] :
% 13.25/7.60 ( ( ord_less_eq_nat @ N @ N3 )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.60 => ( ( ord_less_eq_real @ A @ one_one_real )
% 13.25/7.60 => ( ord_less_eq_real @ ( power_power_real @ A @ N3 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2379_power__decreasing,axiom,
% 13.25/7.60 ! [N: nat,N3: nat,A: rat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ N @ N3 )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.25/7.60 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 13.25/7.60 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N3 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2380_power__decreasing,axiom,
% 13.25/7.60 ! [N: nat,N3: nat,A: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ N @ N3 )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.25/7.60 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 13.25/7.60 => ( ord_less_eq_nat @ ( power_power_nat @ A @ N3 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2381_power__decreasing,axiom,
% 13.25/7.60 ! [N: nat,N3: nat,A: int] :
% 13.25/7.60 ( ( ord_less_eq_nat @ N @ N3 )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.25/7.60 => ( ( ord_less_eq_int @ A @ one_one_int )
% 13.25/7.60 => ( ord_less_eq_int @ ( power_power_int @ A @ N3 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2382_zero__power2,axiom,
% 13.25/7.60 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_zero_rat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2383_zero__power2,axiom,
% 13.25/7.60 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_zero_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2384_zero__power2,axiom,
% 13.25/7.60 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_zero_real ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2385_zero__power2,axiom,
% 13.25/7.60 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_zero_int ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2386_zero__power2,axiom,
% 13.25/7.60 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_zero_complex ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2387_self__le__power,axiom,
% 13.25/7.60 ! [A: real,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_real @ one_one_real @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2388_self__le__power,axiom,
% 13.25/7.60 ! [A: rat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_rat @ one_one_rat @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2389_self__le__power,axiom,
% 13.25/7.60 ! [A: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ one_one_nat @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2390_self__le__power,axiom,
% 13.25/7.60 ! [A: int,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_int @ one_one_int @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2391_one__less__power,axiom,
% 13.25/7.60 ! [A: real,N: nat] :
% 13.25/7.60 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2392_one__less__power,axiom,
% 13.25/7.60 ! [A: rat,N: nat] :
% 13.25/7.60 ( ( ord_less_rat @ one_one_rat @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2393_one__less__power,axiom,
% 13.25/7.60 ! [A: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ one_one_nat @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2394_one__less__power,axiom,
% 13.25/7.60 ! [A: int,N: nat] :
% 13.25/7.60 ( ( ord_less_int @ one_one_int @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2395_pos2,axiom,
% 13.25/7.60 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2396_power__diff,axiom,
% 13.25/7.60 ! [A: complex,N: nat,M: nat] :
% 13.25/7.60 ( ( A != zero_zero_complex )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.60 = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2397_power__diff,axiom,
% 13.25/7.60 ! [A: real,N: nat,M: nat] :
% 13.25/7.60 ( ( A != zero_zero_real )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.60 = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2398_power__diff,axiom,
% 13.25/7.60 ! [A: rat,N: nat,M: nat] :
% 13.25/7.60 ( ( A != zero_zero_rat )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.60 = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2399_power__diff,axiom,
% 13.25/7.60 ! [A: nat,N: nat,M: nat] :
% 13.25/7.60 ( ( A != zero_zero_nat )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.60 = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2400_power__diff,axiom,
% 13.25/7.60 ! [A: int,N: nat,M: nat] :
% 13.25/7.60 ( ( A != zero_zero_int )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.60 = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2401_pos__zdiv__mult__2,axiom,
% 13.25/7.60 ! [A: int,B: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.25/7.60 => ( ( 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 ) )
% 13.25/7.60 = ( divide_divide_int @ B @ A ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2402_neg__zdiv__mult__2,axiom,
% 13.25/7.60 ! [A: int,B: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.25/7.60 => ( ( 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 ) )
% 13.25/7.60 = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2403_less__eq__div__iff__mult__less__eq,axiom,
% 13.25/7.60 ! [Q2: nat,M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ Q2 )
% 13.25/7.60 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q2 ) )
% 13.25/7.60 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q2 ) @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2404_dividend__less__times__div,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2405_dividend__less__div__times,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2406_split__div,axiom,
% 13.25/7.60 ! [P: nat > $o,M: nat,N: nat] :
% 13.25/7.60 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 13.25/7.60 = ( ( ( N = zero_zero_nat )
% 13.25/7.60 => ( P @ zero_zero_nat ) )
% 13.25/7.60 & ( ( N != zero_zero_nat )
% 13.25/7.60 => ! [I4: nat,J3: nat] :
% 13.25/7.60 ( ( ord_less_nat @ J3 @ N )
% 13.25/7.60 => ( ( M
% 13.25/7.60 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 13.25/7.60 => ( P @ I4 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2407_mult__eq__if,axiom,
% 13.25/7.60 ( times_times_nat
% 13.25/7.60 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2408_split__mod,axiom,
% 13.25/7.60 ! [P: nat > $o,M: nat,N: nat] :
% 13.25/7.60 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 13.25/7.60 = ( ( ( N = zero_zero_nat )
% 13.25/7.60 => ( P @ M ) )
% 13.25/7.60 & ( ( N != zero_zero_nat )
% 13.25/7.60 => ! [I4: nat,J3: nat] :
% 13.25/7.60 ( ( ord_less_nat @ J3 @ N )
% 13.25/7.60 => ( ( M
% 13.25/7.60 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 13.25/7.60 => ( P @ J3 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2409_convex__bound__lt,axiom,
% 13.25/7.60 ! [X: real,A: real,Y3: real,U: real,V: real] :
% 13.25/7.60 ( ( ord_less_real @ X @ A )
% 13.25/7.60 => ( ( ord_less_real @ Y3 @ A )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 13.25/7.60 => ( ( ( plus_plus_real @ U @ V )
% 13.25/7.60 = one_one_real )
% 13.25/7.60 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2410_convex__bound__lt,axiom,
% 13.25/7.60 ! [X: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 13.25/7.60 ( ( ord_less_rat @ X @ A )
% 13.25/7.60 => ( ( ord_less_rat @ Y3 @ A )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 13.25/7.60 => ( ( ( plus_plus_rat @ U @ V )
% 13.25/7.60 = one_one_rat )
% 13.25/7.60 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2411_convex__bound__lt,axiom,
% 13.25/7.60 ! [X: int,A: int,Y3: int,U: int,V: int] :
% 13.25/7.60 ( ( ord_less_int @ X @ A )
% 13.25/7.60 => ( ( ord_less_int @ Y3 @ A )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 13.25/7.60 => ( ( ( plus_plus_int @ U @ V )
% 13.25/7.60 = one_one_int )
% 13.25/7.60 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2412_divide__le__eq__numeral_I1_J,axiom,
% 13.25/7.60 ! [B: real,C: real,W: num] :
% 13.25/7.60 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
% 13.25/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.60 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 13.25/7.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.60 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 13.25/7.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.60 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2413_divide__le__eq__numeral_I1_J,axiom,
% 13.25/7.60 ! [B: rat,C: rat,W: num] :
% 13.25/7.60 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
% 13.25/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.60 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 13.25/7.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.60 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 13.25/7.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.60 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2414_le__divide__eq__numeral_I1_J,axiom,
% 13.25/7.60 ! [W: num,B: real,C: real] :
% 13.25/7.60 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
% 13.25/7.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.60 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
% 13.25/7.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.60 => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
% 13.25/7.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.60 => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2415_le__divide__eq__numeral_I1_J,axiom,
% 13.25/7.60 ! [W: num,B: rat,C: rat] :
% 13.25/7.60 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
% 13.25/7.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.60 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
% 13.25/7.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.60 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
% 13.25/7.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.60 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2416_half__gt__zero__iff,axiom,
% 13.25/7.60 ! [A: real] :
% 13.25/7.60 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.60 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2417_half__gt__zero__iff,axiom,
% 13.25/7.60 ! [A: rat] :
% 13.25/7.60 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2418_half__gt__zero,axiom,
% 13.25/7.60 ! [A: real] :
% 13.25/7.60 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.60 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2419_half__gt__zero,axiom,
% 13.25/7.60 ! [A: rat] :
% 13.25/7.60 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.25/7.60 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2420_scaling__mono,axiom,
% 13.25/7.60 ! [U: real,V: real,R: real,S: real] :
% 13.25/7.60 ( ( ord_less_eq_real @ U @ V )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ R )
% 13.25/7.60 => ( ( ord_less_eq_real @ R @ S )
% 13.25/7.60 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2421_scaling__mono,axiom,
% 13.25/7.60 ! [U: rat,V: rat,R: rat,S: rat] :
% 13.25/7.60 ( ( ord_less_eq_rat @ U @ V )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ R )
% 13.25/7.60 => ( ( ord_less_eq_rat @ R @ S )
% 13.25/7.60 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2422_power2__le__imp__le,axiom,
% 13.25/7.60 ! [X: real,Y3: real] :
% 13.25/7.60 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.60 => ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2423_power2__le__imp__le,axiom,
% 13.25/7.60 ! [X: rat,Y3: rat] :
% 13.25/7.60 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.25/7.60 => ( ord_less_eq_rat @ X @ Y3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2424_power2__le__imp__le,axiom,
% 13.25/7.60 ! [X: nat,Y3: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 13.25/7.60 => ( ord_less_eq_nat @ X @ Y3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2425_power2__le__imp__le,axiom,
% 13.25/7.60 ! [X: int,Y3: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.60 => ( ord_less_eq_int @ X @ Y3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2426_power2__eq__imp__eq,axiom,
% 13.25/7.60 ! [X: real,Y3: real] :
% 13.25/7.60 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.60 => ( X = Y3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2427_power2__eq__imp__eq,axiom,
% 13.25/7.60 ! [X: rat,Y3: rat] :
% 13.25/7.60 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.25/7.60 => ( X = Y3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2428_power2__eq__imp__eq,axiom,
% 13.25/7.60 ! [X: nat,Y3: nat] :
% 13.25/7.60 ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 13.25/7.60 => ( X = Y3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2429_power2__eq__imp__eq,axiom,
% 13.25/7.60 ! [X: int,Y3: int] :
% 13.25/7.60 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.60 => ( X = Y3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2430_zero__le__power2,axiom,
% 13.25/7.60 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2431_zero__le__power2,axiom,
% 13.25/7.60 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2432_zero__le__power2,axiom,
% 13.25/7.60 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2433_power2__less__0,axiom,
% 13.25/7.60 ! [A: real] :
% 13.25/7.60 ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2434_power2__less__0,axiom,
% 13.25/7.60 ! [A: rat] :
% 13.25/7.60 ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2435_power2__less__0,axiom,
% 13.25/7.60 ! [A: int] :
% 13.25/7.60 ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2436_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 13.25/7.60 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.25/7.60 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 13.25/7.60 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 13.25/7.60 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2437_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 13.25/7.60 ! [C: nat,A: nat,B: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 13.25/7.60 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 13.25/7.60 = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2438_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 13.25/7.60 ! [C: int,A: int,B: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 13.25/7.60 => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2439_exp__add__not__zero__imp__right,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.60 != zero_zero_nat )
% 13.25/7.60 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.60 != zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2440_exp__add__not__zero__imp__right,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.60 != zero_zero_int )
% 13.25/7.60 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 13.25/7.60 != zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2441_exp__add__not__zero__imp__left,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.60 != zero_zero_nat )
% 13.25/7.60 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 13.25/7.60 != zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2442_exp__add__not__zero__imp__left,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.60 != zero_zero_int )
% 13.25/7.60 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 13.25/7.60 != zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2443_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.60 != zero_zero_nat )
% 13.25/7.60 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 13.25/7.60 != zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2444_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 13.25/7.60 != zero_zero_int )
% 13.25/7.60 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 13.25/7.60 != zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2445_power__diff__power__eq,axiom,
% 13.25/7.60 ! [A: nat,N: nat,M: nat] :
% 13.25/7.60 ( ( A != zero_zero_nat )
% 13.25/7.60 => ( ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 13.25/7.60 = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 13.25/7.60 & ( ~ ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
% 13.25/7.60 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2446_power__diff__power__eq,axiom,
% 13.25/7.60 ! [A: int,N: nat,M: nat] :
% 13.25/7.60 ( ( A != zero_zero_int )
% 13.25/7.60 => ( ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 13.25/7.60 = ( power_power_int @ A @ ( minus_minus_nat @ M @ N ) ) ) )
% 13.25/7.60 & ( ~ ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
% 13.25/7.60 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2447_power__eq__if,axiom,
% 13.25/7.60 ( power_power_complex
% 13.25/7.60 = ( ^ [P4: complex,M3: nat] : ( if_complex @ ( M3 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P4 @ ( power_power_complex @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2448_power__eq__if,axiom,
% 13.25/7.60 ( power_power_real
% 13.25/7.60 = ( ^ [P4: real,M3: nat] : ( if_real @ ( M3 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P4 @ ( power_power_real @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2449_power__eq__if,axiom,
% 13.25/7.60 ( power_power_rat
% 13.25/7.60 = ( ^ [P4: rat,M3: nat] : ( if_rat @ ( M3 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P4 @ ( power_power_rat @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2450_power__eq__if,axiom,
% 13.25/7.60 ( power_power_nat
% 13.25/7.60 = ( ^ [P4: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P4 @ ( power_power_nat @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2451_power__eq__if,axiom,
% 13.25/7.60 ( power_power_int
% 13.25/7.60 = ( ^ [P4: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P4 @ ( power_power_int @ P4 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2452_power__minus__mult,axiom,
% 13.25/7.60 ! [N: nat,A: complex] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 13.25/7.60 = ( power_power_complex @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2453_power__minus__mult,axiom,
% 13.25/7.60 ! [N: nat,A: real] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 13.25/7.60 = ( power_power_real @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2454_power__minus__mult,axiom,
% 13.25/7.60 ! [N: nat,A: rat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 13.25/7.60 = ( power_power_rat @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2455_power__minus__mult,axiom,
% 13.25/7.60 ! [N: nat,A: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 13.25/7.60 = ( power_power_nat @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2456_power__minus__mult,axiom,
% 13.25/7.60 ! [N: nat,A: int] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
% 13.25/7.60 = ( power_power_int @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2457_power2__less__imp__less,axiom,
% 13.25/7.60 ! [X: real,Y3: real] :
% 13.25/7.60 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.60 => ( ord_less_real @ X @ Y3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2458_power2__less__imp__less,axiom,
% 13.25/7.60 ! [X: rat,Y3: rat] :
% 13.25/7.60 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.25/7.60 => ( ord_less_rat @ X @ Y3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2459_power2__less__imp__less,axiom,
% 13.25/7.60 ! [X: nat,Y3: nat] :
% 13.25/7.60 ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 13.25/7.60 => ( ord_less_nat @ X @ Y3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2460_power2__less__imp__less,axiom,
% 13.25/7.60 ! [X: int,Y3: int] :
% 13.25/7.60 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.60 => ( ord_less_int @ X @ Y3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2461_sum__power2__le__zero__iff,axiom,
% 13.25/7.60 ! [X: real,Y3: real] :
% 13.25/7.60 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 13.25/7.60 = ( ( X = zero_zero_real )
% 13.25/7.60 & ( Y3 = zero_zero_real ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2462_sum__power2__le__zero__iff,axiom,
% 13.25/7.60 ! [X: rat,Y3: rat] :
% 13.25/7.60 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 13.25/7.60 = ( ( X = zero_zero_rat )
% 13.25/7.60 & ( Y3 = zero_zero_rat ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2463_sum__power2__le__zero__iff,axiom,
% 13.25/7.60 ! [X: int,Y3: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 13.25/7.60 = ( ( X = zero_zero_int )
% 13.25/7.60 & ( Y3 = zero_zero_int ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2464_sum__power2__ge__zero,axiom,
% 13.25/7.60 ! [X: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2465_sum__power2__ge__zero,axiom,
% 13.25/7.60 ! [X: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2466_sum__power2__ge__zero,axiom,
% 13.25/7.60 ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2467_sum__power2__gt__zero__iff,axiom,
% 13.25/7.60 ! [X: real,Y3: real] :
% 13.25/7.60 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 = ( ( X != zero_zero_real )
% 13.25/7.60 | ( Y3 != zero_zero_real ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2468_sum__power2__gt__zero__iff,axiom,
% 13.25/7.60 ! [X: rat,Y3: rat] :
% 13.25/7.60 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 = ( ( X != zero_zero_rat )
% 13.25/7.60 | ( Y3 != zero_zero_rat ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2469_sum__power2__gt__zero__iff,axiom,
% 13.25/7.60 ! [X: int,Y3: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 = ( ( X != zero_zero_int )
% 13.25/7.60 | ( Y3 != zero_zero_int ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2470_not__sum__power2__lt__zero,axiom,
% 13.25/7.60 ! [X: real,Y3: real] :
% 13.25/7.60 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2471_not__sum__power2__lt__zero,axiom,
% 13.25/7.60 ! [X: rat,Y3: rat] :
% 13.25/7.60 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2472_not__sum__power2__lt__zero,axiom,
% 13.25/7.60 ! [X: int,Y3: int] :
% 13.25/7.60 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2473_divmod__digit__0_I2_J,axiom,
% 13.25/7.60 ! [B: nat,A: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.25/7.60 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 13.25/7.60 = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2474_divmod__digit__0_I2_J,axiom,
% 13.25/7.60 ! [B: int,A: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.60 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 13.25/7.60 = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2475_divmod__digit__0_I2_J,axiom,
% 13.25/7.60 ! [B: code_integer,A: code_integer] :
% 13.25/7.60 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 13.25/7.60 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 13.25/7.60 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2476_bits__stable__imp__add__self,axiom,
% 13.25/7.60 ! [A: nat] :
% 13.25/7.60 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = A )
% 13.25/7.60 => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.60 = zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2477_bits__stable__imp__add__self,axiom,
% 13.25/7.60 ! [A: int] :
% 13.25/7.60 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.60 = A )
% 13.25/7.60 => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 13.25/7.60 = zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2478_bits__stable__imp__add__self,axiom,
% 13.25/7.60 ! [A: code_integer] :
% 13.25/7.60 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.60 = A )
% 13.25/7.60 => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 13.25/7.60 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2479_zero__le__even__power_H,axiom,
% 13.25/7.60 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2480_zero__le__even__power_H,axiom,
% 13.25/7.60 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2481_zero__le__even__power_H,axiom,
% 13.25/7.60 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2482_verit__le__mono__div,axiom,
% 13.25/7.60 ! [A2: nat,B5: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ A2 @ B5 )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_eq_nat
% 13.25/7.60 @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N )
% 13.25/7.60 @ ( if_nat
% 13.25/7.60 @ ( ( modulo_modulo_nat @ B5 @ N )
% 13.25/7.60 = zero_zero_nat )
% 13.25/7.60 @ one_one_nat
% 13.25/7.60 @ zero_zero_nat ) )
% 13.25/7.60 @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2483_divmod__digit__0_I1_J,axiom,
% 13.25/7.60 ! [B: nat,A: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.25/7.60 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 13.25/7.60 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2484_divmod__digit__0_I1_J,axiom,
% 13.25/7.60 ! [B: int,A: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.60 => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 13.25/7.60 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2485_divmod__digit__0_I1_J,axiom,
% 13.25/7.60 ! [B: code_integer,A: code_integer] :
% 13.25/7.60 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 13.25/7.60 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 13.25/7.60 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2486_mod__double__modulus,axiom,
% 13.25/7.60 ! [M: code_integer,X: code_integer] :
% 13.25/7.60 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 13.25/7.60 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 13.25/7.60 => ( ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 = ( modulo364778990260209775nteger @ X @ M ) )
% 13.25/7.60 | ( ( modulo364778990260209775nteger @ X @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X @ M ) @ M ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2487_mod__double__modulus,axiom,
% 13.25/7.60 ! [M: nat,X: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
% 13.25/7.60 => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 = ( modulo_modulo_nat @ X @ M ) )
% 13.25/7.60 | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M ) @ M ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2488_mod__double__modulus,axiom,
% 13.25/7.60 ! [M: int,X: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ M )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.60 => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 = ( modulo_modulo_int @ X @ M ) )
% 13.25/7.60 | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 = ( plus_plus_int @ ( modulo_modulo_int @ X @ M ) @ M ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2489_divmod__digit__1_I2_J,axiom,
% 13.25/7.60 ! [A: code_integer,B: code_integer] :
% 13.25/7.60 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 13.25/7.60 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 13.25/7.60 => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 13.25/7.60 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2490_divmod__digit__1_I2_J,axiom,
% 13.25/7.60 ! [A: nat,B: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ B )
% 13.25/7.60 => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 13.25/7.60 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2491_divmod__digit__1_I2_J,axiom,
% 13.25/7.60 ! [A: int,B: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.25/7.60 => ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.60 => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 13.25/7.60 => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 13.25/7.60 = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2492_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 13.25/7.60 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 13.25/7.60 ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 13.25/7.60 = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2493_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 13.25/7.60 ! [F: num > num > num,A: num,B: num] :
% 13.25/7.60 ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 13.25/7.60 = ( some_num @ ( F @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2494_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 13.25/7.60 ! [F: nat > nat > nat,A: nat,B: nat] :
% 13.25/7.60 ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 13.25/7.60 = ( some_nat @ ( F @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2495_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 13.25/7.60 ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 13.25/7.60 ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 13.25/7.60 = none_P5556105721700978146at_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2496_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 13.25/7.60 ! [Uu: num > num > num,Uv: option_num] :
% 13.25/7.60 ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 13.25/7.60 = none_num ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2497_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 13.25/7.60 ! [Uu: nat > nat > nat,Uv: option_nat] :
% 13.25/7.60 ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 13.25/7.60 = none_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2498_arith__geo__mean,axiom,
% 13.25/7.60 ! [U: real,X: real,Y3: real] :
% 13.25/7.60 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( times_times_real @ X @ Y3 ) )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.60 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2499_arith__geo__mean,axiom,
% 13.25/7.60 ! [U: rat,X: rat,Y3: rat] :
% 13.25/7.60 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( times_times_rat @ X @ Y3 ) )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.25/7.60 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2500_in__children__def,axiom,
% 13.25/7.60 ( vEBT_V5917875025757280293ildren
% 13.25/7.60 = ( ^ [N2: nat,TreeList: list_VEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X2 @ N2 ) ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2501_not__mod__2__eq__1__eq__0,axiom,
% 13.25/7.60 ! [A: nat] :
% 13.25/7.60 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 != one_one_nat )
% 13.25/7.60 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2502_not__mod__2__eq__1__eq__0,axiom,
% 13.25/7.60 ! [A: int] :
% 13.25/7.60 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.60 != one_one_int )
% 13.25/7.60 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2503_not__mod__2__eq__1__eq__0,axiom,
% 13.25/7.60 ! [A: code_integer] :
% 13.25/7.60 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.60 != one_one_Code_integer )
% 13.25/7.60 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2504_not__mod__2__eq__0__eq__1,axiom,
% 13.25/7.60 ! [A: nat] :
% 13.25/7.60 ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 != zero_zero_nat )
% 13.25/7.60 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = one_one_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2505_not__mod__2__eq__0__eq__1,axiom,
% 13.25/7.60 ! [A: int] :
% 13.25/7.60 ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.60 != zero_zero_int )
% 13.25/7.60 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.60 = one_one_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2506_not__mod__2__eq__0__eq__1,axiom,
% 13.25/7.60 ! [A: code_integer] :
% 13.25/7.60 ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.60 != zero_z3403309356797280102nteger )
% 13.25/7.60 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.60 = one_one_Code_integer ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2507_both__member__options__ding,axiom,
% 13.25/7.60 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 13.25/7.60 => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 13.25/7.60 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) )
% 13.25/7.60 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2508_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 13.25/7.60 ! [X: nat,N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.60 => ( ord_less_nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2509_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 13.25/7.60 ! [X: nat,N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.60 => ( ord_less_nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2510_set__n__deg__not__0,axiom,
% 13.25/7.60 ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
% 13.25/7.60 ( ! [X3: vEBT_VEBT] :
% 13.25/7.60 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.60 => ( vEBT_invar_vebt @ X3 @ N ) )
% 13.25/7.60 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 13.25/7.60 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2511_inrange,axiom,
% 13.25/7.60 ! [T: vEBT_VEBT,N: nat] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ T @ N )
% 13.25/7.60 => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2512_set__bit__0,axiom,
% 13.25/7.60 ! [A: int] :
% 13.25/7.60 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 13.25/7.60 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2513_set__bit__0,axiom,
% 13.25/7.60 ! [A: nat] :
% 13.25/7.60 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 13.25/7.60 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2514_nat__induct2,axiom,
% 13.25/7.60 ! [P: nat > $o,N: nat] :
% 13.25/7.60 ( ( P @ zero_zero_nat )
% 13.25/7.60 => ( ( P @ one_one_nat )
% 13.25/7.60 => ( ! [N4: nat] :
% 13.25/7.60 ( ( P @ N4 )
% 13.25/7.60 => ( P @ ( plus_plus_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 => ( P @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2515_deg__deg__n,axiom,
% 13.25/7.60 ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N )
% 13.25/7.60 => ( Deg = N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2516_inthall,axiom,
% 13.25/7.60 ! [Xs2: list_real,P: real > $o,N: nat] :
% 13.25/7.60 ( ! [X3: real] :
% 13.25/7.60 ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
% 13.25/7.60 => ( P @ X3 ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 13.25/7.60 => ( P @ ( nth_real @ Xs2 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2517_inthall,axiom,
% 13.25/7.60 ! [Xs2: list_complex,P: complex > $o,N: nat] :
% 13.25/7.60 ( ! [X3: complex] :
% 13.25/7.60 ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
% 13.25/7.60 => ( P @ X3 ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 13.25/7.60 => ( P @ ( nth_complex @ Xs2 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2518_inthall,axiom,
% 13.25/7.60 ! [Xs2: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,N: nat] :
% 13.25/7.60 ( ! [X3: produc859450856879609959at_nat] :
% 13.25/7.60 ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 13.25/7.60 => ( P @ X3 ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 13.25/7.60 => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2519_inthall,axiom,
% 13.25/7.60 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 13.25/7.60 ( ! [X3: vEBT_VEBT] :
% 13.25/7.60 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 13.25/7.60 => ( P @ X3 ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.60 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2520_inthall,axiom,
% 13.25/7.60 ! [Xs2: list_o,P: $o > $o,N: nat] :
% 13.25/7.60 ( ! [X3: $o] :
% 13.25/7.60 ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 13.25/7.60 => ( P @ X3 ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.60 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2521_inthall,axiom,
% 13.25/7.60 ! [Xs2: list_nat,P: nat > $o,N: nat] :
% 13.25/7.60 ( ! [X3: nat] :
% 13.25/7.60 ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 13.25/7.60 => ( P @ X3 ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.60 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2522_inthall,axiom,
% 13.25/7.60 ! [Xs2: list_int,P: int > $o,N: nat] :
% 13.25/7.60 ( ! [X3: int] :
% 13.25/7.60 ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 13.25/7.60 => ( P @ X3 ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.60 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2523__C4_Ohyps_C_I8_J,axiom,
% 13.25/7.60 ( ( mi = ma )
% 13.25/7.60 => ! [X5: vEBT_VEBT] :
% 13.25/7.60 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
% 13.25/7.60 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2524_atLeastAtMost__iff,axiom,
% 13.25/7.60 ! [I: set_int,L2: set_int,U: set_int] :
% 13.25/7.60 ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L2 @ U ) )
% 13.25/7.60 = ( ( ord_less_eq_set_int @ L2 @ I )
% 13.25/7.60 & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2525_atLeastAtMost__iff,axiom,
% 13.25/7.60 ! [I: rat,L2: rat,U: rat] :
% 13.25/7.60 ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L2 @ U ) )
% 13.25/7.60 = ( ( ord_less_eq_rat @ L2 @ I )
% 13.25/7.60 & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2526_atLeastAtMost__iff,axiom,
% 13.25/7.60 ! [I: num,L2: num,U: num] :
% 13.25/7.60 ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L2 @ U ) )
% 13.25/7.60 = ( ( ord_less_eq_num @ L2 @ I )
% 13.25/7.60 & ( ord_less_eq_num @ I @ U ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2527_atLeastAtMost__iff,axiom,
% 13.25/7.60 ! [I: nat,L2: nat,U: nat] :
% 13.25/7.60 ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 13.25/7.60 = ( ( ord_less_eq_nat @ L2 @ I )
% 13.25/7.60 & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2528_atLeastAtMost__iff,axiom,
% 13.25/7.60 ! [I: int,L2: int,U: int] :
% 13.25/7.60 ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 13.25/7.60 = ( ( ord_less_eq_int @ L2 @ I )
% 13.25/7.60 & ( ord_less_eq_int @ I @ U ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2529_atLeastAtMost__iff,axiom,
% 13.25/7.60 ! [I: real,L2: real,U: real] :
% 13.25/7.60 ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L2 @ U ) )
% 13.25/7.60 = ( ( ord_less_eq_real @ L2 @ I )
% 13.25/7.60 & ( ord_less_eq_real @ I @ U ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2530_Icc__eq__Icc,axiom,
% 13.25/7.60 ! [L2: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 13.25/7.60 ( ( ( set_or370866239135849197et_int @ L2 @ H2 )
% 13.25/7.60 = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 13.25/7.60 = ( ( ( L2 = L3 )
% 13.25/7.60 & ( H2 = H3 ) )
% 13.25/7.60 | ( ~ ( ord_less_eq_set_int @ L2 @ H2 )
% 13.25/7.60 & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2531_Icc__eq__Icc,axiom,
% 13.25/7.60 ! [L2: rat,H2: rat,L3: rat,H3: rat] :
% 13.25/7.60 ( ( ( set_or633870826150836451st_rat @ L2 @ H2 )
% 13.25/7.60 = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 13.25/7.60 = ( ( ( L2 = L3 )
% 13.25/7.60 & ( H2 = H3 ) )
% 13.25/7.60 | ( ~ ( ord_less_eq_rat @ L2 @ H2 )
% 13.25/7.60 & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2532_Icc__eq__Icc,axiom,
% 13.25/7.60 ! [L2: num,H2: num,L3: num,H3: num] :
% 13.25/7.60 ( ( ( set_or7049704709247886629st_num @ L2 @ H2 )
% 13.25/7.60 = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 13.25/7.60 = ( ( ( L2 = L3 )
% 13.25/7.60 & ( H2 = H3 ) )
% 13.25/7.60 | ( ~ ( ord_less_eq_num @ L2 @ H2 )
% 13.25/7.60 & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2533_Icc__eq__Icc,axiom,
% 13.25/7.60 ! [L2: nat,H2: nat,L3: nat,H3: nat] :
% 13.25/7.60 ( ( ( set_or1269000886237332187st_nat @ L2 @ H2 )
% 13.25/7.60 = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 13.25/7.60 = ( ( ( L2 = L3 )
% 13.25/7.60 & ( H2 = H3 ) )
% 13.25/7.60 | ( ~ ( ord_less_eq_nat @ L2 @ H2 )
% 13.25/7.60 & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2534_Icc__eq__Icc,axiom,
% 13.25/7.60 ! [L2: int,H2: int,L3: int,H3: int] :
% 13.25/7.60 ( ( ( set_or1266510415728281911st_int @ L2 @ H2 )
% 13.25/7.60 = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 13.25/7.60 = ( ( ( L2 = L3 )
% 13.25/7.60 & ( H2 = H3 ) )
% 13.25/7.60 | ( ~ ( ord_less_eq_int @ L2 @ H2 )
% 13.25/7.60 & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2535_Icc__eq__Icc,axiom,
% 13.25/7.60 ! [L2: real,H2: real,L3: real,H3: real] :
% 13.25/7.60 ( ( ( set_or1222579329274155063t_real @ L2 @ H2 )
% 13.25/7.60 = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 13.25/7.60 = ( ( ( L2 = L3 )
% 13.25/7.60 & ( H2 = H3 ) )
% 13.25/7.60 | ( ~ ( ord_less_eq_real @ L2 @ H2 )
% 13.25/7.60 & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2536_set__bit__nonnegative__int__iff,axiom,
% 13.25/7.60 ! [N: nat,K: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 13.25/7.60 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2537_set__bit__negative__int__iff,axiom,
% 13.25/7.60 ! [N: nat,K: int] :
% 13.25/7.60 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 13.25/7.60 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2538_atLeastatMost__empty__iff2,axiom,
% 13.25/7.60 ! [A: set_int,B: set_int] :
% 13.25/7.60 ( ( bot_bot_set_set_int
% 13.25/7.60 = ( set_or370866239135849197et_int @ A @ B ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2539_atLeastatMost__empty__iff2,axiom,
% 13.25/7.60 ! [A: rat,B: rat] :
% 13.25/7.60 ( ( bot_bot_set_rat
% 13.25/7.60 = ( set_or633870826150836451st_rat @ A @ B ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2540_atLeastatMost__empty__iff2,axiom,
% 13.25/7.60 ! [A: num,B: num] :
% 13.25/7.60 ( ( bot_bot_set_num
% 13.25/7.60 = ( set_or7049704709247886629st_num @ A @ B ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2541_atLeastatMost__empty__iff2,axiom,
% 13.25/7.60 ! [A: nat,B: nat] :
% 13.25/7.60 ( ( bot_bot_set_nat
% 13.25/7.60 = ( set_or1269000886237332187st_nat @ A @ B ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2542_atLeastatMost__empty__iff2,axiom,
% 13.25/7.60 ! [A: int,B: int] :
% 13.25/7.60 ( ( bot_bot_set_int
% 13.25/7.60 = ( set_or1266510415728281911st_int @ A @ B ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2543_atLeastatMost__empty__iff2,axiom,
% 13.25/7.60 ! [A: real,B: real] :
% 13.25/7.60 ( ( bot_bot_set_real
% 13.25/7.60 = ( set_or1222579329274155063t_real @ A @ B ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2544_atLeastatMost__empty__iff,axiom,
% 13.25/7.60 ! [A: set_int,B: set_int] :
% 13.25/7.60 ( ( ( set_or370866239135849197et_int @ A @ B )
% 13.25/7.60 = bot_bot_set_set_int )
% 13.25/7.60 = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2545_atLeastatMost__empty__iff,axiom,
% 13.25/7.60 ! [A: rat,B: rat] :
% 13.25/7.60 ( ( ( set_or633870826150836451st_rat @ A @ B )
% 13.25/7.60 = bot_bot_set_rat )
% 13.25/7.60 = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2546_atLeastatMost__empty__iff,axiom,
% 13.25/7.60 ! [A: num,B: num] :
% 13.25/7.60 ( ( ( set_or7049704709247886629st_num @ A @ B )
% 13.25/7.60 = bot_bot_set_num )
% 13.25/7.60 = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2547_atLeastatMost__empty__iff,axiom,
% 13.25/7.60 ! [A: nat,B: nat] :
% 13.25/7.60 ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 13.25/7.60 = bot_bot_set_nat )
% 13.25/7.60 = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2548_atLeastatMost__empty__iff,axiom,
% 13.25/7.60 ! [A: int,B: int] :
% 13.25/7.60 ( ( ( set_or1266510415728281911st_int @ A @ B )
% 13.25/7.60 = bot_bot_set_int )
% 13.25/7.60 = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2549_atLeastatMost__empty__iff,axiom,
% 13.25/7.60 ! [A: real,B: real] :
% 13.25/7.60 ( ( ( set_or1222579329274155063t_real @ A @ B )
% 13.25/7.60 = bot_bot_set_real )
% 13.25/7.60 = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2550_atLeastatMost__subset__iff,axiom,
% 13.25/7.60 ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 13.25/7.60 ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_set_int @ C @ A )
% 13.25/7.60 & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2551_atLeastatMost__subset__iff,axiom,
% 13.25/7.60 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.25/7.60 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_rat @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_rat @ C @ A )
% 13.25/7.60 & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2552_atLeastatMost__subset__iff,axiom,
% 13.25/7.60 ! [A: num,B: num,C: num,D: num] :
% 13.25/7.60 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_num @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_num @ C @ A )
% 13.25/7.60 & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2553_atLeastatMost__subset__iff,axiom,
% 13.25/7.60 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.25/7.60 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_nat @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_nat @ C @ A )
% 13.25/7.60 & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2554_atLeastatMost__subset__iff,axiom,
% 13.25/7.60 ! [A: int,B: int,C: int,D: int] :
% 13.25/7.60 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_int @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_int @ C @ A )
% 13.25/7.60 & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2555_atLeastatMost__subset__iff,axiom,
% 13.25/7.60 ! [A: real,B: real,C: real,D: real] :
% 13.25/7.60 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 13.25/7.60 = ( ~ ( ord_less_eq_real @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_real @ C @ A )
% 13.25/7.60 & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2556_atLeastatMost__empty,axiom,
% 13.25/7.60 ! [B: rat,A: rat] :
% 13.25/7.60 ( ( ord_less_rat @ B @ A )
% 13.25/7.60 => ( ( set_or633870826150836451st_rat @ A @ B )
% 13.25/7.60 = bot_bot_set_rat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2557_atLeastatMost__empty,axiom,
% 13.25/7.60 ! [B: num,A: num] :
% 13.25/7.60 ( ( ord_less_num @ B @ A )
% 13.25/7.60 => ( ( set_or7049704709247886629st_num @ A @ B )
% 13.25/7.60 = bot_bot_set_num ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2558_atLeastatMost__empty,axiom,
% 13.25/7.60 ! [B: nat,A: nat] :
% 13.25/7.60 ( ( ord_less_nat @ B @ A )
% 13.25/7.60 => ( ( set_or1269000886237332187st_nat @ A @ B )
% 13.25/7.60 = bot_bot_set_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2559_atLeastatMost__empty,axiom,
% 13.25/7.60 ! [B: int,A: int] :
% 13.25/7.60 ( ( ord_less_int @ B @ A )
% 13.25/7.60 => ( ( set_or1266510415728281911st_int @ A @ B )
% 13.25/7.60 = bot_bot_set_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2560_atLeastatMost__empty,axiom,
% 13.25/7.60 ! [B: real,A: real] :
% 13.25/7.60 ( ( ord_less_real @ B @ A )
% 13.25/7.60 => ( ( set_or1222579329274155063t_real @ A @ B )
% 13.25/7.60 = bot_bot_set_real ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2561_infinite__Icc__iff,axiom,
% 13.25/7.60 ! [A: rat,B: rat] :
% 13.25/7.60 ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 13.25/7.60 = ( ord_less_rat @ A @ B ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2562_infinite__Icc__iff,axiom,
% 13.25/7.60 ! [A: real,B: real] :
% 13.25/7.60 ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 13.25/7.60 = ( ord_less_real @ A @ B ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2563_mod__neg__neg__trivial,axiom,
% 13.25/7.60 ! [K: int,L2: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 13.25/7.60 => ( ( ord_less_int @ L2 @ K )
% 13.25/7.60 => ( ( modulo_modulo_int @ K @ L2 )
% 13.25/7.60 = K ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2564_mod__pos__pos__trivial,axiom,
% 13.25/7.60 ! [K: int,L2: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.60 => ( ( ord_less_int @ K @ L2 )
% 13.25/7.60 => ( ( modulo_modulo_int @ K @ L2 )
% 13.25/7.60 = K ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2565_split__zmod,axiom,
% 13.25/7.60 ! [P: int > $o,N: int,K: int] :
% 13.25/7.60 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 13.25/7.60 = ( ( ( K = zero_zero_int )
% 13.25/7.60 => ( P @ N ) )
% 13.25/7.60 & ( ( ord_less_int @ zero_zero_int @ K )
% 13.25/7.60 => ! [I4: int,J3: int] :
% 13.25/7.60 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 13.25/7.60 & ( ord_less_int @ J3 @ K )
% 13.25/7.60 & ( N
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 13.25/7.60 => ( P @ J3 ) ) )
% 13.25/7.60 & ( ( ord_less_int @ K @ zero_zero_int )
% 13.25/7.60 => ! [I4: int,J3: int] :
% 13.25/7.60 ( ( ( ord_less_int @ K @ J3 )
% 13.25/7.60 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 13.25/7.60 & ( N
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 13.25/7.60 => ( P @ J3 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2566_mod__pos__geq,axiom,
% 13.25/7.60 ! [L2: int,K: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ L2 )
% 13.25/7.60 => ( ( ord_less_eq_int @ L2 @ K )
% 13.25/7.60 => ( ( modulo_modulo_int @ K @ L2 )
% 13.25/7.60 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L2 ) @ L2 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2567_q__pos__lemma,axiom,
% 13.25/7.60 ! [B2: int,Q4: int,R3: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) )
% 13.25/7.60 => ( ( ord_less_int @ R3 @ B2 )
% 13.25/7.60 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 13.25/7.60 => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2568_neg__mod__conj,axiom,
% 13.25/7.60 ! [B: int,A: int] :
% 13.25/7.60 ( ( ord_less_int @ B @ zero_zero_int )
% 13.25/7.60 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 13.25/7.60 & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2569_pos__mod__conj,axiom,
% 13.25/7.60 ! [B: int,A: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 13.25/7.60 & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2570_int__mod__neg__eq,axiom,
% 13.25/7.60 ! [A: int,B: int,Q2: int,R: int] :
% 13.25/7.60 ( ( A
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 13.25/7.60 => ( ( ord_less_int @ B @ R )
% 13.25/7.60 => ( ( modulo_modulo_int @ A @ B )
% 13.25/7.60 = R ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2571_int__mod__pos__eq,axiom,
% 13.25/7.60 ! [A: int,B: int,Q2: int,R: int] :
% 13.25/7.60 ( ( A
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 13.25/7.60 => ( ( ord_less_int @ R @ B )
% 13.25/7.60 => ( ( modulo_modulo_int @ A @ B )
% 13.25/7.60 = R ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2572_zdiv__mono2__lemma,axiom,
% 13.25/7.60 ! [B: int,Q2: int,R: int,B2: int,Q4: int,R3: int] :
% 13.25/7.60 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) )
% 13.25/7.60 => ( ( ord_less_int @ R3 @ B2 )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 13.25/7.60 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 13.25/7.60 => ( ( ord_less_eq_int @ B2 @ B )
% 13.25/7.60 => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2573_zmod__trivial__iff,axiom,
% 13.25/7.60 ! [I: int,K: int] :
% 13.25/7.60 ( ( ( modulo_modulo_int @ I @ K )
% 13.25/7.60 = I )
% 13.25/7.60 = ( ( K = zero_zero_int )
% 13.25/7.60 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 13.25/7.60 & ( ord_less_int @ I @ K ) )
% 13.25/7.60 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 13.25/7.60 & ( ord_less_int @ K @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2574_zdiv__mono2__neg__lemma,axiom,
% 13.25/7.60 ! [B: int,Q2: int,R: int,B2: int,Q4: int,R3: int] :
% 13.25/7.60 ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R )
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) )
% 13.25/7.60 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R3 ) @ zero_zero_int )
% 13.25/7.60 => ( ( ord_less_int @ R @ B )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
% 13.25/7.60 => ( ( ord_less_int @ zero_zero_int @ B2 )
% 13.25/7.60 => ( ( ord_less_eq_int @ B2 @ B )
% 13.25/7.60 => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2575_unique__quotient__lemma,axiom,
% 13.25/7.60 ! [B: int,Q4: int,R3: int,Q2: int,R: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
% 13.25/7.60 => ( ( ord_less_int @ R3 @ B )
% 13.25/7.60 => ( ( ord_less_int @ R @ B )
% 13.25/7.60 => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2576_neg__mod__sign,axiom,
% 13.25/7.60 ! [L2: int,K: int] :
% 13.25/7.60 ( ( ord_less_int @ L2 @ zero_zero_int )
% 13.25/7.60 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L2 ) @ zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2577_Euclidean__Division_Opos__mod__sign,axiom,
% 13.25/7.60 ! [L2: int,K: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ L2 )
% 13.25/7.60 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2578_unique__quotient__lemma__neg,axiom,
% 13.25/7.60 ! [B: int,Q4: int,R3: int,Q2: int,R: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ R @ zero_zero_int )
% 13.25/7.60 => ( ( ord_less_int @ B @ R )
% 13.25/7.60 => ( ( ord_less_int @ B @ R3 )
% 13.25/7.60 => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2579_mod__pos__neg__trivial,axiom,
% 13.25/7.60 ! [K: int,L2: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ K )
% 13.25/7.60 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 13.25/7.60 => ( ( modulo_modulo_int @ K @ L2 )
% 13.25/7.60 = ( plus_plus_int @ K @ L2 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2580_zmod__eq__0__iff,axiom,
% 13.25/7.60 ! [M: int,D: int] :
% 13.25/7.60 ( ( ( modulo_modulo_int @ M @ D )
% 13.25/7.60 = zero_zero_int )
% 13.25/7.60 = ( ? [Q5: int] :
% 13.25/7.60 ( M
% 13.25/7.60 = ( times_times_int @ D @ Q5 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2581_zmod__eq__0D,axiom,
% 13.25/7.60 ! [M: int,D: int] :
% 13.25/7.60 ( ( ( modulo_modulo_int @ M @ D )
% 13.25/7.60 = zero_zero_int )
% 13.25/7.60 => ? [Q3: int] :
% 13.25/7.60 ( M
% 13.25/7.60 = ( times_times_int @ D @ Q3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2582_zmod__le__nonneg__dividend,axiom,
% 13.25/7.60 ! [M: int,K: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 13.25/7.60 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2583_bot__enat__def,axiom,
% 13.25/7.60 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2584_Euclidean__Division_Opos__mod__bound,axiom,
% 13.25/7.60 ! [L2: int,K: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ L2 )
% 13.25/7.60 => ( ord_less_int @ ( modulo_modulo_int @ K @ L2 ) @ L2 ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2585_neg__mod__bound,axiom,
% 13.25/7.60 ! [L2: int,K: int] :
% 13.25/7.60 ( ( ord_less_int @ L2 @ zero_zero_int )
% 13.25/7.60 => ( ord_less_int @ L2 @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2586_bot__nat__def,axiom,
% 13.25/7.60 bot_bot_nat = zero_zero_nat ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2587_imult__is__0,axiom,
% 13.25/7.60 ! [M: extended_enat,N: extended_enat] :
% 13.25/7.60 ( ( ( times_7803423173614009249d_enat @ M @ N )
% 13.25/7.60 = zero_z5237406670263579293d_enat )
% 13.25/7.60 = ( ( M = zero_z5237406670263579293d_enat )
% 13.25/7.60 | ( N = zero_z5237406670263579293d_enat ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2588_zero__one__enat__neq_I1_J,axiom,
% 13.25/7.60 zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2589_set__bit__greater__eq,axiom,
% 13.25/7.60 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2590_verit__la__generic,axiom,
% 13.25/7.60 ! [A: int,X: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ A @ X )
% 13.25/7.60 | ( A = X )
% 13.25/7.60 | ( ord_less_eq_int @ X @ A ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2591_subset__eq__atLeast0__atMost__finite,axiom,
% 13.25/7.60 ! [N3: set_nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_set_nat @ N3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.60 => ( finite_finite_nat @ N3 ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2592_atLeastatMost__psubset__iff,axiom,
% 13.25/7.60 ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 13.25/7.60 ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 13.25/7.60 = ( ( ~ ( ord_less_eq_set_int @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_set_int @ C @ A )
% 13.25/7.60 & ( ord_less_eq_set_int @ B @ D )
% 13.25/7.60 & ( ( ord_less_set_int @ C @ A )
% 13.25/7.60 | ( ord_less_set_int @ B @ D ) ) ) )
% 13.25/7.60 & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2593_atLeastatMost__psubset__iff,axiom,
% 13.25/7.60 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.25/7.60 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 13.25/7.60 = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_rat @ C @ A )
% 13.25/7.60 & ( ord_less_eq_rat @ B @ D )
% 13.25/7.60 & ( ( ord_less_rat @ C @ A )
% 13.25/7.60 | ( ord_less_rat @ B @ D ) ) ) )
% 13.25/7.60 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2594_atLeastatMost__psubset__iff,axiom,
% 13.25/7.60 ! [A: num,B: num,C: num,D: num] :
% 13.25/7.60 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 13.25/7.60 = ( ( ~ ( ord_less_eq_num @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_num @ C @ A )
% 13.25/7.60 & ( ord_less_eq_num @ B @ D )
% 13.25/7.60 & ( ( ord_less_num @ C @ A )
% 13.25/7.60 | ( ord_less_num @ B @ D ) ) ) )
% 13.25/7.60 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2595_atLeastatMost__psubset__iff,axiom,
% 13.25/7.60 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.25/7.60 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 13.25/7.60 = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_nat @ C @ A )
% 13.25/7.60 & ( ord_less_eq_nat @ B @ D )
% 13.25/7.60 & ( ( ord_less_nat @ C @ A )
% 13.25/7.60 | ( ord_less_nat @ B @ D ) ) ) )
% 13.25/7.60 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2596_atLeastatMost__psubset__iff,axiom,
% 13.25/7.60 ! [A: int,B: int,C: int,D: int] :
% 13.25/7.60 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 13.25/7.60 = ( ( ~ ( ord_less_eq_int @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_int @ C @ A )
% 13.25/7.60 & ( ord_less_eq_int @ B @ D )
% 13.25/7.60 & ( ( ord_less_int @ C @ A )
% 13.25/7.60 | ( ord_less_int @ B @ D ) ) ) )
% 13.25/7.60 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2597_atLeastatMost__psubset__iff,axiom,
% 13.25/7.60 ! [A: real,B: real,C: real,D: real] :
% 13.25/7.60 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 13.25/7.60 = ( ( ~ ( ord_less_eq_real @ A @ B )
% 13.25/7.60 | ( ( ord_less_eq_real @ C @ A )
% 13.25/7.60 & ( ord_less_eq_real @ B @ D )
% 13.25/7.60 & ( ( ord_less_real @ C @ A )
% 13.25/7.60 | ( ord_less_real @ B @ D ) ) ) )
% 13.25/7.60 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2598_infinite__Icc,axiom,
% 13.25/7.60 ! [A: rat,B: rat] :
% 13.25/7.60 ( ( ord_less_rat @ A @ B )
% 13.25/7.60 => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2599_infinite__Icc,axiom,
% 13.25/7.60 ! [A: real,B: real] :
% 13.25/7.60 ( ( ord_less_real @ A @ B )
% 13.25/7.60 => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2600_num_Osize_I4_J,axiom,
% 13.25/7.60 ( ( size_size_num @ one )
% 13.25/7.60 = zero_zero_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2601_vebt__succ_Osimps_I3_J,axiom,
% 13.25/7.60 ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 13.25/7.60 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 13.25/7.60 = none_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2602_vebt__succ_Osimps_I4_J,axiom,
% 13.25/7.60 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
% 13.25/7.60 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
% 13.25/7.60 = none_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2603_vebt__mint_Osimps_I2_J,axiom,
% 13.25/7.60 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 13.25/7.60 ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 13.25/7.60 = none_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2604_vebt__maxt_Osimps_I2_J,axiom,
% 13.25/7.60 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 13.25/7.60 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 13.25/7.60 = none_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2605_invar__vebt_Ointros_I2_J,axiom,
% 13.25/7.60 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 13.25/7.60 ( ! [X3: vEBT_VEBT] :
% 13.25/7.60 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.60 => ( vEBT_invar_vebt @ X3 @ N ) )
% 13.25/7.60 => ( ( vEBT_invar_vebt @ Summary @ M )
% 13.25/7.60 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 13.25/7.60 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 => ( ( M = N )
% 13.25/7.60 => ( ( Deg
% 13.25/7.60 = ( plus_plus_nat @ N @ M ) )
% 13.25/7.60 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 13.25/7.60 => ( ! [X3: vEBT_VEBT] :
% 13.25/7.60 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.60 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 13.25/7.60 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2606_zle__diff1__eq,axiom,
% 13.25/7.60 ! [W: int,Z: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
% 13.25/7.60 = ( ord_less_int @ W @ Z ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2607_zle__add1__eq__le,axiom,
% 13.25/7.60 ! [W: int,Z: int] :
% 13.25/7.60 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 13.25/7.60 = ( ord_less_eq_int @ W @ Z ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2608_both__member__options__from__chilf__to__complete__tree,axiom,
% 13.25/7.60 ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 13.25/7.60 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 13.25/7.60 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) )
% 13.25/7.60 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2609_member__inv,axiom,
% 13.25/7.60 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 13.25/7.60 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.60 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 13.25/7.60 & ( ( X = Mi )
% 13.25/7.60 | ( X = Ma )
% 13.25/7.60 | ( ( ord_less_nat @ X @ Ma )
% 13.25/7.60 & ( ord_less_nat @ Mi @ X )
% 13.25/7.60 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.60 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2610_mintlistlength,axiom,
% 13.25/7.60 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 13.25/7.60 => ( ( Mi != Ma )
% 13.25/7.60 => ( ( ord_less_nat @ Mi @ Ma )
% 13.25/7.60 & ? [M4: nat] :
% 13.25/7.60 ( ( ( some_nat @ M4 )
% 13.25/7.60 = ( vEBT_vebt_mint @ Summary ) )
% 13.25/7.60 & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2611_both__member__options__from__complete__tree__to__child,axiom,
% 13.25/7.60 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 13.25/7.60 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.60 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) )
% 13.25/7.60 | ( X = Mi )
% 13.25/7.60 | ( X = Ma ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2612_succ__list__to__short,axiom,
% 13.25/7.60 ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 13.25/7.60 => ( ( ord_less_eq_nat @ Mi @ X )
% 13.25/7.60 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.60 = none_nat ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2613__C1_C,axiom,
% 13.25/7.60 ( ( ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.60 != none_nat )
% 13.25/7.60 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 13.25/7.60 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 13.25/7.60 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 13.25/7.60 & ( ~ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.60 != none_nat )
% 13.25/7.60 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 13.25/7.60 => ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 = none_nat )
% 13.25/7.60 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 13.25/7.60 = none_nat ) )
% 13.25/7.60 & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 != none_nat )
% 13.25/7.60 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 13.25/7.60 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2614_succ__min,axiom,
% 13.25/7.60 ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 13.25/7.60 => ( ( ord_less_nat @ X @ Mi )
% 13.25/7.60 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.60 = ( some_nat @ Mi ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2615_mi__eq__ma__no__ch,axiom,
% 13.25/7.60 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 13.25/7.60 => ( ( Mi = Ma )
% 13.25/7.60 => ( ! [X5: vEBT_VEBT] :
% 13.25/7.60 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.60 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
% 13.25/7.60 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2616_double__eq__0__iff,axiom,
% 13.25/7.60 ! [A: real] :
% 13.25/7.60 ( ( ( plus_plus_real @ A @ A )
% 13.25/7.60 = zero_zero_real )
% 13.25/7.60 = ( A = zero_zero_real ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2617_double__eq__0__iff,axiom,
% 13.25/7.60 ! [A: rat] :
% 13.25/7.60 ( ( ( plus_plus_rat @ A @ A )
% 13.25/7.60 = zero_zero_rat )
% 13.25/7.60 = ( A = zero_zero_rat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2618_double__eq__0__iff,axiom,
% 13.25/7.60 ! [A: int] :
% 13.25/7.60 ( ( ( plus_plus_int @ A @ A )
% 13.25/7.60 = zero_zero_int )
% 13.25/7.60 = ( A = zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2619_insert__simp__mima,axiom,
% 13.25/7.60 ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 13.25/7.60 ( ( ( X = Mi )
% 13.25/7.60 | ( X = Ma ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 13.25/7.60 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.60 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2620_mi__ma__2__deg,axiom,
% 13.25/7.60 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 13.25/7.60 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 13.25/7.60 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2621_summaxma,axiom,
% 13.25/7.60 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 13.25/7.60 => ( ( Mi != Ma )
% 13.25/7.60 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 13.25/7.60 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2622_option_Ocollapse,axiom,
% 13.25/7.60 ! [Option: option_nat] :
% 13.25/7.60 ( ( Option != none_nat )
% 13.25/7.60 => ( ( some_nat @ ( the_nat @ Option ) )
% 13.25/7.60 = Option ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2623_option_Ocollapse,axiom,
% 13.25/7.60 ! [Option: option4927543243414619207at_nat] :
% 13.25/7.60 ( ( Option != none_P5556105721700978146at_nat )
% 13.25/7.60 => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 13.25/7.60 = Option ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2624_option_Ocollapse,axiom,
% 13.25/7.60 ! [Option: option_num] :
% 13.25/7.60 ( ( Option != none_num )
% 13.25/7.60 => ( ( some_num @ ( the_num @ Option ) )
% 13.25/7.60 = Option ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2625__092_060open_062vebt__succ_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ASome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
% 13.25/7.60 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 13.25/7.60 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2626__C2_C,axiom,
% 13.25/7.60 ( ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 = none_nat )
% 13.25/7.60 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 13.25/7.60 = none_nat ) )
% 13.25/7.60 & ( ( ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.60 != none_nat )
% 13.25/7.60 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
% 13.25/7.60 = ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ treeList @ ( the_nat @ ( vEBT_vebt_succ @ summary @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2627_option_Osel,axiom,
% 13.25/7.60 ! [X22: nat] :
% 13.25/7.60 ( ( the_nat @ ( some_nat @ X22 ) )
% 13.25/7.60 = X22 ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2628_option_Osel,axiom,
% 13.25/7.60 ! [X22: product_prod_nat_nat] :
% 13.25/7.60 ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 13.25/7.60 = X22 ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2629_option_Osel,axiom,
% 13.25/7.60 ! [X22: num] :
% 13.25/7.60 ( ( the_num @ ( some_num @ X22 ) )
% 13.25/7.60 = X22 ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2630_option_Oexpand,axiom,
% 13.25/7.60 ! [Option: option_nat,Option2: option_nat] :
% 13.25/7.60 ( ( ( Option = none_nat )
% 13.25/7.60 = ( Option2 = none_nat ) )
% 13.25/7.60 => ( ( ( Option != none_nat )
% 13.25/7.60 => ( ( Option2 != none_nat )
% 13.25/7.60 => ( ( the_nat @ Option )
% 13.25/7.60 = ( the_nat @ Option2 ) ) ) )
% 13.25/7.60 => ( Option = Option2 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2631_option_Oexpand,axiom,
% 13.25/7.60 ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 13.25/7.60 ( ( ( Option = none_P5556105721700978146at_nat )
% 13.25/7.60 = ( Option2 = none_P5556105721700978146at_nat ) )
% 13.25/7.60 => ( ( ( Option != none_P5556105721700978146at_nat )
% 13.25/7.60 => ( ( Option2 != none_P5556105721700978146at_nat )
% 13.25/7.60 => ( ( the_Pr8591224930841456533at_nat @ Option )
% 13.25/7.60 = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 13.25/7.60 => ( Option = Option2 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2632_option_Oexpand,axiom,
% 13.25/7.60 ! [Option: option_num,Option2: option_num] :
% 13.25/7.60 ( ( ( Option = none_num )
% 13.25/7.60 = ( Option2 = none_num ) )
% 13.25/7.60 => ( ( ( Option != none_num )
% 13.25/7.60 => ( ( Option2 != none_num )
% 13.25/7.60 => ( ( the_num @ Option )
% 13.25/7.60 = ( the_num @ Option2 ) ) ) )
% 13.25/7.60 => ( Option = Option2 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2633_option_Oexhaust__sel,axiom,
% 13.25/7.60 ! [Option: option_nat] :
% 13.25/7.60 ( ( Option != none_nat )
% 13.25/7.60 => ( Option
% 13.25/7.60 = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2634_option_Oexhaust__sel,axiom,
% 13.25/7.60 ! [Option: option4927543243414619207at_nat] :
% 13.25/7.60 ( ( Option != none_P5556105721700978146at_nat )
% 13.25/7.60 => ( Option
% 13.25/7.60 = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2635_option_Oexhaust__sel,axiom,
% 13.25/7.60 ! [Option: option_num] :
% 13.25/7.60 ( ( Option != none_num )
% 13.25/7.60 => ( Option
% 13.25/7.60 = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2636_vebt__mint_Osimps_I3_J,axiom,
% 13.25/7.60 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 13.25/7.60 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 13.25/7.60 = ( some_nat @ Mi ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2637_vebt__maxt_Osimps_I3_J,axiom,
% 13.25/7.60 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 13.25/7.60 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 13.25/7.60 = ( some_nat @ Ma ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2638_less__eq__int__code_I1_J,axiom,
% 13.25/7.60 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2639_less__int__code_I1_J,axiom,
% 13.25/7.60 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2640_times__int__code_I2_J,axiom,
% 13.25/7.60 ! [L2: int] :
% 13.25/7.60 ( ( times_times_int @ zero_zero_int @ L2 )
% 13.25/7.60 = zero_zero_int ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2641_times__int__code_I1_J,axiom,
% 13.25/7.60 ! [K: int] :
% 13.25/7.60 ( ( times_times_int @ K @ zero_zero_int )
% 13.25/7.60 = zero_zero_int ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2642_plus__int__code_I1_J,axiom,
% 13.25/7.60 ! [K: int] :
% 13.25/7.60 ( ( plus_plus_int @ K @ zero_zero_int )
% 13.25/7.60 = K ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2643_plus__int__code_I2_J,axiom,
% 13.25/7.60 ! [L2: int] :
% 13.25/7.60 ( ( plus_plus_int @ zero_zero_int @ L2 )
% 13.25/7.60 = L2 ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2644_minus__int__code_I1_J,axiom,
% 13.25/7.60 ! [K: int] :
% 13.25/7.60 ( ( minus_minus_int @ K @ zero_zero_int )
% 13.25/7.60 = K ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2645_int__distrib_I1_J,axiom,
% 13.25/7.60 ! [Z1: int,Z22: int,W: int] :
% 13.25/7.60 ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2646_int__distrib_I2_J,axiom,
% 13.25/7.60 ! [W: int,Z1: int,Z22: int] :
% 13.25/7.60 ( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
% 13.25/7.60 = ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2647_int__distrib_I3_J,axiom,
% 13.25/7.60 ! [Z1: int,Z22: int,W: int] :
% 13.25/7.60 ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
% 13.25/7.60 = ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2648_int__distrib_I4_J,axiom,
% 13.25/7.60 ! [W: int,Z1: int,Z22: int] :
% 13.25/7.60 ( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
% 13.25/7.60 = ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2649_zmult__zless__mono2,axiom,
% 13.25/7.60 ! [I: int,J: int,K: int] :
% 13.25/7.60 ( ( ord_less_int @ I @ J )
% 13.25/7.60 => ( ( ord_less_int @ zero_zero_int @ K )
% 13.25/7.60 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2650_odd__nonzero,axiom,
% 13.25/7.60 ! [Z: int] :
% 13.25/7.60 ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
% 13.25/7.60 != zero_zero_int ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2651_int__ge__induct,axiom,
% 13.25/7.60 ! [K: int,I: int,P: int > $o] :
% 13.25/7.60 ( ( ord_less_eq_int @ K @ I )
% 13.25/7.60 => ( ( P @ K )
% 13.25/7.60 => ( ! [I2: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ K @ I2 )
% 13.25/7.60 => ( ( P @ I2 )
% 13.25/7.60 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 13.25/7.60 => ( P @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2652_zless__add1__eq,axiom,
% 13.25/7.60 ! [W: int,Z: int] :
% 13.25/7.60 ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
% 13.25/7.60 = ( ( ord_less_int @ W @ Z )
% 13.25/7.60 | ( W = Z ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2653_int__gr__induct,axiom,
% 13.25/7.60 ! [K: int,I: int,P: int > $o] :
% 13.25/7.60 ( ( ord_less_int @ K @ I )
% 13.25/7.60 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 13.25/7.60 => ( ! [I2: int] :
% 13.25/7.60 ( ( ord_less_int @ K @ I2 )
% 13.25/7.60 => ( ( P @ I2 )
% 13.25/7.60 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 13.25/7.60 => ( P @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2654_int__le__induct,axiom,
% 13.25/7.60 ! [I: int,K: int,P: int > $o] :
% 13.25/7.60 ( ( ord_less_eq_int @ I @ K )
% 13.25/7.60 => ( ( P @ K )
% 13.25/7.60 => ( ! [I2: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ I2 @ K )
% 13.25/7.60 => ( ( P @ I2 )
% 13.25/7.60 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 13.25/7.60 => ( P @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2655_int__less__induct,axiom,
% 13.25/7.60 ! [I: int,K: int,P: int > $o] :
% 13.25/7.60 ( ( ord_less_int @ I @ K )
% 13.25/7.60 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 13.25/7.60 => ( ! [I2: int] :
% 13.25/7.60 ( ( ord_less_int @ I2 @ K )
% 13.25/7.60 => ( ( P @ I2 )
% 13.25/7.60 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 13.25/7.60 => ( P @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2656_int__one__le__iff__zero__less,axiom,
% 13.25/7.60 ! [Z: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ one_one_int @ Z )
% 13.25/7.60 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2657_pos__zmult__eq__1__iff,axiom,
% 13.25/7.60 ! [M: int,N: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ M )
% 13.25/7.60 => ( ( ( times_times_int @ M @ N )
% 13.25/7.60 = one_one_int )
% 13.25/7.60 = ( ( M = one_one_int )
% 13.25/7.60 & ( N = one_one_int ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2658_odd__less__0__iff,axiom,
% 13.25/7.60 ! [Z: int] :
% 13.25/7.60 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 13.25/7.60 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2659_add1__zle__eq,axiom,
% 13.25/7.60 ! [W: int,Z: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
% 13.25/7.60 = ( ord_less_int @ W @ Z ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2660_zless__imp__add1__zle,axiom,
% 13.25/7.60 ! [W: int,Z: int] :
% 13.25/7.60 ( ( ord_less_int @ W @ Z )
% 13.25/7.60 => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2661_int__induct,axiom,
% 13.25/7.60 ! [P: int > $o,K: int,I: int] :
% 13.25/7.60 ( ( P @ K )
% 13.25/7.60 => ( ! [I2: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ K @ I2 )
% 13.25/7.60 => ( ( P @ I2 )
% 13.25/7.60 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 13.25/7.60 => ( ! [I2: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ I2 @ K )
% 13.25/7.60 => ( ( P @ I2 )
% 13.25/7.60 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 13.25/7.60 => ( P @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2662_invar__vebt_Ointros_I4_J,axiom,
% 13.25/7.60 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 13.25/7.60 ( ! [X3: vEBT_VEBT] :
% 13.25/7.60 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.60 => ( vEBT_invar_vebt @ X3 @ N ) )
% 13.25/7.60 => ( ( vEBT_invar_vebt @ Summary @ M )
% 13.25/7.60 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 13.25/7.60 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 => ( ( M = N )
% 13.25/7.60 => ( ( Deg
% 13.25/7.60 = ( plus_plus_nat @ N @ M ) )
% 13.25/7.60 => ( ! [I2: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
% 13.25/7.60 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 13.25/7.60 => ( ( ( Mi = Ma )
% 13.25/7.60 => ! [X3: vEBT_VEBT] :
% 13.25/7.60 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.60 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 13.25/7.60 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 13.25/7.60 => ( ( ( Mi != Ma )
% 13.25/7.60 => ! [I2: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.60 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 13.25/7.60 = I2 )
% 13.25/7.60 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 13.25/7.60 & ! [X3: nat] :
% 13.25/7.60 ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 13.25/7.60 = I2 )
% 13.25/7.60 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ Mi @ X3 )
% 13.25/7.60 & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 13.25/7.60 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2663_le__imp__0__less,axiom,
% 13.25/7.60 ! [Z: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.60 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2664_nested__mint,axiom,
% 13.25/7.60 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 13.25/7.60 => ( ( N
% 13.25/7.60 = ( suc @ ( suc @ Va ) ) )
% 13.25/7.60 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 13.25/7.60 => ( ( Ma != Mi )
% 13.25/7.60 => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2665_divmod__step__eq,axiom,
% 13.25/7.60 ! [L2: num,R: nat,Q2: nat] :
% 13.25/7.60 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
% 13.25/7.60 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 13.25/7.60 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L2 ) ) ) ) )
% 13.25/7.60 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R )
% 13.25/7.60 => ( ( unique5026877609467782581ep_nat @ L2 @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 13.25/7.60 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2666_divmod__step__eq,axiom,
% 13.25/7.60 ! [L2: num,R: int,Q2: int] :
% 13.25/7.60 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
% 13.25/7.60 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.60 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L2 ) ) ) ) )
% 13.25/7.60 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R )
% 13.25/7.60 => ( ( unique5024387138958732305ep_int @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.60 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2667_divmod__step__eq,axiom,
% 13.25/7.60 ! [L2: num,R: code_integer,Q2: code_integer] :
% 13.25/7.60 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
% 13.25/7.60 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 13.25/7.60 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L2 ) ) ) ) )
% 13.25/7.60 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R )
% 13.25/7.60 => ( ( unique4921790084139445826nteger @ L2 @ ( produc1086072967326762835nteger @ Q2 @ R ) )
% 13.25/7.60 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2668_cpmi,axiom,
% 13.25/7.60 ! [D4: int,P: int > $o,P5: int > $o,B5: set_int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.60 => ( ? [Z3: int] :
% 13.25/7.60 ! [X3: int] :
% 13.25/7.60 ( ( ord_less_int @ X3 @ Z3 )
% 13.25/7.60 => ( ( P @ X3 )
% 13.25/7.60 = ( P5 @ X3 ) ) )
% 13.25/7.60 => ( ! [X3: int] :
% 13.25/7.60 ( ! [Xa2: int] :
% 13.25/7.60 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 => ! [Xb2: int] :
% 13.25/7.60 ( ( member_int @ Xb2 @ B5 )
% 13.25/7.60 => ( X3
% 13.25/7.60 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.60 => ( ( P @ X3 )
% 13.25/7.60 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 13.25/7.60 => ( ! [X3: int,K3: int] :
% 13.25/7.60 ( ( P5 @ X3 )
% 13.25/7.60 = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 13.25/7.60 => ( ( ? [X6: int] : ( P @ X6 ) )
% 13.25/7.60 = ( ? [X2: int] :
% 13.25/7.60 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 & ( P5 @ X2 ) )
% 13.25/7.60 | ? [X2: int] :
% 13.25/7.60 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 & ? [Y: int] :
% 13.25/7.60 ( ( member_int @ Y @ B5 )
% 13.25/7.60 & ( P @ ( plus_plus_int @ Y @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2669_cppi,axiom,
% 13.25/7.60 ! [D4: int,P: int > $o,P5: int > $o,A2: set_int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.60 => ( ? [Z3: int] :
% 13.25/7.60 ! [X3: int] :
% 13.25/7.60 ( ( ord_less_int @ Z3 @ X3 )
% 13.25/7.60 => ( ( P @ X3 )
% 13.25/7.60 = ( P5 @ X3 ) ) )
% 13.25/7.60 => ( ! [X3: int] :
% 13.25/7.60 ( ! [Xa2: int] :
% 13.25/7.60 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 => ! [Xb2: int] :
% 13.25/7.60 ( ( member_int @ Xb2 @ A2 )
% 13.25/7.60 => ( X3
% 13.25/7.60 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.60 => ( ( P @ X3 )
% 13.25/7.60 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 13.25/7.60 => ( ! [X3: int,K3: int] :
% 13.25/7.60 ( ( P5 @ X3 )
% 13.25/7.60 = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 13.25/7.60 => ( ( ? [X6: int] : ( P @ X6 ) )
% 13.25/7.60 = ( ? [X2: int] :
% 13.25/7.60 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 & ( P5 @ X2 ) )
% 13.25/7.60 | ? [X2: int] :
% 13.25/7.60 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 & ? [Y: int] :
% 13.25/7.60 ( ( member_int @ Y @ A2 )
% 13.25/7.60 & ( P @ ( minus_minus_int @ Y @ X2 ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2670_bset_I6_J,axiom,
% 13.25/7.60 ! [D4: int,B5: set_int,T: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.60 => ! [X5: int] :
% 13.25/7.60 ( ! [Xa3: int] :
% 13.25/7.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 => ! [Xb3: int] :
% 13.25/7.60 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.60 => ( X5
% 13.25/7.60 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ X5 @ T )
% 13.25/7.60 => ( ord_less_eq_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2671_bset_I8_J,axiom,
% 13.25/7.60 ! [D4: int,T: int,B5: set_int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.60 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 13.25/7.60 => ! [X5: int] :
% 13.25/7.60 ( ! [Xa3: int] :
% 13.25/7.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 => ! [Xb3: int] :
% 13.25/7.60 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.60 => ( X5
% 13.25/7.60 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ T @ X5 )
% 13.25/7.60 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2672_aset_I6_J,axiom,
% 13.25/7.60 ! [D4: int,T: int,A2: set_int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.60 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 13.25/7.60 => ! [X5: int] :
% 13.25/7.60 ( ! [Xa3: int] :
% 13.25/7.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 => ! [Xb3: int] :
% 13.25/7.60 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.60 => ( X5
% 13.25/7.60 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ X5 @ T )
% 13.25/7.60 => ( ord_less_eq_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2673_aset_I8_J,axiom,
% 13.25/7.60 ! [D4: int,A2: set_int,T: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.60 => ! [X5: int] :
% 13.25/7.60 ( ! [Xa3: int] :
% 13.25/7.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 => ! [Xb3: int] :
% 13.25/7.60 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.60 => ( X5
% 13.25/7.60 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ T @ X5 )
% 13.25/7.60 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2674_aset_I7_J,axiom,
% 13.25/7.60 ! [D4: int,A2: set_int,T: int] :
% 13.25/7.60 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.60 => ! [X5: int] :
% 13.25/7.60 ( ! [Xa3: int] :
% 13.25/7.60 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.60 => ! [Xb3: int] :
% 13.25/7.60 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.60 => ( X5
% 13.25/7.60 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.60 => ( ( ord_less_int @ T @ X5 )
% 13.25/7.60 => ( ord_less_int @ T @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2675_even__odd__cases,axiom,
% 13.25/7.60 ! [X: nat] :
% 13.25/7.60 ( ! [N4: nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( plus_plus_nat @ N4 @ N4 ) )
% 13.25/7.60 => ~ ! [N4: nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2676_deg__SUcn__Node,axiom,
% 13.25/7.60 ! [Tree: vEBT_VEBT,N: nat] :
% 13.25/7.60 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 13.25/7.60 => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 13.25/7.60 ( Tree
% 13.25/7.60 = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList3 @ S3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2677_nat_Oinject,axiom,
% 13.25/7.60 ! [X22: nat,Y2: nat] :
% 13.25/7.60 ( ( ( suc @ X22 )
% 13.25/7.60 = ( suc @ Y2 ) )
% 13.25/7.60 = ( X22 = Y2 ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2678_old_Onat_Oinject,axiom,
% 13.25/7.60 ! [Nat: nat,Nat2: nat] :
% 13.25/7.60 ( ( ( suc @ Nat )
% 13.25/7.60 = ( suc @ Nat2 ) )
% 13.25/7.60 = ( Nat = Nat2 ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2679_Suc__less__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 13.25/7.60 = ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2680_Suc__mono,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ N )
% 13.25/7.60 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2681_lessI,axiom,
% 13.25/7.60 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2682_Suc__le__mono,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 13.25/7.60 = ( ord_less_eq_nat @ N @ M ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2683_add__Suc__right,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 13.25/7.60 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2684_Suc__diff__diff,axiom,
% 13.25/7.60 ! [M: nat,N: nat,K: nat] :
% 13.25/7.60 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 13.25/7.60 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2685_diff__Suc__Suc,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 13.25/7.60 = ( minus_minus_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2686_power__0__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 13.25/7.60 = zero_zero_rat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2687_power__0__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 13.25/7.60 = zero_zero_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2688_power__0__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 13.25/7.60 = zero_zero_real ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2689_power__0__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 13.25/7.60 = zero_zero_int ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2690_power__0__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 13.25/7.60 = zero_zero_complex ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2691_power__Suc0__right,axiom,
% 13.25/7.60 ! [A: nat] :
% 13.25/7.60 ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 13.25/7.60 = A ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2692_power__Suc0__right,axiom,
% 13.25/7.60 ! [A: real] :
% 13.25/7.60 ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 13.25/7.60 = A ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2693_power__Suc0__right,axiom,
% 13.25/7.60 ! [A: int] :
% 13.25/7.60 ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 13.25/7.60 = A ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2694_power__Suc0__right,axiom,
% 13.25/7.60 ! [A: complex] :
% 13.25/7.60 ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 13.25/7.60 = A ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2695_less__Suc0,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 13.25/7.60 = ( N = zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2696_zero__less__Suc,axiom,
% 13.25/7.60 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2697_one__eq__mult__iff,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( suc @ zero_zero_nat )
% 13.25/7.60 = ( times_times_nat @ M @ N ) )
% 13.25/7.60 = ( ( M
% 13.25/7.60 = ( suc @ zero_zero_nat ) )
% 13.25/7.60 & ( N
% 13.25/7.60 = ( suc @ zero_zero_nat ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2698_mult__eq__1__iff,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( times_times_nat @ M @ N )
% 13.25/7.60 = ( suc @ zero_zero_nat ) )
% 13.25/7.60 = ( ( M
% 13.25/7.60 = ( suc @ zero_zero_nat ) )
% 13.25/7.60 & ( N
% 13.25/7.60 = ( suc @ zero_zero_nat ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2699_div__by__Suc__0,axiom,
% 13.25/7.60 ! [M: nat] :
% 13.25/7.60 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 13.25/7.60 = M ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2700_mult__Suc__right,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( times_times_nat @ M @ ( suc @ N ) )
% 13.25/7.60 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2701_power__Suc__0,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.60 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2702_nat__power__eq__Suc__0__iff,axiom,
% 13.25/7.60 ! [X: nat,M: nat] :
% 13.25/7.60 ( ( ( power_power_nat @ X @ M )
% 13.25/7.60 = ( suc @ zero_zero_nat ) )
% 13.25/7.60 = ( ( M = zero_zero_nat )
% 13.25/7.60 | ( X
% 13.25/7.60 = ( suc @ zero_zero_nat ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2703_diff__Suc__1,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 13.25/7.60 = N ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2704_mod__by__Suc__0,axiom,
% 13.25/7.60 ! [M: nat] :
% 13.25/7.60 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 13.25/7.60 = zero_zero_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2705_Suc__pred,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 13.25/7.60 = N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2706_one__le__mult__iff,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 13.25/7.60 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 13.25/7.60 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2707_diff__Suc__diff__eq1,axiom,
% 13.25/7.60 ! [K: nat,J: nat,I: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ K @ J )
% 13.25/7.60 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 13.25/7.60 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2708_diff__Suc__diff__eq2,axiom,
% 13.25/7.60 ! [K: nat,J: nat,I: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ K @ J )
% 13.25/7.60 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 13.25/7.60 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2709_Suc__numeral,axiom,
% 13.25/7.60 ! [N: num] :
% 13.25/7.60 ( ( suc @ ( numeral_numeral_nat @ N ) )
% 13.25/7.60 = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2710_Suc__mod__mult__self4,axiom,
% 13.25/7.60 ! [N: nat,K: nat,M: nat] :
% 13.25/7.60 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K ) @ M ) ) @ N )
% 13.25/7.60 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2711_Suc__mod__mult__self3,axiom,
% 13.25/7.60 ! [K: nat,N: nat,M: nat] :
% 13.25/7.60 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N ) @ M ) ) @ N )
% 13.25/7.60 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2712_Suc__mod__mult__self2,axiom,
% 13.25/7.60 ! [M: nat,N: nat,K: nat] :
% 13.25/7.60 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N @ K ) ) ) @ N )
% 13.25/7.60 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2713_Suc__mod__mult__self1,axiom,
% 13.25/7.60 ! [M: nat,K: nat,N: nat] :
% 13.25/7.60 ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N ) ) ) @ N )
% 13.25/7.60 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2714_add__2__eq__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.60 = ( suc @ ( suc @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2715_add__2__eq__Suc_H,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( suc @ ( suc @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2716_div2__Suc__Suc,axiom,
% 13.25/7.60 ! [M: nat] :
% 13.25/7.60 ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2717_Suc__1,axiom,
% 13.25/7.60 ( ( suc @ one_one_nat )
% 13.25/7.60 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2718_mod2__Suc__Suc,axiom,
% 13.25/7.60 ! [M: nat] :
% 13.25/7.60 ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2719_Suc__diff__1,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 13.25/7.60 = N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2720_Suc__times__numeral__mod__eq,axiom,
% 13.25/7.60 ! [K: num,N: nat] :
% 13.25/7.60 ( ( ( numeral_numeral_nat @ K )
% 13.25/7.60 != one_one_nat )
% 13.25/7.60 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N ) ) @ ( numeral_numeral_nat @ K ) )
% 13.25/7.60 = one_one_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2721_not__mod2__eq__Suc__0__eq__0,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 != ( suc @ zero_zero_nat ) )
% 13.25/7.60 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.60 = zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2722_Suc__inject,axiom,
% 13.25/7.60 ! [X: nat,Y3: nat] :
% 13.25/7.60 ( ( ( suc @ X )
% 13.25/7.60 = ( suc @ Y3 ) )
% 13.25/7.60 => ( X = Y3 ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2723_n__not__Suc__n,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( N
% 13.25/7.60 != ( suc @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2724_VEBT__internal_Ooption__shift_Ocases,axiom,
% 13.25/7.60 ! [X: produc8306885398267862888on_nat] :
% 13.25/7.60 ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 13.25/7.60 => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 13.25/7.60 => ~ ! [F2: nat > nat > nat,A5: nat,B3: nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A5 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2725_VEBT__internal_Ooption__shift_Ocases,axiom,
% 13.25/7.60 ! [X: produc5542196010084753463at_nat] :
% 13.25/7.60 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 13.25/7.60 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 13.25/7.60 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A5: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A5 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2726_VEBT__internal_Ooption__shift_Ocases,axiom,
% 13.25/7.60 ! [X: produc1193250871479095198on_num] :
% 13.25/7.60 ( ! [Uu2: num > num > num,Uv2: option_num] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 13.25/7.60 => ( ! [Uw2: num > num > num,V2: num] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 13.25/7.60 => ~ ! [F2: num > num > num,A5: num,B3: num] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A5 ) @ ( some_num @ B3 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2727_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 13.25/7.60 ! [X: produc2233624965454879586on_nat] :
% 13.25/7.60 ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 13.25/7.60 => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 13.25/7.60 => ~ ! [F2: nat > nat > $o,X3: nat,Y4: nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2728_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 13.25/7.60 ! [X: produc5491161045314408544at_nat] :
% 13.25/7.60 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 13.25/7.60 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 13.25/7.60 => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2729_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 13.25/7.60 ! [X: produc7036089656553540234on_num] :
% 13.25/7.60 ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 13.25/7.60 => ( ! [Uw2: num > num > $o,V2: num] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 13.25/7.60 => ~ ! [F2: num > num > $o,X3: num,Y4: num] :
% 13.25/7.60 ( X
% 13.25/7.60 != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2730_not0__implies__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( N != zero_zero_nat )
% 13.25/7.60 => ? [M4: nat] :
% 13.25/7.60 ( N
% 13.25/7.60 = ( suc @ M4 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2731_Zero__not__Suc,axiom,
% 13.25/7.60 ! [M: nat] :
% 13.25/7.60 ( zero_zero_nat
% 13.25/7.60 != ( suc @ M ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2732_Zero__neq__Suc,axiom,
% 13.25/7.60 ! [M: nat] :
% 13.25/7.60 ( zero_zero_nat
% 13.25/7.60 != ( suc @ M ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2733_Suc__neq__Zero,axiom,
% 13.25/7.60 ! [M: nat] :
% 13.25/7.60 ( ( suc @ M )
% 13.25/7.60 != zero_zero_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2734_zero__induct,axiom,
% 13.25/7.60 ! [P: nat > $o,K: nat] :
% 13.25/7.60 ( ( P @ K )
% 13.25/7.60 => ( ! [N4: nat] :
% 13.25/7.60 ( ( P @ ( suc @ N4 ) )
% 13.25/7.60 => ( P @ N4 ) )
% 13.25/7.60 => ( P @ zero_zero_nat ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2735_diff__induct,axiom,
% 13.25/7.60 ! [P: nat > nat > $o,M: nat,N: nat] :
% 13.25/7.60 ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
% 13.25/7.60 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 13.25/7.60 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.60 ( ( P @ X3 @ Y4 )
% 13.25/7.60 => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
% 13.25/7.60 => ( P @ M @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2736_nat__induct,axiom,
% 13.25/7.60 ! [P: nat > $o,N: nat] :
% 13.25/7.60 ( ( P @ zero_zero_nat )
% 13.25/7.60 => ( ! [N4: nat] :
% 13.25/7.60 ( ( P @ N4 )
% 13.25/7.60 => ( P @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( P @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2737_old_Onat_Oexhaust,axiom,
% 13.25/7.60 ! [Y3: nat] :
% 13.25/7.60 ( ( Y3 != zero_zero_nat )
% 13.25/7.60 => ~ ! [Nat3: nat] :
% 13.25/7.60 ( Y3
% 13.25/7.60 != ( suc @ Nat3 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2738_nat_OdiscI,axiom,
% 13.25/7.60 ! [Nat: nat,X22: nat] :
% 13.25/7.60 ( ( Nat
% 13.25/7.60 = ( suc @ X22 ) )
% 13.25/7.60 => ( Nat != zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2739_old_Onat_Odistinct_I1_J,axiom,
% 13.25/7.60 ! [Nat2: nat] :
% 13.25/7.60 ( zero_zero_nat
% 13.25/7.60 != ( suc @ Nat2 ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2740_old_Onat_Odistinct_I2_J,axiom,
% 13.25/7.60 ! [Nat2: nat] :
% 13.25/7.60 ( ( suc @ Nat2 )
% 13.25/7.60 != zero_zero_nat ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2741_nat_Odistinct_I1_J,axiom,
% 13.25/7.60 ! [X22: nat] :
% 13.25/7.60 ( zero_zero_nat
% 13.25/7.60 != ( suc @ X22 ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2742_not__less__less__Suc__eq,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ~ ( ord_less_nat @ N @ M )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 13.25/7.60 = ( N = M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2743_strict__inc__induct,axiom,
% 13.25/7.60 ! [I: nat,J: nat,P: nat > $o] :
% 13.25/7.60 ( ( ord_less_nat @ I @ J )
% 13.25/7.60 => ( ! [I2: nat] :
% 13.25/7.60 ( ( J
% 13.25/7.60 = ( suc @ I2 ) )
% 13.25/7.60 => ( P @ I2 ) )
% 13.25/7.60 => ( ! [I2: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I2 @ J )
% 13.25/7.60 => ( ( P @ ( suc @ I2 ) )
% 13.25/7.60 => ( P @ I2 ) ) )
% 13.25/7.60 => ( P @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2744_less__Suc__induct,axiom,
% 13.25/7.60 ! [I: nat,J: nat,P: nat > nat > $o] :
% 13.25/7.60 ( ( ord_less_nat @ I @ J )
% 13.25/7.60 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 13.25/7.60 => ( ! [I2: nat,J2: nat,K3: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I2 @ J2 )
% 13.25/7.60 => ( ( ord_less_nat @ J2 @ K3 )
% 13.25/7.60 => ( ( P @ I2 @ J2 )
% 13.25/7.60 => ( ( P @ J2 @ K3 )
% 13.25/7.60 => ( P @ I2 @ K3 ) ) ) ) )
% 13.25/7.60 => ( P @ I @ J ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2745_less__trans__Suc,axiom,
% 13.25/7.60 ! [I: nat,J: nat,K: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I @ J )
% 13.25/7.60 => ( ( ord_less_nat @ J @ K )
% 13.25/7.60 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2746_Suc__less__SucD,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 13.25/7.60 => ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2747_less__antisym,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ~ ( ord_less_nat @ N @ M )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 13.25/7.60 => ( M = N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2748_Suc__less__eq2,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.60 = ( ? [M6: nat] :
% 13.25/7.60 ( ( M
% 13.25/7.60 = ( suc @ M6 ) )
% 13.25/7.60 & ( ord_less_nat @ N @ M6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2749_All__less__Suc,axiom,
% 13.25/7.60 ! [N: nat,P: nat > $o] :
% 13.25/7.60 ( ( ! [I4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 13.25/7.60 => ( P @ I4 ) ) )
% 13.25/7.60 = ( ( P @ N )
% 13.25/7.60 & ! [I4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I4 @ N )
% 13.25/7.60 => ( P @ I4 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2750_not__less__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ~ ( ord_less_nat @ M @ N ) )
% 13.25/7.60 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2751_less__Suc__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 13.25/7.60 = ( ( ord_less_nat @ M @ N )
% 13.25/7.60 | ( M = N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2752_Ex__less__Suc,axiom,
% 13.25/7.60 ! [N: nat,P: nat > $o] :
% 13.25/7.60 ( ( ? [I4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 13.25/7.60 & ( P @ I4 ) ) )
% 13.25/7.60 = ( ( P @ N )
% 13.25/7.60 | ? [I4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I4 @ N )
% 13.25/7.60 & ( P @ I4 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2753_less__SucI,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ N )
% 13.25/7.60 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2754_less__SucE,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 13.25/7.60 => ( ~ ( ord_less_nat @ M @ N )
% 13.25/7.60 => ( M = N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2755_Suc__lessI,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ N )
% 13.25/7.60 => ( ( ( suc @ M )
% 13.25/7.60 != N )
% 13.25/7.60 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2756_Suc__lessE,axiom,
% 13.25/7.60 ! [I: nat,K: nat] :
% 13.25/7.60 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 13.25/7.60 => ~ ! [J2: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I @ J2 )
% 13.25/7.60 => ( K
% 13.25/7.60 != ( suc @ J2 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2757_Suc__lessD,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 13.25/7.60 => ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2758_Nat_OlessE,axiom,
% 13.25/7.60 ! [I: nat,K: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I @ K )
% 13.25/7.60 => ( ( K
% 13.25/7.60 != ( suc @ I ) )
% 13.25/7.60 => ~ ! [J2: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I @ J2 )
% 13.25/7.60 => ( K
% 13.25/7.60 != ( suc @ J2 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2759_transitive__stepwise__le,axiom,
% 13.25/7.60 ! [M: nat,N: nat,R4: nat > nat > $o] :
% 13.25/7.60 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.60 => ( ! [X3: nat] : ( R4 @ X3 @ X3 )
% 13.25/7.60 => ( ! [X3: nat,Y4: nat,Z5: nat] :
% 13.25/7.60 ( ( R4 @ X3 @ Y4 )
% 13.25/7.60 => ( ( R4 @ Y4 @ Z5 )
% 13.25/7.60 => ( R4 @ X3 @ Z5 ) ) )
% 13.25/7.60 => ( ! [N4: nat] : ( R4 @ N4 @ ( suc @ N4 ) )
% 13.25/7.60 => ( R4 @ M @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2760_nat__induct__at__least,axiom,
% 13.25/7.60 ! [M: nat,N: nat,P: nat > $o] :
% 13.25/7.60 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.60 => ( ( P @ M )
% 13.25/7.60 => ( ! [N4: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ M @ N4 )
% 13.25/7.60 => ( ( P @ N4 )
% 13.25/7.60 => ( P @ ( suc @ N4 ) ) ) )
% 13.25/7.60 => ( P @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2761_full__nat__induct,axiom,
% 13.25/7.60 ! [P: nat > $o,N: nat] :
% 13.25/7.60 ( ! [N4: nat] :
% 13.25/7.60 ( ! [M2: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N4 )
% 13.25/7.60 => ( P @ M2 ) )
% 13.25/7.60 => ( P @ N4 ) )
% 13.25/7.60 => ( P @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2762_not__less__eq__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 13.25/7.60 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2763_Suc__n__not__le__n,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2764_le__Suc__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.60 = ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.60 | ( M
% 13.25/7.60 = ( suc @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2765_Suc__le__D,axiom,
% 13.25/7.60 ! [N: nat,M7: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
% 13.25/7.60 => ? [M4: nat] :
% 13.25/7.60 ( M7
% 13.25/7.60 = ( suc @ M4 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2766_le__SucI,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.60 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2767_le__SucE,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.60 => ( ~ ( ord_less_eq_nat @ M @ N )
% 13.25/7.60 => ( M
% 13.25/7.60 = ( suc @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2768_Suc__leD,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 13.25/7.60 => ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2769_add__Suc__shift,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 13.25/7.60 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2770_add__Suc,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 13.25/7.60 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2771_nat__arith_Osuc1,axiom,
% 13.25/7.60 ! [A2: nat,K: nat,A: nat] :
% 13.25/7.60 ( ( A2
% 13.25/7.60 = ( plus_plus_nat @ K @ A ) )
% 13.25/7.60 => ( ( suc @ A2 )
% 13.25/7.60 = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2772_zero__induct__lemma,axiom,
% 13.25/7.60 ! [P: nat > $o,K: nat,I: nat] :
% 13.25/7.60 ( ( P @ K )
% 13.25/7.60 => ( ! [N4: nat] :
% 13.25/7.60 ( ( P @ ( suc @ N4 ) )
% 13.25/7.60 => ( P @ N4 ) )
% 13.25/7.60 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2773_Suc__mult__cancel1,axiom,
% 13.25/7.60 ! [K: nat,M: nat,N: nat] :
% 13.25/7.60 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 13.25/7.60 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 13.25/7.60 = ( M = N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2774_mod__Suc__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) @ N )
% 13.25/7.60 = ( modulo_modulo_nat @ ( suc @ M ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2775_mod__Suc__Suc__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) @ N )
% 13.25/7.60 = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2776_lift__Suc__mono__less__iff,axiom,
% 13.25/7.60 ! [F: nat > real,N: nat,M: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 13.25/7.60 = ( ord_less_nat @ N @ M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2777_lift__Suc__mono__less__iff,axiom,
% 13.25/7.60 ! [F: nat > rat,N: nat,M: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 13.25/7.60 = ( ord_less_nat @ N @ M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2778_lift__Suc__mono__less__iff,axiom,
% 13.25/7.60 ! [F: nat > num,N: nat,M: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 13.25/7.60 = ( ord_less_nat @ N @ M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2779_lift__Suc__mono__less__iff,axiom,
% 13.25/7.60 ! [F: nat > nat,N: nat,M: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 13.25/7.60 = ( ord_less_nat @ N @ M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2780_lift__Suc__mono__less__iff,axiom,
% 13.25/7.60 ! [F: nat > int,N: nat,M: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 13.25/7.60 = ( ord_less_nat @ N @ M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2781_lift__Suc__mono__less,axiom,
% 13.25/7.60 ! [F: nat > real,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_real @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2782_lift__Suc__mono__less,axiom,
% 13.25/7.60 ! [F: nat > rat,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_rat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2783_lift__Suc__mono__less,axiom,
% 13.25/7.60 ! [F: nat > num,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_num @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2784_lift__Suc__mono__less,axiom,
% 13.25/7.60 ! [F: nat > nat,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2785_lift__Suc__mono__less,axiom,
% 13.25/7.60 ! [F: nat > int,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2786_power__Suc,axiom,
% 13.25/7.60 ! [A: complex,N: nat] :
% 13.25/7.60 ( ( power_power_complex @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_complex @ A @ ( power_power_complex @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2787_power__Suc,axiom,
% 13.25/7.60 ! [A: real,N: nat] :
% 13.25/7.60 ( ( power_power_real @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_real @ A @ ( power_power_real @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2788_power__Suc,axiom,
% 13.25/7.60 ! [A: rat,N: nat] :
% 13.25/7.60 ( ( power_power_rat @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_rat @ A @ ( power_power_rat @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2789_power__Suc,axiom,
% 13.25/7.60 ! [A: nat,N: nat] :
% 13.25/7.60 ( ( power_power_nat @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2790_power__Suc,axiom,
% 13.25/7.60 ! [A: int,N: nat] :
% 13.25/7.60 ( ( power_power_int @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2791_power__Suc2,axiom,
% 13.25/7.60 ! [A: complex,N: nat] :
% 13.25/7.60 ( ( power_power_complex @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_complex @ ( power_power_complex @ A @ N ) @ A ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2792_power__Suc2,axiom,
% 13.25/7.60 ! [A: real,N: nat] :
% 13.25/7.60 ( ( power_power_real @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_real @ ( power_power_real @ A @ N ) @ A ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2793_power__Suc2,axiom,
% 13.25/7.60 ! [A: rat,N: nat] :
% 13.25/7.60 ( ( power_power_rat @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_rat @ ( power_power_rat @ A @ N ) @ A ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2794_power__Suc2,axiom,
% 13.25/7.60 ! [A: nat,N: nat] :
% 13.25/7.60 ( ( power_power_nat @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2795_power__Suc2,axiom,
% 13.25/7.60 ! [A: int,N: nat] :
% 13.25/7.60 ( ( power_power_int @ A @ ( suc @ N ) )
% 13.25/7.60 = ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2796_lift__Suc__antimono__le,axiom,
% 13.25/7.60 ! [F: nat > set_int,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_set_int @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2797_lift__Suc__antimono__le,axiom,
% 13.25/7.60 ! [F: nat > rat,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_rat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2798_lift__Suc__antimono__le,axiom,
% 13.25/7.60 ! [F: nat > num,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_num @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2799_lift__Suc__antimono__le,axiom,
% 13.25/7.60 ! [F: nat > nat,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2800_lift__Suc__antimono__le,axiom,
% 13.25/7.60 ! [F: nat > int,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_int @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2801_lift__Suc__mono__le,axiom,
% 13.25/7.60 ! [F: nat > set_int,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2802_lift__Suc__mono__le,axiom,
% 13.25/7.60 ! [F: nat > rat,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2803_lift__Suc__mono__le,axiom,
% 13.25/7.60 ! [F: nat > num,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2804_lift__Suc__mono__le,axiom,
% 13.25/7.60 ! [F: nat > nat,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2805_lift__Suc__mono__le,axiom,
% 13.25/7.60 ! [F: nat > int,N: nat,N6: nat] :
% 13.25/7.60 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 13.25/7.60 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2806_Ex__less__Suc2,axiom,
% 13.25/7.60 ! [N: nat,P: nat > $o] :
% 13.25/7.60 ( ( ? [I4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 13.25/7.60 & ( P @ I4 ) ) )
% 13.25/7.60 = ( ( P @ zero_zero_nat )
% 13.25/7.60 | ? [I4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I4 @ N )
% 13.25/7.60 & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2807_gr0__conv__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 = ( ? [M3: nat] :
% 13.25/7.60 ( N
% 13.25/7.60 = ( suc @ M3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2808_All__less__Suc2,axiom,
% 13.25/7.60 ! [N: nat,P: nat > $o] :
% 13.25/7.60 ( ( ! [I4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 13.25/7.60 => ( P @ I4 ) ) )
% 13.25/7.60 = ( ( P @ zero_zero_nat )
% 13.25/7.60 & ! [I4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ I4 @ N )
% 13.25/7.60 => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2809_gr0__implies__Suc,axiom,
% 13.25/7.60 ! [N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ? [M4: nat] :
% 13.25/7.60 ( N
% 13.25/7.60 = ( suc @ M4 ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2810_less__Suc__eq__0__disj,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 13.25/7.60 = ( ( M = zero_zero_nat )
% 13.25/7.60 | ? [J3: nat] :
% 13.25/7.60 ( ( M
% 13.25/7.60 = ( suc @ J3 ) )
% 13.25/7.60 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2811_Suc__leI,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ N )
% 13.25/7.60 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2812_Suc__le__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 13.25/7.60 = ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2813_dec__induct,axiom,
% 13.25/7.60 ! [I: nat,J: nat,P: nat > $o] :
% 13.25/7.60 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.60 => ( ( P @ I )
% 13.25/7.60 => ( ! [N4: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ I @ N4 )
% 13.25/7.60 => ( ( ord_less_nat @ N4 @ J )
% 13.25/7.60 => ( ( P @ N4 )
% 13.25/7.60 => ( P @ ( suc @ N4 ) ) ) ) )
% 13.25/7.60 => ( P @ J ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2814_inc__induct,axiom,
% 13.25/7.60 ! [I: nat,J: nat,P: nat > $o] :
% 13.25/7.60 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.60 => ( ( P @ J )
% 13.25/7.60 => ( ! [N4: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ I @ N4 )
% 13.25/7.60 => ( ( ord_less_nat @ N4 @ J )
% 13.25/7.60 => ( ( P @ ( suc @ N4 ) )
% 13.25/7.60 => ( P @ N4 ) ) ) )
% 13.25/7.60 => ( P @ I ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2815_Suc__le__lessD,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 13.25/7.60 => ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2816_le__less__Suc__eq,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.60 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 13.25/7.60 = ( N = M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2817_less__Suc__eq__le,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 13.25/7.60 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2818_less__eq__Suc__le,axiom,
% 13.25/7.60 ( ord_less_nat
% 13.25/7.60 = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2819_le__imp__less__Suc,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.60 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2820_add__is__1,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( plus_plus_nat @ M @ N )
% 13.25/7.60 = ( suc @ zero_zero_nat ) )
% 13.25/7.60 = ( ( ( M
% 13.25/7.60 = ( suc @ zero_zero_nat ) )
% 13.25/7.60 & ( N = zero_zero_nat ) )
% 13.25/7.60 | ( ( M = zero_zero_nat )
% 13.25/7.60 & ( N
% 13.25/7.60 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2821_one__is__add,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( suc @ zero_zero_nat )
% 13.25/7.60 = ( plus_plus_nat @ M @ N ) )
% 13.25/7.60 = ( ( ( M
% 13.25/7.60 = ( suc @ zero_zero_nat ) )
% 13.25/7.60 & ( N = zero_zero_nat ) )
% 13.25/7.60 | ( ( M = zero_zero_nat )
% 13.25/7.60 & ( N
% 13.25/7.60 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2822_less__imp__Suc__add,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ N )
% 13.25/7.60 => ? [K3: nat] :
% 13.25/7.60 ( N
% 13.25/7.60 = ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2823_less__iff__Suc__add,axiom,
% 13.25/7.60 ( ord_less_nat
% 13.25/7.60 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.60 ? [K2: nat] :
% 13.25/7.60 ( N2
% 13.25/7.60 = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2824_less__add__Suc2,axiom,
% 13.25/7.60 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2825_less__add__Suc1,axiom,
% 13.25/7.60 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2826_less__natE,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ M @ N )
% 13.25/7.60 => ~ ! [Q3: nat] :
% 13.25/7.60 ( N
% 13.25/7.60 != ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2827_diff__less__Suc,axiom,
% 13.25/7.60 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2828_Suc__diff__Suc,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ N @ M )
% 13.25/7.60 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 13.25/7.60 = ( minus_minus_nat @ M @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2829_Suc__mult__less__cancel1,axiom,
% 13.25/7.60 ! [K: nat,M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 13.25/7.60 = ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2830_Suc__diff__le,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.60 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 13.25/7.60 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2831_One__nat__def,axiom,
% 13.25/7.60 ( one_one_nat
% 13.25/7.60 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2832_Suc__mult__le__cancel1,axiom,
% 13.25/7.60 ! [K: nat,M: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 13.25/7.60 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2833_Suc__div__le__mono,axiom,
% 13.25/7.60 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2834_mult__Suc,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( times_times_nat @ ( suc @ M ) @ N )
% 13.25/7.60 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2835_Suc__eq__plus1__left,axiom,
% 13.25/7.60 ( suc
% 13.25/7.60 = ( plus_plus_nat @ one_one_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2836_plus__1__eq__Suc,axiom,
% 13.25/7.60 ( ( plus_plus_nat @ one_one_nat )
% 13.25/7.60 = suc ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2837_Suc__eq__plus1,axiom,
% 13.25/7.60 ( suc
% 13.25/7.60 = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2838_diff__Suc__eq__diff__pred,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 13.25/7.60 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2839_mod__Suc,axiom,
% 13.25/7.60 ! [M: nat,N: nat] :
% 13.25/7.60 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 13.25/7.60 = N )
% 13.25/7.60 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 13.25/7.60 = zero_zero_nat ) )
% 13.25/7.60 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 13.25/7.60 != N )
% 13.25/7.60 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 13.25/7.60 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2840_mod__induct,axiom,
% 13.25/7.60 ! [P: nat > $o,N: nat,P6: nat,M: nat] :
% 13.25/7.60 ( ( P @ N )
% 13.25/7.60 => ( ( ord_less_nat @ N @ P6 )
% 13.25/7.60 => ( ( ord_less_nat @ M @ P6 )
% 13.25/7.60 => ( ! [N4: nat] :
% 13.25/7.60 ( ( ord_less_nat @ N4 @ P6 )
% 13.25/7.60 => ( ( P @ N4 )
% 13.25/7.60 => ( P @ ( modulo_modulo_nat @ ( suc @ N4 ) @ P6 ) ) ) )
% 13.25/7.60 => ( P @ M ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2841_mod__Suc__le__divisor,axiom,
% 13.25/7.60 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2842_power__le__imp__le__base,axiom,
% 13.25/7.60 ! [A: real,N: nat,B: real] :
% 13.25/7.60 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.25/7.60 => ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2843_power__le__imp__le__base,axiom,
% 13.25/7.60 ! [A: rat,N: nat,B: rat] :
% 13.25/7.60 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.25/7.60 => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2844_power__le__imp__le__base,axiom,
% 13.25/7.60 ! [A: nat,N: nat,B: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.25/7.60 => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2845_power__le__imp__le__base,axiom,
% 13.25/7.60 ! [A: int,N: nat,B: int] :
% 13.25/7.60 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.25/7.60 => ( ord_less_eq_int @ A @ B ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2846_power__inject__base,axiom,
% 13.25/7.60 ! [A: real,N: nat,B: real] :
% 13.25/7.60 ( ( ( power_power_real @ A @ ( suc @ N ) )
% 13.25/7.60 = ( power_power_real @ B @ ( suc @ N ) ) )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.60 => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.25/7.60 => ( A = B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2847_power__inject__base,axiom,
% 13.25/7.60 ! [A: rat,N: nat,B: rat] :
% 13.25/7.60 ( ( ( power_power_rat @ A @ ( suc @ N ) )
% 13.25/7.60 = ( power_power_rat @ B @ ( suc @ N ) ) )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.25/7.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.25/7.60 => ( A = B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2848_power__inject__base,axiom,
% 13.25/7.60 ! [A: nat,N: nat,B: nat] :
% 13.25/7.60 ( ( ( power_power_nat @ A @ ( suc @ N ) )
% 13.25/7.60 = ( power_power_nat @ B @ ( suc @ N ) ) )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.25/7.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 13.25/7.60 => ( A = B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2849_power__inject__base,axiom,
% 13.25/7.60 ! [A: int,N: nat,B: int] :
% 13.25/7.60 ( ( ( power_power_int @ A @ ( suc @ N ) )
% 13.25/7.60 = ( power_power_int @ B @ ( suc @ N ) ) )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.25/7.60 => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.25/7.60 => ( A = B ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2850_power__gt1,axiom,
% 13.25/7.60 ! [A: real,N: nat] :
% 13.25/7.60 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.60 => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2851_power__gt1,axiom,
% 13.25/7.60 ! [A: rat,N: nat] :
% 13.25/7.60 ( ( ord_less_rat @ one_one_rat @ A )
% 13.25/7.60 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2852_power__gt1,axiom,
% 13.25/7.60 ! [A: nat,N: nat] :
% 13.25/7.60 ( ( ord_less_nat @ one_one_nat @ A )
% 13.25/7.60 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2853_power__gt1,axiom,
% 13.25/7.60 ! [A: int,N: nat] :
% 13.25/7.60 ( ( ord_less_int @ one_one_int @ A )
% 13.25/7.60 => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2854_numeral__1__eq__Suc__0,axiom,
% 13.25/7.60 ( ( numeral_numeral_nat @ one )
% 13.25/7.60 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2855_ex__least__nat__less,axiom,
% 13.25/7.60 ! [P: nat > $o,N: nat] :
% 13.25/7.60 ( ( P @ N )
% 13.25/7.60 => ( ~ ( P @ zero_zero_nat )
% 13.25/7.60 => ? [K3: nat] :
% 13.25/7.60 ( ( ord_less_nat @ K3 @ N )
% 13.25/7.60 & ! [I3: nat] :
% 13.25/7.60 ( ( ord_less_eq_nat @ I3 @ K3 )
% 13.25/7.60 => ~ ( P @ I3 ) )
% 13.25/7.60 & ( P @ ( suc @ K3 ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2856_diff__Suc__less,axiom,
% 13.25/7.60 ! [N: nat,I: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2857_one__less__mult,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.60 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 13.25/7.60 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2858_n__less__m__mult__n,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 13.25/7.60 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2859_n__less__n__mult__m,axiom,
% 13.25/7.60 ! [N: nat,M: nat] :
% 13.25/7.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.60 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 13.25/7.60 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2860_pinf_I1_J,axiom,
% 13.25/7.60 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 13.25/7.60 ( ? [Z3: real] :
% 13.25/7.60 ! [X3: real] :
% 13.25/7.60 ( ( ord_less_real @ Z3 @ X3 )
% 13.25/7.60 => ( ( P @ X3 )
% 13.25/7.60 = ( P5 @ X3 ) ) )
% 13.25/7.60 => ( ? [Z3: real] :
% 13.25/7.60 ! [X3: real] :
% 13.25/7.60 ( ( ord_less_real @ Z3 @ X3 )
% 13.25/7.60 => ( ( Q @ X3 )
% 13.25/7.60 = ( Q6 @ X3 ) ) )
% 13.25/7.60 => ? [Z5: real] :
% 13.25/7.60 ! [X5: real] :
% 13.25/7.60 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.60 => ( ( ( P @ X5 )
% 13.25/7.60 & ( Q @ X5 ) )
% 13.25/7.60 = ( ( P5 @ X5 )
% 13.25/7.60 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2861_pinf_I1_J,axiom,
% 13.25/7.60 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 13.25/7.60 ( ? [Z3: rat] :
% 13.25/7.60 ! [X3: rat] :
% 13.25/7.60 ( ( ord_less_rat @ Z3 @ X3 )
% 13.25/7.60 => ( ( P @ X3 )
% 13.25/7.60 = ( P5 @ X3 ) ) )
% 13.25/7.60 => ( ? [Z3: rat] :
% 13.25/7.60 ! [X3: rat] :
% 13.25/7.60 ( ( ord_less_rat @ Z3 @ X3 )
% 13.25/7.60 => ( ( Q @ X3 )
% 13.25/7.60 = ( Q6 @ X3 ) ) )
% 13.25/7.60 => ? [Z5: rat] :
% 13.25/7.60 ! [X5: rat] :
% 13.25/7.60 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.60 => ( ( ( P @ X5 )
% 13.25/7.60 & ( Q @ X5 ) )
% 13.25/7.60 = ( ( P5 @ X5 )
% 13.25/7.60 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.60
% 13.25/7.60 thf(fact_2862_pinf_I1_J,axiom,
% 13.25/7.60 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 13.25/7.60 ( ? [Z3: num] :
% 13.25/7.60 ! [X3: num] :
% 13.25/7.60 ( ( ord_less_num @ Z3 @ X3 )
% 13.25/7.60 => ( ( P @ X3 )
% 13.25/7.60 = ( P5 @ X3 ) ) )
% 13.25/7.60 => ( ? [Z3: num] :
% 13.25/7.60 ! [X3: num] :
% 13.25/7.60 ( ( ord_less_num @ Z3 @ X3 )
% 13.25/7.60 => ( ( Q @ X3 )
% 13.25/7.60 = ( Q6 @ X3 ) ) )
% 13.25/7.60 => ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ Z5 @ X5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2863_pinf_I1_J,axiom,
% 13.25/7.61 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 13.25/7.61 ( ? [Z3: nat] :
% 13.25/7.61 ! [X3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z3 @ X3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: nat] :
% 13.25/7.61 ! [X3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z3 @ X3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2864_pinf_I1_J,axiom,
% 13.25/7.61 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 13.25/7.61 ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ Z3 @ X3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ Z3 @ X3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2865_pinf_I2_J,axiom,
% 13.25/7.61 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 13.25/7.61 ( ? [Z3: real] :
% 13.25/7.61 ! [X3: real] :
% 13.25/7.61 ( ( ord_less_real @ Z3 @ X3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: real] :
% 13.25/7.61 ! [X3: real] :
% 13.25/7.61 ( ( ord_less_real @ Z3 @ X3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2866_pinf_I2_J,axiom,
% 13.25/7.61 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 13.25/7.61 ( ? [Z3: rat] :
% 13.25/7.61 ! [X3: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z3 @ X3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: rat] :
% 13.25/7.61 ! [X3: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z3 @ X3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2867_pinf_I2_J,axiom,
% 13.25/7.61 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 13.25/7.61 ( ? [Z3: num] :
% 13.25/7.61 ! [X3: num] :
% 13.25/7.61 ( ( ord_less_num @ Z3 @ X3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: num] :
% 13.25/7.61 ! [X3: num] :
% 13.25/7.61 ( ( ord_less_num @ Z3 @ X3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ Z5 @ X5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2868_pinf_I2_J,axiom,
% 13.25/7.61 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 13.25/7.61 ( ? [Z3: nat] :
% 13.25/7.61 ! [X3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z3 @ X3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: nat] :
% 13.25/7.61 ! [X3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z3 @ X3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2869_pinf_I2_J,axiom,
% 13.25/7.61 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 13.25/7.61 ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ Z3 @ X3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ Z3 @ X3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2870_pinf_I3_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2871_pinf_I3_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2872_pinf_I3_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2873_pinf_I3_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2874_pinf_I3_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2875_pinf_I4_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2876_pinf_I4_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2877_pinf_I4_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2878_pinf_I4_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2879_pinf_I4_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2880_pinf_I5_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_real @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2881_pinf_I5_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_rat @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2882_pinf_I5_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_num @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2883_pinf_I5_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_nat @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2884_pinf_I5_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_int @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2885_pinf_I7_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_real @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2886_pinf_I7_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_rat @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2887_pinf_I7_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_num @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2888_pinf_I7_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_nat @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2889_pinf_I7_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_int @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2890_minf_I1_J,axiom,
% 13.25/7.61 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 13.25/7.61 ( ? [Z3: real] :
% 13.25/7.61 ! [X3: real] :
% 13.25/7.61 ( ( ord_less_real @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: real] :
% 13.25/7.61 ! [X3: real] :
% 13.25/7.61 ( ( ord_less_real @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2891_minf_I1_J,axiom,
% 13.25/7.61 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 13.25/7.61 ( ? [Z3: rat] :
% 13.25/7.61 ! [X3: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: rat] :
% 13.25/7.61 ! [X3: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2892_minf_I1_J,axiom,
% 13.25/7.61 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 13.25/7.61 ( ? [Z3: num] :
% 13.25/7.61 ! [X3: num] :
% 13.25/7.61 ( ( ord_less_num @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: num] :
% 13.25/7.61 ! [X3: num] :
% 13.25/7.61 ( ( ord_less_num @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2893_minf_I1_J,axiom,
% 13.25/7.61 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 13.25/7.61 ( ? [Z3: nat] :
% 13.25/7.61 ! [X3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: nat] :
% 13.25/7.61 ! [X3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2894_minf_I1_J,axiom,
% 13.25/7.61 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 13.25/7.61 ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 & ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2895_minf_I2_J,axiom,
% 13.25/7.61 ! [P: real > $o,P5: real > $o,Q: real > $o,Q6: real > $o] :
% 13.25/7.61 ( ? [Z3: real] :
% 13.25/7.61 ! [X3: real] :
% 13.25/7.61 ( ( ord_less_real @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: real] :
% 13.25/7.61 ! [X3: real] :
% 13.25/7.61 ( ( ord_less_real @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2896_minf_I2_J,axiom,
% 13.25/7.61 ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q6: rat > $o] :
% 13.25/7.61 ( ? [Z3: rat] :
% 13.25/7.61 ! [X3: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: rat] :
% 13.25/7.61 ! [X3: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2897_minf_I2_J,axiom,
% 13.25/7.61 ! [P: num > $o,P5: num > $o,Q: num > $o,Q6: num > $o] :
% 13.25/7.61 ( ? [Z3: num] :
% 13.25/7.61 ! [X3: num] :
% 13.25/7.61 ( ( ord_less_num @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: num] :
% 13.25/7.61 ! [X3: num] :
% 13.25/7.61 ( ( ord_less_num @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2898_minf_I2_J,axiom,
% 13.25/7.61 ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q6: nat > $o] :
% 13.25/7.61 ( ? [Z3: nat] :
% 13.25/7.61 ! [X3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: nat] :
% 13.25/7.61 ! [X3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2899_minf_I2_J,axiom,
% 13.25/7.61 ! [P: int > $o,P5: int > $o,Q: int > $o,Q6: int > $o] :
% 13.25/7.61 ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ X3 @ Z3 )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 = ( Q6 @ X3 ) ) )
% 13.25/7.61 => ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P5 @ X5 )
% 13.25/7.61 | ( Q6 @ X5 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2900_minf_I3_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2901_minf_I3_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2902_minf_I3_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2903_minf_I3_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2904_minf_I3_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2905_minf_I4_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2906_minf_I4_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2907_minf_I4_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2908_minf_I4_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2909_minf_I4_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.61 => ( X5 != T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2910_minf_I5_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_real @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2911_minf_I5_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_rat @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2912_minf_I5_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_num @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2913_minf_I5_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_nat @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2914_minf_I5_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_int @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2915_minf_I7_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_real @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2916_minf_I7_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_rat @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2917_minf_I7_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_num @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2918_minf_I7_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_nat @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2919_minf_I7_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_int @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2920_nat__induct__non__zero,axiom,
% 13.25/7.61 ! [N: nat,P: nat > $o] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.61 => ( ( P @ one_one_nat )
% 13.25/7.61 => ( ! [N4: nat] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 13.25/7.61 => ( ( P @ N4 )
% 13.25/7.61 => ( P @ ( suc @ N4 ) ) ) )
% 13.25/7.61 => ( P @ N ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2921_realpow__pos__nth2,axiom,
% 13.25/7.61 ! [A: real,N: nat] :
% 13.25/7.61 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.61 => ? [R2: real] :
% 13.25/7.61 ( ( ord_less_real @ zero_zero_real @ R2 )
% 13.25/7.61 & ( ( power_power_real @ R2 @ ( suc @ N ) )
% 13.25/7.61 = A ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2922_power__gt__expt,axiom,
% 13.25/7.61 ! [N: nat,K: nat] :
% 13.25/7.61 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.61 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2923_nat__one__le__power,axiom,
% 13.25/7.61 ! [I: nat,N: nat] :
% 13.25/7.61 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 13.25/7.61 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2924_power__Suc__le__self,axiom,
% 13.25/7.61 ! [A: real,N: nat] :
% 13.25/7.61 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.61 => ( ( ord_less_eq_real @ A @ one_one_real )
% 13.25/7.61 => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2925_power__Suc__le__self,axiom,
% 13.25/7.61 ! [A: rat,N: nat] :
% 13.25/7.61 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.25/7.61 => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 13.25/7.61 => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2926_power__Suc__le__self,axiom,
% 13.25/7.61 ! [A: nat,N: nat] :
% 13.25/7.61 ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 13.25/7.61 => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 13.25/7.61 => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2927_power__Suc__le__self,axiom,
% 13.25/7.61 ! [A: int,N: nat] :
% 13.25/7.61 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.25/7.61 => ( ( ord_less_eq_int @ A @ one_one_int )
% 13.25/7.61 => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2928_power__Suc__less__one,axiom,
% 13.25/7.61 ! [A: real,N: nat] :
% 13.25/7.61 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.61 => ( ( ord_less_real @ A @ one_one_real )
% 13.25/7.61 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2929_power__Suc__less__one,axiom,
% 13.25/7.61 ! [A: rat,N: nat] :
% 13.25/7.61 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.25/7.61 => ( ( ord_less_rat @ A @ one_one_rat )
% 13.25/7.61 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2930_power__Suc__less__one,axiom,
% 13.25/7.61 ! [A: nat,N: nat] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ A )
% 13.25/7.61 => ( ( ord_less_nat @ A @ one_one_nat )
% 13.25/7.61 => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2931_power__Suc__less__one,axiom,
% 13.25/7.61 ! [A: int,N: nat] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ A )
% 13.25/7.61 => ( ( ord_less_int @ A @ one_one_int )
% 13.25/7.61 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2932_option_Osize_I4_J,axiom,
% 13.25/7.61 ! [X22: nat] :
% 13.25/7.61 ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2933_option_Osize_I4_J,axiom,
% 13.25/7.61 ! [X22: product_prod_nat_nat] :
% 13.25/7.61 ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2934_option_Osize_I4_J,axiom,
% 13.25/7.61 ! [X22: num] :
% 13.25/7.61 ( ( size_size_option_num @ ( some_num @ X22 ) )
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2935_option_Osize_I3_J,axiom,
% 13.25/7.61 ( ( size_size_option_nat @ none_nat )
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2936_option_Osize_I3_J,axiom,
% 13.25/7.61 ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2937_option_Osize_I3_J,axiom,
% 13.25/7.61 ( ( size_size_option_num @ none_num )
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2938_numeral__2__eq__2,axiom,
% 13.25/7.61 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 13.25/7.61 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2939_Suc__double__not__eq__double,axiom,
% 13.25/7.61 ! [M: nat,N: nat] :
% 13.25/7.61 ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.61 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2940_double__not__eq__Suc__double,axiom,
% 13.25/7.61 ! [M: nat,N: nat] :
% 13.25/7.61 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 13.25/7.61 != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2941_div__geq,axiom,
% 13.25/7.61 ! [N: nat,M: nat] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.61 => ( ~ ( ord_less_nat @ M @ N )
% 13.25/7.61 => ( ( divide_divide_nat @ M @ N )
% 13.25/7.61 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2942_div__if,axiom,
% 13.25/7.61 ( divide_divide_nat
% 13.25/7.61 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.61 ( if_nat
% 13.25/7.61 @ ( ( ord_less_nat @ M3 @ N2 )
% 13.25/7.61 | ( N2 = zero_zero_nat ) )
% 13.25/7.61 @ zero_zero_nat
% 13.25/7.61 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2943_Suc__diff__eq__diff__pred,axiom,
% 13.25/7.61 ! [N: nat,M: nat] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.61 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 13.25/7.61 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2944_Suc__pred_H,axiom,
% 13.25/7.61 ! [N: nat] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.61 => ( N
% 13.25/7.61 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2945_div__nat__eqI,axiom,
% 13.25/7.61 ! [N: nat,Q2: nat,M: nat] :
% 13.25/7.61 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M )
% 13.25/7.61 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
% 13.25/7.61 => ( ( divide_divide_nat @ M @ N )
% 13.25/7.61 = Q2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2946_add__eq__if,axiom,
% 13.25/7.61 ( plus_plus_nat
% 13.25/7.61 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2947_Suc__nat__number__of__add,axiom,
% 13.25/7.61 ! [V: num,N: nat] :
% 13.25/7.61 ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
% 13.25/7.61 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2948_num_Osize_I5_J,axiom,
% 13.25/7.61 ! [X22: num] :
% 13.25/7.61 ( ( size_size_num @ ( bit0 @ X22 ) )
% 13.25/7.61 = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2949_less__2__cases,axiom,
% 13.25/7.61 ! [N: nat] :
% 13.25/7.61 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.61 => ( ( N = zero_zero_nat )
% 13.25/7.61 | ( N
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2950_less__2__cases__iff,axiom,
% 13.25/7.61 ! [N: nat] :
% 13.25/7.61 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.61 = ( ( N = zero_zero_nat )
% 13.25/7.61 | ( N
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2951_le__div__geq,axiom,
% 13.25/7.61 ! [N: nat,M: nat] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.61 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.61 => ( ( divide_divide_nat @ M @ N )
% 13.25/7.61 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2952_split__div_H,axiom,
% 13.25/7.61 ! [P: nat > $o,M: nat,N: nat] :
% 13.25/7.61 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 13.25/7.61 = ( ( ( N = zero_zero_nat )
% 13.25/7.61 & ( P @ zero_zero_nat ) )
% 13.25/7.61 | ? [Q5: nat] :
% 13.25/7.61 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M )
% 13.25/7.61 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
% 13.25/7.61 & ( P @ Q5 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2953_Suc__times__mod__eq,axiom,
% 13.25/7.61 ! [M: nat,N: nat] :
% 13.25/7.61 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 13.25/7.61 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 13.25/7.61 = one_one_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2954_vebt__succ_Osimps_I5_J,axiom,
% 13.25/7.61 ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 13.25/7.61 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
% 13.25/7.61 = none_nat ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2955_power__odd__eq,axiom,
% 13.25/7.61 ! [A: complex,N: nat] :
% 13.25/7.61 ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.61 = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2956_power__odd__eq,axiom,
% 13.25/7.61 ! [A: real,N: nat] :
% 13.25/7.61 ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.61 = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2957_power__odd__eq,axiom,
% 13.25/7.61 ! [A: rat,N: nat] :
% 13.25/7.61 ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.61 = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2958_power__odd__eq,axiom,
% 13.25/7.61 ! [A: nat,N: nat] :
% 13.25/7.61 ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.61 = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2959_power__odd__eq,axiom,
% 13.25/7.61 ! [A: int,N: nat] :
% 13.25/7.61 ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.61 = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2960_nat__bit__induct,axiom,
% 13.25/7.61 ! [P: nat > $o,N: nat] :
% 13.25/7.61 ( ( P @ zero_zero_nat )
% 13.25/7.61 => ( ! [N4: nat] :
% 13.25/7.61 ( ( P @ N4 )
% 13.25/7.61 => ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 13.25/7.61 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 13.25/7.61 => ( ! [N4: nat] :
% 13.25/7.61 ( ( P @ N4 )
% 13.25/7.61 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
% 13.25/7.61 => ( P @ N ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2961_div__2__gt__zero,axiom,
% 13.25/7.61 ! [N: nat] :
% 13.25/7.61 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.61 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2962_Suc__n__div__2__gt__zero,axiom,
% 13.25/7.61 ! [N: nat] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.61 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2963_pinf_I6_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_eq_real @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2964_pinf_I6_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2965_pinf_I6_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2966_pinf_I6_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2967_pinf_I6_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.61 => ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2968_pinf_I8_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_eq_real @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2969_pinf_I8_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_eq_rat @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2970_pinf_I8_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_eq_num @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2971_pinf_I8_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_eq_nat @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2972_pinf_I8_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.61 => ( ord_less_eq_int @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2973_minf_I6_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_eq_real @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2974_minf_I6_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_eq_rat @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2975_minf_I6_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_eq_num @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2976_minf_I6_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_eq_nat @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2977_minf_I6_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.61 => ( ord_less_eq_int @ X5 @ T ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2978_minf_I8_J,axiom,
% 13.25/7.61 ! [T: real] :
% 13.25/7.61 ? [Z5: real] :
% 13.25/7.61 ! [X5: real] :
% 13.25/7.61 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_eq_real @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2979_minf_I8_J,axiom,
% 13.25/7.61 ! [T: rat] :
% 13.25/7.61 ? [Z5: rat] :
% 13.25/7.61 ! [X5: rat] :
% 13.25/7.61 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2980_minf_I8_J,axiom,
% 13.25/7.61 ! [T: num] :
% 13.25/7.61 ? [Z5: num] :
% 13.25/7.61 ! [X5: num] :
% 13.25/7.61 ( ( ord_less_num @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2981_minf_I8_J,axiom,
% 13.25/7.61 ! [T: nat] :
% 13.25/7.61 ? [Z5: nat] :
% 13.25/7.61 ! [X5: nat] :
% 13.25/7.61 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2982_minf_I8_J,axiom,
% 13.25/7.61 ! [T: int] :
% 13.25/7.61 ? [Z5: int] :
% 13.25/7.61 ! [X5: int] :
% 13.25/7.61 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.61 => ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2983_inf__period_I1_J,axiom,
% 13.25/7.61 ! [P: real > $o,D4: real,Q: real > $o] :
% 13.25/7.61 ( ! [X3: real,K3: real] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: real,K3: real] :
% 13.25/7.61 ( ( Q @ X3 )
% 13.25/7.61 = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: real,K4: real] :
% 13.25/7.61 ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
% 13.25/7.61 & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2984_inf__period_I1_J,axiom,
% 13.25/7.61 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 13.25/7.61 ( ! [X3: rat,K3: rat] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: rat,K3: rat] :
% 13.25/7.61 ( ( Q @ X3 )
% 13.25/7.61 = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: rat,K4: rat] :
% 13.25/7.61 ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
% 13.25/7.61 & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2985_inf__period_I1_J,axiom,
% 13.25/7.61 ! [P: int > $o,D4: int,Q: int > $o] :
% 13.25/7.61 ( ! [X3: int,K3: int] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: int,K3: int] :
% 13.25/7.61 ( ( Q @ X3 )
% 13.25/7.61 = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: int,K4: int] :
% 13.25/7.61 ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
% 13.25/7.61 & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2986_inf__period_I2_J,axiom,
% 13.25/7.61 ! [P: real > $o,D4: real,Q: real > $o] :
% 13.25/7.61 ( ! [X3: real,K3: real] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 = ( P @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: real,K3: real] :
% 13.25/7.61 ( ( Q @ X3 )
% 13.25/7.61 = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: real,K4: real] :
% 13.25/7.61 ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) )
% 13.25/7.61 | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2987_inf__period_I2_J,axiom,
% 13.25/7.61 ! [P: rat > $o,D4: rat,Q: rat > $o] :
% 13.25/7.61 ( ! [X3: rat,K3: rat] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 = ( P @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: rat,K3: rat] :
% 13.25/7.61 ( ( Q @ X3 )
% 13.25/7.61 = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: rat,K4: rat] :
% 13.25/7.61 ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) )
% 13.25/7.61 | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2988_inf__period_I2_J,axiom,
% 13.25/7.61 ! [P: int > $o,D4: int,Q: int > $o] :
% 13.25/7.61 ( ! [X3: int,K3: int] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: int,K3: int] :
% 13.25/7.61 ( ( Q @ X3 )
% 13.25/7.61 = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: int,K4: int] :
% 13.25/7.61 ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 = ( ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) )
% 13.25/7.61 | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2989_odd__0__le__power__imp__0__le,axiom,
% 13.25/7.61 ! [A: real,N: nat] :
% 13.25/7.61 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 13.25/7.61 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2990_odd__0__le__power__imp__0__le,axiom,
% 13.25/7.61 ! [A: rat,N: nat] :
% 13.25/7.61 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 13.25/7.61 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2991_odd__0__le__power__imp__0__le,axiom,
% 13.25/7.61 ! [A: int,N: nat] :
% 13.25/7.61 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 13.25/7.61 => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2992_odd__power__less__zero,axiom,
% 13.25/7.61 ! [A: real,N: nat] :
% 13.25/7.61 ( ( ord_less_real @ A @ zero_zero_real )
% 13.25/7.61 => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2993_odd__power__less__zero,axiom,
% 13.25/7.61 ! [A: rat,N: nat] :
% 13.25/7.61 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.25/7.61 => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2994_odd__power__less__zero,axiom,
% 13.25/7.61 ! [A: int,N: nat] :
% 13.25/7.61 ( ( ord_less_int @ A @ zero_zero_int )
% 13.25/7.61 => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2995_imp__le__cong,axiom,
% 13.25/7.61 ! [X: int,X8: int,P: $o,P5: $o] :
% 13.25/7.61 ( ( X = X8 )
% 13.25/7.61 => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 13.25/7.61 => ( P = P5 ) )
% 13.25/7.61 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.61 => P )
% 13.25/7.61 = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 13.25/7.61 => P5 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2996_conj__le__cong,axiom,
% 13.25/7.61 ! [X: int,X8: int,P: $o,P5: $o] :
% 13.25/7.61 ( ( X = X8 )
% 13.25/7.61 => ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 13.25/7.61 => ( P = P5 ) )
% 13.25/7.61 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.61 & P )
% 13.25/7.61 = ( ( ord_less_eq_int @ zero_zero_int @ X8 )
% 13.25/7.61 & P5 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2997_int__power__div__base,axiom,
% 13.25/7.61 ! [M: nat,K: int] :
% 13.25/7.61 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.61 => ( ( ord_less_int @ zero_zero_int @ K )
% 13.25/7.61 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 13.25/7.61 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2998_set__bit__Suc,axiom,
% 13.25/7.61 ! [N: nat,A: code_integer] :
% 13.25/7.61 ( ( bit_se2793503036327961859nteger @ ( suc @ N ) @ A )
% 13.25/7.61 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_2999_set__bit__Suc,axiom,
% 13.25/7.61 ! [N: nat,A: int] :
% 13.25/7.61 ( ( bit_se7879613467334960850it_int @ ( suc @ N ) @ A )
% 13.25/7.61 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3000_set__bit__Suc,axiom,
% 13.25/7.61 ! [N: nat,A: nat] :
% 13.25/7.61 ( ( bit_se7882103937844011126it_nat @ ( suc @ N ) @ A )
% 13.25/7.61 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3001_aset_I2_J,axiom,
% 13.25/7.61 ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
% 13.25/7.61 ( ! [X3: int] :
% 13.25/7.61 ( ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb2: int] :
% 13.25/7.61 ( ( member_int @ Xb2 @ A2 )
% 13.25/7.61 => ( X3
% 13.25/7.61 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: int] :
% 13.25/7.61 ( ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb2: int] :
% 13.25/7.61 ( ( member_int @ Xb2 @ A2 )
% 13.25/7.61 => ( X3
% 13.25/7.61 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
% 13.25/7.61 | ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3002_aset_I1_J,axiom,
% 13.25/7.61 ! [D4: int,A2: set_int,P: int > $o,Q: int > $o] :
% 13.25/7.61 ( ! [X3: int] :
% 13.25/7.61 ( ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb2: int] :
% 13.25/7.61 ( ( member_int @ Xb2 @ A2 )
% 13.25/7.61 => ( X3
% 13.25/7.61 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 => ( P @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: int] :
% 13.25/7.61 ( ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb2: int] :
% 13.25/7.61 ( ( member_int @ Xb2 @ A2 )
% 13.25/7.61 => ( X3
% 13.25/7.61 != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 => ( Q @ ( plus_plus_int @ X3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 => ( ( P @ ( plus_plus_int @ X5 @ D4 ) )
% 13.25/7.61 & ( Q @ ( plus_plus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3003_bset_I2_J,axiom,
% 13.25/7.61 ! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
% 13.25/7.61 ( ! [X3: int] :
% 13.25/7.61 ( ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb2: int] :
% 13.25/7.61 ( ( member_int @ Xb2 @ B5 )
% 13.25/7.61 => ( X3
% 13.25/7.61 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: int] :
% 13.25/7.61 ( ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb2: int] :
% 13.25/7.61 ( ( member_int @ Xb2 @ B5 )
% 13.25/7.61 => ( X3
% 13.25/7.61 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 | ( Q @ X5 ) )
% 13.25/7.61 => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
% 13.25/7.61 | ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3004_bset_I1_J,axiom,
% 13.25/7.61 ! [D4: int,B5: set_int,P: int > $o,Q: int > $o] :
% 13.25/7.61 ( ! [X3: int] :
% 13.25/7.61 ( ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb2: int] :
% 13.25/7.61 ( ( member_int @ Xb2 @ B5 )
% 13.25/7.61 => ( X3
% 13.25/7.61 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 => ( P @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 13.25/7.61 => ( ! [X3: int] :
% 13.25/7.61 ( ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb2: int] :
% 13.25/7.61 ( ( member_int @ Xb2 @ B5 )
% 13.25/7.61 => ( X3
% 13.25/7.61 != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
% 13.25/7.61 => ( ( Q @ X3 )
% 13.25/7.61 => ( Q @ ( minus_minus_int @ X3 @ D4 ) ) ) )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( ( P @ X5 )
% 13.25/7.61 & ( Q @ X5 ) )
% 13.25/7.61 => ( ( P @ ( minus_minus_int @ X5 @ D4 ) )
% 13.25/7.61 & ( Q @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3005_invar__vebt_Ointros_I3_J,axiom,
% 13.25/7.61 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 13.25/7.61 ( ! [X3: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X3 @ N ) )
% 13.25/7.61 => ( ( vEBT_invar_vebt @ Summary @ M )
% 13.25/7.61 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.61 => ( ( M
% 13.25/7.61 = ( suc @ N ) )
% 13.25/7.61 => ( ( Deg
% 13.25/7.61 = ( plus_plus_nat @ N @ M ) )
% 13.25/7.61 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 13.25/7.61 => ( ! [X3: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.61 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3006_minusinfinity,axiom,
% 13.25/7.61 ! [D: int,P1: int > $o,P: int > $o] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D )
% 13.25/7.61 => ( ! [X3: int,K3: int] :
% 13.25/7.61 ( ( P1 @ X3 )
% 13.25/7.61 = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 13.25/7.61 => ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ X3 @ Z3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P1 @ X3 ) ) )
% 13.25/7.61 => ( ? [X_12: int] : ( P1 @ X_12 )
% 13.25/7.61 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3007_plusinfinity,axiom,
% 13.25/7.61 ! [D: int,P5: int > $o,P: int > $o] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D )
% 13.25/7.61 => ( ! [X3: int,K3: int] :
% 13.25/7.61 ( ( P5 @ X3 )
% 13.25/7.61 = ( P5 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 13.25/7.61 => ( ? [Z3: int] :
% 13.25/7.61 ! [X3: int] :
% 13.25/7.61 ( ( ord_less_int @ Z3 @ X3 )
% 13.25/7.61 => ( ( P @ X3 )
% 13.25/7.61 = ( P5 @ X3 ) ) )
% 13.25/7.61 => ( ? [X_12: int] : ( P5 @ X_12 )
% 13.25/7.61 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3008_incr__mult__lemma,axiom,
% 13.25/7.61 ! [D: int,P: int > $o,K: int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D )
% 13.25/7.61 => ( ! [X3: int] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 => ( P @ ( plus_plus_int @ X3 @ D ) ) )
% 13.25/7.61 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ( P @ X5 )
% 13.25/7.61 => ( P @ ( plus_plus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3009_invar__vebt_Ointros_I5_J,axiom,
% 13.25/7.61 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 13.25/7.61 ( ! [X3: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X3 @ N ) )
% 13.25/7.61 => ( ( vEBT_invar_vebt @ Summary @ M )
% 13.25/7.61 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.61 => ( ( M
% 13.25/7.61 = ( suc @ N ) )
% 13.25/7.61 => ( ( Deg
% 13.25/7.61 = ( plus_plus_nat @ N @ M ) )
% 13.25/7.61 => ( ! [I2: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.61 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X6 ) )
% 13.25/7.61 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 13.25/7.61 => ( ( ( Mi = Ma )
% 13.25/7.61 => ! [X3: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.61 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
% 13.25/7.61 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 13.25/7.61 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 13.25/7.61 => ( ( ( Mi != Ma )
% 13.25/7.61 => ! [I2: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 13.25/7.61 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 13.25/7.61 = I2 )
% 13.25/7.61 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 13.25/7.61 & ! [X3: nat] :
% 13.25/7.61 ( ( ( ( vEBT_VEBT_high @ X3 @ N )
% 13.25/7.61 = I2 )
% 13.25/7.61 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
% 13.25/7.61 => ( ( ord_less_nat @ Mi @ X3 )
% 13.25/7.61 & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3010_decr__mult__lemma,axiom,
% 13.25/7.61 ! [D: int,P: int > $o,K: int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D )
% 13.25/7.61 => ( ! [X3: int] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 => ( P @ ( minus_minus_int @ X3 @ D ) ) )
% 13.25/7.61 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ( P @ X5 )
% 13.25/7.61 => ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3011_periodic__finite__ex,axiom,
% 13.25/7.61 ! [D: int,P: int > $o] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D )
% 13.25/7.61 => ( ! [X3: int,K3: int] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 = ( P @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
% 13.25/7.61 => ( ( ? [X6: int] : ( P @ X6 ) )
% 13.25/7.61 = ( ? [X2: int] :
% 13.25/7.61 ( ( member_int @ X2 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 13.25/7.61 & ( P @ X2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3012_bset_I3_J,axiom,
% 13.25/7.61 ! [D4: int,T: int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.61 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B5 )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( X5 = T )
% 13.25/7.61 => ( ( minus_minus_int @ X5 @ D4 )
% 13.25/7.61 = T ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3013_bset_I4_J,axiom,
% 13.25/7.61 ! [D4: int,T: int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.61 => ( ( member_int @ T @ B5 )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( X5 != T )
% 13.25/7.61 => ( ( minus_minus_int @ X5 @ D4 )
% 13.25/7.61 != T ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3014_bset_I5_J,axiom,
% 13.25/7.61 ! [D4: int,B5: set_int,T: int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( ord_less_int @ X5 @ T )
% 13.25/7.61 => ( ord_less_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3015_bset_I7_J,axiom,
% 13.25/7.61 ! [D4: int,T: int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.61 => ( ( member_int @ T @ B5 )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( ord_less_int @ T @ X5 )
% 13.25/7.61 => ( ord_less_int @ T @ ( minus_minus_int @ X5 @ D4 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3016_aset_I3_J,axiom,
% 13.25/7.61 ! [D4: int,T: int,A2: set_int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.61 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( X5 = T )
% 13.25/7.61 => ( ( plus_plus_int @ X5 @ D4 )
% 13.25/7.61 = T ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3017_aset_I4_J,axiom,
% 13.25/7.61 ! [D4: int,T: int,A2: set_int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.61 => ( ( member_int @ T @ A2 )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( X5 != T )
% 13.25/7.61 => ( ( plus_plus_int @ X5 @ D4 )
% 13.25/7.61 != T ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3018_aset_I5_J,axiom,
% 13.25/7.61 ! [D4: int,T: int,A2: set_int] :
% 13.25/7.61 ( ( ord_less_int @ zero_zero_int @ D4 )
% 13.25/7.61 => ( ( member_int @ T @ A2 )
% 13.25/7.61 => ! [X5: int] :
% 13.25/7.61 ( ! [Xa3: int] :
% 13.25/7.61 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.61 => ! [Xb3: int] :
% 13.25/7.61 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.61 => ( X5
% 13.25/7.61 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.61 => ( ( ord_less_int @ X5 @ T )
% 13.25/7.61 => ( ord_less_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3019_vebt__insert_Osimps_I4_J,axiom,
% 13.25/7.61 ! [V: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 13.25/7.61 ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) @ X )
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3020_invar__vebt_Ocases,axiom,
% 13.25/7.61 ! [A1: vEBT_VEBT,A22: nat] :
% 13.25/7.61 ( ( vEBT_invar_vebt @ A1 @ A22 )
% 13.25/7.61 => ( ( ? [A5: $o,B3: $o] :
% 13.25/7.61 ( A1
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ( A22
% 13.25/7.61 != ( suc @ zero_zero_nat ) ) )
% 13.25/7.61 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 13.25/7.61 ( ( A1
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.61 => ( ( A22 = Deg2 )
% 13.25/7.61 => ( ! [X5: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 13.25/7.61 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 13.25/7.61 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 13.25/7.61 => ( ( M4 = N4 )
% 13.25/7.61 => ( ( Deg2
% 13.25/7.61 = ( plus_plus_nat @ N4 @ M4 ) )
% 13.25/7.61 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 13.25/7.61 => ~ ! [X5: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.61 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 13.25/7.61 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 13.25/7.61 ( ( A1
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.61 => ( ( A22 = Deg2 )
% 13.25/7.61 => ( ! [X5: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 13.25/7.61 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 13.25/7.61 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 13.25/7.61 => ( ( M4
% 13.25/7.61 = ( suc @ N4 ) )
% 13.25/7.61 => ( ( Deg2
% 13.25/7.61 = ( plus_plus_nat @ N4 @ M4 ) )
% 13.25/7.61 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 13.25/7.61 => ~ ! [X5: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.61 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) ) ) ) ) ) ) ) )
% 13.25/7.61 => ( ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 13.25/7.61 ( ( A1
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.61 => ( ( A22 = Deg2 )
% 13.25/7.61 => ( ! [X5: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 13.25/7.61 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 13.25/7.61 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 13.25/7.61 => ( ( M4 = N4 )
% 13.25/7.61 => ( ( Deg2
% 13.25/7.61 = ( plus_plus_nat @ N4 @ M4 ) )
% 13.25/7.61 => ( ! [I3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 13.25/7.61 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 13.25/7.61 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 13.25/7.61 => ( ( ( Mi2 = Ma2 )
% 13.25/7.61 => ! [X5: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.61 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 13.25/7.61 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 13.25/7.61 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.61 => ~ ( ( Mi2 != Ma2 )
% 13.25/7.61 => ! [I3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 13.25/7.61 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 13.25/7.61 = I3 )
% 13.25/7.61 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 13.25/7.61 & ! [X5: nat] :
% 13.25/7.61 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 13.25/7.61 = I3 )
% 13.25/7.61 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 13.25/7.61 => ( ( ord_less_nat @ Mi2 @ X5 )
% 13.25/7.61 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 13.25/7.61 => ~ ! [TreeList3: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 13.25/7.61 ( ( A1
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.61 => ( ( A22 = Deg2 )
% 13.25/7.61 => ( ! [X5: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X5 @ N4 ) )
% 13.25/7.61 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 13.25/7.61 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 13.25/7.61 => ( ( M4
% 13.25/7.61 = ( suc @ N4 ) )
% 13.25/7.61 => ( ( Deg2
% 13.25/7.61 = ( plus_plus_nat @ N4 @ M4 ) )
% 13.25/7.61 => ( ! [I3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 13.25/7.61 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X6 ) )
% 13.25/7.61 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 13.25/7.61 => ( ( ( Mi2 = Ma2 )
% 13.25/7.61 => ! [X5: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.61 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
% 13.25/7.61 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 13.25/7.61 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.61 => ~ ( ( Mi2 != Ma2 )
% 13.25/7.61 => ! [I3: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 13.25/7.61 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N4 )
% 13.25/7.61 = I3 )
% 13.25/7.61 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N4 ) ) )
% 13.25/7.61 & ! [X5: nat] :
% 13.25/7.61 ( ( ( ( vEBT_VEBT_high @ X5 @ N4 )
% 13.25/7.61 = I3 )
% 13.25/7.61 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N4 ) ) )
% 13.25/7.61 => ( ( ord_less_nat @ Mi2 @ X5 )
% 13.25/7.61 & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3021_invar__vebt_Osimps,axiom,
% 13.25/7.61 ( vEBT_invar_vebt
% 13.25/7.61 = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 13.25/7.61 ( ( ? [A3: $o,B4: $o] :
% 13.25/7.61 ( A12
% 13.25/7.61 = ( vEBT_Leaf @ A3 @ B4 ) )
% 13.25/7.61 & ( A23
% 13.25/7.61 = ( suc @ zero_zero_nat ) ) )
% 13.25/7.61 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 13.25/7.61 ( ( A12
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary3 ) )
% 13.25/7.61 & ! [X2: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 13.25/7.61 & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 13.25/7.61 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 13.25/7.61 & ( A23
% 13.25/7.61 = ( plus_plus_nat @ N2 @ N2 ) )
% 13.25/7.61 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 13.25/7.61 & ! [X2: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 13.25/7.61 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.61 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
% 13.25/7.61 ( ( A12
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList @ Summary3 ) )
% 13.25/7.61 & ! [X2: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 13.25/7.61 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 13.25/7.61 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 13.25/7.61 & ( A23
% 13.25/7.61 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 13.25/7.61 & ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
% 13.25/7.61 & ! [X2: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 13.25/7.61 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.61 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 13.25/7.61 ( ( A12
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
% 13.25/7.61 & ! [X2: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 13.25/7.61 & ( vEBT_invar_vebt @ Summary3 @ N2 )
% 13.25/7.61 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 13.25/7.61 & ( A23
% 13.25/7.61 = ( plus_plus_nat @ N2 @ N2 ) )
% 13.25/7.61 & ! [I4: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 13.25/7.61 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
% 13.25/7.61 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 13.25/7.61 & ( ( Mi3 = Ma3 )
% 13.25/7.61 => ! [X2: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 13.25/7.61 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.61 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.61 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 13.25/7.61 & ( ( Mi3 != Ma3 )
% 13.25/7.61 => ! [I4: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 13.25/7.61 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 13.25/7.61 = I4 )
% 13.25/7.61 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 13.25/7.61 & ! [X2: nat] :
% 13.25/7.61 ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
% 13.25/7.61 = I4 )
% 13.25/7.61 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
% 13.25/7.61 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.61 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) )
% 13.25/7.61 | ? [TreeList: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 13.25/7.61 ( ( A12
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList @ Summary3 ) )
% 13.25/7.61 & ! [X2: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 13.25/7.61 => ( vEBT_invar_vebt @ X2 @ N2 ) )
% 13.25/7.61 & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
% 13.25/7.61 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 13.25/7.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 13.25/7.61 & ( A23
% 13.25/7.61 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 13.25/7.61 & ! [I4: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 13.25/7.61 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X6 ) )
% 13.25/7.61 = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
% 13.25/7.61 & ( ( Mi3 = Ma3 )
% 13.25/7.61 => ! [X2: vEBT_VEBT] :
% 13.25/7.61 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 13.25/7.61 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.61 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.61 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 13.25/7.61 & ( ( Mi3 != Ma3 )
% 13.25/7.61 => ! [I4: nat] :
% 13.25/7.61 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 13.25/7.61 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 13.25/7.61 = I4 )
% 13.25/7.61 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 13.25/7.61 & ! [X2: nat] :
% 13.25/7.61 ( ( ( ( vEBT_VEBT_high @ X2 @ N2 )
% 13.25/7.61 = I4 )
% 13.25/7.61 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N2 ) ) )
% 13.25/7.61 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.61 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3022_vebt__member_Osimps_I4_J,axiom,
% 13.25/7.61 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 13.25/7.61 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3023_psubsetI,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( A2 != B5 )
% 13.25/7.61 => ( ord_less_set_int @ A2 @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3024_Diff__eq__empty__iff,axiom,
% 13.25/7.61 ! [A2: set_real,B5: set_real] :
% 13.25/7.61 ( ( ( minus_minus_set_real @ A2 @ B5 )
% 13.25/7.61 = bot_bot_set_real )
% 13.25/7.61 = ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3025_Diff__eq__empty__iff,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( ( minus_minus_set_nat @ A2 @ B5 )
% 13.25/7.61 = bot_bot_set_nat )
% 13.25/7.61 = ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3026_Diff__eq__empty__iff,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( ( minus_minus_set_int @ A2 @ B5 )
% 13.25/7.61 = bot_bot_set_int )
% 13.25/7.61 = ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3027_empty__subsetI,axiom,
% 13.25/7.61 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3028_empty__subsetI,axiom,
% 13.25/7.61 ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3029_empty__subsetI,axiom,
% 13.25/7.61 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3030_Leaf__0__not,axiom,
% 13.25/7.61 ! [A: $o,B: $o] :
% 13.25/7.61 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3031_deg1Leaf,axiom,
% 13.25/7.61 ! [T: vEBT_VEBT] :
% 13.25/7.61 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 13.25/7.61 = ( ? [A3: $o,B4: $o] :
% 13.25/7.61 ( T
% 13.25/7.61 = ( vEBT_Leaf @ A3 @ B4 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3032_deg__1__Leaf,axiom,
% 13.25/7.61 ! [T: vEBT_VEBT] :
% 13.25/7.61 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 13.25/7.61 => ? [A5: $o,B3: $o] :
% 13.25/7.61 ( T
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3033_deg__1__Leafy,axiom,
% 13.25/7.61 ! [T: vEBT_VEBT,N: nat] :
% 13.25/7.61 ( ( vEBT_invar_vebt @ T @ N )
% 13.25/7.61 => ( ( N = one_one_nat )
% 13.25/7.61 => ? [A5: $o,B3: $o] :
% 13.25/7.61 ( T
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3034_subsetI,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ! [X3: nat] :
% 13.25/7.61 ( ( member_nat @ X3 @ A2 )
% 13.25/7.61 => ( member_nat @ X3 @ B5 ) )
% 13.25/7.61 => ( ord_less_eq_set_nat @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3035_subsetI,axiom,
% 13.25/7.61 ! [A2: set_real,B5: set_real] :
% 13.25/7.61 ( ! [X3: real] :
% 13.25/7.61 ( ( member_real @ X3 @ A2 )
% 13.25/7.61 => ( member_real @ X3 @ B5 ) )
% 13.25/7.61 => ( ord_less_eq_set_real @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3036_subsetI,axiom,
% 13.25/7.61 ! [A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ! [X3: complex] :
% 13.25/7.61 ( ( member_complex @ X3 @ A2 )
% 13.25/7.61 => ( member_complex @ X3 @ B5 ) )
% 13.25/7.61 => ( ord_le211207098394363844omplex @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3037_subsetI,axiom,
% 13.25/7.61 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ! [X3: produc859450856879609959at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ X3 @ A2 )
% 13.25/7.61 => ( member8206827879206165904at_nat @ X3 @ B5 ) )
% 13.25/7.61 => ( ord_le3000389064537975527at_nat @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3038_subsetI,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ! [X3: int] :
% 13.25/7.61 ( ( member_int @ X3 @ A2 )
% 13.25/7.61 => ( member_int @ X3 @ B5 ) )
% 13.25/7.61 => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3039_subset__antisym,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.61 => ( A2 = B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3040_Diff__idemp,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ B5 )
% 13.25/7.61 = ( minus_minus_set_nat @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3041_Diff__iff,axiom,
% 13.25/7.61 ! [C: real,A2: set_real,B5: set_real] :
% 13.25/7.61 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
% 13.25/7.61 = ( ( member_real @ C @ A2 )
% 13.25/7.61 & ~ ( member_real @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3042_Diff__iff,axiom,
% 13.25/7.61 ! [C: int,A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.61 = ( ( member_int @ C @ A2 )
% 13.25/7.61 & ~ ( member_int @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3043_Diff__iff,axiom,
% 13.25/7.61 ! [C: complex,A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.61 = ( ( member_complex @ C @ A2 )
% 13.25/7.61 & ~ ( member_complex @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3044_Diff__iff,axiom,
% 13.25/7.61 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) )
% 13.25/7.61 = ( ( member8206827879206165904at_nat @ C @ A2 )
% 13.25/7.61 & ~ ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3045_Diff__iff,axiom,
% 13.25/7.61 ! [C: nat,A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.61 = ( ( member_nat @ C @ A2 )
% 13.25/7.61 & ~ ( member_nat @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3046_DiffI,axiom,
% 13.25/7.61 ! [C: real,A2: set_real,B5: set_real] :
% 13.25/7.61 ( ( member_real @ C @ A2 )
% 13.25/7.61 => ( ~ ( member_real @ C @ B5 )
% 13.25/7.61 => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3047_DiffI,axiom,
% 13.25/7.61 ! [C: int,A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( member_int @ C @ A2 )
% 13.25/7.61 => ( ~ ( member_int @ C @ B5 )
% 13.25/7.61 => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3048_DiffI,axiom,
% 13.25/7.61 ! [C: complex,A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ( member_complex @ C @ A2 )
% 13.25/7.61 => ( ~ ( member_complex @ C @ B5 )
% 13.25/7.61 => ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3049_DiffI,axiom,
% 13.25/7.61 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ C @ A2 )
% 13.25/7.61 => ( ~ ( member8206827879206165904at_nat @ C @ B5 )
% 13.25/7.61 => ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3050_DiffI,axiom,
% 13.25/7.61 ! [C: nat,A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( member_nat @ C @ A2 )
% 13.25/7.61 => ( ~ ( member_nat @ C @ B5 )
% 13.25/7.61 => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3051_subset__empty,axiom,
% 13.25/7.61 ! [A2: set_nat] :
% 13.25/7.61 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 13.25/7.61 = ( A2 = bot_bot_set_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3052_subset__empty,axiom,
% 13.25/7.61 ! [A2: set_real] :
% 13.25/7.61 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 13.25/7.61 = ( A2 = bot_bot_set_real ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3053_subset__empty,axiom,
% 13.25/7.61 ! [A2: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 13.25/7.61 = ( A2 = bot_bot_set_int ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3054_Diff__cancel,axiom,
% 13.25/7.61 ! [A2: set_int] :
% 13.25/7.61 ( ( minus_minus_set_int @ A2 @ A2 )
% 13.25/7.61 = bot_bot_set_int ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3055_Diff__cancel,axiom,
% 13.25/7.61 ! [A2: set_real] :
% 13.25/7.61 ( ( minus_minus_set_real @ A2 @ A2 )
% 13.25/7.61 = bot_bot_set_real ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3056_Diff__cancel,axiom,
% 13.25/7.61 ! [A2: set_nat] :
% 13.25/7.61 ( ( minus_minus_set_nat @ A2 @ A2 )
% 13.25/7.61 = bot_bot_set_nat ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3057_empty__Diff,axiom,
% 13.25/7.61 ! [A2: set_int] :
% 13.25/7.61 ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 13.25/7.61 = bot_bot_set_int ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3058_empty__Diff,axiom,
% 13.25/7.61 ! [A2: set_real] :
% 13.25/7.61 ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 13.25/7.61 = bot_bot_set_real ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3059_empty__Diff,axiom,
% 13.25/7.61 ! [A2: set_nat] :
% 13.25/7.61 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 13.25/7.61 = bot_bot_set_nat ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3060_Diff__empty,axiom,
% 13.25/7.61 ! [A2: set_int] :
% 13.25/7.61 ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 13.25/7.61 = A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3061_Diff__empty,axiom,
% 13.25/7.61 ! [A2: set_real] :
% 13.25/7.61 ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 13.25/7.61 = A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3062_Diff__empty,axiom,
% 13.25/7.61 ! [A2: set_nat] :
% 13.25/7.61 ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 13.25/7.61 = A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3063_DiffD2,axiom,
% 13.25/7.61 ! [C: real,A2: set_real,B5: set_real] :
% 13.25/7.61 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( member_real @ C @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3064_DiffD2,axiom,
% 13.25/7.61 ! [C: int,A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( member_int @ C @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3065_DiffD2,axiom,
% 13.25/7.61 ! [C: complex,A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( member_complex @ C @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3066_DiffD2,axiom,
% 13.25/7.61 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( member8206827879206165904at_nat @ C @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3067_DiffD2,axiom,
% 13.25/7.61 ! [C: nat,A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( member_nat @ C @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3068_DiffD1,axiom,
% 13.25/7.61 ! [C: real,A2: set_real,B5: set_real] :
% 13.25/7.61 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
% 13.25/7.61 => ( member_real @ C @ A2 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3069_DiffD1,axiom,
% 13.25/7.61 ! [C: int,A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.61 => ( member_int @ C @ A2 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3070_DiffD1,axiom,
% 13.25/7.61 ! [C: complex,A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.61 => ( member_complex @ C @ A2 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3071_DiffD1,axiom,
% 13.25/7.61 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) )
% 13.25/7.61 => ( member8206827879206165904at_nat @ C @ A2 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3072_DiffD1,axiom,
% 13.25/7.61 ! [C: nat,A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.61 => ( member_nat @ C @ A2 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3073_DiffE,axiom,
% 13.25/7.61 ! [C: real,A2: set_real,B5: set_real] :
% 13.25/7.61 ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( ( member_real @ C @ A2 )
% 13.25/7.61 => ( member_real @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3074_DiffE,axiom,
% 13.25/7.61 ! [C: int,A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( ( member_int @ C @ A2 )
% 13.25/7.61 => ( member_int @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3075_DiffE,axiom,
% 13.25/7.61 ! [C: complex,A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( ( member_complex @ C @ A2 )
% 13.25/7.61 => ( member_complex @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3076_DiffE,axiom,
% 13.25/7.61 ! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( ( member8206827879206165904at_nat @ C @ A2 )
% 13.25/7.61 => ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3077_DiffE,axiom,
% 13.25/7.61 ! [C: nat,A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.61 => ~ ( ( member_nat @ C @ A2 )
% 13.25/7.61 => ( member_nat @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3078_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 13.25/7.61 ! [Uu: $o] :
% 13.25/7.61 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3079_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 13.25/7.61 ! [Uv: $o] :
% 13.25/7.61 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3080_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 13.25/7.61 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3081_vebt__member_Osimps_I1_J,axiom,
% 13.25/7.61 ! [A: $o,B: $o,X: nat] :
% 13.25/7.61 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
% 13.25/7.61 = ( ( ( X = zero_zero_nat )
% 13.25/7.61 => A )
% 13.25/7.61 & ( ( X != zero_zero_nat )
% 13.25/7.61 => ( ( ( X = one_one_nat )
% 13.25/7.61 => B )
% 13.25/7.61 & ( X = one_one_nat ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3082_vebt__insert_Osimps_I1_J,axiom,
% 13.25/7.61 ! [X: nat,A: $o,B: $o] :
% 13.25/7.61 ( ( ( X = zero_zero_nat )
% 13.25/7.61 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 13.25/7.61 = ( vEBT_Leaf @ $true @ B ) ) )
% 13.25/7.61 & ( ( X != zero_zero_nat )
% 13.25/7.61 => ( ( ( X = one_one_nat )
% 13.25/7.61 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 13.25/7.61 = ( vEBT_Leaf @ A @ $true ) ) )
% 13.25/7.61 & ( ( X != one_one_nat )
% 13.25/7.61 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
% 13.25/7.61 = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3083_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT] :
% 13.25/7.61 ( ~ ( vEBT_VEBT_minNull @ X )
% 13.25/7.61 => ( ! [Uv2: $o] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 13.25/7.61 => ( ! [Uu2: $o] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 13.25/7.61 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3084_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT] :
% 13.25/7.61 ( ( vEBT_VEBT_minNull @ X )
% 13.25/7.61 => ( ( X
% 13.25/7.61 != ( vEBT_Leaf @ $false @ $false ) )
% 13.25/7.61 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3085_invar__vebt_Ointros_I1_J,axiom,
% 13.25/7.61 ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3086_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Y3: $o] :
% 13.25/7.61 ( ( ( vEBT_VEBT_minNull @ X )
% 13.25/7.61 = Y3 )
% 13.25/7.61 => ( ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ $false @ $false ) )
% 13.25/7.61 => ~ Y3 )
% 13.25/7.61 => ( ( ? [Uv2: $o] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 13.25/7.61 => Y3 )
% 13.25/7.61 => ( ( ? [Uu2: $o] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 13.25/7.61 => Y3 )
% 13.25/7.61 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 13.25/7.61 => ~ Y3 )
% 13.25/7.61 => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 13.25/7.61 => Y3 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3087_vebt__succ_Osimps_I2_J,axiom,
% 13.25/7.61 ! [Uv: $o,Uw: $o,N: nat] :
% 13.25/7.61 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
% 13.25/7.61 = none_nat ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3088_Diff__mono,axiom,
% 13.25/7.61 ! [A2: set_nat,C4: set_nat,D4: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 13.25/7.61 => ( ( ord_less_eq_set_nat @ D4 @ B5 )
% 13.25/7.61 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3089_Diff__mono,axiom,
% 13.25/7.61 ! [A2: set_int,C4: set_int,D4: set_int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ C4 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ D4 @ B5 )
% 13.25/7.61 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3090_Diff__subset,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B5 ) @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3091_Diff__subset,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B5 ) @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3092_double__diff,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat,C4: set_nat] :
% 13.25/7.61 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.61 => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 13.25/7.61 => ( ( minus_minus_set_nat @ B5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 13.25/7.61 = A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3093_double__diff,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int,C4: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 13.25/7.61 => ( ( minus_minus_set_int @ B5 @ ( minus_minus_set_int @ C4 @ A2 ) )
% 13.25/7.61 = A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3094_in__mono,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat,X: nat] :
% 13.25/7.61 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.61 => ( ( member_nat @ X @ A2 )
% 13.25/7.61 => ( member_nat @ X @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3095_in__mono,axiom,
% 13.25/7.61 ! [A2: set_real,B5: set_real,X: real] :
% 13.25/7.61 ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.61 => ( ( member_real @ X @ A2 )
% 13.25/7.61 => ( member_real @ X @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3096_in__mono,axiom,
% 13.25/7.61 ! [A2: set_complex,B5: set_complex,X: complex] :
% 13.25/7.61 ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.61 => ( ( member_complex @ X @ A2 )
% 13.25/7.61 => ( member_complex @ X @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3097_in__mono,axiom,
% 13.25/7.61 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat] :
% 13.25/7.61 ( ( ord_le3000389064537975527at_nat @ A2 @ B5 )
% 13.25/7.61 => ( ( member8206827879206165904at_nat @ X @ A2 )
% 13.25/7.61 => ( member8206827879206165904at_nat @ X @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3098_in__mono,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int,X: int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( member_int @ X @ A2 )
% 13.25/7.61 => ( member_int @ X @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3099_subsetD,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat,C: nat] :
% 13.25/7.61 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.61 => ( ( member_nat @ C @ A2 )
% 13.25/7.61 => ( member_nat @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3100_subsetD,axiom,
% 13.25/7.61 ! [A2: set_real,B5: set_real,C: real] :
% 13.25/7.61 ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.61 => ( ( member_real @ C @ A2 )
% 13.25/7.61 => ( member_real @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3101_subsetD,axiom,
% 13.25/7.61 ! [A2: set_complex,B5: set_complex,C: complex] :
% 13.25/7.61 ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.61 => ( ( member_complex @ C @ A2 )
% 13.25/7.61 => ( member_complex @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3102_subsetD,axiom,
% 13.25/7.61 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
% 13.25/7.61 ( ( ord_le3000389064537975527at_nat @ A2 @ B5 )
% 13.25/7.61 => ( ( member8206827879206165904at_nat @ C @ A2 )
% 13.25/7.61 => ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3103_subsetD,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int,C: int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( member_int @ C @ A2 )
% 13.25/7.61 => ( member_int @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3104_equalityE,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( A2 = B5 )
% 13.25/7.61 => ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ~ ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3105_subset__eq,axiom,
% 13.25/7.61 ( ord_less_eq_set_nat
% 13.25/7.61 = ( ^ [A6: set_nat,B6: set_nat] :
% 13.25/7.61 ! [X2: nat] :
% 13.25/7.61 ( ( member_nat @ X2 @ A6 )
% 13.25/7.61 => ( member_nat @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3106_subset__eq,axiom,
% 13.25/7.61 ( ord_less_eq_set_real
% 13.25/7.61 = ( ^ [A6: set_real,B6: set_real] :
% 13.25/7.61 ! [X2: real] :
% 13.25/7.61 ( ( member_real @ X2 @ A6 )
% 13.25/7.61 => ( member_real @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3107_subset__eq,axiom,
% 13.25/7.61 ( ord_le211207098394363844omplex
% 13.25/7.61 = ( ^ [A6: set_complex,B6: set_complex] :
% 13.25/7.61 ! [X2: complex] :
% 13.25/7.61 ( ( member_complex @ X2 @ A6 )
% 13.25/7.61 => ( member_complex @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3108_subset__eq,axiom,
% 13.25/7.61 ( ord_le3000389064537975527at_nat
% 13.25/7.61 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ! [X2: produc859450856879609959at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ X2 @ A6 )
% 13.25/7.61 => ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3109_subset__eq,axiom,
% 13.25/7.61 ( ord_less_eq_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ! [X2: int] :
% 13.25/7.61 ( ( member_int @ X2 @ A6 )
% 13.25/7.61 => ( member_int @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3110_equalityD1,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( A2 = B5 )
% 13.25/7.61 => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3111_equalityD2,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( A2 = B5 )
% 13.25/7.61 => ( ord_less_eq_set_int @ B5 @ A2 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3112_subset__iff,axiom,
% 13.25/7.61 ( ord_less_eq_set_nat
% 13.25/7.61 = ( ^ [A6: set_nat,B6: set_nat] :
% 13.25/7.61 ! [T2: nat] :
% 13.25/7.61 ( ( member_nat @ T2 @ A6 )
% 13.25/7.61 => ( member_nat @ T2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3113_subset__iff,axiom,
% 13.25/7.61 ( ord_less_eq_set_real
% 13.25/7.61 = ( ^ [A6: set_real,B6: set_real] :
% 13.25/7.61 ! [T2: real] :
% 13.25/7.61 ( ( member_real @ T2 @ A6 )
% 13.25/7.61 => ( member_real @ T2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3114_subset__iff,axiom,
% 13.25/7.61 ( ord_le211207098394363844omplex
% 13.25/7.61 = ( ^ [A6: set_complex,B6: set_complex] :
% 13.25/7.61 ! [T2: complex] :
% 13.25/7.61 ( ( member_complex @ T2 @ A6 )
% 13.25/7.61 => ( member_complex @ T2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3115_subset__iff,axiom,
% 13.25/7.61 ( ord_le3000389064537975527at_nat
% 13.25/7.61 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ! [T2: produc859450856879609959at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ T2 @ A6 )
% 13.25/7.61 => ( member8206827879206165904at_nat @ T2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3116_subset__iff,axiom,
% 13.25/7.61 ( ord_less_eq_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ! [T2: int] :
% 13.25/7.61 ( ( member_int @ T2 @ A6 )
% 13.25/7.61 => ( member_int @ T2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3117_subset__refl,axiom,
% 13.25/7.61 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3118_Collect__mono,axiom,
% 13.25/7.61 ! [P: complex > $o,Q: complex > $o] :
% 13.25/7.61 ( ! [X3: complex] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 => ( Q @ X3 ) )
% 13.25/7.61 => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3119_Collect__mono,axiom,
% 13.25/7.61 ! [P: real > $o,Q: real > $o] :
% 13.25/7.61 ( ! [X3: real] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 => ( Q @ X3 ) )
% 13.25/7.61 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3120_Collect__mono,axiom,
% 13.25/7.61 ! [P: list_nat > $o,Q: list_nat > $o] :
% 13.25/7.61 ( ! [X3: list_nat] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 => ( Q @ X3 ) )
% 13.25/7.61 => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3121_Collect__mono,axiom,
% 13.25/7.61 ! [P: nat > $o,Q: nat > $o] :
% 13.25/7.61 ( ! [X3: nat] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 => ( Q @ X3 ) )
% 13.25/7.61 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3122_Collect__mono,axiom,
% 13.25/7.61 ! [P: int > $o,Q: int > $o] :
% 13.25/7.61 ( ! [X3: int] :
% 13.25/7.61 ( ( P @ X3 )
% 13.25/7.61 => ( Q @ X3 ) )
% 13.25/7.61 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3123_subset__trans,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int,C4: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 13.25/7.61 => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3124_set__eq__subset,axiom,
% 13.25/7.61 ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 13.25/7.61 & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3125_Collect__mono__iff,axiom,
% 13.25/7.61 ! [P: complex > $o,Q: complex > $o] :
% 13.25/7.61 ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
% 13.25/7.61 = ( ! [X2: complex] :
% 13.25/7.61 ( ( P @ X2 )
% 13.25/7.61 => ( Q @ X2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3126_Collect__mono__iff,axiom,
% 13.25/7.61 ! [P: real > $o,Q: real > $o] :
% 13.25/7.61 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 13.25/7.61 = ( ! [X2: real] :
% 13.25/7.61 ( ( P @ X2 )
% 13.25/7.61 => ( Q @ X2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3127_Collect__mono__iff,axiom,
% 13.25/7.61 ! [P: list_nat > $o,Q: list_nat > $o] :
% 13.25/7.61 ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 13.25/7.61 = ( ! [X2: list_nat] :
% 13.25/7.61 ( ( P @ X2 )
% 13.25/7.61 => ( Q @ X2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3128_Collect__mono__iff,axiom,
% 13.25/7.61 ! [P: nat > $o,Q: nat > $o] :
% 13.25/7.61 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 13.25/7.61 = ( ! [X2: nat] :
% 13.25/7.61 ( ( P @ X2 )
% 13.25/7.61 => ( Q @ X2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3129_Collect__mono__iff,axiom,
% 13.25/7.61 ! [P: int > $o,Q: int > $o] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 13.25/7.61 = ( ! [X2: int] :
% 13.25/7.61 ( ( P @ X2 )
% 13.25/7.61 => ( Q @ X2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3130_psubsetD,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat,C: nat] :
% 13.25/7.61 ( ( ord_less_set_nat @ A2 @ B5 )
% 13.25/7.61 => ( ( member_nat @ C @ A2 )
% 13.25/7.61 => ( member_nat @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3131_psubsetD,axiom,
% 13.25/7.61 ! [A2: set_real,B5: set_real,C: real] :
% 13.25/7.61 ( ( ord_less_set_real @ A2 @ B5 )
% 13.25/7.61 => ( ( member_real @ C @ A2 )
% 13.25/7.61 => ( member_real @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3132_psubsetD,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int,C: int] :
% 13.25/7.61 ( ( ord_less_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( member_int @ C @ A2 )
% 13.25/7.61 => ( member_int @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3133_psubsetD,axiom,
% 13.25/7.61 ! [A2: set_complex,B5: set_complex,C: complex] :
% 13.25/7.61 ( ( ord_less_set_complex @ A2 @ B5 )
% 13.25/7.61 => ( ( member_complex @ C @ A2 )
% 13.25/7.61 => ( member_complex @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3134_psubsetD,axiom,
% 13.25/7.61 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
% 13.25/7.61 ( ( ord_le6428140832669894131at_nat @ A2 @ B5 )
% 13.25/7.61 => ( ( member8206827879206165904at_nat @ C @ A2 )
% 13.25/7.61 => ( member8206827879206165904at_nat @ C @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3135_psubset__imp__ex__mem,axiom,
% 13.25/7.61 ! [A2: set_real,B5: set_real] :
% 13.25/7.61 ( ( ord_less_set_real @ A2 @ B5 )
% 13.25/7.61 => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3136_psubset__imp__ex__mem,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_set_int @ A2 @ B5 )
% 13.25/7.61 => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3137_psubset__imp__ex__mem,axiom,
% 13.25/7.61 ! [A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ( ord_less_set_complex @ A2 @ B5 )
% 13.25/7.61 => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3138_psubset__imp__ex__mem,axiom,
% 13.25/7.61 ! [A2: set_Pr8693737435421807431at_nat,B5: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ( ord_le6428140832669894131at_nat @ A2 @ B5 )
% 13.25/7.61 => ? [B3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ B5 @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3139_psubset__imp__ex__mem,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( ord_less_set_nat @ A2 @ B5 )
% 13.25/7.61 => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3140_vebt__succ_Osimps_I1_J,axiom,
% 13.25/7.61 ! [B: $o,Uu: $o] :
% 13.25/7.61 ( ( B
% 13.25/7.61 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 13.25/7.61 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.61 & ( ~ B
% 13.25/7.61 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
% 13.25/7.61 = none_nat ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3141_vebt__mint_Osimps_I1_J,axiom,
% 13.25/7.61 ! [A: $o,B: $o] :
% 13.25/7.61 ( ( A
% 13.25/7.61 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 13.25/7.61 = ( some_nat @ zero_zero_nat ) ) )
% 13.25/7.61 & ( ~ A
% 13.25/7.61 => ( ( B
% 13.25/7.61 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 13.25/7.61 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.61 & ( ~ B
% 13.25/7.61 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 13.25/7.61 = none_nat ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3142_vebt__maxt_Osimps_I1_J,axiom,
% 13.25/7.61 ! [B: $o,A: $o] :
% 13.25/7.61 ( ( B
% 13.25/7.61 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 13.25/7.61 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.61 & ( ~ B
% 13.25/7.61 => ( ( A
% 13.25/7.61 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 13.25/7.61 = ( some_nat @ zero_zero_nat ) ) )
% 13.25/7.61 & ( ~ A
% 13.25/7.61 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 13.25/7.61 = none_nat ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3143_vebt__succ_Ocases,axiom,
% 13.25/7.61 ! [X: produc9072475918466114483BT_nat] :
% 13.25/7.61 ( ! [Uu2: $o,B3: $o] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) )
% 13.25/7.61 => ( ! [Uv2: $o,Uw2: $o,N4: nat] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) )
% 13.25/7.61 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
% 13.25/7.61 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
% 13.25/7.61 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
% 13.25/7.61 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3144_not__psubset__empty,axiom,
% 13.25/7.61 ! [A2: set_nat] :
% 13.25/7.61 ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3145_not__psubset__empty,axiom,
% 13.25/7.61 ! [A2: set_int] :
% 13.25/7.61 ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3146_not__psubset__empty,axiom,
% 13.25/7.61 ! [A2: set_real] :
% 13.25/7.61 ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3147_subset__iff__psubset__eq,axiom,
% 13.25/7.61 ( ord_less_eq_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ( ( ord_less_set_int @ A6 @ B6 )
% 13.25/7.61 | ( A6 = B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3148_subset__psubset__trans,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int,C4: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( ord_less_set_int @ B5 @ C4 )
% 13.25/7.61 => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3149_subset__not__subset__eq,axiom,
% 13.25/7.61 ( ord_less_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 13.25/7.61 & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3150_psubset__subset__trans,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int,C4: set_int] :
% 13.25/7.61 ( ( ord_less_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ B5 @ C4 )
% 13.25/7.61 => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3151_psubset__imp__subset,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3152_psubset__eq,axiom,
% 13.25/7.61 ( ord_less_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 13.25/7.61 & ( A6 != B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3153_psubsetE,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_set_int @ A2 @ B5 )
% 13.25/7.61 => ~ ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3154_vebt__mint_Oelims,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Y3: option_nat] :
% 13.25/7.61 ( ( ( vEBT_vebt_mint @ X )
% 13.25/7.61 = Y3 )
% 13.25/7.61 => ( ! [A5: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ~ ( ( A5
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( some_nat @ zero_zero_nat ) ) )
% 13.25/7.61 & ( ~ A5
% 13.25/7.61 => ( ( B3
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.61 & ( ~ B3
% 13.25/7.61 => ( Y3 = none_nat ) ) ) ) ) )
% 13.25/7.61 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 13.25/7.61 => ( Y3 != none_nat ) )
% 13.25/7.61 => ~ ! [Mi2: nat] :
% 13.25/7.61 ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 13.25/7.61 => ( Y3
% 13.25/7.61 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3155_vebt__maxt_Oelims,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Y3: option_nat] :
% 13.25/7.61 ( ( ( vEBT_vebt_maxt @ X )
% 13.25/7.61 = Y3 )
% 13.25/7.61 => ( ! [A5: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ~ ( ( B3
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.61 & ( ~ B3
% 13.25/7.61 => ( ( A5
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( some_nat @ zero_zero_nat ) ) )
% 13.25/7.61 & ( ~ A5
% 13.25/7.61 => ( Y3 = none_nat ) ) ) ) ) )
% 13.25/7.61 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 13.25/7.61 => ( Y3 != none_nat ) )
% 13.25/7.61 => ~ ! [Mi2: nat,Ma2: nat] :
% 13.25/7.61 ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 13.25/7.61 => ( Y3
% 13.25/7.61 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3156_vebt__insert_Osimps_I2_J,axiom,
% 13.25/7.61 ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 13.25/7.61 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X )
% 13.25/7.61 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3157_vebt__member_Osimps_I2_J,axiom,
% 13.25/7.61 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
% 13.25/7.61 ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3158_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 13.25/7.61 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 13.25/7.61 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3159_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 13.25/7.61 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3160_vebt__insert_Osimps_I3_J,axiom,
% 13.25/7.61 ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 13.25/7.61 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X )
% 13.25/7.61 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3161_vebt__member_Osimps_I3_J,axiom,
% 13.25/7.61 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
% 13.25/7.61 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3162_finite__Diff,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( finite_finite_int @ A2 )
% 13.25/7.61 => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3163_finite__Diff,axiom,
% 13.25/7.61 ! [A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.61 => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3164_finite__Diff,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( finite_finite_nat @ A2 )
% 13.25/7.61 => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3165_finite__Diff2,axiom,
% 13.25/7.61 ! [B5: set_int,A2: set_int] :
% 13.25/7.61 ( ( finite_finite_int @ B5 )
% 13.25/7.61 => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.61 = ( finite_finite_int @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3166_finite__Diff2,axiom,
% 13.25/7.61 ! [B5: set_complex,A2: set_complex] :
% 13.25/7.61 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.61 => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.61 = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3167_finite__Diff2,axiom,
% 13.25/7.61 ! [B5: set_nat,A2: set_nat] :
% 13.25/7.61 ( ( finite_finite_nat @ B5 )
% 13.25/7.61 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.61 = ( finite_finite_nat @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3168_succ__greatereq__min,axiom,
% 13.25/7.61 ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 13.25/7.61 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 13.25/7.61 => ( ( ord_less_eq_nat @ Mi @ X )
% 13.25/7.61 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.61 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.61 @ ( if_option_nat
% 13.25/7.61 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 != none_nat )
% 13.25/7.61 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 13.25/7.61 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 13.25/7.61 @ ( if_option_nat
% 13.25/7.61 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.61 = none_nat )
% 13.25/7.61 @ none_nat
% 13.25/7.61 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 13.25/7.61 @ none_nat ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3169_succ__less__length__list,axiom,
% 13.25/7.61 ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 13.25/7.61 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 13.25/7.61 => ( ( ord_less_eq_nat @ Mi @ X )
% 13.25/7.61 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.61 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.61 = ( if_option_nat
% 13.25/7.61 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 != none_nat )
% 13.25/7.61 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 13.25/7.61 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 13.25/7.61 @ ( if_option_nat
% 13.25/7.61 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.61 = none_nat )
% 13.25/7.61 @ none_nat
% 13.25/7.61 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3170_unset__bit__0,axiom,
% 13.25/7.61 ! [A: int] :
% 13.25/7.61 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 13.25/7.61 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3171_unset__bit__0,axiom,
% 13.25/7.61 ! [A: nat] :
% 13.25/7.61 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 13.25/7.61 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3172_buildup__nothing__in__leaf,axiom,
% 13.25/7.61 ! [N: nat,X: nat] :
% 13.25/7.61 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3173_unset__bit__Suc,axiom,
% 13.25/7.61 ! [N: nat,A: code_integer] :
% 13.25/7.61 ( ( bit_se8260200283734997820nteger @ ( suc @ N ) @ A )
% 13.25/7.61 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3174_unset__bit__Suc,axiom,
% 13.25/7.61 ! [N: nat,A: int] :
% 13.25/7.61 ( ( bit_se4203085406695923979it_int @ ( suc @ N ) @ A )
% 13.25/7.61 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3175_unset__bit__Suc,axiom,
% 13.25/7.61 ! [N: nat,A: nat] :
% 13.25/7.61 ( ( bit_se4205575877204974255it_nat @ ( suc @ N ) @ A )
% 13.25/7.61 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3176_set__vebt_H__def,axiom,
% 13.25/7.61 ( vEBT_VEBT_set_vebt
% 13.25/7.61 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3177_finite__interval__int1,axiom,
% 13.25/7.61 ! [A: int,B: int] :
% 13.25/7.61 ( finite_finite_int
% 13.25/7.61 @ ( collect_int
% 13.25/7.61 @ ^ [I4: int] :
% 13.25/7.61 ( ( ord_less_eq_int @ A @ I4 )
% 13.25/7.61 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3178_finite__interval__int4,axiom,
% 13.25/7.61 ! [A: int,B: int] :
% 13.25/7.61 ( finite_finite_int
% 13.25/7.61 @ ( collect_int
% 13.25/7.61 @ ^ [I4: int] :
% 13.25/7.61 ( ( ord_less_int @ A @ I4 )
% 13.25/7.61 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3179_unset__bit__nonnegative__int__iff,axiom,
% 13.25/7.61 ! [N: nat,K: int] :
% 13.25/7.61 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 13.25/7.61 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3180_unset__bit__negative__int__iff,axiom,
% 13.25/7.61 ! [N: nat,K: int] :
% 13.25/7.61 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 13.25/7.61 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3181_finite__Collect__subsets,axiom,
% 13.25/7.61 ! [A2: set_nat] :
% 13.25/7.61 ( ( finite_finite_nat @ A2 )
% 13.25/7.61 => ( finite1152437895449049373et_nat
% 13.25/7.61 @ ( collect_set_nat
% 13.25/7.61 @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3182_finite__Collect__subsets,axiom,
% 13.25/7.61 ! [A2: set_complex] :
% 13.25/7.61 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.61 => ( finite6551019134538273531omplex
% 13.25/7.61 @ ( collect_set_complex
% 13.25/7.61 @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3183_finite__Collect__subsets,axiom,
% 13.25/7.61 ! [A2: set_int] :
% 13.25/7.61 ( ( finite_finite_int @ A2 )
% 13.25/7.61 => ( finite6197958912794628473et_int
% 13.25/7.61 @ ( collect_set_int
% 13.25/7.61 @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3184_finite__interval__int3,axiom,
% 13.25/7.61 ! [A: int,B: int] :
% 13.25/7.61 ( finite_finite_int
% 13.25/7.61 @ ( collect_int
% 13.25/7.61 @ ^ [I4: int] :
% 13.25/7.61 ( ( ord_less_int @ A @ I4 )
% 13.25/7.61 & ( ord_less_eq_int @ I4 @ B ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3185_finite__interval__int2,axiom,
% 13.25/7.61 ! [A: int,B: int] :
% 13.25/7.61 ( finite_finite_int
% 13.25/7.61 @ ( collect_int
% 13.25/7.61 @ ^ [I4: int] :
% 13.25/7.61 ( ( ord_less_eq_int @ A @ I4 )
% 13.25/7.61 & ( ord_less_int @ I4 @ B ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3186_finite__Collect__less__nat,axiom,
% 13.25/7.61 ! [K: nat] :
% 13.25/7.61 ( finite_finite_nat
% 13.25/7.61 @ ( collect_nat
% 13.25/7.61 @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3187_finite__Collect__le__nat,axiom,
% 13.25/7.61 ! [K: nat] :
% 13.25/7.61 ( finite_finite_nat
% 13.25/7.61 @ ( collect_nat
% 13.25/7.61 @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3188_minus__set__def,axiom,
% 13.25/7.61 ( minus_8321449233255521966at_nat
% 13.25/7.61 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( collec7088162979684241874at_nat
% 13.25/7.61 @ ( minus_4666906530369457071_nat_o
% 13.25/7.61 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3189_minus__set__def,axiom,
% 13.25/7.61 ( minus_811609699411566653omplex
% 13.25/7.61 = ( ^ [A6: set_complex,B6: set_complex] :
% 13.25/7.61 ( collect_complex
% 13.25/7.61 @ ( minus_8727706125548526216plex_o
% 13.25/7.61 @ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3190_minus__set__def,axiom,
% 13.25/7.61 ( minus_minus_set_real
% 13.25/7.61 = ( ^ [A6: set_real,B6: set_real] :
% 13.25/7.61 ( collect_real
% 13.25/7.61 @ ( minus_minus_real_o
% 13.25/7.61 @ ^ [X2: real] : ( member_real @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3191_minus__set__def,axiom,
% 13.25/7.61 ( minus_7954133019191499631st_nat
% 13.25/7.61 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 13.25/7.61 ( collect_list_nat
% 13.25/7.61 @ ( minus_1139252259498527702_nat_o
% 13.25/7.61 @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3192_minus__set__def,axiom,
% 13.25/7.61 ( minus_minus_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ( collect_int
% 13.25/7.61 @ ( minus_minus_int_o
% 13.25/7.61 @ ^ [X2: int] : ( member_int @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3193_minus__set__def,axiom,
% 13.25/7.61 ( minus_minus_set_nat
% 13.25/7.61 = ( ^ [A6: set_nat,B6: set_nat] :
% 13.25/7.61 ( collect_nat
% 13.25/7.61 @ ( minus_minus_nat_o
% 13.25/7.61 @ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3194_set__diff__eq,axiom,
% 13.25/7.61 ( minus_8321449233255521966at_nat
% 13.25/7.61 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( collec7088162979684241874at_nat
% 13.25/7.61 @ ^ [X2: produc859450856879609959at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ X2 @ A6 )
% 13.25/7.61 & ~ ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3195_set__diff__eq,axiom,
% 13.25/7.61 ( minus_811609699411566653omplex
% 13.25/7.61 = ( ^ [A6: set_complex,B6: set_complex] :
% 13.25/7.61 ( collect_complex
% 13.25/7.61 @ ^ [X2: complex] :
% 13.25/7.61 ( ( member_complex @ X2 @ A6 )
% 13.25/7.61 & ~ ( member_complex @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3196_set__diff__eq,axiom,
% 13.25/7.61 ( minus_minus_set_real
% 13.25/7.61 = ( ^ [A6: set_real,B6: set_real] :
% 13.25/7.61 ( collect_real
% 13.25/7.61 @ ^ [X2: real] :
% 13.25/7.61 ( ( member_real @ X2 @ A6 )
% 13.25/7.61 & ~ ( member_real @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3197_set__diff__eq,axiom,
% 13.25/7.61 ( minus_7954133019191499631st_nat
% 13.25/7.61 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 13.25/7.61 ( collect_list_nat
% 13.25/7.61 @ ^ [X2: list_nat] :
% 13.25/7.61 ( ( member_list_nat @ X2 @ A6 )
% 13.25/7.61 & ~ ( member_list_nat @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3198_set__diff__eq,axiom,
% 13.25/7.61 ( minus_minus_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ( collect_int
% 13.25/7.61 @ ^ [X2: int] :
% 13.25/7.61 ( ( member_int @ X2 @ A6 )
% 13.25/7.61 & ~ ( member_int @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3199_set__diff__eq,axiom,
% 13.25/7.61 ( minus_minus_set_nat
% 13.25/7.61 = ( ^ [A6: set_nat,B6: set_nat] :
% 13.25/7.61 ( collect_nat
% 13.25/7.61 @ ^ [X2: nat] :
% 13.25/7.61 ( ( member_nat @ X2 @ A6 )
% 13.25/7.61 & ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3200_less__set__def,axiom,
% 13.25/7.61 ( ord_less_set_nat
% 13.25/7.61 = ( ^ [A6: set_nat,B6: set_nat] :
% 13.25/7.61 ( ord_less_nat_o
% 13.25/7.61 @ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3201_less__set__def,axiom,
% 13.25/7.61 ( ord_less_set_real
% 13.25/7.61 = ( ^ [A6: set_real,B6: set_real] :
% 13.25/7.61 ( ord_less_real_o
% 13.25/7.61 @ ^ [X2: real] : ( member_real @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3202_less__set__def,axiom,
% 13.25/7.61 ( ord_less_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ( ord_less_int_o
% 13.25/7.61 @ ^ [X2: int] : ( member_int @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3203_less__set__def,axiom,
% 13.25/7.61 ( ord_less_set_complex
% 13.25/7.61 = ( ^ [A6: set_complex,B6: set_complex] :
% 13.25/7.61 ( ord_less_complex_o
% 13.25/7.61 @ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3204_less__set__def,axiom,
% 13.25/7.61 ( ord_le6428140832669894131at_nat
% 13.25/7.61 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ord_le7432393201274343594_nat_o
% 13.25/7.61 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3205_Collect__subset,axiom,
% 13.25/7.61 ! [A2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
% 13.25/7.61 ( ord_le3000389064537975527at_nat
% 13.25/7.61 @ ( collec7088162979684241874at_nat
% 13.25/7.61 @ ^ [X2: produc859450856879609959at_nat] :
% 13.25/7.61 ( ( member8206827879206165904at_nat @ X2 @ A2 )
% 13.25/7.61 & ( P @ X2 ) ) )
% 13.25/7.61 @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3206_Collect__subset,axiom,
% 13.25/7.61 ! [A2: set_complex,P: complex > $o] :
% 13.25/7.61 ( ord_le211207098394363844omplex
% 13.25/7.61 @ ( collect_complex
% 13.25/7.61 @ ^ [X2: complex] :
% 13.25/7.61 ( ( member_complex @ X2 @ A2 )
% 13.25/7.61 & ( P @ X2 ) ) )
% 13.25/7.61 @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3207_Collect__subset,axiom,
% 13.25/7.61 ! [A2: set_real,P: real > $o] :
% 13.25/7.61 ( ord_less_eq_set_real
% 13.25/7.61 @ ( collect_real
% 13.25/7.61 @ ^ [X2: real] :
% 13.25/7.61 ( ( member_real @ X2 @ A2 )
% 13.25/7.61 & ( P @ X2 ) ) )
% 13.25/7.61 @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3208_Collect__subset,axiom,
% 13.25/7.61 ! [A2: set_list_nat,P: list_nat > $o] :
% 13.25/7.61 ( ord_le6045566169113846134st_nat
% 13.25/7.61 @ ( collect_list_nat
% 13.25/7.61 @ ^ [X2: list_nat] :
% 13.25/7.61 ( ( member_list_nat @ X2 @ A2 )
% 13.25/7.61 & ( P @ X2 ) ) )
% 13.25/7.61 @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3209_Collect__subset,axiom,
% 13.25/7.61 ! [A2: set_nat,P: nat > $o] :
% 13.25/7.61 ( ord_less_eq_set_nat
% 13.25/7.61 @ ( collect_nat
% 13.25/7.61 @ ^ [X2: nat] :
% 13.25/7.61 ( ( member_nat @ X2 @ A2 )
% 13.25/7.61 & ( P @ X2 ) ) )
% 13.25/7.61 @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3210_Collect__subset,axiom,
% 13.25/7.61 ! [A2: set_int,P: int > $o] :
% 13.25/7.61 ( ord_less_eq_set_int
% 13.25/7.61 @ ( collect_int
% 13.25/7.61 @ ^ [X2: int] :
% 13.25/7.61 ( ( member_int @ X2 @ A2 )
% 13.25/7.61 & ( P @ X2 ) ) )
% 13.25/7.61 @ A2 ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3211_less__eq__set__def,axiom,
% 13.25/7.61 ( ord_less_eq_set_nat
% 13.25/7.61 = ( ^ [A6: set_nat,B6: set_nat] :
% 13.25/7.61 ( ord_less_eq_nat_o
% 13.25/7.61 @ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3212_less__eq__set__def,axiom,
% 13.25/7.61 ( ord_less_eq_set_real
% 13.25/7.61 = ( ^ [A6: set_real,B6: set_real] :
% 13.25/7.61 ( ord_less_eq_real_o
% 13.25/7.61 @ ^ [X2: real] : ( member_real @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: real] : ( member_real @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3213_less__eq__set__def,axiom,
% 13.25/7.61 ( ord_le211207098394363844omplex
% 13.25/7.61 = ( ^ [A6: set_complex,B6: set_complex] :
% 13.25/7.61 ( ord_le4573692005234683329plex_o
% 13.25/7.61 @ ^ [X2: complex] : ( member_complex @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: complex] : ( member_complex @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3214_less__eq__set__def,axiom,
% 13.25/7.61 ( ord_le3000389064537975527at_nat
% 13.25/7.61 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 13.25/7.61 ( ord_le7858099551454983350_nat_o
% 13.25/7.61 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3215_less__eq__set__def,axiom,
% 13.25/7.61 ( ord_less_eq_set_int
% 13.25/7.61 = ( ^ [A6: set_int,B6: set_int] :
% 13.25/7.61 ( ord_less_eq_int_o
% 13.25/7.61 @ ^ [X2: int] : ( member_int @ X2 @ A6 )
% 13.25/7.61 @ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3216_mult__commute__abs,axiom,
% 13.25/7.61 ! [C: real] :
% 13.25/7.61 ( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
% 13.25/7.61 = ( times_times_real @ C ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3217_mult__commute__abs,axiom,
% 13.25/7.61 ! [C: rat] :
% 13.25/7.61 ( ( ^ [X2: rat] : ( times_times_rat @ X2 @ C ) )
% 13.25/7.61 = ( times_times_rat @ C ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3218_mult__commute__abs,axiom,
% 13.25/7.61 ! [C: nat] :
% 13.25/7.61 ( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
% 13.25/7.61 = ( times_times_nat @ C ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3219_mult__commute__abs,axiom,
% 13.25/7.61 ! [C: int] :
% 13.25/7.61 ( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
% 13.25/7.61 = ( times_times_int @ C ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3220_lambda__zero,axiom,
% 13.25/7.61 ( ( ^ [H: complex] : zero_zero_complex )
% 13.25/7.61 = ( times_times_complex @ zero_zero_complex ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3221_lambda__zero,axiom,
% 13.25/7.61 ( ( ^ [H: real] : zero_zero_real )
% 13.25/7.61 = ( times_times_real @ zero_zero_real ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3222_lambda__zero,axiom,
% 13.25/7.61 ( ( ^ [H: rat] : zero_zero_rat )
% 13.25/7.61 = ( times_times_rat @ zero_zero_rat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3223_lambda__zero,axiom,
% 13.25/7.61 ( ( ^ [H: nat] : zero_zero_nat )
% 13.25/7.61 = ( times_times_nat @ zero_zero_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3224_lambda__zero,axiom,
% 13.25/7.61 ( ( ^ [H: int] : zero_zero_int )
% 13.25/7.61 = ( times_times_int @ zero_zero_int ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3225_lambda__one,axiom,
% 13.25/7.61 ( ( ^ [X2: complex] : X2 )
% 13.25/7.61 = ( times_times_complex @ one_one_complex ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3226_lambda__one,axiom,
% 13.25/7.61 ( ( ^ [X2: real] : X2 )
% 13.25/7.61 = ( times_times_real @ one_one_real ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3227_lambda__one,axiom,
% 13.25/7.61 ( ( ^ [X2: rat] : X2 )
% 13.25/7.61 = ( times_times_rat @ one_one_rat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3228_lambda__one,axiom,
% 13.25/7.61 ( ( ^ [X2: nat] : X2 )
% 13.25/7.61 = ( times_times_nat @ one_one_nat ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3229_lambda__one,axiom,
% 13.25/7.61 ( ( ^ [X2: int] : X2 )
% 13.25/7.61 = ( times_times_int @ one_one_int ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3230_finite__M__bounded__by__nat,axiom,
% 13.25/7.61 ! [P: nat > $o,I: nat] :
% 13.25/7.61 ( finite_finite_nat
% 13.25/7.61 @ ( collect_nat
% 13.25/7.61 @ ^ [K2: nat] :
% 13.25/7.61 ( ( P @ K2 )
% 13.25/7.61 & ( ord_less_nat @ K2 @ I ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3231_finite__less__ub,axiom,
% 13.25/7.61 ! [F: nat > nat,U: nat] :
% 13.25/7.61 ( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
% 13.25/7.61 => ( finite_finite_nat
% 13.25/7.61 @ ( collect_nat
% 13.25/7.61 @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3232_numeral__code_I2_J,axiom,
% 13.25/7.61 ! [N: num] :
% 13.25/7.61 ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
% 13.25/7.61 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3233_numeral__code_I2_J,axiom,
% 13.25/7.61 ! [N: num] :
% 13.25/7.61 ( ( numeral_numeral_real @ ( bit0 @ N ) )
% 13.25/7.61 = ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3234_numeral__code_I2_J,axiom,
% 13.25/7.61 ! [N: num] :
% 13.25/7.61 ( ( numeral_numeral_rat @ ( bit0 @ N ) )
% 13.25/7.61 = ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3235_numeral__code_I2_J,axiom,
% 13.25/7.61 ! [N: num] :
% 13.25/7.61 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 13.25/7.61 = ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3236_numeral__code_I2_J,axiom,
% 13.25/7.61 ! [N: num] :
% 13.25/7.61 ( ( numeral_numeral_int @ ( bit0 @ N ) )
% 13.25/7.61 = ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3237_set__vebt__def,axiom,
% 13.25/7.61 ( vEBT_set_vebt
% 13.25/7.61 = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3238_unset__bit__less__eq,axiom,
% 13.25/7.61 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3239_power__numeral__even,axiom,
% 13.25/7.61 ! [Z: complex,W: num] :
% 13.25/7.61 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 13.25/7.61 = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3240_power__numeral__even,axiom,
% 13.25/7.61 ! [Z: real,W: num] :
% 13.25/7.61 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 13.25/7.61 = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3241_power__numeral__even,axiom,
% 13.25/7.61 ! [Z: rat,W: num] :
% 13.25/7.61 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 13.25/7.61 = ( times_times_rat @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3242_power__numeral__even,axiom,
% 13.25/7.61 ! [Z: nat,W: num] :
% 13.25/7.61 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 13.25/7.61 = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3243_power__numeral__even,axiom,
% 13.25/7.61 ! [Z: int,W: num] :
% 13.25/7.61 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
% 13.25/7.61 = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3244_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 13.25/7.61 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X: nat] :
% 13.25/7.61 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X )
% 13.25/7.61 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.61 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 13.25/7.61 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3245_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 13.25/7.61 ! [A: $o,B: $o,X: nat] :
% 13.25/7.61 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
% 13.25/7.61 = ( ( ( X = zero_zero_nat )
% 13.25/7.61 => A )
% 13.25/7.61 & ( ( X != zero_zero_nat )
% 13.25/7.61 => ( ( ( X = one_one_nat )
% 13.25/7.61 => B )
% 13.25/7.61 & ( X = one_one_nat ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3246_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.61 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 13.25/7.61 => ( ! [A5: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ( ( ( Xa = zero_zero_nat )
% 13.25/7.61 => A5 )
% 13.25/7.61 & ( ( Xa != zero_zero_nat )
% 13.25/7.61 => ( ( ( Xa = one_one_nat )
% 13.25/7.61 => B3 )
% 13.25/7.61 & ( Xa = one_one_nat ) ) ) ) )
% 13.25/7.61 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 13.25/7.61 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.61 ( ? [S3: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 13.25/7.61 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3247_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.61 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 13.25/7.61 => ( ! [A5: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ~ ( ( ( Xa = zero_zero_nat )
% 13.25/7.61 => A5 )
% 13.25/7.61 & ( ( Xa != zero_zero_nat )
% 13.25/7.61 => ( ( ( Xa = one_one_nat )
% 13.25/7.61 => B3 )
% 13.25/7.61 & ( Xa = one_one_nat ) ) ) ) )
% 13.25/7.61 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.61 ( ? [S3: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 13.25/7.61 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3248_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 13.25/7.61 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 13.25/7.61 = Y3 )
% 13.25/7.61 => ( ! [A5: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( ~ ( ( ( Xa = zero_zero_nat )
% 13.25/7.61 => A5 )
% 13.25/7.61 & ( ( Xa != zero_zero_nat )
% 13.25/7.61 => ( ( ( Xa = one_one_nat )
% 13.25/7.61 => B3 )
% 13.25/7.61 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 13.25/7.61 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 13.25/7.61 => Y3 )
% 13.25/7.61 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.61 ( ? [S3: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3249_vebt__member_Osimps_I5_J,axiom,
% 13.25/7.61 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 13.25/7.61 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 13.25/7.61 = ( ( X != Mi )
% 13.25/7.61 => ( ( X != Ma )
% 13.25/7.61 => ( ~ ( ord_less_nat @ X @ Mi )
% 13.25/7.61 & ( ~ ( ord_less_nat @ X @ Mi )
% 13.25/7.61 => ( ~ ( ord_less_nat @ Ma @ X )
% 13.25/7.61 & ( ~ ( ord_less_nat @ Ma @ X )
% 13.25/7.61 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.61 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3250_finite__has__minimal2,axiom,
% 13.25/7.61 ! [A2: set_real,A: real] :
% 13.25/7.61 ( ( finite_finite_real @ A2 )
% 13.25/7.61 => ( ( member_real @ A @ A2 )
% 13.25/7.61 => ? [X3: real] :
% 13.25/7.61 ( ( member_real @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_real @ X3 @ A )
% 13.25/7.61 & ! [Xa2: real] :
% 13.25/7.61 ( ( member_real @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_real @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3251_finite__has__minimal2,axiom,
% 13.25/7.61 ! [A2: set_set_int,A: set_int] :
% 13.25/7.61 ( ( finite6197958912794628473et_int @ A2 )
% 13.25/7.61 => ( ( member_set_int @ A @ A2 )
% 13.25/7.61 => ? [X3: set_int] :
% 13.25/7.61 ( ( member_set_int @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_set_int @ X3 @ A )
% 13.25/7.61 & ! [Xa2: set_int] :
% 13.25/7.61 ( ( member_set_int @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3252_finite__has__minimal2,axiom,
% 13.25/7.61 ! [A2: set_rat,A: rat] :
% 13.25/7.61 ( ( finite_finite_rat @ A2 )
% 13.25/7.61 => ( ( member_rat @ A @ A2 )
% 13.25/7.61 => ? [X3: rat] :
% 13.25/7.61 ( ( member_rat @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_rat @ X3 @ A )
% 13.25/7.61 & ! [Xa2: rat] :
% 13.25/7.61 ( ( member_rat @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_rat @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3253_finite__has__minimal2,axiom,
% 13.25/7.61 ! [A2: set_num,A: num] :
% 13.25/7.61 ( ( finite_finite_num @ A2 )
% 13.25/7.61 => ( ( member_num @ A @ A2 )
% 13.25/7.61 => ? [X3: num] :
% 13.25/7.61 ( ( member_num @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_num @ X3 @ A )
% 13.25/7.61 & ! [Xa2: num] :
% 13.25/7.61 ( ( member_num @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_num @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3254_finite__has__minimal2,axiom,
% 13.25/7.61 ! [A2: set_nat,A: nat] :
% 13.25/7.61 ( ( finite_finite_nat @ A2 )
% 13.25/7.61 => ( ( member_nat @ A @ A2 )
% 13.25/7.61 => ? [X3: nat] :
% 13.25/7.61 ( ( member_nat @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_nat @ X3 @ A )
% 13.25/7.61 & ! [Xa2: nat] :
% 13.25/7.61 ( ( member_nat @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3255_finite__has__minimal2,axiom,
% 13.25/7.61 ! [A2: set_int,A: int] :
% 13.25/7.61 ( ( finite_finite_int @ A2 )
% 13.25/7.61 => ( ( member_int @ A @ A2 )
% 13.25/7.61 => ? [X3: int] :
% 13.25/7.61 ( ( member_int @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_int @ X3 @ A )
% 13.25/7.61 & ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_int @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3256_finite__has__maximal2,axiom,
% 13.25/7.61 ! [A2: set_real,A: real] :
% 13.25/7.61 ( ( finite_finite_real @ A2 )
% 13.25/7.61 => ( ( member_real @ A @ A2 )
% 13.25/7.61 => ? [X3: real] :
% 13.25/7.61 ( ( member_real @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_real @ A @ X3 )
% 13.25/7.61 & ! [Xa2: real] :
% 13.25/7.61 ( ( member_real @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_real @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3257_finite__has__maximal2,axiom,
% 13.25/7.61 ! [A2: set_set_int,A: set_int] :
% 13.25/7.61 ( ( finite6197958912794628473et_int @ A2 )
% 13.25/7.61 => ( ( member_set_int @ A @ A2 )
% 13.25/7.61 => ? [X3: set_int] :
% 13.25/7.61 ( ( member_set_int @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_set_int @ A @ X3 )
% 13.25/7.61 & ! [Xa2: set_int] :
% 13.25/7.61 ( ( member_set_int @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3258_finite__has__maximal2,axiom,
% 13.25/7.61 ! [A2: set_rat,A: rat] :
% 13.25/7.61 ( ( finite_finite_rat @ A2 )
% 13.25/7.61 => ( ( member_rat @ A @ A2 )
% 13.25/7.61 => ? [X3: rat] :
% 13.25/7.61 ( ( member_rat @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_rat @ A @ X3 )
% 13.25/7.61 & ! [Xa2: rat] :
% 13.25/7.61 ( ( member_rat @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_rat @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3259_finite__has__maximal2,axiom,
% 13.25/7.61 ! [A2: set_num,A: num] :
% 13.25/7.61 ( ( finite_finite_num @ A2 )
% 13.25/7.61 => ( ( member_num @ A @ A2 )
% 13.25/7.61 => ? [X3: num] :
% 13.25/7.61 ( ( member_num @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_num @ A @ X3 )
% 13.25/7.61 & ! [Xa2: num] :
% 13.25/7.61 ( ( member_num @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_num @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3260_finite__has__maximal2,axiom,
% 13.25/7.61 ! [A2: set_nat,A: nat] :
% 13.25/7.61 ( ( finite_finite_nat @ A2 )
% 13.25/7.61 => ( ( member_nat @ A @ A2 )
% 13.25/7.61 => ? [X3: nat] :
% 13.25/7.61 ( ( member_nat @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_nat @ A @ X3 )
% 13.25/7.61 & ! [Xa2: nat] :
% 13.25/7.61 ( ( member_nat @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_nat @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3261_finite__has__maximal2,axiom,
% 13.25/7.61 ! [A2: set_int,A: int] :
% 13.25/7.61 ( ( finite_finite_int @ A2 )
% 13.25/7.61 => ( ( member_int @ A @ A2 )
% 13.25/7.61 => ? [X3: int] :
% 13.25/7.61 ( ( member_int @ X3 @ A2 )
% 13.25/7.61 & ( ord_less_eq_int @ A @ X3 )
% 13.25/7.61 & ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_int @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3262_rev__finite__subset,axiom,
% 13.25/7.61 ! [B5: set_nat,A2: set_nat] :
% 13.25/7.61 ( ( finite_finite_nat @ B5 )
% 13.25/7.61 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.61 => ( finite_finite_nat @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3263_rev__finite__subset,axiom,
% 13.25/7.61 ! [B5: set_complex,A2: set_complex] :
% 13.25/7.61 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.61 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.61 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3264_rev__finite__subset,axiom,
% 13.25/7.61 ! [B5: set_int,A2: set_int] :
% 13.25/7.61 ( ( finite_finite_int @ B5 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( finite_finite_int @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3265_infinite__super,axiom,
% 13.25/7.61 ! [S2: set_nat,T3: set_nat] :
% 13.25/7.61 ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.61 => ( ~ ( finite_finite_nat @ S2 )
% 13.25/7.61 => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3266_infinite__super,axiom,
% 13.25/7.61 ! [S2: set_complex,T3: set_complex] :
% 13.25/7.61 ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.61 => ( ~ ( finite3207457112153483333omplex @ S2 )
% 13.25/7.61 => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3267_infinite__super,axiom,
% 13.25/7.61 ! [S2: set_int,T3: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ S2 @ T3 )
% 13.25/7.61 => ( ~ ( finite_finite_int @ S2 )
% 13.25/7.61 => ~ ( finite_finite_int @ T3 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3268_finite__subset,axiom,
% 13.25/7.61 ! [A2: set_nat,B5: set_nat] :
% 13.25/7.61 ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.61 => ( ( finite_finite_nat @ B5 )
% 13.25/7.61 => ( finite_finite_nat @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3269_finite__subset,axiom,
% 13.25/7.61 ! [A2: set_complex,B5: set_complex] :
% 13.25/7.61 ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.61 => ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.61 => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3270_finite__subset,axiom,
% 13.25/7.61 ! [A2: set_int,B5: set_int] :
% 13.25/7.61 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.61 => ( ( finite_finite_int @ B5 )
% 13.25/7.61 => ( finite_finite_int @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3271_Diff__infinite__finite,axiom,
% 13.25/7.61 ! [T3: set_int,S2: set_int] :
% 13.25/7.61 ( ( finite_finite_int @ T3 )
% 13.25/7.61 => ( ~ ( finite_finite_int @ S2 )
% 13.25/7.61 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S2 @ T3 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3272_Diff__infinite__finite,axiom,
% 13.25/7.61 ! [T3: set_complex,S2: set_complex] :
% 13.25/7.61 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.61 => ( ~ ( finite3207457112153483333omplex @ S2 )
% 13.25/7.61 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S2 @ T3 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3273_Diff__infinite__finite,axiom,
% 13.25/7.61 ! [T3: set_nat,S2: set_nat] :
% 13.25/7.61 ( ( finite_finite_nat @ T3 )
% 13.25/7.61 => ( ~ ( finite_finite_nat @ S2 )
% 13.25/7.61 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3274_finite__psubset__induct,axiom,
% 13.25/7.61 ! [A2: set_nat,P: set_nat > $o] :
% 13.25/7.61 ( ( finite_finite_nat @ A2 )
% 13.25/7.61 => ( ! [A7: set_nat] :
% 13.25/7.61 ( ( finite_finite_nat @ A7 )
% 13.25/7.61 => ( ! [B7: set_nat] :
% 13.25/7.61 ( ( ord_less_set_nat @ B7 @ A7 )
% 13.25/7.61 => ( P @ B7 ) )
% 13.25/7.61 => ( P @ A7 ) ) )
% 13.25/7.61 => ( P @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3275_finite__psubset__induct,axiom,
% 13.25/7.61 ! [A2: set_int,P: set_int > $o] :
% 13.25/7.61 ( ( finite_finite_int @ A2 )
% 13.25/7.61 => ( ! [A7: set_int] :
% 13.25/7.61 ( ( finite_finite_int @ A7 )
% 13.25/7.61 => ( ! [B7: set_int] :
% 13.25/7.61 ( ( ord_less_set_int @ B7 @ A7 )
% 13.25/7.61 => ( P @ B7 ) )
% 13.25/7.61 => ( P @ A7 ) ) )
% 13.25/7.61 => ( P @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3276_finite__psubset__induct,axiom,
% 13.25/7.61 ! [A2: set_complex,P: set_complex > $o] :
% 13.25/7.61 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.61 => ( ! [A7: set_complex] :
% 13.25/7.61 ( ( finite3207457112153483333omplex @ A7 )
% 13.25/7.61 => ( ! [B7: set_complex] :
% 13.25/7.61 ( ( ord_less_set_complex @ B7 @ A7 )
% 13.25/7.61 => ( P @ B7 ) )
% 13.25/7.61 => ( P @ A7 ) ) )
% 13.25/7.61 => ( P @ A2 ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3277_vebt__member_Oelims_I2_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.61 ( ( vEBT_vebt_member @ X @ Xa )
% 13.25/7.61 => ( ! [A5: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ~ ( ( ( Xa = zero_zero_nat )
% 13.25/7.61 => A5 )
% 13.25/7.61 & ( ( Xa != zero_zero_nat )
% 13.25/7.61 => ( ( ( Xa = one_one_nat )
% 13.25/7.61 => B3 )
% 13.25/7.61 & ( Xa = one_one_nat ) ) ) ) )
% 13.25/7.61 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.61 ( ? [Summary2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.61 => ~ ( ( Xa != Mi2 )
% 13.25/7.61 => ( ( Xa != Ma2 )
% 13.25/7.61 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.61 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.61 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.61 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.61 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3278_vebt__member_Oelims_I3_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.61 ( ~ ( vEBT_vebt_member @ X @ Xa )
% 13.25/7.61 => ( ! [A5: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ( ( ( Xa = zero_zero_nat )
% 13.25/7.61 => A5 )
% 13.25/7.61 & ( ( Xa != zero_zero_nat )
% 13.25/7.61 => ( ( ( Xa = one_one_nat )
% 13.25/7.61 => B3 )
% 13.25/7.61 & ( Xa = one_one_nat ) ) ) ) )
% 13.25/7.61 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 13.25/7.61 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 13.25/7.61 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 13.25/7.61 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.61 ( ? [Summary2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.61 => ( ( Xa != Mi2 )
% 13.25/7.61 => ( ( Xa != Ma2 )
% 13.25/7.61 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.61 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.61 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.61 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.61 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3279_vebt__member_Oelims_I1_J,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 13.25/7.61 ( ( ( vEBT_vebt_member @ X @ Xa )
% 13.25/7.61 = Y3 )
% 13.25/7.61 => ( ! [A5: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( ~ ( ( ( Xa = zero_zero_nat )
% 13.25/7.61 => A5 )
% 13.25/7.61 & ( ( Xa != zero_zero_nat )
% 13.25/7.61 => ( ( ( Xa = one_one_nat )
% 13.25/7.61 => B3 )
% 13.25/7.61 & ( Xa = one_one_nat ) ) ) ) ) ) )
% 13.25/7.61 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 13.25/7.61 => Y3 )
% 13.25/7.61 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 13.25/7.61 => Y3 )
% 13.25/7.61 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 13.25/7.61 => Y3 )
% 13.25/7.61 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.61 ( ? [Summary2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( ~ ( ( Xa != Mi2 )
% 13.25/7.61 => ( ( Xa != Ma2 )
% 13.25/7.61 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.61 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.61 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.61 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.61 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.61 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3280_finite__has__minimal,axiom,
% 13.25/7.61 ! [A2: set_real] :
% 13.25/7.61 ( ( finite_finite_real @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_real )
% 13.25/7.61 => ? [X3: real] :
% 13.25/7.61 ( ( member_real @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: real] :
% 13.25/7.61 ( ( member_real @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_real @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3281_finite__has__minimal,axiom,
% 13.25/7.61 ! [A2: set_set_int] :
% 13.25/7.61 ( ( finite6197958912794628473et_int @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_set_int )
% 13.25/7.61 => ? [X3: set_int] :
% 13.25/7.61 ( ( member_set_int @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: set_int] :
% 13.25/7.61 ( ( member_set_int @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3282_finite__has__minimal,axiom,
% 13.25/7.61 ! [A2: set_rat] :
% 13.25/7.61 ( ( finite_finite_rat @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_rat )
% 13.25/7.61 => ? [X3: rat] :
% 13.25/7.61 ( ( member_rat @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: rat] :
% 13.25/7.61 ( ( member_rat @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_rat @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3283_finite__has__minimal,axiom,
% 13.25/7.61 ! [A2: set_num] :
% 13.25/7.61 ( ( finite_finite_num @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_num )
% 13.25/7.61 => ? [X3: num] :
% 13.25/7.61 ( ( member_num @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: num] :
% 13.25/7.61 ( ( member_num @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_num @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3284_finite__has__minimal,axiom,
% 13.25/7.61 ! [A2: set_nat] :
% 13.25/7.61 ( ( finite_finite_nat @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_nat )
% 13.25/7.61 => ? [X3: nat] :
% 13.25/7.61 ( ( member_nat @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: nat] :
% 13.25/7.61 ( ( member_nat @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3285_finite__has__minimal,axiom,
% 13.25/7.61 ! [A2: set_int] :
% 13.25/7.61 ( ( finite_finite_int @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_int )
% 13.25/7.61 => ? [X3: int] :
% 13.25/7.61 ( ( member_int @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_int @ Xa2 @ X3 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3286_finite__has__maximal,axiom,
% 13.25/7.61 ! [A2: set_real] :
% 13.25/7.61 ( ( finite_finite_real @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_real )
% 13.25/7.61 => ? [X3: real] :
% 13.25/7.61 ( ( member_real @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: real] :
% 13.25/7.61 ( ( member_real @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_real @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3287_finite__has__maximal,axiom,
% 13.25/7.61 ! [A2: set_set_int] :
% 13.25/7.61 ( ( finite6197958912794628473et_int @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_set_int )
% 13.25/7.61 => ? [X3: set_int] :
% 13.25/7.61 ( ( member_set_int @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: set_int] :
% 13.25/7.61 ( ( member_set_int @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_set_int @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3288_finite__has__maximal,axiom,
% 13.25/7.61 ! [A2: set_rat] :
% 13.25/7.61 ( ( finite_finite_rat @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_rat )
% 13.25/7.61 => ? [X3: rat] :
% 13.25/7.61 ( ( member_rat @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: rat] :
% 13.25/7.61 ( ( member_rat @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_rat @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3289_finite__has__maximal,axiom,
% 13.25/7.61 ! [A2: set_num] :
% 13.25/7.61 ( ( finite_finite_num @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_num )
% 13.25/7.61 => ? [X3: num] :
% 13.25/7.61 ( ( member_num @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: num] :
% 13.25/7.61 ( ( member_num @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_num @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3290_finite__has__maximal,axiom,
% 13.25/7.61 ! [A2: set_nat] :
% 13.25/7.61 ( ( finite_finite_nat @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_nat )
% 13.25/7.61 => ? [X3: nat] :
% 13.25/7.61 ( ( member_nat @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: nat] :
% 13.25/7.61 ( ( member_nat @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_nat @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3291_finite__has__maximal,axiom,
% 13.25/7.61 ! [A2: set_int] :
% 13.25/7.61 ( ( finite_finite_int @ A2 )
% 13.25/7.61 => ( ( A2 != bot_bot_set_int )
% 13.25/7.61 => ? [X3: int] :
% 13.25/7.61 ( ( member_int @ X3 @ A2 )
% 13.25/7.61 & ! [Xa2: int] :
% 13.25/7.61 ( ( member_int @ Xa2 @ A2 )
% 13.25/7.61 => ( ( ord_less_eq_int @ X3 @ Xa2 )
% 13.25/7.61 => ( X3 = Xa2 ) ) ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3292_vebt__succ_Osimps_I6_J,axiom,
% 13.25/7.61 ! [X: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 13.25/7.61 ( ( ( ord_less_nat @ X @ Mi )
% 13.25/7.61 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 13.25/7.61 = ( some_nat @ Mi ) ) )
% 13.25/7.61 & ( ~ ( ord_less_nat @ X @ Mi )
% 13.25/7.61 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 13.25/7.61 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.61 @ ( if_option_nat
% 13.25/7.61 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 != none_nat )
% 13.25/7.61 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 13.25/7.61 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.61 @ ( if_option_nat
% 13.25/7.61 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.61 = none_nat )
% 13.25/7.61 @ none_nat
% 13.25/7.61 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 13.25/7.61 @ none_nat ) ) ) ) ).
% 13.25/7.61
% 13.25/7.61 thf(fact_3293_vebt__succ_Oelims,axiom,
% 13.25/7.61 ! [X: vEBT_VEBT,Xa: nat,Y3: option_nat] :
% 13.25/7.61 ( ( ( vEBT_vebt_succ @ X @ Xa )
% 13.25/7.61 = Y3 )
% 13.25/7.61 => ( ! [Uu2: $o,B3: $o] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Leaf @ Uu2 @ B3 ) )
% 13.25/7.61 => ( ( Xa = zero_zero_nat )
% 13.25/7.61 => ~ ( ( B3
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.61 & ( ~ B3
% 13.25/7.61 => ( Y3 = none_nat ) ) ) ) )
% 13.25/7.61 => ( ( ? [Uv2: $o,Uw2: $o] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 13.25/7.61 => ( ? [N4: nat] :
% 13.25/7.61 ( Xa
% 13.25/7.61 = ( suc @ N4 ) )
% 13.25/7.61 => ( Y3 != none_nat ) ) )
% 13.25/7.61 => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 13.25/7.61 => ( Y3 != none_nat ) )
% 13.25/7.61 => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 13.25/7.61 => ( Y3 != none_nat ) )
% 13.25/7.61 => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 13.25/7.61 ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 13.25/7.61 => ( Y3 != none_nat ) )
% 13.25/7.61 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.61 ( ( X
% 13.25/7.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.61 => ~ ( ( ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.61 => ( Y3
% 13.25/7.61 = ( some_nat @ Mi2 ) ) )
% 13.25/7.61 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.61 => ( Y3
% 13.25/7.62 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 @ ( if_option_nat
% 13.25/7.62 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 != none_nat )
% 13.25/7.62 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 13.25/7.62 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 @ ( if_option_nat
% 13.25/7.62 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.62 = none_nat )
% 13.25/7.62 @ none_nat
% 13.25/7.62 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 13.25/7.62 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3294_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 13.25/7.62 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 13.25/7.62 = Y3 )
% 13.25/7.62 => ( ( ? [Uu2: $o,Uv2: $o] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.62 => Y3 )
% 13.25/7.62 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 13.25/7.62 => Y3 )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat] :
% 13.25/7.62 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 13.25/7.62 => ( Y3
% 13.25/7.62 = ( ~ ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 ) ) ) ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.62 ( ? [Vc2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 13.25/7.62 => ( Y3
% 13.25/7.62 = ( ~ ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 )
% 13.25/7.62 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 13.25/7.62 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.62 ( ? [Vd2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 13.25/7.62 => ( Y3
% 13.25/7.62 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3295_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.62 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 13.25/7.62 => ( ! [Uu2: $o,Uv2: $o] :
% 13.25/7.62 ( X
% 13.25/7.62 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.62 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat] :
% 13.25/7.62 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 13.25/7.62 => ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 ) ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.62 ( ? [Vc2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 13.25/7.62 => ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 )
% 13.25/7.62 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 13.25/7.62 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.62 ( ? [Vd2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 13.25/7.62 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3296_finite__nth__roots,axiom,
% 13.25/7.62 ! [N: nat,C: complex] :
% 13.25/7.62 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [Z2: complex] :
% 13.25/7.62 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.62 = C ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3297_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.62 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat] :
% 13.25/7.62 ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 13.25/7.62 => ~ ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 ) ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.62 ( ? [Vc2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 13.25/7.62 => ~ ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 )
% 13.25/7.62 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 13.25/7.62 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 13.25/7.62 ( ? [Vd2: vEBT_VEBT] :
% 13.25/7.62 ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 13.25/7.62 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3298_member__valid__both__member__options,axiom,
% 13.25/7.62 ! [Tree: vEBT_VEBT,N: nat,X: nat] :
% 13.25/7.62 ( ( vEBT_invar_vebt @ Tree @ N )
% 13.25/7.62 => ( ( vEBT_vebt_member @ Tree @ X )
% 13.25/7.62 => ( ( vEBT_V5719532721284313246member @ Tree @ X )
% 13.25/7.62 | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3299_set__conv__nth,axiom,
% 13.25/7.62 ( set_complex2
% 13.25/7.62 = ( ^ [Xs: list_complex] :
% 13.25/7.62 ( collect_complex
% 13.25/7.62 @ ^ [Uu3: complex] :
% 13.25/7.62 ? [I4: nat] :
% 13.25/7.62 ( ( Uu3
% 13.25/7.62 = ( nth_complex @ Xs @ I4 ) )
% 13.25/7.62 & ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3300_set__conv__nth,axiom,
% 13.25/7.62 ( set_real2
% 13.25/7.62 = ( ^ [Xs: list_real] :
% 13.25/7.62 ( collect_real
% 13.25/7.62 @ ^ [Uu3: real] :
% 13.25/7.62 ? [I4: nat] :
% 13.25/7.62 ( ( Uu3
% 13.25/7.62 = ( nth_real @ Xs @ I4 ) )
% 13.25/7.62 & ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3301_set__conv__nth,axiom,
% 13.25/7.62 ( set_list_nat2
% 13.25/7.62 = ( ^ [Xs: list_list_nat] :
% 13.25/7.62 ( collect_list_nat
% 13.25/7.62 @ ^ [Uu3: list_nat] :
% 13.25/7.62 ? [I4: nat] :
% 13.25/7.62 ( ( Uu3
% 13.25/7.62 = ( nth_list_nat @ Xs @ I4 ) )
% 13.25/7.62 & ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3302_set__conv__nth,axiom,
% 13.25/7.62 ( set_VEBT_VEBT2
% 13.25/7.62 = ( ^ [Xs: list_VEBT_VEBT] :
% 13.25/7.62 ( collect_VEBT_VEBT
% 13.25/7.62 @ ^ [Uu3: vEBT_VEBT] :
% 13.25/7.62 ? [I4: nat] :
% 13.25/7.62 ( ( Uu3
% 13.25/7.62 = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 13.25/7.62 & ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3303_set__conv__nth,axiom,
% 13.25/7.62 ( set_o2
% 13.25/7.62 = ( ^ [Xs: list_o] :
% 13.25/7.62 ( collect_o
% 13.25/7.62 @ ^ [Uu3: $o] :
% 13.25/7.62 ? [I4: nat] :
% 13.25/7.62 ( ( Uu3
% 13.25/7.62 = ( nth_o @ Xs @ I4 ) )
% 13.25/7.62 & ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3304_set__conv__nth,axiom,
% 13.25/7.62 ( set_nat2
% 13.25/7.62 = ( ^ [Xs: list_nat] :
% 13.25/7.62 ( collect_nat
% 13.25/7.62 @ ^ [Uu3: nat] :
% 13.25/7.62 ? [I4: nat] :
% 13.25/7.62 ( ( Uu3
% 13.25/7.62 = ( nth_nat @ Xs @ I4 ) )
% 13.25/7.62 & ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3305_set__conv__nth,axiom,
% 13.25/7.62 ( set_int2
% 13.25/7.62 = ( ^ [Xs: list_int] :
% 13.25/7.62 ( collect_int
% 13.25/7.62 @ ^ [Uu3: int] :
% 13.25/7.62 ? [I4: nat] :
% 13.25/7.62 ( ( Uu3
% 13.25/7.62 = ( nth_int @ Xs @ I4 ) )
% 13.25/7.62 & ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3306_finite__lists__length__le,axiom,
% 13.25/7.62 ! [A2: set_complex,N: nat] :
% 13.25/7.62 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.62 => ( finite8712137658972009173omplex
% 13.25/7.62 @ ( collect_list_complex
% 13.25/7.62 @ ^ [Xs: list_complex] :
% 13.25/7.62 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3307_finite__lists__length__le,axiom,
% 13.25/7.62 ! [A2: set_VEBT_VEBT,N: nat] :
% 13.25/7.62 ( ( finite5795047828879050333T_VEBT @ A2 )
% 13.25/7.62 => ( finite3004134309566078307T_VEBT
% 13.25/7.62 @ ( collec5608196760682091941T_VEBT
% 13.25/7.62 @ ^ [Xs: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3308_finite__lists__length__le,axiom,
% 13.25/7.62 ! [A2: set_o,N: nat] :
% 13.25/7.62 ( ( finite_finite_o @ A2 )
% 13.25/7.62 => ( finite_finite_list_o
% 13.25/7.62 @ ( collect_list_o
% 13.25/7.62 @ ^ [Xs: list_o] :
% 13.25/7.62 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3309_finite__lists__length__le,axiom,
% 13.25/7.62 ! [A2: set_nat,N: nat] :
% 13.25/7.62 ( ( finite_finite_nat @ A2 )
% 13.25/7.62 => ( finite8100373058378681591st_nat
% 13.25/7.62 @ ( collect_list_nat
% 13.25/7.62 @ ^ [Xs: list_nat] :
% 13.25/7.62 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3310_finite__lists__length__le,axiom,
% 13.25/7.62 ! [A2: set_int,N: nat] :
% 13.25/7.62 ( ( finite_finite_int @ A2 )
% 13.25/7.62 => ( finite3922522038869484883st_int
% 13.25/7.62 @ ( collect_list_int
% 13.25/7.62 @ ^ [Xs: list_int] :
% 13.25/7.62 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3311_vebt__succ_Opelims,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat,Y3: option_nat] :
% 13.25/7.62 ( ( ( vEBT_vebt_succ @ X @ Xa )
% 13.25/7.62 = Y3 )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [Uu2: $o,B3: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ Uu2 @ B3 ) )
% 13.25/7.62 => ( ( Xa = zero_zero_nat )
% 13.25/7.62 => ( ( ( B3
% 13.25/7.62 => ( Y3
% 13.25/7.62 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.62 & ( ~ B3
% 13.25/7.62 => ( Y3 = none_nat ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
% 13.25/7.62 => ( ! [Uv2: $o,Uw2: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 13.25/7.62 => ! [N4: nat] :
% 13.25/7.62 ( ( Xa
% 13.25/7.62 = ( suc @ N4 ) )
% 13.25/7.62 => ( ( Y3 = none_nat )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N4 ) ) ) ) ) )
% 13.25/7.62 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 13.25/7.62 => ( ( Y3 = none_nat )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 13.25/7.62 => ( ( Y3 = none_nat )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 13.25/7.62 => ( ( Y3 = none_nat )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa ) ) ) )
% 13.25/7.62 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.62 => ( ( ( ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.62 => ( Y3
% 13.25/7.62 = ( some_nat @ Mi2 ) ) )
% 13.25/7.62 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.62 => ( Y3
% 13.25/7.62 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 @ ( if_option_nat
% 13.25/7.62 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 != none_nat )
% 13.25/7.62 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 13.25/7.62 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 @ ( if_option_nat
% 13.25/7.62 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.62 = none_nat )
% 13.25/7.62 @ none_nat
% 13.25/7.62 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList3 @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 13.25/7.62 @ none_nat ) ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3312_buildup__nothing__in__min__max,axiom,
% 13.25/7.62 ! [N: nat,X: nat] :
% 13.25/7.62 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3313_both__member__options__def,axiom,
% 13.25/7.62 ( vEBT_V8194947554948674370ptions
% 13.25/7.62 = ( ^ [T2: vEBT_VEBT,X2: nat] :
% 13.25/7.62 ( ( vEBT_V5719532721284313246member @ T2 @ X2 )
% 13.25/7.62 | ( vEBT_VEBT_membermima @ T2 @ X2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3314_subset__code_I1_J,axiom,
% 13.25/7.62 ! [Xs2: list_nat,B5: set_nat] :
% 13.25/7.62 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B5 )
% 13.25/7.62 = ( ! [X2: nat] :
% 13.25/7.62 ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 13.25/7.62 => ( member_nat @ X2 @ B5 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3315_subset__code_I1_J,axiom,
% 13.25/7.62 ! [Xs2: list_real,B5: set_real] :
% 13.25/7.62 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B5 )
% 13.25/7.62 = ( ! [X2: real] :
% 13.25/7.62 ( ( member_real @ X2 @ ( set_real2 @ Xs2 ) )
% 13.25/7.62 => ( member_real @ X2 @ B5 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3316_subset__code_I1_J,axiom,
% 13.25/7.62 ! [Xs2: list_complex,B5: set_complex] :
% 13.25/7.62 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B5 )
% 13.25/7.62 = ( ! [X2: complex] :
% 13.25/7.62 ( ( member_complex @ X2 @ ( set_complex2 @ Xs2 ) )
% 13.25/7.62 => ( member_complex @ X2 @ B5 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3317_subset__code_I1_J,axiom,
% 13.25/7.62 ! [Xs2: list_P8469869581646625389at_nat,B5: set_Pr8693737435421807431at_nat] :
% 13.25/7.62 ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs2 ) @ B5 )
% 13.25/7.62 = ( ! [X2: produc859450856879609959at_nat] :
% 13.25/7.62 ( ( member8206827879206165904at_nat @ X2 @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 13.25/7.62 => ( member8206827879206165904at_nat @ X2 @ B5 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3318_subset__code_I1_J,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,B5: set_VEBT_VEBT] :
% 13.25/7.62 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B5 )
% 13.25/7.62 = ( ! [X2: vEBT_VEBT] :
% 13.25/7.62 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 13.25/7.62 => ( member_VEBT_VEBT @ X2 @ B5 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3319_subset__code_I1_J,axiom,
% 13.25/7.62 ! [Xs2: list_int,B5: set_int] :
% 13.25/7.62 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B5 )
% 13.25/7.62 = ( ! [X2: int] :
% 13.25/7.62 ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 13.25/7.62 => ( member_int @ X2 @ B5 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3320_length__induct,axiom,
% 13.25/7.62 ! [P: list_VEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
% 13.25/7.62 ( ! [Xs3: list_VEBT_VEBT] :
% 13.25/7.62 ( ! [Ys: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 13.25/7.62 => ( P @ Ys ) )
% 13.25/7.62 => ( P @ Xs3 ) )
% 13.25/7.62 => ( P @ Xs2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3321_length__induct,axiom,
% 13.25/7.62 ! [P: list_o > $o,Xs2: list_o] :
% 13.25/7.62 ( ! [Xs3: list_o] :
% 13.25/7.62 ( ! [Ys: list_o] :
% 13.25/7.62 ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
% 13.25/7.62 => ( P @ Ys ) )
% 13.25/7.62 => ( P @ Xs3 ) )
% 13.25/7.62 => ( P @ Xs2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3322_length__induct,axiom,
% 13.25/7.62 ! [P: list_nat > $o,Xs2: list_nat] :
% 13.25/7.62 ( ! [Xs3: list_nat] :
% 13.25/7.62 ( ! [Ys: list_nat] :
% 13.25/7.62 ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
% 13.25/7.62 => ( P @ Ys ) )
% 13.25/7.62 => ( P @ Xs3 ) )
% 13.25/7.62 => ( P @ Xs2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3323_length__induct,axiom,
% 13.25/7.62 ! [P: list_int > $o,Xs2: list_int] :
% 13.25/7.62 ( ! [Xs3: list_int] :
% 13.25/7.62 ( ! [Ys: list_int] :
% 13.25/7.62 ( ( ord_less_nat @ ( size_size_list_int @ Ys ) @ ( size_size_list_int @ Xs3 ) )
% 13.25/7.62 => ( P @ Ys ) )
% 13.25/7.62 => ( P @ Xs3 ) )
% 13.25/7.62 => ( P @ Xs2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3324_finite__maxlen,axiom,
% 13.25/7.62 ! [M5: set_list_VEBT_VEBT] :
% 13.25/7.62 ( ( finite3004134309566078307T_VEBT @ M5 )
% 13.25/7.62 => ? [N4: nat] :
% 13.25/7.62 ! [X5: list_VEBT_VEBT] :
% 13.25/7.62 ( ( member2936631157270082147T_VEBT @ X5 @ M5 )
% 13.25/7.62 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3325_finite__maxlen,axiom,
% 13.25/7.62 ! [M5: set_list_o] :
% 13.25/7.62 ( ( finite_finite_list_o @ M5 )
% 13.25/7.62 => ? [N4: nat] :
% 13.25/7.62 ! [X5: list_o] :
% 13.25/7.62 ( ( member_list_o @ X5 @ M5 )
% 13.25/7.62 => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3326_finite__maxlen,axiom,
% 13.25/7.62 ! [M5: set_list_nat] :
% 13.25/7.62 ( ( finite8100373058378681591st_nat @ M5 )
% 13.25/7.62 => ? [N4: nat] :
% 13.25/7.62 ! [X5: list_nat] :
% 13.25/7.62 ( ( member_list_nat @ X5 @ M5 )
% 13.25/7.62 => ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3327_finite__maxlen,axiom,
% 13.25/7.62 ! [M5: set_list_int] :
% 13.25/7.62 ( ( finite3922522038869484883st_int @ M5 )
% 13.25/7.62 => ? [N4: nat] :
% 13.25/7.62 ! [X5: list_int] :
% 13.25/7.62 ( ( member_list_int @ X5 @ M5 )
% 13.25/7.62 => ( ord_less_nat @ ( size_size_list_int @ X5 ) @ N4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3328_list__eq__iff__nth__eq,axiom,
% 13.25/7.62 ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : ( Y6 = Z4 ) )
% 13.25/7.62 = ( ^ [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 13.25/7.62 = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 13.25/7.62 & ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 13.25/7.62 => ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 13.25/7.62 = ( nth_VEBT_VEBT @ Ys2 @ I4 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3329_list__eq__iff__nth__eq,axiom,
% 13.25/7.62 ( ( ^ [Y6: list_o,Z4: list_o] : ( Y6 = Z4 ) )
% 13.25/7.62 = ( ^ [Xs: list_o,Ys2: list_o] :
% 13.25/7.62 ( ( ( size_size_list_o @ Xs )
% 13.25/7.62 = ( size_size_list_o @ Ys2 ) )
% 13.25/7.62 & ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 13.25/7.62 => ( ( nth_o @ Xs @ I4 )
% 13.25/7.62 = ( nth_o @ Ys2 @ I4 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3330_list__eq__iff__nth__eq,axiom,
% 13.25/7.62 ( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
% 13.25/7.62 = ( ^ [Xs: list_nat,Ys2: list_nat] :
% 13.25/7.62 ( ( ( size_size_list_nat @ Xs )
% 13.25/7.62 = ( size_size_list_nat @ Ys2 ) )
% 13.25/7.62 & ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 13.25/7.62 => ( ( nth_nat @ Xs @ I4 )
% 13.25/7.62 = ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3331_list__eq__iff__nth__eq,axiom,
% 13.25/7.62 ( ( ^ [Y6: list_int,Z4: list_int] : ( Y6 = Z4 ) )
% 13.25/7.62 = ( ^ [Xs: list_int,Ys2: list_int] :
% 13.25/7.62 ( ( ( size_size_list_int @ Xs )
% 13.25/7.62 = ( size_size_list_int @ Ys2 ) )
% 13.25/7.62 & ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 13.25/7.62 => ( ( nth_int @ Xs @ I4 )
% 13.25/7.62 = ( nth_int @ Ys2 @ I4 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3332_Skolem__list__nth,axiom,
% 13.25/7.62 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 13.25/7.62 ( ( ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ K )
% 13.25/7.62 => ? [X6: vEBT_VEBT] : ( P @ I4 @ X6 ) ) )
% 13.25/7.62 = ( ? [Xs: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 13.25/7.62 = K )
% 13.25/7.62 & ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ K )
% 13.25/7.62 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3333_Skolem__list__nth,axiom,
% 13.25/7.62 ! [K: nat,P: nat > $o > $o] :
% 13.25/7.62 ( ( ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ K )
% 13.25/7.62 => ? [X6: $o] : ( P @ I4 @ X6 ) ) )
% 13.25/7.62 = ( ? [Xs: list_o] :
% 13.25/7.62 ( ( ( size_size_list_o @ Xs )
% 13.25/7.62 = K )
% 13.25/7.62 & ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ K )
% 13.25/7.62 => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3334_Skolem__list__nth,axiom,
% 13.25/7.62 ! [K: nat,P: nat > nat > $o] :
% 13.25/7.62 ( ( ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ K )
% 13.25/7.62 => ? [X6: nat] : ( P @ I4 @ X6 ) ) )
% 13.25/7.62 = ( ? [Xs: list_nat] :
% 13.25/7.62 ( ( ( size_size_list_nat @ Xs )
% 13.25/7.62 = K )
% 13.25/7.62 & ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ K )
% 13.25/7.62 => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3335_Skolem__list__nth,axiom,
% 13.25/7.62 ! [K: nat,P: nat > int > $o] :
% 13.25/7.62 ( ( ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ K )
% 13.25/7.62 => ? [X6: int] : ( P @ I4 @ X6 ) ) )
% 13.25/7.62 = ( ? [Xs: list_int] :
% 13.25/7.62 ( ( ( size_size_list_int @ Xs )
% 13.25/7.62 = K )
% 13.25/7.62 & ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ K )
% 13.25/7.62 => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3336_nth__equalityI,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 13.25/7.62 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 13.25/7.62 => ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
% 13.25/7.62 = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) )
% 13.25/7.62 => ( Xs2 = Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3337_nth__equalityI,axiom,
% 13.25/7.62 ! [Xs2: list_o,Ys3: list_o] :
% 13.25/7.62 ( ( ( size_size_list_o @ Xs2 )
% 13.25/7.62 = ( size_size_list_o @ Ys3 ) )
% 13.25/7.62 => ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 => ( ( nth_o @ Xs2 @ I2 )
% 13.25/7.62 = ( nth_o @ Ys3 @ I2 ) ) )
% 13.25/7.62 => ( Xs2 = Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3338_nth__equalityI,axiom,
% 13.25/7.62 ! [Xs2: list_nat,Ys3: list_nat] :
% 13.25/7.62 ( ( ( size_size_list_nat @ Xs2 )
% 13.25/7.62 = ( size_size_list_nat @ Ys3 ) )
% 13.25/7.62 => ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 => ( ( nth_nat @ Xs2 @ I2 )
% 13.25/7.62 = ( nth_nat @ Ys3 @ I2 ) ) )
% 13.25/7.62 => ( Xs2 = Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3339_nth__equalityI,axiom,
% 13.25/7.62 ! [Xs2: list_int,Ys3: list_int] :
% 13.25/7.62 ( ( ( size_size_list_int @ Xs2 )
% 13.25/7.62 = ( size_size_list_int @ Ys3 ) )
% 13.25/7.62 => ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 => ( ( nth_int @ Xs2 @ I2 )
% 13.25/7.62 = ( nth_int @ Ys3 @ I2 ) ) )
% 13.25/7.62 => ( Xs2 = Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3340_ex__nat__less,axiom,
% 13.25/7.62 ! [N: nat,P: nat > $o] :
% 13.25/7.62 ( ( ? [M3: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ M3 @ N )
% 13.25/7.62 & ( P @ M3 ) ) )
% 13.25/7.62 = ( ? [X2: nat] :
% 13.25/7.62 ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.62 & ( P @ X2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3341_all__nat__less,axiom,
% 13.25/7.62 ! [N: nat,P: nat > $o] :
% 13.25/7.62 ( ( ! [M3: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ M3 @ N )
% 13.25/7.62 => ( P @ M3 ) ) )
% 13.25/7.62 = ( ! [X2: nat] :
% 13.25/7.62 ( ( member_nat @ X2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.62 => ( P @ X2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3342_length__pos__if__in__set,axiom,
% 13.25/7.62 ! [X: real,Xs2: list_real] :
% 13.25/7.62 ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 13.25/7.62 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3343_length__pos__if__in__set,axiom,
% 13.25/7.62 ! [X: complex,Xs2: list_complex] :
% 13.25/7.62 ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 13.25/7.62 => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3344_length__pos__if__in__set,axiom,
% 13.25/7.62 ! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
% 13.25/7.62 ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 13.25/7.62 => ( ord_less_nat @ zero_zero_nat @ ( size_s3679842834875189465at_nat @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3345_length__pos__if__in__set,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 13.25/7.62 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 13.25/7.62 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3346_length__pos__if__in__set,axiom,
% 13.25/7.62 ! [X: $o,Xs2: list_o] :
% 13.25/7.62 ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 13.25/7.62 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3347_length__pos__if__in__set,axiom,
% 13.25/7.62 ! [X: nat,Xs2: list_nat] :
% 13.25/7.62 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 13.25/7.62 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3348_length__pos__if__in__set,axiom,
% 13.25/7.62 ! [X: int,Xs2: list_int] :
% 13.25/7.62 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 13.25/7.62 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3349_all__set__conv__all__nth,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 13.25/7.62 ( ( ! [X2: vEBT_VEBT] :
% 13.25/7.62 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X2 ) ) )
% 13.25/7.62 = ( ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3350_all__set__conv__all__nth,axiom,
% 13.25/7.62 ! [Xs2: list_o,P: $o > $o] :
% 13.25/7.62 ( ( ! [X2: $o] :
% 13.25/7.62 ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X2 ) ) )
% 13.25/7.62 = ( ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3351_all__set__conv__all__nth,axiom,
% 13.25/7.62 ! [Xs2: list_nat,P: nat > $o] :
% 13.25/7.62 ( ( ! [X2: nat] :
% 13.25/7.62 ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X2 ) ) )
% 13.25/7.62 = ( ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3352_all__set__conv__all__nth,axiom,
% 13.25/7.62 ! [Xs2: list_int,P: int > $o] :
% 13.25/7.62 ( ( ! [X2: int] :
% 13.25/7.62 ( ( member_int @ X2 @ ( set_int2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X2 ) ) )
% 13.25/7.62 = ( ! [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3353_all__nth__imp__all__set,axiom,
% 13.25/7.62 ! [Xs2: list_real,P: real > $o,X: real] :
% 13.25/7.62 ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_real @ Xs2 @ I2 ) ) )
% 13.25/7.62 => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3354_all__nth__imp__all__set,axiom,
% 13.25/7.62 ! [Xs2: list_complex,P: complex > $o,X: complex] :
% 13.25/7.62 ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_complex @ Xs2 @ I2 ) ) )
% 13.25/7.62 => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3355_all__nth__imp__all__set,axiom,
% 13.25/7.62 ! [Xs2: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
% 13.25/7.62 ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ I2 ) ) )
% 13.25/7.62 => ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 13.25/7.62 => ( P @ X ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3356_all__nth__imp__all__set,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 13.25/7.62 ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
% 13.25/7.62 => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3357_all__nth__imp__all__set,axiom,
% 13.25/7.62 ! [Xs2: list_o,P: $o > $o,X: $o] :
% 13.25/7.62 ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_o @ Xs2 @ I2 ) ) )
% 13.25/7.62 => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3358_all__nth__imp__all__set,axiom,
% 13.25/7.62 ! [Xs2: list_nat,P: nat > $o,X: nat] :
% 13.25/7.62 ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
% 13.25/7.62 => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3359_all__nth__imp__all__set,axiom,
% 13.25/7.62 ! [Xs2: list_int,P: int > $o,X: int] :
% 13.25/7.62 ( ! [I2: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 => ( P @ ( nth_int @ Xs2 @ I2 ) ) )
% 13.25/7.62 => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3360_in__set__conv__nth,axiom,
% 13.25/7.62 ! [X: real,Xs2: list_real] :
% 13.25/7.62 ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
% 13.25/7.62 = ( ? [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs2 ) )
% 13.25/7.62 & ( ( nth_real @ Xs2 @ I4 )
% 13.25/7.62 = X ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3361_in__set__conv__nth,axiom,
% 13.25/7.62 ! [X: complex,Xs2: list_complex] :
% 13.25/7.62 ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
% 13.25/7.62 = ( ? [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 13.25/7.62 & ( ( nth_complex @ Xs2 @ I4 )
% 13.25/7.62 = X ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3362_in__set__conv__nth,axiom,
% 13.25/7.62 ! [X: produc859450856879609959at_nat,Xs2: list_P8469869581646625389at_nat] :
% 13.25/7.62 ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs2 ) )
% 13.25/7.62 = ( ? [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 13.25/7.62 & ( ( nth_Pr6744343527793145070at_nat @ Xs2 @ I4 )
% 13.25/7.62 = X ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3363_in__set__conv__nth,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
% 13.25/7.62 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 13.25/7.62 = ( ? [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 & ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 13.25/7.62 = X ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3364_in__set__conv__nth,axiom,
% 13.25/7.62 ! [X: $o,Xs2: list_o] :
% 13.25/7.62 ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
% 13.25/7.62 = ( ? [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 & ( ( nth_o @ Xs2 @ I4 )
% 13.25/7.62 = X ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3365_in__set__conv__nth,axiom,
% 13.25/7.62 ! [X: nat,Xs2: list_nat] :
% 13.25/7.62 ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
% 13.25/7.62 = ( ? [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 & ( ( nth_nat @ Xs2 @ I4 )
% 13.25/7.62 = X ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3366_in__set__conv__nth,axiom,
% 13.25/7.62 ! [X: int,Xs2: list_int] :
% 13.25/7.62 ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
% 13.25/7.62 = ( ? [I4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 & ( ( nth_int @ Xs2 @ I4 )
% 13.25/7.62 = X ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3367_list__ball__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 => ( ! [X3: vEBT_VEBT] :
% 13.25/7.62 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X3 ) )
% 13.25/7.62 => ( P @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3368_list__ball__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_o,P: $o > $o] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 => ( ! [X3: $o] :
% 13.25/7.62 ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X3 ) )
% 13.25/7.62 => ( P @ ( nth_o @ Xs2 @ N ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3369_list__ball__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_nat,P: nat > $o] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 => ( ! [X3: nat] :
% 13.25/7.62 ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X3 ) )
% 13.25/7.62 => ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3370_list__ball__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_int,P: int > $o] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 => ( ! [X3: int] :
% 13.25/7.62 ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
% 13.25/7.62 => ( P @ X3 ) )
% 13.25/7.62 => ( P @ ( nth_int @ Xs2 @ N ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3371_nth__mem,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_real] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 13.25/7.62 => ( member_real @ ( nth_real @ Xs2 @ N ) @ ( set_real2 @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3372_nth__mem,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_complex] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 13.25/7.62 => ( member_complex @ ( nth_complex @ Xs2 @ N ) @ ( set_complex2 @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3373_nth__mem,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_P8469869581646625389at_nat] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 13.25/7.62 => ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs2 @ N ) @ ( set_Pr5518436109238095868at_nat @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3374_nth__mem,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3375_nth__mem,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_o] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 => ( member_o @ ( nth_o @ Xs2 @ N ) @ ( set_o2 @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3376_nth__mem,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_nat] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 => ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3377_nth__mem,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_int] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 => ( member_int @ ( nth_int @ Xs2 @ N ) @ ( set_int2 @ Xs2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3378_finite__lists__length__eq,axiom,
% 13.25/7.62 ! [A2: set_complex,N: nat] :
% 13.25/7.62 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.62 => ( finite8712137658972009173omplex
% 13.25/7.62 @ ( collect_list_complex
% 13.25/7.62 @ ^ [Xs: list_complex] :
% 13.25/7.62 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ( size_s3451745648224563538omplex @ Xs )
% 13.25/7.62 = N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3379_finite__lists__length__eq,axiom,
% 13.25/7.62 ! [A2: set_VEBT_VEBT,N: nat] :
% 13.25/7.62 ( ( finite5795047828879050333T_VEBT @ A2 )
% 13.25/7.62 => ( finite3004134309566078307T_VEBT
% 13.25/7.62 @ ( collec5608196760682091941T_VEBT
% 13.25/7.62 @ ^ [Xs: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ( size_s6755466524823107622T_VEBT @ Xs )
% 13.25/7.62 = N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3380_finite__lists__length__eq,axiom,
% 13.25/7.62 ! [A2: set_o,N: nat] :
% 13.25/7.62 ( ( finite_finite_o @ A2 )
% 13.25/7.62 => ( finite_finite_list_o
% 13.25/7.62 @ ( collect_list_o
% 13.25/7.62 @ ^ [Xs: list_o] :
% 13.25/7.62 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ( size_size_list_o @ Xs )
% 13.25/7.62 = N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3381_finite__lists__length__eq,axiom,
% 13.25/7.62 ! [A2: set_nat,N: nat] :
% 13.25/7.62 ( ( finite_finite_nat @ A2 )
% 13.25/7.62 => ( finite8100373058378681591st_nat
% 13.25/7.62 @ ( collect_list_nat
% 13.25/7.62 @ ^ [Xs: list_nat] :
% 13.25/7.62 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ( size_size_list_nat @ Xs )
% 13.25/7.62 = N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3382_finite__lists__length__eq,axiom,
% 13.25/7.62 ! [A2: set_int,N: nat] :
% 13.25/7.62 ( ( finite_finite_int @ A2 )
% 13.25/7.62 => ( finite3922522038869484883st_int
% 13.25/7.62 @ ( collect_list_int
% 13.25/7.62 @ ^ [Xs: list_int] :
% 13.25/7.62 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 13.25/7.62 & ( ( size_size_list_int @ Xs )
% 13.25/7.62 = N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3383_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 13.25/7.62 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X: nat] :
% 13.25/7.62 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X )
% 13.25/7.62 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3384_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 13.25/7.62 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
% 13.25/7.62 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X )
% 13.25/7.62 = ( ( X = Mi )
% 13.25/7.62 | ( X = Ma )
% 13.25/7.62 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3385_vebt__member_Opelims_I3_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.62 ( ~ ( vEBT_vebt_member @ X @ Xa )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [A5: $o,B3: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) )
% 13.25/7.62 => ( ( ( Xa = zero_zero_nat )
% 13.25/7.62 => A5 )
% 13.25/7.62 & ( ( Xa != zero_zero_nat )
% 13.25/7.62 => ( ( ( Xa = one_one_nat )
% 13.25/7.62 => B3 )
% 13.25/7.62 & ( Xa = one_one_nat ) ) ) ) ) )
% 13.25/7.62 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
% 13.25/7.62 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) )
% 13.25/7.62 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) )
% 13.25/7.62 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
% 13.25/7.62 => ( ( Xa != Mi2 )
% 13.25/7.62 => ( ( Xa != Ma2 )
% 13.25/7.62 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.62 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.62 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.62 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.62 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3386_vebt__member_Opelims_I1_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 13.25/7.62 ( ( ( vEBT_vebt_member @ X @ Xa )
% 13.25/7.62 = Y3 )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [A5: $o,B3: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.62 => ( ( Y3
% 13.25/7.62 = ( ( ( Xa = zero_zero_nat )
% 13.25/7.62 => A5 )
% 13.25/7.62 & ( ( Xa != zero_zero_nat )
% 13.25/7.62 => ( ( ( Xa = one_one_nat )
% 13.25/7.62 => B3 )
% 13.25/7.62 & ( Xa = one_one_nat ) ) ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 13.25/7.62 => ( ~ Y3
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 13.25/7.62 => ( ~ Y3
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 13.25/7.62 => ( ~ Y3
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
% 13.25/7.62 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.62 => ( ( Y3
% 13.25/7.62 = ( ( Xa != Mi2 )
% 13.25/7.62 => ( ( Xa != Ma2 )
% 13.25/7.62 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.62 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.62 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.62 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.62 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3387_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.62 ( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [A5: $o,B3: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) )
% 13.25/7.62 => ( ( ( Xa = zero_zero_nat )
% 13.25/7.62 => A5 )
% 13.25/7.62 & ( ( Xa != zero_zero_nat )
% 13.25/7.62 => ( ( ( Xa = one_one_nat )
% 13.25/7.62 => B3 )
% 13.25/7.62 & ( Xa = one_one_nat ) ) ) ) ) )
% 13.25/7.62 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
% 13.25/7.62 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
% 13.25/7.62 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3388_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.62 ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [A5: $o,B3: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) )
% 13.25/7.62 => ~ ( ( ( Xa = zero_zero_nat )
% 13.25/7.62 => A5 )
% 13.25/7.62 & ( ( Xa != zero_zero_nat )
% 13.25/7.62 => ( ( ( Xa = one_one_nat )
% 13.25/7.62 => B3 )
% 13.25/7.62 & ( Xa = one_one_nat ) ) ) ) ) )
% 13.25/7.62 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
% 13.25/7.62 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3389_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 13.25/7.62 ( ( ( vEBT_V5719532721284313246member @ X @ Xa )
% 13.25/7.62 = Y3 )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [A5: $o,B3: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.62 => ( ( Y3
% 13.25/7.62 = ( ( ( Xa = zero_zero_nat )
% 13.25/7.62 => A5 )
% 13.25/7.62 & ( ( Xa != zero_zero_nat )
% 13.25/7.62 => ( ( ( Xa = one_one_nat )
% 13.25/7.62 => B3 )
% 13.25/7.62 & ( Xa = one_one_nat ) ) ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 13.25/7.62 => ( ~ Y3
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
% 13.25/7.62 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
% 13.25/7.62 => ( ( Y3
% 13.25/7.62 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3390_vebt__member_Opelims_I2_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.62 ( ( vEBT_vebt_member @ X @ Xa )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [A5: $o,B3: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) )
% 13.25/7.62 => ~ ( ( ( Xa = zero_zero_nat )
% 13.25/7.62 => A5 )
% 13.25/7.62 & ( ( Xa != zero_zero_nat )
% 13.25/7.62 => ( ( ( Xa = one_one_nat )
% 13.25/7.62 => B3 )
% 13.25/7.62 & ( Xa = one_one_nat ) ) ) ) ) )
% 13.25/7.62 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) )
% 13.25/7.62 => ~ ( ( Xa != Mi2 )
% 13.25/7.62 => ( ( Xa != Ma2 )
% 13.25/7.62 => ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.62 & ( ~ ( ord_less_nat @ Xa @ Mi2 )
% 13.25/7.62 => ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.62 & ( ~ ( ord_less_nat @ Ma2 @ Xa )
% 13.25/7.62 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3391_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 13.25/7.62 ( ( ( vEBT_VEBT_membermima @ X @ Xa )
% 13.25/7.62 = Y3 )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [Uu2: $o,Uv2: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.62 => ( ~ Y3
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 13.25/7.62 => ( ~ Y3
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 13.25/7.62 => ( ( Y3
% 13.25/7.62 = ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) ) ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 13.25/7.62 => ( ( Y3
% 13.25/7.62 = ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 )
% 13.25/7.62 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) ) ) )
% 13.25/7.62 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 13.25/7.62 => ( ( Y3
% 13.25/7.62 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3392_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.62 ( ~ ( vEBT_VEBT_membermima @ X @ Xa )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [Uu2: $o,Uv2: $o] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
% 13.25/7.62 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 13.25/7.62 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) )
% 13.25/7.62 => ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 ) ) ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 13.25/7.62 => ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 )
% 13.25/7.62 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 13.25/7.62 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 13.25/7.62 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3393_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 13.25/7.62 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.62 ( ( vEBT_VEBT_membermima @ X @ Xa )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa ) )
% 13.25/7.62 => ~ ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 ) ) ) )
% 13.25/7.62 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa ) )
% 13.25/7.62 => ~ ( ( Xa = Mi2 )
% 13.25/7.62 | ( Xa = Ma2 )
% 13.25/7.62 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 13.25/7.62 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 13.25/7.62 ( ( X
% 13.25/7.62 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) )
% 13.25/7.62 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd2 ) @ Xa ) )
% 13.25/7.62 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.62 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.62 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3394_finite__roots__unity,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 13.25/7.62 => ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [Z2: real] :
% 13.25/7.62 ( ( power_power_real @ Z2 @ N )
% 13.25/7.62 = one_one_real ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3395_finite__roots__unity,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 13.25/7.62 => ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [Z2: complex] :
% 13.25/7.62 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.62 = one_one_complex ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3396_flip__bit__Suc,axiom,
% 13.25/7.62 ! [N: nat,A: code_integer] :
% 13.25/7.62 ( ( bit_se1345352211410354436nteger @ ( suc @ N ) @ A )
% 13.25/7.62 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3397_flip__bit__Suc,axiom,
% 13.25/7.62 ! [N: nat,A: int] :
% 13.25/7.62 ( ( bit_se2159334234014336723it_int @ ( suc @ N ) @ A )
% 13.25/7.62 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3398_flip__bit__Suc,axiom,
% 13.25/7.62 ! [N: nat,A: nat] :
% 13.25/7.62 ( ( bit_se2161824704523386999it_nat @ ( suc @ N ) @ A )
% 13.25/7.62 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3399_Bolzano,axiom,
% 13.25/7.62 ! [A: real,B: real,P: real > real > $o] :
% 13.25/7.62 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.62 => ( ! [A5: real,B3: real,C2: real] :
% 13.25/7.62 ( ( P @ A5 @ B3 )
% 13.25/7.62 => ( ( P @ B3 @ C2 )
% 13.25/7.62 => ( ( ord_less_eq_real @ A5 @ B3 )
% 13.25/7.62 => ( ( ord_less_eq_real @ B3 @ C2 )
% 13.25/7.62 => ( P @ A5 @ C2 ) ) ) ) )
% 13.25/7.62 => ( ! [X3: real] :
% 13.25/7.62 ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.62 => ( ( ord_less_eq_real @ X3 @ B )
% 13.25/7.62 => ? [D5: real] :
% 13.25/7.62 ( ( ord_less_real @ zero_zero_real @ D5 )
% 13.25/7.62 & ! [A5: real,B3: real] :
% 13.25/7.62 ( ( ( ord_less_eq_real @ A5 @ X3 )
% 13.25/7.62 & ( ord_less_eq_real @ X3 @ B3 )
% 13.25/7.62 & ( ord_less_real @ ( minus_minus_real @ B3 @ A5 ) @ D5 ) )
% 13.25/7.62 => ( P @ A5 @ B3 ) ) ) ) )
% 13.25/7.62 => ( P @ A @ B ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3400_mult__le__cancel__iff1,axiom,
% 13.25/7.62 ! [Z: real,X: real,Y3: real] :
% 13.25/7.62 ( ( ord_less_real @ zero_zero_real @ Z )
% 13.25/7.62 => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 13.25/7.62 = ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3401_mult__le__cancel__iff1,axiom,
% 13.25/7.62 ! [Z: rat,X: rat,Y3: rat] :
% 13.25/7.62 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 13.25/7.62 => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 13.25/7.62 = ( ord_less_eq_rat @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3402_mult__le__cancel__iff1,axiom,
% 13.25/7.62 ! [Z: int,X: int,Y3: int] :
% 13.25/7.62 ( ( ord_less_int @ zero_zero_int @ Z )
% 13.25/7.62 => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 13.25/7.62 = ( ord_less_eq_int @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3403_mult__le__cancel__iff2,axiom,
% 13.25/7.62 ! [Z: real,X: real,Y3: real] :
% 13.25/7.62 ( ( ord_less_real @ zero_zero_real @ Z )
% 13.25/7.62 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y3 ) )
% 13.25/7.62 = ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3404_mult__le__cancel__iff2,axiom,
% 13.25/7.62 ! [Z: rat,X: rat,Y3: rat] :
% 13.25/7.62 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 13.25/7.62 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ ( times_times_rat @ Z @ Y3 ) )
% 13.25/7.62 = ( ord_less_eq_rat @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3405_mult__le__cancel__iff2,axiom,
% 13.25/7.62 ! [Z: int,X: int,Y3: int] :
% 13.25/7.62 ( ( ord_less_int @ zero_zero_int @ Z )
% 13.25/7.62 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y3 ) )
% 13.25/7.62 = ( ord_less_eq_int @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3406_divides__aux__eq,axiom,
% 13.25/7.62 ! [Q2: nat,R: nat] :
% 13.25/7.62 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R ) )
% 13.25/7.62 = ( R = zero_zero_nat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3407_divides__aux__eq,axiom,
% 13.25/7.62 ! [Q2: int,R: int] :
% 13.25/7.62 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.62 = ( R = zero_zero_int ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3408_flip__bit__nonnegative__int__iff,axiom,
% 13.25/7.62 ! [N: nat,K: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 13.25/7.62 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3409_flip__bit__negative__int__iff,axiom,
% 13.25/7.62 ! [N: nat,K: int] :
% 13.25/7.62 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 13.25/7.62 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3410_mult__less__iff1,axiom,
% 13.25/7.62 ! [Z: real,X: real,Y3: real] :
% 13.25/7.62 ( ( ord_less_real @ zero_zero_real @ Z )
% 13.25/7.62 => ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 13.25/7.62 = ( ord_less_real @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3411_mult__less__iff1,axiom,
% 13.25/7.62 ! [Z: rat,X: rat,Y3: rat] :
% 13.25/7.62 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 13.25/7.62 => ( ( ord_less_rat @ ( times_times_rat @ X @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 13.25/7.62 = ( ord_less_rat @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3412_mult__less__iff1,axiom,
% 13.25/7.62 ! [Z: int,X: int,Y3: int] :
% 13.25/7.62 ( ( ord_less_int @ zero_zero_int @ Z )
% 13.25/7.62 => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 13.25/7.62 = ( ord_less_int @ X @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3413_arcosh__1,axiom,
% 13.25/7.62 ( ( arcosh_real @ one_one_real )
% 13.25/7.62 = zero_zero_real ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3414_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_num,Ys3: list_num] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys3 ) ) ) @ ( nth_num @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3415_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3416_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_VEBT_VEBT,Ys3: list_o] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3417_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_VEBT_VEBT,Ys3: list_nat] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3418_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_VEBT_VEBT,Ys3: list_int] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3419_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_o,Ys3: list_VEBT_VEBT] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) @ ( nth_VEBT_VEBT @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3420_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_o,Ys3: list_o] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( product_Pair_o_o @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) @ ( nth_o @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3421_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_o,Ys3: list_nat] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) @ ( nth_nat @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3422_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_o,Ys3: list_int] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( product_Pair_o_int @ ( nth_o @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) @ ( nth_int @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3423_product__nth,axiom,
% 13.25/7.62 ! [N: nat,Xs2: list_nat,Ys3: list_num] :
% 13.25/7.62 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_num @ Ys3 ) ) )
% 13.25/7.62 => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs2 @ Ys3 ) @ N )
% 13.25/7.62 = ( product_Pair_nat_num @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys3 ) ) ) @ ( nth_num @ Ys3 @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys3 ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3424_neg__eucl__rel__int__mult__2,axiom,
% 13.25/7.62 ! [B: int,A: int,Q2: int,R: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ B @ zero_zero_int )
% 13.25/7.62 => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.62 => ( 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 ) ) @ R ) @ one_one_int ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3425_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_real,X: real > complex,Y3: real > complex] :
% 13.25/7.62 ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_complex ) ) ) )
% 13.25/7.62 => ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_complex ) ) ) )
% 13.25/7.62 => ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_complex ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3426_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_nat,X: nat > complex,Y3: nat > complex] :
% 13.25/7.62 ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_complex ) ) ) )
% 13.25/7.62 => ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_complex ) ) ) )
% 13.25/7.62 => ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_complex ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3427_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_int,X: int > complex,Y3: int > complex] :
% 13.25/7.62 ( ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_complex ) ) ) )
% 13.25/7.62 => ( ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_complex ) ) ) )
% 13.25/7.62 => ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_complex ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3428_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_complex,X: complex > complex,Y3: complex > complex] :
% 13.25/7.62 ( ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_complex ) ) ) )
% 13.25/7.62 => ( ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_complex ) ) ) )
% 13.25/7.62 => ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_complex ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3429_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_real,X: real > real,Y3: real > real] :
% 13.25/7.62 ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_real ) ) ) )
% 13.25/7.62 => ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_real ) ) ) )
% 13.25/7.62 => ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_real ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3430_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_nat,X: nat > real,Y3: nat > real] :
% 13.25/7.62 ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_real ) ) ) )
% 13.25/7.62 => ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_real ) ) ) )
% 13.25/7.62 => ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_real ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3431_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_int,X: int > real,Y3: int > real] :
% 13.25/7.62 ( ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_real ) ) ) )
% 13.25/7.62 => ( ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_real ) ) ) )
% 13.25/7.62 => ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_real ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3432_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_complex,X: complex > real,Y3: complex > real] :
% 13.25/7.62 ( ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_real ) ) ) )
% 13.25/7.62 => ( ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_real ) ) ) )
% 13.25/7.62 => ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_real ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3433_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_real,X: real > rat,Y3: real > rat] :
% 13.25/7.62 ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_rat ) ) ) )
% 13.25/7.62 => ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_rat ) ) ) )
% 13.25/7.62 => ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_rat @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_rat ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3434_prod_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_nat,X: nat > rat,Y3: nat > rat] :
% 13.25/7.62 ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != one_one_rat ) ) ) )
% 13.25/7.62 => ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != one_one_rat ) ) ) )
% 13.25/7.62 => ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( times_times_rat @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != one_one_rat ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3435_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_real,X: real > complex,Y3: real > complex] :
% 13.25/7.62 ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_complex ) ) ) )
% 13.25/7.62 => ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_complex ) ) ) )
% 13.25/7.62 => ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_complex ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3436_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_nat,X: nat > complex,Y3: nat > complex] :
% 13.25/7.62 ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_complex ) ) ) )
% 13.25/7.62 => ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_complex ) ) ) )
% 13.25/7.62 => ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_complex ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3437_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_int,X: int > complex,Y3: int > complex] :
% 13.25/7.62 ( ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_complex ) ) ) )
% 13.25/7.62 => ( ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_complex ) ) ) )
% 13.25/7.62 => ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_complex ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3438_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_complex,X: complex > complex,Y3: complex > complex] :
% 13.25/7.62 ( ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_complex ) ) ) )
% 13.25/7.62 => ( ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_complex ) ) ) )
% 13.25/7.62 => ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_complex @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_complex ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3439_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_real,X: real > real,Y3: real > real] :
% 13.25/7.62 ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_real ) ) ) )
% 13.25/7.62 => ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_real ) ) ) )
% 13.25/7.62 => ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3440_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_nat,X: nat > real,Y3: nat > real] :
% 13.25/7.62 ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_real ) ) ) )
% 13.25/7.62 => ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_real ) ) ) )
% 13.25/7.62 => ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3441_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_int,X: int > real,Y3: int > real] :
% 13.25/7.62 ( ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_real ) ) ) )
% 13.25/7.62 => ( ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_real ) ) ) )
% 13.25/7.62 => ( finite_finite_int
% 13.25/7.62 @ ( collect_int
% 13.25/7.62 @ ^ [I4: int] :
% 13.25/7.62 ( ( member_int @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3442_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_complex,X: complex > real,Y3: complex > real] :
% 13.25/7.62 ( ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_real ) ) ) )
% 13.25/7.62 => ( ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_real ) ) ) )
% 13.25/7.62 => ( finite3207457112153483333omplex
% 13.25/7.62 @ ( collect_complex
% 13.25/7.62 @ ^ [I4: complex] :
% 13.25/7.62 ( ( member_complex @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_real @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3443_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_real,X: real > rat,Y3: real > rat] :
% 13.25/7.62 ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_rat ) ) ) )
% 13.25/7.62 => ( ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_rat ) ) ) )
% 13.25/7.62 => ( finite_finite_real
% 13.25/7.62 @ ( collect_real
% 13.25/7.62 @ ^ [I4: real] :
% 13.25/7.62 ( ( member_real @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3444_sum_Ofinite__Collect__op,axiom,
% 13.25/7.62 ! [I5: set_nat,X: nat > rat,Y3: nat > rat] :
% 13.25/7.62 ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( X @ I4 )
% 13.25/7.62 != zero_zero_rat ) ) ) )
% 13.25/7.62 => ( ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( Y3 @ I4 )
% 13.25/7.62 != zero_zero_rat ) ) ) )
% 13.25/7.62 => ( finite_finite_nat
% 13.25/7.62 @ ( collect_nat
% 13.25/7.62 @ ^ [I4: nat] :
% 13.25/7.62 ( ( member_nat @ I4 @ I5 )
% 13.25/7.62 & ( ( plus_plus_rat @ ( X @ I4 ) @ ( Y3 @ I4 ) )
% 13.25/7.62 != zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3445_pos__eucl__rel__int__mult__2,axiom,
% 13.25/7.62 ! [B: int,A: int,Q2: int,R: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.25/7.62 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.62 => ( 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 ) ) @ R ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3446_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 13.25/7.62 ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3447_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,Ys3: list_o] :
% 13.25/7.62 ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3448_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,Ys3: list_nat] :
% 13.25/7.62 ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3449_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,Ys3: list_int] :
% 13.25/7.62 ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3450_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_o,Ys3: list_VEBT_VEBT] :
% 13.25/7.62 ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3451_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_o,Ys3: list_o] :
% 13.25/7.62 ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3452_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_o,Ys3: list_nat] :
% 13.25/7.62 ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3453_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_o,Ys3: list_int] :
% 13.25/7.62 ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3454_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_nat,Ys3: list_VEBT_VEBT] :
% 13.25/7.62 ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3455_length__product,axiom,
% 13.25/7.62 ! [Xs2: list_nat,Ys3: list_o] :
% 13.25/7.62 ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs2 @ Ys3 ) )
% 13.25/7.62 = ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_o @ Ys3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3456_unique__quotient,axiom,
% 13.25/7.62 ! [A: int,B: int,Q2: int,R: int,Q4: int,R3: int] :
% 13.25/7.62 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.62 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
% 13.25/7.62 => ( Q2 = Q4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3457_unique__remainder,axiom,
% 13.25/7.62 ! [A: int,B: int,Q2: int,R: int,Q4: int,R3: int] :
% 13.25/7.62 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.62 => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R3 ) )
% 13.25/7.62 => ( R = R3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3458_eucl__rel__int__by0,axiom,
% 13.25/7.62 ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3459_div__int__unique,axiom,
% 13.25/7.62 ! [K: int,L2: int,Q2: int,R: int] :
% 13.25/7.62 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.62 => ( ( divide_divide_int @ K @ L2 )
% 13.25/7.62 = Q2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3460_mod__int__unique,axiom,
% 13.25/7.62 ! [K: int,L2: int,Q2: int,R: int] :
% 13.25/7.62 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.62 => ( ( modulo_modulo_int @ K @ L2 )
% 13.25/7.62 = R ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3461_eucl__rel__int__dividesI,axiom,
% 13.25/7.62 ! [L2: int,K: int,Q2: int] :
% 13.25/7.62 ( ( L2 != zero_zero_int )
% 13.25/7.62 => ( ( K
% 13.25/7.62 = ( times_times_int @ Q2 @ L2 ) )
% 13.25/7.62 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3462_eucl__rel__int,axiom,
% 13.25/7.62 ! [K: int,L2: int] : ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L2 ) @ ( modulo_modulo_int @ K @ L2 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3463_eucl__rel__int__iff,axiom,
% 13.25/7.62 ! [K: int,L2: int,Q2: int,R: int] :
% 13.25/7.62 ( ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.62 = ( ( K
% 13.25/7.62 = ( plus_plus_int @ ( times_times_int @ L2 @ Q2 ) @ R ) )
% 13.25/7.62 & ( ( ord_less_int @ zero_zero_int @ L2 )
% 13.25/7.62 => ( ( ord_less_eq_int @ zero_zero_int @ R )
% 13.25/7.62 & ( ord_less_int @ R @ L2 ) ) )
% 13.25/7.62 & ( ~ ( ord_less_int @ zero_zero_int @ L2 )
% 13.25/7.62 => ( ( ( ord_less_int @ L2 @ zero_zero_int )
% 13.25/7.62 => ( ( ord_less_int @ L2 @ R )
% 13.25/7.62 & ( ord_less_eq_int @ R @ zero_zero_int ) ) )
% 13.25/7.62 & ( ~ ( ord_less_int @ L2 @ zero_zero_int )
% 13.25/7.62 => ( Q2 = zero_zero_int ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3464_insert__simp__norm,axiom,
% 13.25/7.62 ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 13.25/7.62 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.62 => ( ( ord_less_nat @ Mi @ X )
% 13.25/7.62 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 13.25/7.62 => ( ( X != Ma )
% 13.25/7.62 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3465_insert__simp__excp,axiom,
% 13.25/7.62 ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
% 13.25/7.62 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.62 => ( ( ord_less_nat @ X @ Mi )
% 13.25/7.62 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 13.25/7.62 => ( ( X != Ma )
% 13.25/7.62 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
% 13.25/7.62 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3466_gcd__nat__induct,axiom,
% 13.25/7.62 ! [P: nat > nat > $o,M: nat,N: nat] :
% 13.25/7.62 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 13.25/7.62 => ( ! [M4: nat,N4: nat] :
% 13.25/7.62 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 13.25/7.62 => ( ( P @ N4 @ ( modulo_modulo_nat @ M4 @ N4 ) )
% 13.25/7.62 => ( P @ M4 @ N4 ) ) )
% 13.25/7.62 => ( P @ M @ N ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3467_concat__bit__Suc,axiom,
% 13.25/7.62 ! [N: nat,K: int,L2: int] :
% 13.25/7.62 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L2 )
% 13.25/7.62 = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3468_even__succ__mod__exp,axiom,
% 13.25/7.62 ! [A: nat,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.62 = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3469_even__succ__mod__exp,axiom,
% 13.25/7.62 ! [A: int,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.62 = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3470_even__succ__mod__exp,axiom,
% 13.25/7.62 ! [A: code_integer,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.62 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3471_option_Osize__gen_I2_J,axiom,
% 13.25/7.62 ! [X: nat > nat,X22: nat] :
% 13.25/7.62 ( ( size_option_nat @ X @ ( some_nat @ X22 ) )
% 13.25/7.62 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3472_option_Osize__gen_I2_J,axiom,
% 13.25/7.62 ! [X: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 13.25/7.62 ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X22 ) )
% 13.25/7.62 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3473_option_Osize__gen_I2_J,axiom,
% 13.25/7.62 ! [X: num > nat,X22: num] :
% 13.25/7.62 ( ( size_option_num @ X @ ( some_num @ X22 ) )
% 13.25/7.62 = ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3474_order__refl,axiom,
% 13.25/7.62 ! [X: set_int] : ( ord_less_eq_set_int @ X @ X ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3475_order__refl,axiom,
% 13.25/7.62 ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3476_order__refl,axiom,
% 13.25/7.62 ! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3477_order__refl,axiom,
% 13.25/7.62 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3478_order__refl,axiom,
% 13.25/7.62 ! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3479_dual__order_Orefl,axiom,
% 13.25/7.62 ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3480_dual__order_Orefl,axiom,
% 13.25/7.62 ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3481_dual__order_Orefl,axiom,
% 13.25/7.62 ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3482_dual__order_Orefl,axiom,
% 13.25/7.62 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3483_dual__order_Orefl,axiom,
% 13.25/7.62 ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3484_nat__dvd__1__iff__1,axiom,
% 13.25/7.62 ! [M: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 13.25/7.62 = ( M = one_one_nat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3485_dvd__0__left__iff,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 13.25/7.62 = ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3486_dvd__0__left__iff,axiom,
% 13.25/7.62 ! [A: complex] :
% 13.25/7.62 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 13.25/7.62 = ( A = zero_zero_complex ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3487_dvd__0__left__iff,axiom,
% 13.25/7.62 ! [A: real] :
% 13.25/7.62 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 13.25/7.62 = ( A = zero_zero_real ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3488_dvd__0__left__iff,axiom,
% 13.25/7.62 ! [A: rat] :
% 13.25/7.62 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 13.25/7.62 = ( A = zero_zero_rat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3489_dvd__0__left__iff,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 13.25/7.62 = ( A = zero_zero_nat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3490_dvd__0__left__iff,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 13.25/7.62 = ( A = zero_zero_int ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3491_dvd__0__right,axiom,
% 13.25/7.62 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3492_dvd__0__right,axiom,
% 13.25/7.62 ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3493_dvd__0__right,axiom,
% 13.25/7.62 ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3494_dvd__0__right,axiom,
% 13.25/7.62 ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3495_dvd__0__right,axiom,
% 13.25/7.62 ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3496_dvd__0__right,axiom,
% 13.25/7.62 ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3497_dvd__add__triv__left__iff,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3498_dvd__add__triv__left__iff,axiom,
% 13.25/7.62 ! [A: real,B: real] :
% 13.25/7.62 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 13.25/7.62 = ( dvd_dvd_real @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3499_dvd__add__triv__left__iff,axiom,
% 13.25/7.62 ! [A: rat,B: rat] :
% 13.25/7.62 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 13.25/7.62 = ( dvd_dvd_rat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3500_dvd__add__triv__left__iff,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 13.25/7.62 = ( dvd_dvd_nat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3501_dvd__add__triv__left__iff,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 13.25/7.62 = ( dvd_dvd_int @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3502_dvd__add__triv__right__iff,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3503_dvd__add__triv__right__iff,axiom,
% 13.25/7.62 ! [A: real,B: real] :
% 13.25/7.62 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 13.25/7.62 = ( dvd_dvd_real @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3504_dvd__add__triv__right__iff,axiom,
% 13.25/7.62 ! [A: rat,B: rat] :
% 13.25/7.62 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 13.25/7.62 = ( dvd_dvd_rat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3505_dvd__add__triv__right__iff,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 13.25/7.62 = ( dvd_dvd_nat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3506_dvd__add__triv__right__iff,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 13.25/7.62 = ( dvd_dvd_int @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3507_dvd__1__iff__1,axiom,
% 13.25/7.62 ! [M: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 13.25/7.62 = ( M
% 13.25/7.62 = ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3508_dvd__1__left,axiom,
% 13.25/7.62 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3509_div__dvd__div,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ A @ C )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3510_div__dvd__div,axiom,
% 13.25/7.62 ! [A: nat,B: nat,C: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.62 => ( ( dvd_dvd_nat @ A @ C )
% 13.25/7.62 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 13.25/7.62 = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3511_div__dvd__div,axiom,
% 13.25/7.62 ! [A: int,B: int,C: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.62 => ( ( dvd_dvd_int @ A @ C )
% 13.25/7.62 => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 13.25/7.62 = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3512_max_Oabsorb1,axiom,
% 13.25/7.62 ! [B: extended_enat,A: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 13.25/7.62 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3513_max_Oabsorb1,axiom,
% 13.25/7.62 ! [B: rat,A: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.62 => ( ( ord_max_rat @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3514_max_Oabsorb1,axiom,
% 13.25/7.62 ! [B: num,A: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ B @ A )
% 13.25/7.62 => ( ( ord_max_num @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3515_max_Oabsorb1,axiom,
% 13.25/7.62 ! [B: nat,A: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.62 => ( ( ord_max_nat @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3516_max_Oabsorb1,axiom,
% 13.25/7.62 ! [B: int,A: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ B @ A )
% 13.25/7.62 => ( ( ord_max_int @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3517_max_Oabsorb2,axiom,
% 13.25/7.62 ! [A: extended_enat,B: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ A @ B )
% 13.25/7.62 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3518_max_Oabsorb2,axiom,
% 13.25/7.62 ! [A: rat,B: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.62 => ( ( ord_max_rat @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3519_max_Oabsorb2,axiom,
% 13.25/7.62 ! [A: num,B: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.62 => ( ( ord_max_num @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3520_max_Oabsorb2,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.62 => ( ( ord_max_nat @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3521_max_Oabsorb2,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ A @ B )
% 13.25/7.62 => ( ( ord_max_int @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3522_max_Obounded__iff,axiom,
% 13.25/7.62 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 13.25/7.62 = ( ( ord_le2932123472753598470d_enat @ B @ A )
% 13.25/7.62 & ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3523_max_Obounded__iff,axiom,
% 13.25/7.62 ! [B: rat,C: rat,A: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 13.25/7.62 = ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.62 & ( ord_less_eq_rat @ C @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3524_max_Obounded__iff,axiom,
% 13.25/7.62 ! [B: num,C: num,A: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 13.25/7.62 = ( ( ord_less_eq_num @ B @ A )
% 13.25/7.62 & ( ord_less_eq_num @ C @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3525_max_Obounded__iff,axiom,
% 13.25/7.62 ! [B: nat,C: nat,A: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 13.25/7.62 = ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.62 & ( ord_less_eq_nat @ C @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3526_max_Obounded__iff,axiom,
% 13.25/7.62 ! [B: int,C: int,A: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 13.25/7.62 = ( ( ord_less_eq_int @ B @ A )
% 13.25/7.62 & ( ord_less_eq_int @ C @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3527_nat__mult__dvd__cancel__disj,axiom,
% 13.25/7.62 ! [K: nat,M: nat,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.25/7.62 = ( ( K = zero_zero_nat )
% 13.25/7.62 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3528_max_Oabsorb3,axiom,
% 13.25/7.62 ! [B: extended_enat,A: extended_enat] :
% 13.25/7.62 ( ( ord_le72135733267957522d_enat @ B @ A )
% 13.25/7.62 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3529_max_Oabsorb3,axiom,
% 13.25/7.62 ! [B: real,A: real] :
% 13.25/7.62 ( ( ord_less_real @ B @ A )
% 13.25/7.62 => ( ( ord_max_real @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3530_max_Oabsorb3,axiom,
% 13.25/7.62 ! [B: rat,A: rat] :
% 13.25/7.62 ( ( ord_less_rat @ B @ A )
% 13.25/7.62 => ( ( ord_max_rat @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3531_max_Oabsorb3,axiom,
% 13.25/7.62 ! [B: num,A: num] :
% 13.25/7.62 ( ( ord_less_num @ B @ A )
% 13.25/7.62 => ( ( ord_max_num @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3532_max_Oabsorb3,axiom,
% 13.25/7.62 ! [B: nat,A: nat] :
% 13.25/7.62 ( ( ord_less_nat @ B @ A )
% 13.25/7.62 => ( ( ord_max_nat @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3533_max_Oabsorb3,axiom,
% 13.25/7.62 ! [B: int,A: int] :
% 13.25/7.62 ( ( ord_less_int @ B @ A )
% 13.25/7.62 => ( ( ord_max_int @ A @ B )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3534_max_Oabsorb4,axiom,
% 13.25/7.62 ! [A: extended_enat,B: extended_enat] :
% 13.25/7.62 ( ( ord_le72135733267957522d_enat @ A @ B )
% 13.25/7.62 => ( ( ord_ma741700101516333627d_enat @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3535_max_Oabsorb4,axiom,
% 13.25/7.62 ! [A: real,B: real] :
% 13.25/7.62 ( ( ord_less_real @ A @ B )
% 13.25/7.62 => ( ( ord_max_real @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3536_max_Oabsorb4,axiom,
% 13.25/7.62 ! [A: rat,B: rat] :
% 13.25/7.62 ( ( ord_less_rat @ A @ B )
% 13.25/7.62 => ( ( ord_max_rat @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3537_max_Oabsorb4,axiom,
% 13.25/7.62 ! [A: num,B: num] :
% 13.25/7.62 ( ( ord_less_num @ A @ B )
% 13.25/7.62 => ( ( ord_max_num @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3538_max_Oabsorb4,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( ord_less_nat @ A @ B )
% 13.25/7.62 => ( ( ord_max_nat @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3539_max_Oabsorb4,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( ord_less_int @ A @ B )
% 13.25/7.62 => ( ( ord_max_int @ A @ B )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3540_max__less__iff__conj,axiom,
% 13.25/7.62 ! [X: extended_enat,Y3: extended_enat,Z: extended_enat] :
% 13.25/7.62 ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ X @ Y3 ) @ Z )
% 13.25/7.62 = ( ( ord_le72135733267957522d_enat @ X @ Z )
% 13.25/7.62 & ( ord_le72135733267957522d_enat @ Y3 @ Z ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3541_max__less__iff__conj,axiom,
% 13.25/7.62 ! [X: real,Y3: real,Z: real] :
% 13.25/7.62 ( ( ord_less_real @ ( ord_max_real @ X @ Y3 ) @ Z )
% 13.25/7.62 = ( ( ord_less_real @ X @ Z )
% 13.25/7.62 & ( ord_less_real @ Y3 @ Z ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3542_max__less__iff__conj,axiom,
% 13.25/7.62 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.62 ( ( ord_less_rat @ ( ord_max_rat @ X @ Y3 ) @ Z )
% 13.25/7.62 = ( ( ord_less_rat @ X @ Z )
% 13.25/7.62 & ( ord_less_rat @ Y3 @ Z ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3543_max__less__iff__conj,axiom,
% 13.25/7.62 ! [X: num,Y3: num,Z: num] :
% 13.25/7.62 ( ( ord_less_num @ ( ord_max_num @ X @ Y3 ) @ Z )
% 13.25/7.62 = ( ( ord_less_num @ X @ Z )
% 13.25/7.62 & ( ord_less_num @ Y3 @ Z ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3544_max__less__iff__conj,axiom,
% 13.25/7.62 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.62 ( ( ord_less_nat @ ( ord_max_nat @ X @ Y3 ) @ Z )
% 13.25/7.62 = ( ( ord_less_nat @ X @ Z )
% 13.25/7.62 & ( ord_less_nat @ Y3 @ Z ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3545_max__less__iff__conj,axiom,
% 13.25/7.62 ! [X: int,Y3: int,Z: int] :
% 13.25/7.62 ( ( ord_less_int @ ( ord_max_int @ X @ Y3 ) @ Z )
% 13.25/7.62 = ( ( ord_less_int @ X @ Z )
% 13.25/7.62 & ( ord_less_int @ Y3 @ Z ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3546_max__Suc__Suc,axiom,
% 13.25/7.62 ! [M: nat,N: nat] :
% 13.25/7.62 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 13.25/7.62 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3547_max__nat_Oeq__neutr__iff,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( ( ord_max_nat @ A @ B )
% 13.25/7.62 = zero_zero_nat )
% 13.25/7.62 = ( ( A = zero_zero_nat )
% 13.25/7.62 & ( B = zero_zero_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3548_max__nat_Oleft__neutral,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( ord_max_nat @ zero_zero_nat @ A )
% 13.25/7.62 = A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3549_max__nat_Oneutr__eq__iff,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( zero_zero_nat
% 13.25/7.62 = ( ord_max_nat @ A @ B ) )
% 13.25/7.62 = ( ( A = zero_zero_nat )
% 13.25/7.62 & ( B = zero_zero_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3550_max__nat_Oright__neutral,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( ord_max_nat @ A @ zero_zero_nat )
% 13.25/7.62 = A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3551_max__0L,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( ord_max_nat @ zero_zero_nat @ N )
% 13.25/7.62 = N ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3552_max__0R,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( ord_max_nat @ N @ zero_zero_nat )
% 13.25/7.62 = N ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3553_concat__bit__0,axiom,
% 13.25/7.62 ! [K: int,L2: int] :
% 13.25/7.62 ( ( bit_concat_bit @ zero_zero_nat @ K @ L2 )
% 13.25/7.62 = L2 ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3554_dvd__mult__cancel__left,axiom,
% 13.25/7.62 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 13.25/7.62 = ( ( C = zero_z3403309356797280102nteger )
% 13.25/7.62 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3555_dvd__mult__cancel__left,axiom,
% 13.25/7.62 ! [C: complex,A: complex,B: complex] :
% 13.25/7.62 ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 13.25/7.62 = ( ( C = zero_zero_complex )
% 13.25/7.62 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3556_dvd__mult__cancel__left,axiom,
% 13.25/7.62 ! [C: real,A: real,B: real] :
% 13.25/7.62 ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 13.25/7.62 = ( ( C = zero_zero_real )
% 13.25/7.62 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3557_dvd__mult__cancel__left,axiom,
% 13.25/7.62 ! [C: rat,A: rat,B: rat] :
% 13.25/7.62 ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 13.25/7.62 = ( ( C = zero_zero_rat )
% 13.25/7.62 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3558_dvd__mult__cancel__left,axiom,
% 13.25/7.62 ! [C: int,A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.25/7.62 = ( ( C = zero_zero_int )
% 13.25/7.62 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3559_dvd__mult__cancel__right,axiom,
% 13.25/7.62 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 13.25/7.62 = ( ( C = zero_z3403309356797280102nteger )
% 13.25/7.62 | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3560_dvd__mult__cancel__right,axiom,
% 13.25/7.62 ! [A: complex,C: complex,B: complex] :
% 13.25/7.62 ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 13.25/7.62 = ( ( C = zero_zero_complex )
% 13.25/7.62 | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3561_dvd__mult__cancel__right,axiom,
% 13.25/7.62 ! [A: real,C: real,B: real] :
% 13.25/7.62 ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 13.25/7.62 = ( ( C = zero_zero_real )
% 13.25/7.62 | ( dvd_dvd_real @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3562_dvd__mult__cancel__right,axiom,
% 13.25/7.62 ! [A: rat,C: rat,B: rat] :
% 13.25/7.62 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 13.25/7.62 = ( ( C = zero_zero_rat )
% 13.25/7.62 | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3563_dvd__mult__cancel__right,axiom,
% 13.25/7.62 ! [A: int,C: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 13.25/7.62 = ( ( C = zero_zero_int )
% 13.25/7.62 | ( dvd_dvd_int @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3564_dvd__times__left__cancel__iff,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.62 ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3565_dvd__times__left__cancel__iff,axiom,
% 13.25/7.62 ! [A: nat,B: nat,C: nat] :
% 13.25/7.62 ( ( A != zero_zero_nat )
% 13.25/7.62 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 13.25/7.62 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3566_dvd__times__left__cancel__iff,axiom,
% 13.25/7.62 ! [A: int,B: int,C: int] :
% 13.25/7.62 ( ( A != zero_zero_int )
% 13.25/7.62 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 13.25/7.62 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3567_dvd__times__right__cancel__iff,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.62 ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3568_dvd__times__right__cancel__iff,axiom,
% 13.25/7.62 ! [A: nat,B: nat,C: nat] :
% 13.25/7.62 ( ( A != zero_zero_nat )
% 13.25/7.62 => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 13.25/7.62 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3569_dvd__times__right__cancel__iff,axiom,
% 13.25/7.62 ! [A: int,B: int,C: int] :
% 13.25/7.62 ( ( A != zero_zero_int )
% 13.25/7.62 => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 13.25/7.62 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3570_unit__prod,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.62 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3571_unit__prod,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.62 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.62 => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3572_unit__prod,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.62 => ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.62 => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3573_dvd__add__times__triv__right__iff,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3574_dvd__add__times__triv__right__iff,axiom,
% 13.25/7.62 ! [A: real,B: real,C: real] :
% 13.25/7.62 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 13.25/7.62 = ( dvd_dvd_real @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3575_dvd__add__times__triv__right__iff,axiom,
% 13.25/7.62 ! [A: rat,B: rat,C: rat] :
% 13.25/7.62 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 13.25/7.62 = ( dvd_dvd_rat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3576_dvd__add__times__triv__right__iff,axiom,
% 13.25/7.62 ! [A: nat,B: nat,C: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 13.25/7.62 = ( dvd_dvd_nat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3577_dvd__add__times__triv__right__iff,axiom,
% 13.25/7.62 ! [A: int,B: int,C: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 13.25/7.62 = ( dvd_dvd_int @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3578_dvd__add__times__triv__left__iff,axiom,
% 13.25/7.62 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3579_dvd__add__times__triv__left__iff,axiom,
% 13.25/7.62 ! [A: real,C: real,B: real] :
% 13.25/7.62 ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 13.25/7.62 = ( dvd_dvd_real @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3580_dvd__add__times__triv__left__iff,axiom,
% 13.25/7.62 ! [A: rat,C: rat,B: rat] :
% 13.25/7.62 ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 13.25/7.62 = ( dvd_dvd_rat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3581_dvd__add__times__triv__left__iff,axiom,
% 13.25/7.62 ! [A: nat,C: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 13.25/7.62 = ( dvd_dvd_nat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3582_dvd__add__times__triv__left__iff,axiom,
% 13.25/7.62 ! [A: int,C: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 13.25/7.62 = ( dvd_dvd_int @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3583_dvd__div__mult__self,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.62 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3584_dvd__div__mult__self,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.62 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3585_dvd__div__mult__self,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.62 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3586_dvd__mult__div__cancel,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.62 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3587_dvd__mult__div__cancel,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.62 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3588_dvd__mult__div__cancel,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.62 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3589_unit__div__1__div__1,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.62 => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3590_unit__div__1__div__1,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.62 => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3591_unit__div__1__div__1,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.62 => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3592_unit__div__1__unit,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.62 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3593_unit__div__1__unit,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.62 => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3594_unit__div__1__unit,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.62 => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3595_unit__div,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.62 => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3596_unit__div,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.62 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.62 => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3597_unit__div,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.62 => ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.62 => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3598_div__add,axiom,
% 13.25/7.62 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ C @ A )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.62 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 13.25/7.62 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3599_div__add,axiom,
% 13.25/7.62 ! [C: nat,A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ C @ A )
% 13.25/7.62 => ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.62 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.25/7.62 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3600_div__add,axiom,
% 13.25/7.62 ! [C: int,A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ C @ A )
% 13.25/7.62 => ( ( dvd_dvd_int @ C @ B )
% 13.25/7.62 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.25/7.62 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3601_div__diff,axiom,
% 13.25/7.62 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ C @ A )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.62 => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 13.25/7.62 = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3602_div__diff,axiom,
% 13.25/7.62 ! [C: int,A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ C @ A )
% 13.25/7.62 => ( ( dvd_dvd_int @ C @ B )
% 13.25/7.62 => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 13.25/7.62 = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3603_dvd__imp__mod__0,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.62 => ( ( modulo_modulo_nat @ B @ A )
% 13.25/7.62 = zero_zero_nat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3604_dvd__imp__mod__0,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.62 => ( ( modulo_modulo_int @ B @ A )
% 13.25/7.62 = zero_zero_int ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3605_dvd__imp__mod__0,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.62 => ( ( modulo364778990260209775nteger @ B @ A )
% 13.25/7.62 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3606_max__number__of_I1_J,axiom,
% 13.25/7.62 ! [U: num,V: num] :
% 13.25/7.62 ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 13.25/7.62 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 13.25/7.62 = ( numera1916890842035813515d_enat @ V ) ) )
% 13.25/7.62 & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 13.25/7.62 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 13.25/7.62 = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3607_max__number__of_I1_J,axiom,
% 13.25/7.62 ! [U: num,V: num] :
% 13.25/7.62 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 13.25/7.62 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 13.25/7.62 = ( numeral_numeral_real @ V ) ) )
% 13.25/7.62 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 13.25/7.62 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 13.25/7.62 = ( numeral_numeral_real @ U ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3608_max__number__of_I1_J,axiom,
% 13.25/7.62 ! [U: num,V: num] :
% 13.25/7.62 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 13.25/7.62 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 13.25/7.62 = ( numeral_numeral_rat @ V ) ) )
% 13.25/7.62 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 13.25/7.62 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 13.25/7.62 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3609_max__number__of_I1_J,axiom,
% 13.25/7.62 ! [U: num,V: num] :
% 13.25/7.62 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 13.25/7.62 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 13.25/7.62 = ( numeral_numeral_nat @ V ) ) )
% 13.25/7.62 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 13.25/7.62 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 13.25/7.62 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3610_max__number__of_I1_J,axiom,
% 13.25/7.62 ! [U: num,V: num] :
% 13.25/7.62 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 13.25/7.62 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 13.25/7.62 = ( numeral_numeral_int @ V ) ) )
% 13.25/7.62 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 13.25/7.62 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 13.25/7.62 = ( numeral_numeral_int @ U ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3611_max__0__1_I4_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
% 13.25/7.62 = ( numera1916890842035813515d_enat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3612_max__0__1_I4_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
% 13.25/7.62 = ( numeral_numeral_real @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3613_max__0__1_I4_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
% 13.25/7.62 = ( numeral_numeral_rat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3614_max__0__1_I4_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
% 13.25/7.62 = ( numeral_numeral_nat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3615_max__0__1_I4_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
% 13.25/7.62 = ( numeral_numeral_int @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3616_max__0__1_I3_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 13.25/7.62 = ( numera1916890842035813515d_enat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3617_max__0__1_I3_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
% 13.25/7.62 = ( numeral_numeral_real @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3618_max__0__1_I3_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
% 13.25/7.62 = ( numeral_numeral_rat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3619_max__0__1_I3_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
% 13.25/7.62 = ( numeral_numeral_nat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3620_max__0__1_I3_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
% 13.25/7.62 = ( numeral_numeral_int @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3621_max__0__1_I1_J,axiom,
% 13.25/7.62 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 13.25/7.62 = one_one_real ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3622_max__0__1_I1_J,axiom,
% 13.25/7.62 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 13.25/7.62 = one_one_rat ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3623_max__0__1_I1_J,axiom,
% 13.25/7.62 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 13.25/7.62 = one_one_nat ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3624_max__0__1_I1_J,axiom,
% 13.25/7.62 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat )
% 13.25/7.62 = one_on7984719198319812577d_enat ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3625_max__0__1_I1_J,axiom,
% 13.25/7.62 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 13.25/7.62 = one_one_int ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3626_max__0__1_I2_J,axiom,
% 13.25/7.62 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 13.25/7.62 = one_one_real ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3627_max__0__1_I2_J,axiom,
% 13.25/7.62 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 13.25/7.62 = one_one_rat ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3628_max__0__1_I2_J,axiom,
% 13.25/7.62 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 13.25/7.62 = one_one_nat ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3629_max__0__1_I2_J,axiom,
% 13.25/7.62 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat )
% 13.25/7.62 = one_on7984719198319812577d_enat ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3630_max__0__1_I2_J,axiom,
% 13.25/7.62 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 13.25/7.62 = one_one_int ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3631_max__0__1_I6_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
% 13.25/7.62 = ( numera1916890842035813515d_enat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3632_max__0__1_I6_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
% 13.25/7.62 = ( numeral_numeral_real @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3633_max__0__1_I6_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat )
% 13.25/7.62 = ( numeral_numeral_rat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3634_max__0__1_I6_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
% 13.25/7.62 = ( numeral_numeral_nat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3635_max__0__1_I6_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
% 13.25/7.62 = ( numeral_numeral_int @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3636_max__0__1_I5_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
% 13.25/7.62 = ( numera1916890842035813515d_enat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3637_max__0__1_I5_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
% 13.25/7.62 = ( numeral_numeral_real @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3638_max__0__1_I5_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
% 13.25/7.62 = ( numeral_numeral_rat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3639_max__0__1_I5_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
% 13.25/7.62 = ( numeral_numeral_nat @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3640_max__0__1_I5_J,axiom,
% 13.25/7.62 ! [X: num] :
% 13.25/7.62 ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
% 13.25/7.62 = ( numeral_numeral_int @ X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3641_list__update__beyond,axiom,
% 13.25/7.62 ! [Xs2: list_VEBT_VEBT,I: nat,X: vEBT_VEBT] :
% 13.25/7.62 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I )
% 13.25/7.62 => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
% 13.25/7.62 = Xs2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3642_list__update__beyond,axiom,
% 13.25/7.62 ! [Xs2: list_o,I: nat,X: $o] :
% 13.25/7.62 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I )
% 13.25/7.62 => ( ( list_update_o @ Xs2 @ I @ X )
% 13.25/7.62 = Xs2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3643_list__update__beyond,axiom,
% 13.25/7.62 ! [Xs2: list_nat,I: nat,X: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
% 13.25/7.62 => ( ( list_update_nat @ Xs2 @ I @ X )
% 13.25/7.62 = Xs2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3644_list__update__beyond,axiom,
% 13.25/7.62 ! [Xs2: list_int,I: nat,X: int] :
% 13.25/7.62 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
% 13.25/7.62 => ( ( list_update_int @ Xs2 @ I @ X )
% 13.25/7.62 = Xs2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3645_concat__bit__nonnegative__iff,axiom,
% 13.25/7.62 ! [N: nat,K: int,L2: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L2 ) )
% 13.25/7.62 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3646_concat__bit__negative__iff,axiom,
% 13.25/7.62 ! [N: nat,K: int,L2: int] :
% 13.25/7.62 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L2 ) @ zero_zero_int )
% 13.25/7.62 = ( ord_less_int @ L2 @ zero_zero_int ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3647_unit__div__mult__self,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.62 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3648_unit__div__mult__self,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.62 => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3649_unit__div__mult__self,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.62 => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 13.25/7.62 = B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3650_unit__mult__div__div,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.62 => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 13.25/7.62 = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3651_unit__mult__div__div,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.62 => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 13.25/7.62 = ( divide_divide_nat @ B @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3652_unit__mult__div__div,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.62 => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 13.25/7.62 = ( divide_divide_int @ B @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3653_even__Suc__Suc__iff,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N ) ) )
% 13.25/7.62 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3654_even__Suc,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) )
% 13.25/7.62 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3655_pow__divides__pow__iff,axiom,
% 13.25/7.62 ! [N: nat,A: nat,B: nat] :
% 13.25/7.62 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
% 13.25/7.62 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3656_pow__divides__pow__iff,axiom,
% 13.25/7.62 ! [N: nat,A: int,B: int] :
% 13.25/7.62 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
% 13.25/7.62 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3657_nth__list__update__eq,axiom,
% 13.25/7.62 ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
% 13.25/7.62 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ I )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3658_nth__list__update__eq,axiom,
% 13.25/7.62 ! [I: nat,Xs2: list_o,X: $o] :
% 13.25/7.62 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ I )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3659_nth__list__update__eq,axiom,
% 13.25/7.62 ! [I: nat,Xs2: list_nat,X: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3660_nth__list__update__eq,axiom,
% 13.25/7.62 ! [I: nat,Xs2: list_int,X: int] :
% 13.25/7.62 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ I )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3661_even__mult__iff,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
% 13.25/7.62 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3662_even__mult__iff,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
% 13.25/7.62 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3663_even__mult__iff,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
% 13.25/7.62 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3664_even__add,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 13.25/7.62 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3665_even__add,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 13.25/7.62 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3666_even__add,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 13.25/7.62 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3667_odd__add,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 13.25/7.62 = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.62 != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3668_odd__add,axiom,
% 13.25/7.62 ! [A: nat,B: nat] :
% 13.25/7.62 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 13.25/7.62 = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.62 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3669_odd__add,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 13.25/7.62 = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.62 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3670_even__mod__2__iff,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.62 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3671_even__mod__2__iff,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 13.25/7.62 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3672_even__mod__2__iff,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3673_even__Suc__div__two,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.62 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3674_odd__Suc__div__two,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.62 => ( ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3675_set__swap,axiom,
% 13.25/7.62 ! [I: nat,Xs2: list_VEBT_VEBT,J: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.62 => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) )
% 13.25/7.62 = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3676_set__swap,axiom,
% 13.25/7.62 ! [I: nat,Xs2: list_o,J: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.62 => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I @ ( nth_o @ Xs2 @ J ) ) @ J @ ( nth_o @ Xs2 @ I ) ) )
% 13.25/7.62 = ( set_o2 @ Xs2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3677_set__swap,axiom,
% 13.25/7.62 ! [I: nat,Xs2: list_nat,J: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.62 => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
% 13.25/7.62 = ( set_nat2 @ Xs2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3678_set__swap,axiom,
% 13.25/7.62 ! [I: nat,Xs2: list_int,J: nat] :
% 13.25/7.62 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.62 => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I @ ( nth_int @ Xs2 @ J ) ) @ J @ ( nth_int @ Xs2 @ I ) ) )
% 13.25/7.62 = ( set_int2 @ Xs2 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3679_zero__le__power__eq__numeral,axiom,
% 13.25/7.62 ! [A: real,W: num] :
% 13.25/7.62 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 13.25/7.62 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3680_zero__le__power__eq__numeral,axiom,
% 13.25/7.62 ! [A: rat,W: num] :
% 13.25/7.62 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 13.25/7.62 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3681_zero__le__power__eq__numeral,axiom,
% 13.25/7.62 ! [A: int,W: num] :
% 13.25/7.62 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 13.25/7.62 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3682_power__less__zero__eq__numeral,axiom,
% 13.25/7.62 ! [A: real,W: num] :
% 13.25/7.62 ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 13.25/7.62 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3683_power__less__zero__eq__numeral,axiom,
% 13.25/7.62 ! [A: rat,W: num] :
% 13.25/7.62 ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 13.25/7.62 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3684_power__less__zero__eq__numeral,axiom,
% 13.25/7.62 ! [A: int,W: num] :
% 13.25/7.62 ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 13.25/7.62 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3685_power__less__zero__eq,axiom,
% 13.25/7.62 ! [A: real,N: nat] :
% 13.25/7.62 ( ( ord_less_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 13.25/7.62 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.62 & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3686_power__less__zero__eq,axiom,
% 13.25/7.62 ! [A: rat,N: nat] :
% 13.25/7.62 ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 13.25/7.62 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.62 & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3687_power__less__zero__eq,axiom,
% 13.25/7.62 ! [A: int,N: nat] :
% 13.25/7.62 ( ( ord_less_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 13.25/7.62 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.62 & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3688_even__plus__one__iff,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 13.25/7.62 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3689_even__plus__one__iff,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 13.25/7.62 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3690_even__plus__one__iff,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 13.25/7.62 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3691_even__diff,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 13.25/7.62 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3692_even__diff,axiom,
% 13.25/7.62 ! [A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 13.25/7.62 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3693_odd__Suc__minus__one,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.62 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 13.25/7.62 = N ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3694_even__diff__nat,axiom,
% 13.25/7.62 ! [M: nat,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.62 = ( ( ord_less_nat @ M @ N )
% 13.25/7.62 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3695_zero__less__power__eq__numeral,axiom,
% 13.25/7.62 ! [A: real,W: num] :
% 13.25/7.62 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
% 13.25/7.62 = ( ( ( numeral_numeral_nat @ W )
% 13.25/7.62 = zero_zero_nat )
% 13.25/7.62 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( A != zero_zero_real ) )
% 13.25/7.62 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3696_zero__less__power__eq__numeral,axiom,
% 13.25/7.62 ! [A: rat,W: num] :
% 13.25/7.62 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
% 13.25/7.62 = ( ( ( numeral_numeral_nat @ W )
% 13.25/7.62 = zero_zero_nat )
% 13.25/7.62 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( A != zero_zero_rat ) )
% 13.25/7.62 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3697_zero__less__power__eq__numeral,axiom,
% 13.25/7.62 ! [A: int,W: num] :
% 13.25/7.62 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
% 13.25/7.62 = ( ( ( numeral_numeral_nat @ W )
% 13.25/7.62 = zero_zero_nat )
% 13.25/7.62 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( A != zero_zero_int ) )
% 13.25/7.62 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3698_even__succ__div__2,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3699_even__succ__div__2,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3700_even__succ__div__2,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3701_even__succ__div__two,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3702_even__succ__div__two,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3703_even__succ__div__two,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3704_odd__succ__div__two,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3705_odd__succ__div__two,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3706_odd__succ__div__two,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.62 = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3707_even__power,axiom,
% 13.25/7.62 ! [A: code_integer,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N ) )
% 13.25/7.62 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3708_even__power,axiom,
% 13.25/7.62 ! [A: nat,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N ) )
% 13.25/7.62 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3709_even__power,axiom,
% 13.25/7.62 ! [A: int,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N ) )
% 13.25/7.62 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3710_odd__two__times__div__two__nat,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.62 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.62 = ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3711_odd__two__times__div__two__succ,axiom,
% 13.25/7.62 ! [A: code_integer] :
% 13.25/7.62 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3712_odd__two__times__div__two__succ,axiom,
% 13.25/7.62 ! [A: nat] :
% 13.25/7.62 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3713_odd__two__times__div__two__succ,axiom,
% 13.25/7.62 ! [A: int] :
% 13.25/7.62 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 13.25/7.62 = A ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3714_power__le__zero__eq__numeral,axiom,
% 13.25/7.62 ! [A: real,W: num] :
% 13.25/7.62 ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
% 13.25/7.62 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 13.25/7.62 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( A = zero_zero_real ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3715_power__le__zero__eq__numeral,axiom,
% 13.25/7.62 ! [A: rat,W: num] :
% 13.25/7.62 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
% 13.25/7.62 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 13.25/7.62 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( A = zero_zero_rat ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3716_power__le__zero__eq__numeral,axiom,
% 13.25/7.62 ! [A: int,W: num] :
% 13.25/7.62 ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
% 13.25/7.62 = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 13.25/7.62 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.62 & ( A = zero_zero_int ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3717_semiring__parity__class_Oeven__mask__iff,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) )
% 13.25/7.62 = ( N = zero_zero_nat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3718_semiring__parity__class_Oeven__mask__iff,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) )
% 13.25/7.62 = ( N = zero_zero_nat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3719_semiring__parity__class_Oeven__mask__iff,axiom,
% 13.25/7.62 ! [N: nat] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 13.25/7.62 = ( N = zero_zero_nat ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3720_even__succ__div__exp,axiom,
% 13.25/7.62 ! [A: code_integer,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.62 = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3721_even__succ__div__exp,axiom,
% 13.25/7.62 ! [A: nat,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.62 = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3722_even__succ__div__exp,axiom,
% 13.25/7.62 ! [A: int,N: nat] :
% 13.25/7.62 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.62 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.62 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.62 = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3723_division__decomp,axiom,
% 13.25/7.62 ! [A: nat,B: nat,C: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 13.25/7.62 => ? [B8: nat,C5: nat] :
% 13.25/7.62 ( ( A
% 13.25/7.62 = ( times_times_nat @ B8 @ C5 ) )
% 13.25/7.62 & ( dvd_dvd_nat @ B8 @ B )
% 13.25/7.62 & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3724_division__decomp,axiom,
% 13.25/7.62 ! [A: int,B: int,C: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 13.25/7.62 => ? [B8: int,C5: int] :
% 13.25/7.62 ( ( A
% 13.25/7.62 = ( times_times_int @ B8 @ C5 ) )
% 13.25/7.62 & ( dvd_dvd_int @ B8 @ B )
% 13.25/7.62 & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3725_dvd__productE,axiom,
% 13.25/7.62 ! [P6: nat,A: nat,B: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ P6 @ ( times_times_nat @ A @ B ) )
% 13.25/7.62 => ~ ! [X3: nat,Y4: nat] :
% 13.25/7.62 ( ( P6
% 13.25/7.62 = ( times_times_nat @ X3 @ Y4 ) )
% 13.25/7.62 => ( ( dvd_dvd_nat @ X3 @ A )
% 13.25/7.62 => ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3726_dvd__productE,axiom,
% 13.25/7.62 ! [P6: int,A: int,B: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ P6 @ ( times_times_int @ A @ B ) )
% 13.25/7.62 => ~ ! [X3: int,Y4: int] :
% 13.25/7.62 ( ( P6
% 13.25/7.62 = ( times_times_int @ X3 @ Y4 ) )
% 13.25/7.62 => ( ( dvd_dvd_int @ X3 @ A )
% 13.25/7.62 => ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3727_dvd__trans,axiom,
% 13.25/7.62 ! [A: nat,B: nat,C: nat] :
% 13.25/7.62 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.62 => ( ( dvd_dvd_nat @ B @ C )
% 13.25/7.62 => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3728_dvd__trans,axiom,
% 13.25/7.62 ! [A: int,B: int,C: int] :
% 13.25/7.62 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.62 => ( ( dvd_dvd_int @ B @ C )
% 13.25/7.62 => ( dvd_dvd_int @ A @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3729_dvd__trans,axiom,
% 13.25/7.62 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.62 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.62 => ( ( dvd_dvd_Code_integer @ B @ C )
% 13.25/7.62 => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3730_dvd__refl,axiom,
% 13.25/7.62 ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3731_dvd__refl,axiom,
% 13.25/7.62 ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3732_dvd__refl,axiom,
% 13.25/7.62 ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3733_max__absorb2,axiom,
% 13.25/7.62 ! [X: extended_enat,Y3: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ X @ Y3 )
% 13.25/7.62 => ( ( ord_ma741700101516333627d_enat @ X @ Y3 )
% 13.25/7.62 = Y3 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3734_max__absorb2,axiom,
% 13.25/7.62 ! [X: set_int,Y3: set_int] :
% 13.25/7.62 ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.62 => ( ( ord_max_set_int @ X @ Y3 )
% 13.25/7.62 = Y3 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3735_max__absorb2,axiom,
% 13.25/7.62 ! [X: rat,Y3: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.62 => ( ( ord_max_rat @ X @ Y3 )
% 13.25/7.62 = Y3 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3736_max__absorb2,axiom,
% 13.25/7.62 ! [X: num,Y3: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.62 => ( ( ord_max_num @ X @ Y3 )
% 13.25/7.62 = Y3 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3737_max__absorb2,axiom,
% 13.25/7.62 ! [X: nat,Y3: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.62 => ( ( ord_max_nat @ X @ Y3 )
% 13.25/7.62 = Y3 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3738_max__absorb2,axiom,
% 13.25/7.62 ! [X: int,Y3: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.62 => ( ( ord_max_int @ X @ Y3 )
% 13.25/7.62 = Y3 ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3739_max__absorb1,axiom,
% 13.25/7.62 ! [Y3: extended_enat,X: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ Y3 @ X )
% 13.25/7.62 => ( ( ord_ma741700101516333627d_enat @ X @ Y3 )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3740_max__absorb1,axiom,
% 13.25/7.62 ! [Y3: set_int,X: set_int] :
% 13.25/7.62 ( ( ord_less_eq_set_int @ Y3 @ X )
% 13.25/7.62 => ( ( ord_max_set_int @ X @ Y3 )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3741_max__absorb1,axiom,
% 13.25/7.62 ! [Y3: rat,X: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ Y3 @ X )
% 13.25/7.62 => ( ( ord_max_rat @ X @ Y3 )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3742_max__absorb1,axiom,
% 13.25/7.62 ! [Y3: num,X: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ Y3 @ X )
% 13.25/7.62 => ( ( ord_max_num @ X @ Y3 )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3743_max__absorb1,axiom,
% 13.25/7.62 ! [Y3: nat,X: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ Y3 @ X )
% 13.25/7.62 => ( ( ord_max_nat @ X @ Y3 )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3744_max__absorb1,axiom,
% 13.25/7.62 ! [Y3: int,X: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ Y3 @ X )
% 13.25/7.62 => ( ( ord_max_int @ X @ Y3 )
% 13.25/7.62 = X ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3745_max__def,axiom,
% 13.25/7.62 ( ord_ma741700101516333627d_enat
% 13.25/7.62 = ( ^ [A3: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3746_max__def,axiom,
% 13.25/7.62 ( ord_max_set_int
% 13.25/7.62 = ( ^ [A3: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3747_max__def,axiom,
% 13.25/7.62 ( ord_max_rat
% 13.25/7.62 = ( ^ [A3: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3748_max__def,axiom,
% 13.25/7.62 ( ord_max_num
% 13.25/7.62 = ( ^ [A3: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3749_max__def,axiom,
% 13.25/7.62 ( ord_max_nat
% 13.25/7.62 = ( ^ [A3: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3750_max__def,axiom,
% 13.25/7.62 ( ord_max_int
% 13.25/7.62 = ( ^ [A3: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3751_max_OcoboundedI2,axiom,
% 13.25/7.62 ! [C: extended_enat,B: extended_enat,A: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ C @ B )
% 13.25/7.62 => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3752_max_OcoboundedI2,axiom,
% 13.25/7.62 ! [C: rat,B: rat,A: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ C @ B )
% 13.25/7.62 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3753_max_OcoboundedI2,axiom,
% 13.25/7.62 ! [C: num,B: num,A: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ C @ B )
% 13.25/7.62 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3754_max_OcoboundedI2,axiom,
% 13.25/7.62 ! [C: nat,B: nat,A: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ C @ B )
% 13.25/7.62 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3755_max_OcoboundedI2,axiom,
% 13.25/7.62 ! [C: int,B: int,A: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ C @ B )
% 13.25/7.62 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3756_max_OcoboundedI1,axiom,
% 13.25/7.62 ! [C: extended_enat,A: extended_enat,B: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ C @ A )
% 13.25/7.62 => ( ord_le2932123472753598470d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3757_max_OcoboundedI1,axiom,
% 13.25/7.62 ! [C: rat,A: rat,B: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ C @ A )
% 13.25/7.62 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3758_max_OcoboundedI1,axiom,
% 13.25/7.62 ! [C: num,A: num,B: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ C @ A )
% 13.25/7.62 => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3759_max_OcoboundedI1,axiom,
% 13.25/7.62 ! [C: nat,A: nat,B: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ C @ A )
% 13.25/7.62 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3760_max_OcoboundedI1,axiom,
% 13.25/7.62 ! [C: int,A: int,B: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ C @ A )
% 13.25/7.62 => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3761_max_Oabsorb__iff2,axiom,
% 13.25/7.62 ( ord_le2932123472753598470d_enat
% 13.25/7.62 = ( ^ [A3: extended_enat,B4: extended_enat] :
% 13.25/7.62 ( ( ord_ma741700101516333627d_enat @ A3 @ B4 )
% 13.25/7.62 = B4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3762_max_Oabsorb__iff2,axiom,
% 13.25/7.62 ( ord_less_eq_rat
% 13.25/7.62 = ( ^ [A3: rat,B4: rat] :
% 13.25/7.62 ( ( ord_max_rat @ A3 @ B4 )
% 13.25/7.62 = B4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3763_max_Oabsorb__iff2,axiom,
% 13.25/7.62 ( ord_less_eq_num
% 13.25/7.62 = ( ^ [A3: num,B4: num] :
% 13.25/7.62 ( ( ord_max_num @ A3 @ B4 )
% 13.25/7.62 = B4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3764_max_Oabsorb__iff2,axiom,
% 13.25/7.62 ( ord_less_eq_nat
% 13.25/7.62 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.62 ( ( ord_max_nat @ A3 @ B4 )
% 13.25/7.62 = B4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3765_max_Oabsorb__iff2,axiom,
% 13.25/7.62 ( ord_less_eq_int
% 13.25/7.62 = ( ^ [A3: int,B4: int] :
% 13.25/7.62 ( ( ord_max_int @ A3 @ B4 )
% 13.25/7.62 = B4 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3766_max_Oabsorb__iff1,axiom,
% 13.25/7.62 ( ord_le2932123472753598470d_enat
% 13.25/7.62 = ( ^ [B4: extended_enat,A3: extended_enat] :
% 13.25/7.62 ( ( ord_ma741700101516333627d_enat @ A3 @ B4 )
% 13.25/7.62 = A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3767_max_Oabsorb__iff1,axiom,
% 13.25/7.62 ( ord_less_eq_rat
% 13.25/7.62 = ( ^ [B4: rat,A3: rat] :
% 13.25/7.62 ( ( ord_max_rat @ A3 @ B4 )
% 13.25/7.62 = A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3768_max_Oabsorb__iff1,axiom,
% 13.25/7.62 ( ord_less_eq_num
% 13.25/7.62 = ( ^ [B4: num,A3: num] :
% 13.25/7.62 ( ( ord_max_num @ A3 @ B4 )
% 13.25/7.62 = A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3769_max_Oabsorb__iff1,axiom,
% 13.25/7.62 ( ord_less_eq_nat
% 13.25/7.62 = ( ^ [B4: nat,A3: nat] :
% 13.25/7.62 ( ( ord_max_nat @ A3 @ B4 )
% 13.25/7.62 = A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3770_max_Oabsorb__iff1,axiom,
% 13.25/7.62 ( ord_less_eq_int
% 13.25/7.62 = ( ^ [B4: int,A3: int] :
% 13.25/7.62 ( ( ord_max_int @ A3 @ B4 )
% 13.25/7.62 = A3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3771_le__max__iff__disj,axiom,
% 13.25/7.62 ! [Z: extended_enat,X: extended_enat,Y3: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y3 ) )
% 13.25/7.62 = ( ( ord_le2932123472753598470d_enat @ Z @ X )
% 13.25/7.62 | ( ord_le2932123472753598470d_enat @ Z @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3772_le__max__iff__disj,axiom,
% 13.25/7.62 ! [Z: rat,X: rat,Y3: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X @ Y3 ) )
% 13.25/7.62 = ( ( ord_less_eq_rat @ Z @ X )
% 13.25/7.62 | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3773_le__max__iff__disj,axiom,
% 13.25/7.62 ! [Z: num,X: num,Y3: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X @ Y3 ) )
% 13.25/7.62 = ( ( ord_less_eq_num @ Z @ X )
% 13.25/7.62 | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3774_le__max__iff__disj,axiom,
% 13.25/7.62 ! [Z: nat,X: nat,Y3: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y3 ) )
% 13.25/7.62 = ( ( ord_less_eq_nat @ Z @ X )
% 13.25/7.62 | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3775_le__max__iff__disj,axiom,
% 13.25/7.62 ! [Z: int,X: int,Y3: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y3 ) )
% 13.25/7.62 = ( ( ord_less_eq_int @ Z @ X )
% 13.25/7.62 | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3776_max_Ocobounded2,axiom,
% 13.25/7.62 ! [B: extended_enat,A: extended_enat] : ( ord_le2932123472753598470d_enat @ B @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3777_max_Ocobounded2,axiom,
% 13.25/7.62 ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3778_max_Ocobounded2,axiom,
% 13.25/7.62 ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3779_max_Ocobounded2,axiom,
% 13.25/7.62 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3780_max_Ocobounded2,axiom,
% 13.25/7.62 ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3781_max_Ocobounded1,axiom,
% 13.25/7.62 ! [A: extended_enat,B: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3782_max_Ocobounded1,axiom,
% 13.25/7.62 ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3783_max_Ocobounded1,axiom,
% 13.25/7.62 ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3784_max_Ocobounded1,axiom,
% 13.25/7.62 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3785_max_Ocobounded1,axiom,
% 13.25/7.62 ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3786_max_Oorder__iff,axiom,
% 13.25/7.62 ( ord_le2932123472753598470d_enat
% 13.25/7.62 = ( ^ [B4: extended_enat,A3: extended_enat] :
% 13.25/7.62 ( A3
% 13.25/7.62 = ( ord_ma741700101516333627d_enat @ A3 @ B4 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3787_max_Oorder__iff,axiom,
% 13.25/7.62 ( ord_less_eq_rat
% 13.25/7.62 = ( ^ [B4: rat,A3: rat] :
% 13.25/7.62 ( A3
% 13.25/7.62 = ( ord_max_rat @ A3 @ B4 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3788_max_Oorder__iff,axiom,
% 13.25/7.62 ( ord_less_eq_num
% 13.25/7.62 = ( ^ [B4: num,A3: num] :
% 13.25/7.62 ( A3
% 13.25/7.62 = ( ord_max_num @ A3 @ B4 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3789_max_Oorder__iff,axiom,
% 13.25/7.62 ( ord_less_eq_nat
% 13.25/7.62 = ( ^ [B4: nat,A3: nat] :
% 13.25/7.62 ( A3
% 13.25/7.62 = ( ord_max_nat @ A3 @ B4 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3790_max_Oorder__iff,axiom,
% 13.25/7.62 ( ord_less_eq_int
% 13.25/7.62 = ( ^ [B4: int,A3: int] :
% 13.25/7.62 ( A3
% 13.25/7.62 = ( ord_max_int @ A3 @ B4 ) ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3791_max_OboundedI,axiom,
% 13.25/7.62 ! [B: extended_enat,A: extended_enat,C: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 13.25/7.62 => ( ( ord_le2932123472753598470d_enat @ C @ A )
% 13.25/7.62 => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3792_max_OboundedI,axiom,
% 13.25/7.62 ! [B: rat,A: rat,C: rat] :
% 13.25/7.62 ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.62 => ( ( ord_less_eq_rat @ C @ A )
% 13.25/7.62 => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3793_max_OboundedI,axiom,
% 13.25/7.62 ! [B: num,A: num,C: num] :
% 13.25/7.62 ( ( ord_less_eq_num @ B @ A )
% 13.25/7.62 => ( ( ord_less_eq_num @ C @ A )
% 13.25/7.62 => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3794_max_OboundedI,axiom,
% 13.25/7.62 ! [B: nat,A: nat,C: nat] :
% 13.25/7.62 ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.62 => ( ( ord_less_eq_nat @ C @ A )
% 13.25/7.62 => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3795_max_OboundedI,axiom,
% 13.25/7.62 ! [B: int,A: int,C: int] :
% 13.25/7.62 ( ( ord_less_eq_int @ B @ A )
% 13.25/7.62 => ( ( ord_less_eq_int @ C @ A )
% 13.25/7.62 => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).
% 13.25/7.62
% 13.25/7.62 thf(fact_3796_max_OboundedE,axiom,
% 13.25/7.62 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 13.25/7.62 ( ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 13.25/7.62 => ~ ( ( ord_le2932123472753598470d_enat @ B @ A )
% 13.25/7.63 => ~ ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3797_max_OboundedE,axiom,
% 13.25/7.63 ! [B: rat,C: rat,A: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.63 => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3798_max_OboundedE,axiom,
% 13.25/7.63 ! [B: num,C: num,A: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_eq_num @ B @ A )
% 13.25/7.63 => ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3799_max_OboundedE,axiom,
% 13.25/7.63 ! [B: nat,C: nat,A: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.63 => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3800_max_OboundedE,axiom,
% 13.25/7.63 ! [B: int,C: int,A: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_eq_int @ B @ A )
% 13.25/7.63 => ~ ( ord_less_eq_int @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3801_max_OorderI,axiom,
% 13.25/7.63 ! [A: extended_enat,B: extended_enat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( ord_ma741700101516333627d_enat @ A @ B ) )
% 13.25/7.63 => ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3802_max_OorderI,axiom,
% 13.25/7.63 ! [A: rat,B: rat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( ord_max_rat @ A @ B ) )
% 13.25/7.63 => ( ord_less_eq_rat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3803_max_OorderI,axiom,
% 13.25/7.63 ! [A: num,B: num] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( ord_max_num @ A @ B ) )
% 13.25/7.63 => ( ord_less_eq_num @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3804_max_OorderI,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( ord_max_nat @ A @ B ) )
% 13.25/7.63 => ( ord_less_eq_nat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3805_max_OorderI,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( ord_max_int @ A @ B ) )
% 13.25/7.63 => ( ord_less_eq_int @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3806_max_OorderE,axiom,
% 13.25/7.63 ! [B: extended_enat,A: extended_enat] :
% 13.25/7.63 ( ( ord_le2932123472753598470d_enat @ B @ A )
% 13.25/7.63 => ( A
% 13.25/7.63 = ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3807_max_OorderE,axiom,
% 13.25/7.63 ! [B: rat,A: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.63 => ( A
% 13.25/7.63 = ( ord_max_rat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3808_max_OorderE,axiom,
% 13.25/7.63 ! [B: num,A: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ B @ A )
% 13.25/7.63 => ( A
% 13.25/7.63 = ( ord_max_num @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3809_max_OorderE,axiom,
% 13.25/7.63 ! [B: nat,A: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.63 => ( A
% 13.25/7.63 = ( ord_max_nat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3810_max_OorderE,axiom,
% 13.25/7.63 ! [B: int,A: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ B @ A )
% 13.25/7.63 => ( A
% 13.25/7.63 = ( ord_max_int @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3811_max_Omono,axiom,
% 13.25/7.63 ! [C: extended_enat,A: extended_enat,D: extended_enat,B: extended_enat] :
% 13.25/7.63 ( ( ord_le2932123472753598470d_enat @ C @ A )
% 13.25/7.63 => ( ( ord_le2932123472753598470d_enat @ D @ B )
% 13.25/7.63 => ( ord_le2932123472753598470d_enat @ ( ord_ma741700101516333627d_enat @ C @ D ) @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3812_max_Omono,axiom,
% 13.25/7.63 ! [C: rat,A: rat,D: rat,B: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ C @ A )
% 13.25/7.63 => ( ( ord_less_eq_rat @ D @ B )
% 13.25/7.63 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3813_max_Omono,axiom,
% 13.25/7.63 ! [C: num,A: num,D: num,B: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ C @ A )
% 13.25/7.63 => ( ( ord_less_eq_num @ D @ B )
% 13.25/7.63 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3814_max_Omono,axiom,
% 13.25/7.63 ! [C: nat,A: nat,D: nat,B: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ C @ A )
% 13.25/7.63 => ( ( ord_less_eq_nat @ D @ B )
% 13.25/7.63 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3815_max_Omono,axiom,
% 13.25/7.63 ! [C: int,A: int,D: int,B: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ C @ A )
% 13.25/7.63 => ( ( ord_less_eq_int @ D @ B )
% 13.25/7.63 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3816_max_Ostrict__coboundedI2,axiom,
% 13.25/7.63 ! [C: extended_enat,B: extended_enat,A: extended_enat] :
% 13.25/7.63 ( ( ord_le72135733267957522d_enat @ C @ B )
% 13.25/7.63 => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3817_max_Ostrict__coboundedI2,axiom,
% 13.25/7.63 ! [C: real,B: real,A: real] :
% 13.25/7.63 ( ( ord_less_real @ C @ B )
% 13.25/7.63 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3818_max_Ostrict__coboundedI2,axiom,
% 13.25/7.63 ! [C: rat,B: rat,A: rat] :
% 13.25/7.63 ( ( ord_less_rat @ C @ B )
% 13.25/7.63 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3819_max_Ostrict__coboundedI2,axiom,
% 13.25/7.63 ! [C: num,B: num,A: num] :
% 13.25/7.63 ( ( ord_less_num @ C @ B )
% 13.25/7.63 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3820_max_Ostrict__coboundedI2,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( ord_less_nat @ C @ B )
% 13.25/7.63 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3821_max_Ostrict__coboundedI2,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( ord_less_int @ C @ B )
% 13.25/7.63 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3822_max_Ostrict__coboundedI1,axiom,
% 13.25/7.63 ! [C: extended_enat,A: extended_enat,B: extended_enat] :
% 13.25/7.63 ( ( ord_le72135733267957522d_enat @ C @ A )
% 13.25/7.63 => ( ord_le72135733267957522d_enat @ C @ ( ord_ma741700101516333627d_enat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3823_max_Ostrict__coboundedI1,axiom,
% 13.25/7.63 ! [C: real,A: real,B: real] :
% 13.25/7.63 ( ( ord_less_real @ C @ A )
% 13.25/7.63 => ( ord_less_real @ C @ ( ord_max_real @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3824_max_Ostrict__coboundedI1,axiom,
% 13.25/7.63 ! [C: rat,A: rat,B: rat] :
% 13.25/7.63 ( ( ord_less_rat @ C @ A )
% 13.25/7.63 => ( ord_less_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3825_max_Ostrict__coboundedI1,axiom,
% 13.25/7.63 ! [C: num,A: num,B: num] :
% 13.25/7.63 ( ( ord_less_num @ C @ A )
% 13.25/7.63 => ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3826_max_Ostrict__coboundedI1,axiom,
% 13.25/7.63 ! [C: nat,A: nat,B: nat] :
% 13.25/7.63 ( ( ord_less_nat @ C @ A )
% 13.25/7.63 => ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3827_max_Ostrict__coboundedI1,axiom,
% 13.25/7.63 ! [C: int,A: int,B: int] :
% 13.25/7.63 ( ( ord_less_int @ C @ A )
% 13.25/7.63 => ( ord_less_int @ C @ ( ord_max_int @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3828_max_Ostrict__order__iff,axiom,
% 13.25/7.63 ( ord_le72135733267957522d_enat
% 13.25/7.63 = ( ^ [B4: extended_enat,A3: extended_enat] :
% 13.25/7.63 ( ( A3
% 13.25/7.63 = ( ord_ma741700101516333627d_enat @ A3 @ B4 ) )
% 13.25/7.63 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3829_max_Ostrict__order__iff,axiom,
% 13.25/7.63 ( ord_less_real
% 13.25/7.63 = ( ^ [B4: real,A3: real] :
% 13.25/7.63 ( ( A3
% 13.25/7.63 = ( ord_max_real @ A3 @ B4 ) )
% 13.25/7.63 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3830_max_Ostrict__order__iff,axiom,
% 13.25/7.63 ( ord_less_rat
% 13.25/7.63 = ( ^ [B4: rat,A3: rat] :
% 13.25/7.63 ( ( A3
% 13.25/7.63 = ( ord_max_rat @ A3 @ B4 ) )
% 13.25/7.63 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3831_max_Ostrict__order__iff,axiom,
% 13.25/7.63 ( ord_less_num
% 13.25/7.63 = ( ^ [B4: num,A3: num] :
% 13.25/7.63 ( ( A3
% 13.25/7.63 = ( ord_max_num @ A3 @ B4 ) )
% 13.25/7.63 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3832_max_Ostrict__order__iff,axiom,
% 13.25/7.63 ( ord_less_nat
% 13.25/7.63 = ( ^ [B4: nat,A3: nat] :
% 13.25/7.63 ( ( A3
% 13.25/7.63 = ( ord_max_nat @ A3 @ B4 ) )
% 13.25/7.63 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3833_max_Ostrict__order__iff,axiom,
% 13.25/7.63 ( ord_less_int
% 13.25/7.63 = ( ^ [B4: int,A3: int] :
% 13.25/7.63 ( ( A3
% 13.25/7.63 = ( ord_max_int @ A3 @ B4 ) )
% 13.25/7.63 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3834_max_Ostrict__boundedE,axiom,
% 13.25/7.63 ! [B: extended_enat,C: extended_enat,A: extended_enat] :
% 13.25/7.63 ( ( ord_le72135733267957522d_enat @ ( ord_ma741700101516333627d_enat @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_le72135733267957522d_enat @ B @ A )
% 13.25/7.63 => ~ ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3835_max_Ostrict__boundedE,axiom,
% 13.25/7.63 ! [B: real,C: real,A: real] :
% 13.25/7.63 ( ( ord_less_real @ ( ord_max_real @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_real @ B @ A )
% 13.25/7.63 => ~ ( ord_less_real @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3836_max_Ostrict__boundedE,axiom,
% 13.25/7.63 ! [B: rat,C: rat,A: rat] :
% 13.25/7.63 ( ( ord_less_rat @ ( ord_max_rat @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_rat @ B @ A )
% 13.25/7.63 => ~ ( ord_less_rat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3837_max_Ostrict__boundedE,axiom,
% 13.25/7.63 ! [B: num,C: num,A: num] :
% 13.25/7.63 ( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_num @ B @ A )
% 13.25/7.63 => ~ ( ord_less_num @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3838_max_Ostrict__boundedE,axiom,
% 13.25/7.63 ! [B: nat,C: nat,A: nat] :
% 13.25/7.63 ( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_nat @ B @ A )
% 13.25/7.63 => ~ ( ord_less_nat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3839_max_Ostrict__boundedE,axiom,
% 13.25/7.63 ! [B: int,C: int,A: int] :
% 13.25/7.63 ( ( ord_less_int @ ( ord_max_int @ B @ C ) @ A )
% 13.25/7.63 => ~ ( ( ord_less_int @ B @ A )
% 13.25/7.63 => ~ ( ord_less_int @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3840_less__max__iff__disj,axiom,
% 13.25/7.63 ! [Z: extended_enat,X: extended_enat,Y3: extended_enat] :
% 13.25/7.63 ( ( ord_le72135733267957522d_enat @ Z @ ( ord_ma741700101516333627d_enat @ X @ Y3 ) )
% 13.25/7.63 = ( ( ord_le72135733267957522d_enat @ Z @ X )
% 13.25/7.63 | ( ord_le72135733267957522d_enat @ Z @ Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3841_less__max__iff__disj,axiom,
% 13.25/7.63 ! [Z: real,X: real,Y3: real] :
% 13.25/7.63 ( ( ord_less_real @ Z @ ( ord_max_real @ X @ Y3 ) )
% 13.25/7.63 = ( ( ord_less_real @ Z @ X )
% 13.25/7.63 | ( ord_less_real @ Z @ Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3842_less__max__iff__disj,axiom,
% 13.25/7.63 ! [Z: rat,X: rat,Y3: rat] :
% 13.25/7.63 ( ( ord_less_rat @ Z @ ( ord_max_rat @ X @ Y3 ) )
% 13.25/7.63 = ( ( ord_less_rat @ Z @ X )
% 13.25/7.63 | ( ord_less_rat @ Z @ Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3843_less__max__iff__disj,axiom,
% 13.25/7.63 ! [Z: num,X: num,Y3: num] :
% 13.25/7.63 ( ( ord_less_num @ Z @ ( ord_max_num @ X @ Y3 ) )
% 13.25/7.63 = ( ( ord_less_num @ Z @ X )
% 13.25/7.63 | ( ord_less_num @ Z @ Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3844_less__max__iff__disj,axiom,
% 13.25/7.63 ! [Z: nat,X: nat,Y3: nat] :
% 13.25/7.63 ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y3 ) )
% 13.25/7.63 = ( ( ord_less_nat @ Z @ X )
% 13.25/7.63 | ( ord_less_nat @ Z @ Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3845_less__max__iff__disj,axiom,
% 13.25/7.63 ! [Z: int,X: int,Y3: int] :
% 13.25/7.63 ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y3 ) )
% 13.25/7.63 = ( ( ord_less_int @ Z @ X )
% 13.25/7.63 | ( ord_less_int @ Z @ Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3846_max__add__distrib__left,axiom,
% 13.25/7.63 ! [X: real,Y3: real,Z: real] :
% 13.25/7.63 ( ( plus_plus_real @ ( ord_max_real @ X @ Y3 ) @ Z )
% 13.25/7.63 = ( ord_max_real @ ( plus_plus_real @ X @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3847_max__add__distrib__left,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.63 ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y3 ) @ Z )
% 13.25/7.63 = ( ord_max_rat @ ( plus_plus_rat @ X @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3848_max__add__distrib__left,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.63 ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y3 ) @ Z )
% 13.25/7.63 = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3849_max__add__distrib__left,axiom,
% 13.25/7.63 ! [X: int,Y3: int,Z: int] :
% 13.25/7.63 ( ( plus_plus_int @ ( ord_max_int @ X @ Y3 ) @ Z )
% 13.25/7.63 = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3850_max__add__distrib__right,axiom,
% 13.25/7.63 ! [X: real,Y3: real,Z: real] :
% 13.25/7.63 ( ( plus_plus_real @ X @ ( ord_max_real @ Y3 @ Z ) )
% 13.25/7.63 = ( ord_max_real @ ( plus_plus_real @ X @ Y3 ) @ ( plus_plus_real @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3851_max__add__distrib__right,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.63 ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y3 @ Z ) )
% 13.25/7.63 = ( ord_max_rat @ ( plus_plus_rat @ X @ Y3 ) @ ( plus_plus_rat @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3852_max__add__distrib__right,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.63 ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y3 @ Z ) )
% 13.25/7.63 = ( ord_max_nat @ ( plus_plus_nat @ X @ Y3 ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3853_max__add__distrib__right,axiom,
% 13.25/7.63 ! [X: int,Y3: int,Z: int] :
% 13.25/7.63 ( ( plus_plus_int @ X @ ( ord_max_int @ Y3 @ Z ) )
% 13.25/7.63 = ( ord_max_int @ ( plus_plus_int @ X @ Y3 ) @ ( plus_plus_int @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3854_max__diff__distrib__left,axiom,
% 13.25/7.63 ! [X: real,Y3: real,Z: real] :
% 13.25/7.63 ( ( minus_minus_real @ ( ord_max_real @ X @ Y3 ) @ Z )
% 13.25/7.63 = ( ord_max_real @ ( minus_minus_real @ X @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3855_max__diff__distrib__left,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.63 ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y3 ) @ Z )
% 13.25/7.63 = ( ord_max_rat @ ( minus_minus_rat @ X @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3856_max__diff__distrib__left,axiom,
% 13.25/7.63 ! [X: int,Y3: int,Z: int] :
% 13.25/7.63 ( ( minus_minus_int @ ( ord_max_int @ X @ Y3 ) @ Z )
% 13.25/7.63 = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3857_nat__add__max__left,axiom,
% 13.25/7.63 ! [M: nat,N: nat,Q2: nat] :
% 13.25/7.63 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 13.25/7.63 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3858_nat__add__max__right,axiom,
% 13.25/7.63 ! [M: nat,N: nat,Q2: nat] :
% 13.25/7.63 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 13.25/7.63 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3859_dvd__0__left,axiom,
% 13.25/7.63 ! [A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 13.25/7.63 => ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3860_dvd__0__left,axiom,
% 13.25/7.63 ! [A: complex] :
% 13.25/7.63 ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 13.25/7.63 => ( A = zero_zero_complex ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3861_dvd__0__left,axiom,
% 13.25/7.63 ! [A: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ zero_zero_real @ A )
% 13.25/7.63 => ( A = zero_zero_real ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3862_dvd__0__left,axiom,
% 13.25/7.63 ! [A: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 13.25/7.63 => ( A = zero_zero_rat ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3863_dvd__0__left,axiom,
% 13.25/7.63 ! [A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 13.25/7.63 => ( A = zero_zero_nat ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3864_dvd__0__left,axiom,
% 13.25/7.63 ! [A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ zero_zero_int @ A )
% 13.25/7.63 => ( A = zero_zero_int ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3865_nat__mult__max__left,axiom,
% 13.25/7.63 ! [M: nat,N: nat,Q2: nat] :
% 13.25/7.63 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
% 13.25/7.63 = ( ord_max_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3866_nat__mult__max__right,axiom,
% 13.25/7.63 ! [M: nat,N: nat,Q2: nat] :
% 13.25/7.63 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
% 13.25/7.63 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3867_dvdE,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.63 => ~ ! [K3: code_integer] :
% 13.25/7.63 ( A
% 13.25/7.63 != ( times_3573771949741848930nteger @ B @ K3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3868_dvdE,axiom,
% 13.25/7.63 ! [B: real,A: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ B @ A )
% 13.25/7.63 => ~ ! [K3: real] :
% 13.25/7.63 ( A
% 13.25/7.63 != ( times_times_real @ B @ K3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3869_dvdE,axiom,
% 13.25/7.63 ! [B: rat,A: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ B @ A )
% 13.25/7.63 => ~ ! [K3: rat] :
% 13.25/7.63 ( A
% 13.25/7.63 != ( times_times_rat @ B @ K3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3870_dvdE,axiom,
% 13.25/7.63 ! [B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ A )
% 13.25/7.63 => ~ ! [K3: nat] :
% 13.25/7.63 ( A
% 13.25/7.63 != ( times_times_nat @ B @ K3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3871_dvdE,axiom,
% 13.25/7.63 ! [B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ A )
% 13.25/7.63 => ~ ! [K3: int] :
% 13.25/7.63 ( A
% 13.25/7.63 != ( times_times_int @ B @ K3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3872_dvdI,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,K: code_integer] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( times_3573771949741848930nteger @ B @ K ) )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3873_dvdI,axiom,
% 13.25/7.63 ! [A: real,B: real,K: real] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( times_times_real @ B @ K ) )
% 13.25/7.63 => ( dvd_dvd_real @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3874_dvdI,axiom,
% 13.25/7.63 ! [A: rat,B: rat,K: rat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( times_times_rat @ B @ K ) )
% 13.25/7.63 => ( dvd_dvd_rat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3875_dvdI,axiom,
% 13.25/7.63 ! [A: nat,B: nat,K: nat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( times_times_nat @ B @ K ) )
% 13.25/7.63 => ( dvd_dvd_nat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3876_dvdI,axiom,
% 13.25/7.63 ! [A: int,B: int,K: int] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( times_times_int @ B @ K ) )
% 13.25/7.63 => ( dvd_dvd_int @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3877_dvd__def,axiom,
% 13.25/7.63 ( dvd_dvd_Code_integer
% 13.25/7.63 = ( ^ [B4: code_integer,A3: code_integer] :
% 13.25/7.63 ? [K2: code_integer] :
% 13.25/7.63 ( A3
% 13.25/7.63 = ( times_3573771949741848930nteger @ B4 @ K2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3878_dvd__def,axiom,
% 13.25/7.63 ( dvd_dvd_real
% 13.25/7.63 = ( ^ [B4: real,A3: real] :
% 13.25/7.63 ? [K2: real] :
% 13.25/7.63 ( A3
% 13.25/7.63 = ( times_times_real @ B4 @ K2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3879_dvd__def,axiom,
% 13.25/7.63 ( dvd_dvd_rat
% 13.25/7.63 = ( ^ [B4: rat,A3: rat] :
% 13.25/7.63 ? [K2: rat] :
% 13.25/7.63 ( A3
% 13.25/7.63 = ( times_times_rat @ B4 @ K2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3880_dvd__def,axiom,
% 13.25/7.63 ( dvd_dvd_nat
% 13.25/7.63 = ( ^ [B4: nat,A3: nat] :
% 13.25/7.63 ? [K2: nat] :
% 13.25/7.63 ( A3
% 13.25/7.63 = ( times_times_nat @ B4 @ K2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3881_dvd__def,axiom,
% 13.25/7.63 ( dvd_dvd_int
% 13.25/7.63 = ( ^ [B4: int,A3: int] :
% 13.25/7.63 ? [K2: int] :
% 13.25/7.63 ( A3
% 13.25/7.63 = ( times_times_int @ B4 @ K2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3882_dvd__mult,axiom,
% 13.25/7.63 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ C )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3883_dvd__mult,axiom,
% 13.25/7.63 ! [A: real,C: real,B: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ A @ C )
% 13.25/7.63 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3884_dvd__mult,axiom,
% 13.25/7.63 ! [A: rat,C: rat,B: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ A @ C )
% 13.25/7.63 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3885_dvd__mult,axiom,
% 13.25/7.63 ! [A: nat,C: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ C )
% 13.25/7.63 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3886_dvd__mult,axiom,
% 13.25/7.63 ! [A: int,C: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ C )
% 13.25/7.63 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3887_dvd__mult2,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3888_dvd__mult2,axiom,
% 13.25/7.63 ! [A: real,B: real,C: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ A @ B )
% 13.25/7.63 => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3889_dvd__mult2,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ A @ B )
% 13.25/7.63 => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3890_dvd__mult2,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.63 => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3891_dvd__mult2,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.63 => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3892_dvd__mult__left,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3893_dvd__mult__left,axiom,
% 13.25/7.63 ! [A: real,B: real,C: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_real @ A @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3894_dvd__mult__left,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_rat @ A @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3895_dvd__mult__left,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_nat @ A @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3896_dvd__mult__left,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_int @ A @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3897_dvd__triv__left,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3898_dvd__triv__left,axiom,
% 13.25/7.63 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3899_dvd__triv__left,axiom,
% 13.25/7.63 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3900_dvd__triv__left,axiom,
% 13.25/7.63 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3901_dvd__triv__left,axiom,
% 13.25/7.63 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3902_mult__dvd__mono,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ D )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3903_mult__dvd__mono,axiom,
% 13.25/7.63 ! [A: real,B: real,C: real,D: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_real @ C @ D )
% 13.25/7.63 => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3904_mult__dvd__mono,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_rat @ C @ D )
% 13.25/7.63 => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3905_mult__dvd__mono,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ D )
% 13.25/7.63 => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3906_mult__dvd__mono,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int,D: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ D )
% 13.25/7.63 => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3907_dvd__mult__right,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3908_dvd__mult__right,axiom,
% 13.25/7.63 ! [A: real,B: real,C: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_real @ B @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3909_dvd__mult__right,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_rat @ B @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3910_dvd__mult__right,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_nat @ B @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3911_dvd__mult__right,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 13.25/7.63 => ( dvd_dvd_int @ B @ C ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3912_dvd__triv__right,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3913_dvd__triv__right,axiom,
% 13.25/7.63 ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3914_dvd__triv__right,axiom,
% 13.25/7.63 ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3915_dvd__triv__right,axiom,
% 13.25/7.63 ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3916_dvd__triv__right,axiom,
% 13.25/7.63 ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3917_one__dvd,axiom,
% 13.25/7.63 ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3918_one__dvd,axiom,
% 13.25/7.63 ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3919_one__dvd,axiom,
% 13.25/7.63 ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3920_one__dvd,axiom,
% 13.25/7.63 ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3921_one__dvd,axiom,
% 13.25/7.63 ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3922_one__dvd,axiom,
% 13.25/7.63 ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3923_unit__imp__dvd,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3924_unit__imp__dvd,axiom,
% 13.25/7.63 ! [B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( dvd_dvd_nat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3925_unit__imp__dvd,axiom,
% 13.25/7.63 ! [B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( dvd_dvd_int @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3926_dvd__unit__imp__unit,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3927_dvd__unit__imp__unit,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3928_dvd__unit__imp__unit,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3929_dvd__add,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ C )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3930_dvd__add,axiom,
% 13.25/7.63 ! [A: real,B: real,C: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_real @ A @ C )
% 13.25/7.63 => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3931_dvd__add,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_rat @ A @ C )
% 13.25/7.63 => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3932_dvd__add,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ C )
% 13.25/7.63 => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3933_dvd__add,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ C )
% 13.25/7.63 => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3934_dvd__add__left__iff,axiom,
% 13.25/7.63 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ C )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3935_dvd__add__left__iff,axiom,
% 13.25/7.63 ! [A: real,C: real,B: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ A @ C )
% 13.25/7.63 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_real @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3936_dvd__add__left__iff,axiom,
% 13.25/7.63 ! [A: rat,C: rat,B: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ A @ C )
% 13.25/7.63 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3937_dvd__add__left__iff,axiom,
% 13.25/7.63 ! [A: nat,C: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ C )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3938_dvd__add__left__iff,axiom,
% 13.25/7.63 ! [A: int,C: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ C )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_int @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3939_dvd__add__right__iff,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3940_dvd__add__right__iff,axiom,
% 13.25/7.63 ! [A: real,B: real,C: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_real @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3941_dvd__add__right__iff,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3942_dvd__add__right__iff,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3943_dvd__add__right__iff,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3944_dvd__diff,axiom,
% 13.25/7.63 ! [X: code_integer,Y3: code_integer,Z: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ X @ Y3 )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ X @ Z )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ X @ ( minus_8373710615458151222nteger @ Y3 @ Z ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3945_dvd__diff,axiom,
% 13.25/7.63 ! [X: real,Y3: real,Z: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ X @ Y3 )
% 13.25/7.63 => ( ( dvd_dvd_real @ X @ Z )
% 13.25/7.63 => ( dvd_dvd_real @ X @ ( minus_minus_real @ Y3 @ Z ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3946_dvd__diff,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ X @ Y3 )
% 13.25/7.63 => ( ( dvd_dvd_rat @ X @ Z )
% 13.25/7.63 => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y3 @ Z ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3947_dvd__diff,axiom,
% 13.25/7.63 ! [X: int,Y3: int,Z: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ X @ Y3 )
% 13.25/7.63 => ( ( dvd_dvd_int @ X @ Z )
% 13.25/7.63 => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y3 @ Z ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3948_dvd__diff__commute,axiom,
% 13.25/7.63 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3949_dvd__diff__commute,axiom,
% 13.25/7.63 ! [A: int,C: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
% 13.25/7.63 = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3950_div__div__div__same,axiom,
% 13.25/7.63 ! [D: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ D @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.63 => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 13.25/7.63 = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3951_div__div__div__same,axiom,
% 13.25/7.63 ! [D: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ D @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ B @ A )
% 13.25/7.63 => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 13.25/7.63 = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3952_div__div__div__same,axiom,
% 13.25/7.63 ! [D: int,B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ D @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ B @ A )
% 13.25/7.63 => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 13.25/7.63 = ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3953_dvd__div__eq__cancel,axiom,
% 13.25/7.63 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.25/7.63 ( ( ( divide6298287555418463151nteger @ A @ C )
% 13.25/7.63 = ( divide6298287555418463151nteger @ B @ C ) )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3954_dvd__div__eq__cancel,axiom,
% 13.25/7.63 ! [A: complex,C: complex,B: complex] :
% 13.25/7.63 ( ( ( divide1717551699836669952omplex @ A @ C )
% 13.25/7.63 = ( divide1717551699836669952omplex @ B @ C ) )
% 13.25/7.63 => ( ( dvd_dvd_complex @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_complex @ C @ B )
% 13.25/7.63 => ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3955_dvd__div__eq__cancel,axiom,
% 13.25/7.63 ! [A: real,C: real,B: real] :
% 13.25/7.63 ( ( ( divide_divide_real @ A @ C )
% 13.25/7.63 = ( divide_divide_real @ B @ C ) )
% 13.25/7.63 => ( ( dvd_dvd_real @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_real @ C @ B )
% 13.25/7.63 => ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3956_dvd__div__eq__cancel,axiom,
% 13.25/7.63 ! [A: rat,C: rat,B: rat] :
% 13.25/7.63 ( ( ( divide_divide_rat @ A @ C )
% 13.25/7.63 = ( divide_divide_rat @ B @ C ) )
% 13.25/7.63 => ( ( dvd_dvd_rat @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_rat @ C @ B )
% 13.25/7.63 => ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3957_dvd__div__eq__cancel,axiom,
% 13.25/7.63 ! [A: nat,C: nat,B: nat] :
% 13.25/7.63 ( ( ( divide_divide_nat @ A @ C )
% 13.25/7.63 = ( divide_divide_nat @ B @ C ) )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3958_dvd__div__eq__cancel,axiom,
% 13.25/7.63 ! [A: int,C: int,B: int] :
% 13.25/7.63 ( ( ( divide_divide_int @ A @ C )
% 13.25/7.63 = ( divide_divide_int @ B @ C ) )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3959_dvd__div__eq__iff,axiom,
% 13.25/7.63 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( ( ( divide6298287555418463151nteger @ A @ C )
% 13.25/7.63 = ( divide6298287555418463151nteger @ B @ C ) )
% 13.25/7.63 = ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3960_dvd__div__eq__iff,axiom,
% 13.25/7.63 ! [C: complex,A: complex,B: complex] :
% 13.25/7.63 ( ( dvd_dvd_complex @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_complex @ C @ B )
% 13.25/7.63 => ( ( ( divide1717551699836669952omplex @ A @ C )
% 13.25/7.63 = ( divide1717551699836669952omplex @ B @ C ) )
% 13.25/7.63 = ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3961_dvd__div__eq__iff,axiom,
% 13.25/7.63 ! [C: real,A: real,B: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_real @ C @ B )
% 13.25/7.63 => ( ( ( divide_divide_real @ A @ C )
% 13.25/7.63 = ( divide_divide_real @ B @ C ) )
% 13.25/7.63 = ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3962_dvd__div__eq__iff,axiom,
% 13.25/7.63 ! [C: rat,A: rat,B: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_rat @ C @ B )
% 13.25/7.63 => ( ( ( divide_divide_rat @ A @ C )
% 13.25/7.63 = ( divide_divide_rat @ B @ C ) )
% 13.25/7.63 = ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3963_dvd__div__eq__iff,axiom,
% 13.25/7.63 ! [C: nat,A: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( ( ( divide_divide_nat @ A @ C )
% 13.25/7.63 = ( divide_divide_nat @ B @ C ) )
% 13.25/7.63 = ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3964_dvd__div__eq__iff,axiom,
% 13.25/7.63 ! [C: int,A: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ A )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( ( ( divide_divide_int @ A @ C )
% 13.25/7.63 = ( divide_divide_int @ B @ C ) )
% 13.25/7.63 = ( A = B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3965_dvd__power__same,axiom,
% 13.25/7.63 ! [X: code_integer,Y3: code_integer,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ X @ Y3 )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y3 @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3966_dvd__power__same,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ X @ Y3 )
% 13.25/7.63 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3967_dvd__power__same,axiom,
% 13.25/7.63 ! [X: real,Y3: real,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_real @ X @ Y3 )
% 13.25/7.63 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y3 @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3968_dvd__power__same,axiom,
% 13.25/7.63 ! [X: int,Y3: int,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_int @ X @ Y3 )
% 13.25/7.63 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3969_dvd__power__same,axiom,
% 13.25/7.63 ! [X: complex,Y3: complex,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_complex @ X @ Y3 )
% 13.25/7.63 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y3 @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3970_mod__mod__cancel,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
% 13.25/7.63 = ( modulo_modulo_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3971_mod__mod__cancel,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
% 13.25/7.63 = ( modulo_modulo_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3972_mod__mod__cancel,axiom,
% 13.25/7.63 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
% 13.25/7.63 = ( modulo364778990260209775nteger @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3973_dvd__mod,axiom,
% 13.25/7.63 ! [K: nat,M: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ K @ M )
% 13.25/7.63 => ( ( dvd_dvd_nat @ K @ N )
% 13.25/7.63 => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3974_dvd__mod,axiom,
% 13.25/7.63 ! [K: int,M: int,N: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ K @ M )
% 13.25/7.63 => ( ( dvd_dvd_int @ K @ N )
% 13.25/7.63 => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3975_dvd__mod,axiom,
% 13.25/7.63 ! [K: code_integer,M: code_integer,N: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ K @ M )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ K @ N )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3976_dvd__mod__imp__dvd,axiom,
% 13.25/7.63 ! [C: nat,A: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3977_dvd__mod__imp__dvd,axiom,
% 13.25/7.63 ! [C: int,A: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( dvd_dvd_int @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3978_dvd__mod__imp__dvd,axiom,
% 13.25/7.63 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3979_dvd__mod__iff,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 13.25/7.63 = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3980_dvd__mod__iff,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 13.25/7.63 = ( dvd_dvd_int @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3981_dvd__mod__iff,axiom,
% 13.25/7.63 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3982_dvd__diff__nat,axiom,
% 13.25/7.63 ! [K: nat,M: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ K @ M )
% 13.25/7.63 => ( ( dvd_dvd_nat @ K @ N )
% 13.25/7.63 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3983_zdvd__zdiffD,axiom,
% 13.25/7.63 ! [K: int,M: int,N: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
% 13.25/7.63 => ( ( dvd_dvd_int @ K @ N )
% 13.25/7.63 => ( dvd_dvd_int @ K @ M ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3984_dvd__pos__nat,axiom,
% 13.25/7.63 ! [N: nat,M: nat] :
% 13.25/7.63 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.63 => ( ( dvd_dvd_nat @ M @ N )
% 13.25/7.63 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3985_max__def__raw,axiom,
% 13.25/7.63 ( ord_ma741700101516333627d_enat
% 13.25/7.63 = ( ^ [A3: extended_enat,B4: extended_enat] : ( if_Extended_enat @ ( ord_le2932123472753598470d_enat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3986_max__def__raw,axiom,
% 13.25/7.63 ( ord_max_set_int
% 13.25/7.63 = ( ^ [A3: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3987_max__def__raw,axiom,
% 13.25/7.63 ( ord_max_rat
% 13.25/7.63 = ( ^ [A3: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3988_max__def__raw,axiom,
% 13.25/7.63 ( ord_max_num
% 13.25/7.63 = ( ^ [A3: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3989_max__def__raw,axiom,
% 13.25/7.63 ( ord_max_nat
% 13.25/7.63 = ( ^ [A3: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3990_max__def__raw,axiom,
% 13.25/7.63 ( ord_max_int
% 13.25/7.63 = ( ^ [A3: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A3 @ B4 ) @ B4 @ A3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3991_bezout__lemma__nat,axiom,
% 13.25/7.63 ! [D: nat,A: nat,B: nat,X: nat,Y3: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ D @ A )
% 13.25/7.63 => ( ( dvd_dvd_nat @ D @ B )
% 13.25/7.63 => ( ( ( ( times_times_nat @ A @ X )
% 13.25/7.63 = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
% 13.25/7.63 | ( ( times_times_nat @ B @ X )
% 13.25/7.63 = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 13.25/7.63 => ? [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ D @ A )
% 13.25/7.63 & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 13.25/7.63 & ( ( ( times_times_nat @ A @ X3 )
% 13.25/7.63 = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
% 13.25/7.63 | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
% 13.25/7.63 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3992_bezout__add__nat,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ? [D3: nat,X3: nat,Y4: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ D3 @ A )
% 13.25/7.63 & ( dvd_dvd_nat @ D3 @ B )
% 13.25/7.63 & ( ( ( times_times_nat @ A @ X3 )
% 13.25/7.63 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) )
% 13.25/7.63 | ( ( times_times_nat @ B @ X3 )
% 13.25/7.63 = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D3 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3993_bezout1__nat,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ? [D3: nat,X3: nat,Y4: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ D3 @ A )
% 13.25/7.63 & ( dvd_dvd_nat @ D3 @ B )
% 13.25/7.63 & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
% 13.25/7.63 = D3 )
% 13.25/7.63 | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 13.25/7.63 = D3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3994_subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: complex,B: complex] :
% 13.25/7.63 ( ( ord_le211207098394363844omplex
% 13.25/7.63 @ ( collect_complex
% 13.25/7.63 @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
% 13.25/7.63 @ ( collect_complex
% 13.25/7.63 @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
% 13.25/7.63 = ( dvd_dvd_complex @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3995_subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: real,B: real] :
% 13.25/7.63 ( ( ord_less_eq_set_real
% 13.25/7.63 @ ( collect_real
% 13.25/7.63 @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 13.25/7.63 @ ( collect_real
% 13.25/7.63 @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
% 13.25/7.63 = ( dvd_dvd_real @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3996_subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( ord_less_eq_set_nat
% 13.25/7.63 @ ( collect_nat
% 13.25/7.63 @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 13.25/7.63 @ ( collect_nat
% 13.25/7.63 @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3997_subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( ord_le7084787975880047091nteger
% 13.25/7.63 @ ( collect_Code_integer
% 13.25/7.63 @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 13.25/7.63 @ ( collect_Code_integer
% 13.25/7.63 @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3998_subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( ord_less_eq_set_int
% 13.25/7.63 @ ( collect_int
% 13.25/7.63 @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 13.25/7.63 @ ( collect_int
% 13.25/7.63 @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
% 13.25/7.63 = ( dvd_dvd_int @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_3999_concat__bit__assoc,axiom,
% 13.25/7.63 ! [N: nat,K: int,M: nat,L2: int,R: int] :
% 13.25/7.63 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L2 @ R ) )
% 13.25/7.63 = ( bit_concat_bit @ ( plus_plus_nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L2 ) @ R ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4000_strict__subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: complex,B: complex] :
% 13.25/7.63 ( ( ord_less_set_complex
% 13.25/7.63 @ ( collect_complex
% 13.25/7.63 @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
% 13.25/7.63 @ ( collect_complex
% 13.25/7.63 @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
% 13.25/7.63 = ( ( dvd_dvd_complex @ A @ B )
% 13.25/7.63 & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4001_strict__subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: real,B: real] :
% 13.25/7.63 ( ( ord_less_set_real
% 13.25/7.63 @ ( collect_real
% 13.25/7.63 @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 13.25/7.63 @ ( collect_real
% 13.25/7.63 @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
% 13.25/7.63 = ( ( dvd_dvd_real @ A @ B )
% 13.25/7.63 & ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4002_strict__subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( ord_less_set_nat
% 13.25/7.63 @ ( collect_nat
% 13.25/7.63 @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 13.25/7.63 @ ( collect_nat
% 13.25/7.63 @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
% 13.25/7.63 = ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.63 & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4003_strict__subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( ord_less_set_int
% 13.25/7.63 @ ( collect_int
% 13.25/7.63 @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 13.25/7.63 @ ( collect_int
% 13.25/7.63 @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
% 13.25/7.63 = ( ( dvd_dvd_int @ A @ B )
% 13.25/7.63 & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4004_strict__subset__divisors__dvd,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( ord_le1307284697595431911nteger
% 13.25/7.63 @ ( collect_Code_integer
% 13.25/7.63 @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 13.25/7.63 @ ( collect_Code_integer
% 13.25/7.63 @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
% 13.25/7.63 = ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.63 & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4005_nat__minus__add__max,axiom,
% 13.25/7.63 ! [N: nat,M: nat] :
% 13.25/7.63 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 13.25/7.63 = ( ord_max_nat @ N @ M ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4006_not__is__unit__0,axiom,
% 13.25/7.63 ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4007_not__is__unit__0,axiom,
% 13.25/7.63 ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4008_not__is__unit__0,axiom,
% 13.25/7.63 ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4009_minf_I10_J,axiom,
% 13.25/7.63 ! [D: code_integer,S: code_integer] :
% 13.25/7.63 ? [Z5: code_integer] :
% 13.25/7.63 ! [X5: code_integer] :
% 13.25/7.63 ( ( ord_le6747313008572928689nteger @ X5 @ Z5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4010_minf_I10_J,axiom,
% 13.25/7.63 ! [D: real,S: real] :
% 13.25/7.63 ? [Z5: real] :
% 13.25/7.63 ! [X5: real] :
% 13.25/7.63 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4011_minf_I10_J,axiom,
% 13.25/7.63 ! [D: rat,S: rat] :
% 13.25/7.63 ? [Z5: rat] :
% 13.25/7.63 ! [X5: rat] :
% 13.25/7.63 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4012_minf_I10_J,axiom,
% 13.25/7.63 ! [D: nat,S: nat] :
% 13.25/7.63 ? [Z5: nat] :
% 13.25/7.63 ! [X5: nat] :
% 13.25/7.63 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4013_minf_I10_J,axiom,
% 13.25/7.63 ! [D: int,S: int] :
% 13.25/7.63 ? [Z5: int] :
% 13.25/7.63 ! [X5: int] :
% 13.25/7.63 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4014_minf_I9_J,axiom,
% 13.25/7.63 ! [D: code_integer,S: code_integer] :
% 13.25/7.63 ? [Z5: code_integer] :
% 13.25/7.63 ! [X5: code_integer] :
% 13.25/7.63 ( ( ord_le6747313008572928689nteger @ X5 @ Z5 )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4015_minf_I9_J,axiom,
% 13.25/7.63 ! [D: real,S: real] :
% 13.25/7.63 ? [Z5: real] :
% 13.25/7.63 ! [X5: real] :
% 13.25/7.63 ( ( ord_less_real @ X5 @ Z5 )
% 13.25/7.63 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4016_minf_I9_J,axiom,
% 13.25/7.63 ! [D: rat,S: rat] :
% 13.25/7.63 ? [Z5: rat] :
% 13.25/7.63 ! [X5: rat] :
% 13.25/7.63 ( ( ord_less_rat @ X5 @ Z5 )
% 13.25/7.63 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4017_minf_I9_J,axiom,
% 13.25/7.63 ! [D: nat,S: nat] :
% 13.25/7.63 ? [Z5: nat] :
% 13.25/7.63 ! [X5: nat] :
% 13.25/7.63 ( ( ord_less_nat @ X5 @ Z5 )
% 13.25/7.63 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4018_minf_I9_J,axiom,
% 13.25/7.63 ! [D: int,S: int] :
% 13.25/7.63 ? [Z5: int] :
% 13.25/7.63 ! [X5: int] :
% 13.25/7.63 ( ( ord_less_int @ X5 @ Z5 )
% 13.25/7.63 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4019_pinf_I10_J,axiom,
% 13.25/7.63 ! [D: code_integer,S: code_integer] :
% 13.25/7.63 ? [Z5: code_integer] :
% 13.25/7.63 ! [X5: code_integer] :
% 13.25/7.63 ( ( ord_le6747313008572928689nteger @ Z5 @ X5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4020_pinf_I10_J,axiom,
% 13.25/7.63 ! [D: real,S: real] :
% 13.25/7.63 ? [Z5: real] :
% 13.25/7.63 ! [X5: real] :
% 13.25/7.63 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4021_pinf_I10_J,axiom,
% 13.25/7.63 ! [D: rat,S: rat] :
% 13.25/7.63 ? [Z5: rat] :
% 13.25/7.63 ! [X5: rat] :
% 13.25/7.63 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4022_pinf_I10_J,axiom,
% 13.25/7.63 ! [D: nat,S: nat] :
% 13.25/7.63 ? [Z5: nat] :
% 13.25/7.63 ! [X5: nat] :
% 13.25/7.63 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4023_pinf_I10_J,axiom,
% 13.25/7.63 ! [D: int,S: int] :
% 13.25/7.63 ? [Z5: int] :
% 13.25/7.63 ! [X5: int] :
% 13.25/7.63 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.63 => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4024_pinf_I9_J,axiom,
% 13.25/7.63 ! [D: code_integer,S: code_integer] :
% 13.25/7.63 ? [Z5: code_integer] :
% 13.25/7.63 ! [X5: code_integer] :
% 13.25/7.63 ( ( ord_le6747313008572928689nteger @ Z5 @ X5 )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4025_pinf_I9_J,axiom,
% 13.25/7.63 ! [D: real,S: real] :
% 13.25/7.63 ? [Z5: real] :
% 13.25/7.63 ! [X5: real] :
% 13.25/7.63 ( ( ord_less_real @ Z5 @ X5 )
% 13.25/7.63 => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4026_pinf_I9_J,axiom,
% 13.25/7.63 ! [D: rat,S: rat] :
% 13.25/7.63 ? [Z5: rat] :
% 13.25/7.63 ! [X5: rat] :
% 13.25/7.63 ( ( ord_less_rat @ Z5 @ X5 )
% 13.25/7.63 => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4027_pinf_I9_J,axiom,
% 13.25/7.63 ! [D: nat,S: nat] :
% 13.25/7.63 ? [Z5: nat] :
% 13.25/7.63 ! [X5: nat] :
% 13.25/7.63 ( ( ord_less_nat @ Z5 @ X5 )
% 13.25/7.63 => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4028_pinf_I9_J,axiom,
% 13.25/7.63 ! [D: int,S: int] :
% 13.25/7.63 ? [Z5: int] :
% 13.25/7.63 ! [X5: int] :
% 13.25/7.63 ( ( ord_less_int @ Z5 @ X5 )
% 13.25/7.63 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) )
% 13.25/7.63 = ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ S ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4029_dvd__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.63 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 13.25/7.63 = zero_z3403309356797280102nteger )
% 13.25/7.63 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4030_dvd__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: complex,A: complex] :
% 13.25/7.63 ( ( dvd_dvd_complex @ B @ A )
% 13.25/7.63 => ( ( ( divide1717551699836669952omplex @ A @ B )
% 13.25/7.63 = zero_zero_complex )
% 13.25/7.63 = ( A = zero_zero_complex ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4031_dvd__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: real,A: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ B @ A )
% 13.25/7.63 => ( ( ( divide_divide_real @ A @ B )
% 13.25/7.63 = zero_zero_real )
% 13.25/7.63 = ( A = zero_zero_real ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4032_dvd__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: rat,A: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ B @ A )
% 13.25/7.63 => ( ( ( divide_divide_rat @ A @ B )
% 13.25/7.63 = zero_zero_rat )
% 13.25/7.63 = ( A = zero_zero_rat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4033_dvd__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ A )
% 13.25/7.63 => ( ( ( divide_divide_nat @ A @ B )
% 13.25/7.63 = zero_zero_nat )
% 13.25/7.63 = ( A = zero_zero_nat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4034_dvd__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ A )
% 13.25/7.63 => ( ( ( divide_divide_int @ A @ B )
% 13.25/7.63 = zero_zero_int )
% 13.25/7.63 = ( A = zero_zero_int ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4035_unit__mult__right__cancel,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.63 => ( ( ( times_3573771949741848930nteger @ B @ A )
% 13.25/7.63 = ( times_3573771949741848930nteger @ C @ A ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4036_unit__mult__right__cancel,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.63 => ( ( ( times_times_nat @ B @ A )
% 13.25/7.63 = ( times_times_nat @ C @ A ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4037_unit__mult__right__cancel,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.63 => ( ( ( times_times_int @ B @ A )
% 13.25/7.63 = ( times_times_int @ C @ A ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4038_unit__mult__left__cancel,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.63 => ( ( ( times_3573771949741848930nteger @ A @ B )
% 13.25/7.63 = ( times_3573771949741848930nteger @ A @ C ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4039_unit__mult__left__cancel,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.63 => ( ( ( times_times_nat @ A @ B )
% 13.25/7.63 = ( times_times_nat @ A @ C ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4040_unit__mult__left__cancel,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.63 => ( ( ( times_times_int @ A @ B )
% 13.25/7.63 = ( times_times_int @ A @ C ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4041_mult__unit__dvd__iff_H,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4042_mult__unit__dvd__iff_H,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4043_mult__unit__dvd__iff_H,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_int @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4044_dvd__mult__unit__iff_H,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4045_dvd__mult__unit__iff_H,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4046_dvd__mult__unit__iff_H,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4047_mult__unit__dvd__iff,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4048_mult__unit__dvd__iff,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4049_mult__unit__dvd__iff,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4050_dvd__mult__unit__iff,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4051_dvd__mult__unit__iff,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4052_dvd__mult__unit__iff,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 13.25/7.63 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4053_is__unit__mult__iff,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 13.25/7.63 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.63 & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4054_is__unit__mult__iff,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 13.25/7.63 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.63 & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4055_is__unit__mult__iff,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 13.25/7.63 = ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.63 & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4056_dvd__div__mult,axiom,
% 13.25/7.63 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4057_dvd__div__mult,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 13.25/7.63 = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4058_dvd__div__mult,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 13.25/7.63 = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4059_div__mult__swap,axiom,
% 13.25/7.63 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4060_div__mult__swap,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 13.25/7.63 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4061_div__mult__swap,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 13.25/7.63 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4062_div__div__eq__right,axiom,
% 13.25/7.63 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.63 => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 13.25/7.63 = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4063_div__div__eq__right,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ B @ A )
% 13.25/7.63 => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 13.25/7.63 = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4064_div__div__eq__right,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ B @ A )
% 13.25/7.63 => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 13.25/7.63 = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4065_dvd__div__mult2__eq,axiom,
% 13.25/7.63 ! [B: code_integer,C: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 13.25/7.63 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4066_dvd__div__mult2__eq,axiom,
% 13.25/7.63 ! [B: nat,C: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 13.25/7.63 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 13.25/7.63 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4067_dvd__div__mult2__eq,axiom,
% 13.25/7.63 ! [B: int,C: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 13.25/7.63 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 13.25/7.63 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4068_dvd__mult__imp__div,axiom,
% 13.25/7.63 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4069_dvd__mult__imp__div,axiom,
% 13.25/7.63 ! [A: nat,C: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 13.25/7.63 => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4070_dvd__mult__imp__div,axiom,
% 13.25/7.63 ! [A: int,C: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 13.25/7.63 => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4071_div__mult__div__if__dvd,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ D @ C )
% 13.25/7.63 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4072_div__mult__div__if__dvd,axiom,
% 13.25/7.63 ! [B: nat,A: nat,D: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ A )
% 13.25/7.63 => ( ( dvd_dvd_nat @ D @ C )
% 13.25/7.63 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 13.25/7.63 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4073_div__mult__div__if__dvd,axiom,
% 13.25/7.63 ! [B: int,A: int,D: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ A )
% 13.25/7.63 => ( ( dvd_dvd_int @ D @ C )
% 13.25/7.63 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 13.25/7.63 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4074_dvd__div__unit__iff,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4075_dvd__div__unit__iff,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4076_dvd__div__unit__iff,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 13.25/7.63 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4077_div__unit__dvd__iff,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4078_div__unit__dvd__iff,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4079_div__unit__dvd__iff,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4080_unit__div__cancel,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.63 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 13.25/7.63 = ( divide6298287555418463151nteger @ C @ A ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4081_unit__div__cancel,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.63 => ( ( ( divide_divide_nat @ B @ A )
% 13.25/7.63 = ( divide_divide_nat @ C @ A ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4082_unit__div__cancel,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.63 => ( ( ( divide_divide_int @ B @ A )
% 13.25/7.63 = ( divide_divide_int @ C @ A ) )
% 13.25/7.63 = ( B = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4083_div__plus__div__distrib__dvd__left,axiom,
% 13.25/7.63 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ A )
% 13.25/7.63 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 13.25/7.63 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4084_div__plus__div__distrib__dvd__left,axiom,
% 13.25/7.63 ! [C: nat,A: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ A )
% 13.25/7.63 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.25/7.63 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4085_div__plus__div__distrib__dvd__left,axiom,
% 13.25/7.63 ! [C: int,A: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ A )
% 13.25/7.63 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.25/7.63 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4086_div__plus__div__distrib__dvd__right,axiom,
% 13.25/7.63 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 13.25/7.63 = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4087_div__plus__div__distrib__dvd__right,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 13.25/7.63 = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4088_div__plus__div__distrib__dvd__right,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 13.25/7.63 = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4089_div__power,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.63 => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4090_div__power,axiom,
% 13.25/7.63 ! [B: nat,A: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ A )
% 13.25/7.63 => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
% 13.25/7.63 = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4091_div__power,axiom,
% 13.25/7.63 ! [B: int,A: int,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ A )
% 13.25/7.63 => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
% 13.25/7.63 = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4092_mod__eq__0__iff__dvd,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( ( modulo_modulo_nat @ A @ B )
% 13.25/7.63 = zero_zero_nat )
% 13.25/7.63 = ( dvd_dvd_nat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4093_mod__eq__0__iff__dvd,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.63 = zero_zero_int )
% 13.25/7.63 = ( dvd_dvd_int @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4094_mod__eq__0__iff__dvd,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( ( modulo364778990260209775nteger @ A @ B )
% 13.25/7.63 = zero_z3403309356797280102nteger )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4095_dvd__eq__mod__eq__0,axiom,
% 13.25/7.63 ( dvd_dvd_nat
% 13.25/7.63 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.63 ( ( modulo_modulo_nat @ B4 @ A3 )
% 13.25/7.63 = zero_zero_nat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4096_dvd__eq__mod__eq__0,axiom,
% 13.25/7.63 ( dvd_dvd_int
% 13.25/7.63 = ( ^ [A3: int,B4: int] :
% 13.25/7.63 ( ( modulo_modulo_int @ B4 @ A3 )
% 13.25/7.63 = zero_zero_int ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4097_dvd__eq__mod__eq__0,axiom,
% 13.25/7.63 ( dvd_dvd_Code_integer
% 13.25/7.63 = ( ^ [A3: code_integer,B4: code_integer] :
% 13.25/7.63 ( ( modulo364778990260209775nteger @ B4 @ A3 )
% 13.25/7.63 = zero_z3403309356797280102nteger ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4098_mod__0__imp__dvd,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( ( modulo_modulo_nat @ A @ B )
% 13.25/7.63 = zero_zero_nat )
% 13.25/7.63 => ( dvd_dvd_nat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4099_mod__0__imp__dvd,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.63 = zero_zero_int )
% 13.25/7.63 => ( dvd_dvd_int @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4100_mod__0__imp__dvd,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( ( modulo364778990260209775nteger @ A @ B )
% 13.25/7.63 = zero_z3403309356797280102nteger )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4101_dvd__power__le,axiom,
% 13.25/7.63 ! [X: code_integer,Y3: code_integer,N: nat,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4102_dvd__power__le,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat,N: nat,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.63 => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4103_dvd__power__le,axiom,
% 13.25/7.63 ! [X: real,Y3: real,N: nat,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_real @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.63 => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4104_dvd__power__le,axiom,
% 13.25/7.63 ! [X: int,Y3: int,N: nat,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_int @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.63 => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4105_dvd__power__le,axiom,
% 13.25/7.63 ! [X: complex,Y3: complex,N: nat,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_complex @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.63 => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4106_power__le__dvd,axiom,
% 13.25/7.63 ! [A: code_integer,N: nat,B: code_integer,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4107_power__le__dvd,axiom,
% 13.25/7.63 ! [A: nat,N: nat,B: nat,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4108_power__le__dvd,axiom,
% 13.25/7.63 ! [A: real,N: nat,B: real,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4109_power__le__dvd,axiom,
% 13.25/7.63 ! [A: int,N: nat,B: int,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4110_power__le__dvd,axiom,
% 13.25/7.63 ! [A: complex,N: nat,B: complex,M: nat] :
% 13.25/7.63 ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4111_le__imp__power__dvd,axiom,
% 13.25/7.63 ! [M: nat,N: nat,A: code_integer] :
% 13.25/7.63 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4112_le__imp__power__dvd,axiom,
% 13.25/7.63 ! [M: nat,N: nat,A: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4113_le__imp__power__dvd,axiom,
% 13.25/7.63 ! [M: nat,N: nat,A: real] :
% 13.25/7.63 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4114_le__imp__power__dvd,axiom,
% 13.25/7.63 ! [M: nat,N: nat,A: int] :
% 13.25/7.63 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4115_le__imp__power__dvd,axiom,
% 13.25/7.63 ! [M: nat,N: nat,A: complex] :
% 13.25/7.63 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4116_dvd__minus__mod,axiom,
% 13.25/7.63 ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4117_dvd__minus__mod,axiom,
% 13.25/7.63 ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4118_dvd__minus__mod,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4119_mod__eq__dvd__iff,axiom,
% 13.25/7.63 ! [A: int,C: int,B: int] :
% 13.25/7.63 ( ( ( modulo_modulo_int @ A @ C )
% 13.25/7.63 = ( modulo_modulo_int @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4120_mod__eq__dvd__iff,axiom,
% 13.25/7.63 ! [A: code_integer,C: code_integer,B: code_integer] :
% 13.25/7.63 ( ( ( modulo364778990260209775nteger @ A @ C )
% 13.25/7.63 = ( modulo364778990260209775nteger @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4121_nat__dvd__not__less,axiom,
% 13.25/7.63 ! [M: nat,N: nat] :
% 13.25/7.63 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.63 => ( ( ord_less_nat @ M @ N )
% 13.25/7.63 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4122_bezout__add__strong__nat,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( A != zero_zero_nat )
% 13.25/7.63 => ? [D3: nat,X3: nat,Y4: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ D3 @ A )
% 13.25/7.63 & ( dvd_dvd_nat @ D3 @ B )
% 13.25/7.63 & ( ( times_times_nat @ A @ X3 )
% 13.25/7.63 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D3 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4123_set__update__subsetI,axiom,
% 13.25/7.63 ! [Xs2: list_nat,A2: set_nat,X: nat,I: nat] :
% 13.25/7.63 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 13.25/7.63 => ( ( member_nat @ X @ A2 )
% 13.25/7.63 => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4124_set__update__subsetI,axiom,
% 13.25/7.63 ! [Xs2: list_real,A2: set_real,X: real,I: nat] :
% 13.25/7.63 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ A2 )
% 13.25/7.63 => ( ( member_real @ X @ A2 )
% 13.25/7.63 => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4125_set__update__subsetI,axiom,
% 13.25/7.63 ! [Xs2: list_complex,A2: set_complex,X: complex,I: nat] :
% 13.25/7.63 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 13.25/7.63 => ( ( member_complex @ X @ A2 )
% 13.25/7.63 => ( ord_le211207098394363844omplex @ ( set_complex2 @ ( list_update_complex @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4126_set__update__subsetI,axiom,
% 13.25/7.63 ! [Xs2: list_P8469869581646625389at_nat,A2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat,I: nat] :
% 13.25/7.63 ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs2 ) @ A2 )
% 13.25/7.63 => ( ( member8206827879206165904at_nat @ X @ A2 )
% 13.25/7.63 => ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4127_set__update__subsetI,axiom,
% 13.25/7.63 ! [Xs2: list_VEBT_VEBT,A2: set_VEBT_VEBT,X: vEBT_VEBT,I: nat] :
% 13.25/7.63 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 13.25/7.63 => ( ( member_VEBT_VEBT @ X @ A2 )
% 13.25/7.63 => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4128_set__update__subsetI,axiom,
% 13.25/7.63 ! [Xs2: list_int,A2: set_int,X: int,I: nat] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 13.25/7.63 => ( ( member_int @ X @ A2 )
% 13.25/7.63 => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4129_dvd__minus__self,axiom,
% 13.25/7.63 ! [M: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 13.25/7.63 = ( ( ord_less_nat @ N @ M )
% 13.25/7.63 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4130_zdvd__antisym__nonneg,axiom,
% 13.25/7.63 ! [M: int,N: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 13.25/7.63 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 13.25/7.63 => ( ( dvd_dvd_int @ M @ N )
% 13.25/7.63 => ( ( dvd_dvd_int @ N @ M )
% 13.25/7.63 => ( M = N ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4131_less__eq__dvd__minus,axiom,
% 13.25/7.63 ! [M: nat,N: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.63 => ( ( dvd_dvd_nat @ M @ N )
% 13.25/7.63 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4132_dvd__diffD1,axiom,
% 13.25/7.63 ! [K: nat,M: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.63 => ( ( dvd_dvd_nat @ K @ M )
% 13.25/7.63 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.63 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4133_dvd__diffD,axiom,
% 13.25/7.63 ! [K: nat,M: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.63 => ( ( dvd_dvd_nat @ K @ N )
% 13.25/7.63 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.63 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4134_zdvd__not__zless,axiom,
% 13.25/7.63 ! [M: int,N: int] :
% 13.25/7.63 ( ( ord_less_int @ zero_zero_int @ M )
% 13.25/7.63 => ( ( ord_less_int @ M @ N )
% 13.25/7.63 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4135_zdvd__mult__cancel,axiom,
% 13.25/7.63 ! [K: int,M: int,N: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
% 13.25/7.63 => ( ( K != zero_zero_int )
% 13.25/7.63 => ( dvd_dvd_int @ M @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4136_zdvd__mono,axiom,
% 13.25/7.63 ! [K: int,M: int,T: int] :
% 13.25/7.63 ( ( K != zero_zero_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ M @ T )
% 13.25/7.63 = ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ T ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4137_zdvd__reduce,axiom,
% 13.25/7.63 ! [K: int,N: int,M: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N @ ( times_times_int @ K @ M ) ) )
% 13.25/7.63 = ( dvd_dvd_int @ K @ N ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4138_zdvd__period,axiom,
% 13.25/7.63 ! [A: int,D: int,X: int,T: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ D )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
% 13.25/7.63 = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4139_finite__divisors__nat,axiom,
% 13.25/7.63 ! [M: nat] :
% 13.25/7.63 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.63 => ( finite_finite_nat
% 13.25/7.63 @ ( collect_nat
% 13.25/7.63 @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4140_div2__even__ext__nat,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat] :
% 13.25/7.63 ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.63 = ( divide_divide_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.63 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
% 13.25/7.63 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y3 ) )
% 13.25/7.63 => ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4141_unit__dvdE,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.63 => ~ ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.63 => ! [C2: code_integer] :
% 13.25/7.63 ( B
% 13.25/7.63 != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4142_unit__dvdE,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.63 => ~ ( ( A != zero_zero_nat )
% 13.25/7.63 => ! [C2: nat] :
% 13.25/7.63 ( B
% 13.25/7.63 != ( times_times_nat @ A @ C2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4143_unit__dvdE,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.63 => ~ ( ( A != zero_zero_int )
% 13.25/7.63 => ! [C2: int] :
% 13.25/7.63 ( B
% 13.25/7.63 != ( times_times_int @ A @ C2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4144_unity__coeff__ex,axiom,
% 13.25/7.63 ! [P: code_integer > $o,L2: code_integer] :
% 13.25/7.63 ( ( ? [X2: code_integer] : ( P @ ( times_3573771949741848930nteger @ L2 @ X2 ) ) )
% 13.25/7.63 = ( ? [X2: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ L2 @ ( plus_p5714425477246183910nteger @ X2 @ zero_z3403309356797280102nteger ) )
% 13.25/7.63 & ( P @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4145_unity__coeff__ex,axiom,
% 13.25/7.63 ! [P: complex > $o,L2: complex] :
% 13.25/7.63 ( ( ? [X2: complex] : ( P @ ( times_times_complex @ L2 @ X2 ) ) )
% 13.25/7.63 = ( ? [X2: complex] :
% 13.25/7.63 ( ( dvd_dvd_complex @ L2 @ ( plus_plus_complex @ X2 @ zero_zero_complex ) )
% 13.25/7.63 & ( P @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4146_unity__coeff__ex,axiom,
% 13.25/7.63 ! [P: real > $o,L2: real] :
% 13.25/7.63 ( ( ? [X2: real] : ( P @ ( times_times_real @ L2 @ X2 ) ) )
% 13.25/7.63 = ( ? [X2: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ L2 @ ( plus_plus_real @ X2 @ zero_zero_real ) )
% 13.25/7.63 & ( P @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4147_unity__coeff__ex,axiom,
% 13.25/7.63 ! [P: rat > $o,L2: rat] :
% 13.25/7.63 ( ( ? [X2: rat] : ( P @ ( times_times_rat @ L2 @ X2 ) ) )
% 13.25/7.63 = ( ? [X2: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ L2 @ ( plus_plus_rat @ X2 @ zero_zero_rat ) )
% 13.25/7.63 & ( P @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4148_unity__coeff__ex,axiom,
% 13.25/7.63 ! [P: nat > $o,L2: nat] :
% 13.25/7.63 ( ( ? [X2: nat] : ( P @ ( times_times_nat @ L2 @ X2 ) ) )
% 13.25/7.63 = ( ? [X2: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ L2 @ ( plus_plus_nat @ X2 @ zero_zero_nat ) )
% 13.25/7.63 & ( P @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4149_unity__coeff__ex,axiom,
% 13.25/7.63 ! [P: int > $o,L2: int] :
% 13.25/7.63 ( ( ? [X2: int] : ( P @ ( times_times_int @ L2 @ X2 ) ) )
% 13.25/7.63 = ( ? [X2: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ L2 @ ( plus_plus_int @ X2 @ zero_zero_int ) )
% 13.25/7.63 & ( P @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4150_dvd__div__eq__mult,axiom,
% 13.25/7.63 ! [A: code_integer,B: code_integer,C: code_integer] :
% 13.25/7.63 ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.63 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 13.25/7.63 = C )
% 13.25/7.63 = ( B
% 13.25/7.63 = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4151_dvd__div__eq__mult,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( A != zero_zero_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.63 => ( ( ( divide_divide_nat @ B @ A )
% 13.25/7.63 = C )
% 13.25/7.63 = ( B
% 13.25/7.63 = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4152_dvd__div__eq__mult,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( A != zero_zero_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ B )
% 13.25/7.63 => ( ( ( divide_divide_int @ B @ A )
% 13.25/7.63 = C )
% 13.25/7.63 = ( B
% 13.25/7.63 = ( times_times_int @ C @ A ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4153_div__dvd__iff__mult,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( B != zero_z3403309356797280102nteger )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4154_div__dvd__iff__mult,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( B != zero_zero_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ B @ A )
% 13.25/7.63 => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4155_div__dvd__iff__mult,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( B != zero_zero_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ B @ A )
% 13.25/7.63 => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 13.25/7.63 = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4156_dvd__div__iff__mult,axiom,
% 13.25/7.63 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( C != zero_z3403309356797280102nteger )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4157_dvd__div__iff__mult,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( C != zero_zero_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4158_dvd__div__iff__mult,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( C != zero_zero_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 13.25/7.63 = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4159_dvd__div__div__eq__mult,axiom,
% 13.25/7.63 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 13.25/7.63 ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.63 => ( ( C != zero_z3403309356797280102nteger )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ D )
% 13.25/7.63 => ( ( ( divide6298287555418463151nteger @ B @ A )
% 13.25/7.63 = ( divide6298287555418463151nteger @ D @ C ) )
% 13.25/7.63 = ( ( times_3573771949741848930nteger @ B @ C )
% 13.25/7.63 = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4160_dvd__div__div__eq__mult,axiom,
% 13.25/7.63 ! [A: nat,C: nat,B: nat,D: nat] :
% 13.25/7.63 ( ( A != zero_zero_nat )
% 13.25/7.63 => ( ( C != zero_zero_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ D )
% 13.25/7.63 => ( ( ( divide_divide_nat @ B @ A )
% 13.25/7.63 = ( divide_divide_nat @ D @ C ) )
% 13.25/7.63 = ( ( times_times_nat @ B @ C )
% 13.25/7.63 = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4161_dvd__div__div__eq__mult,axiom,
% 13.25/7.63 ! [A: int,C: int,B: int,D: int] :
% 13.25/7.63 ( ( A != zero_zero_int )
% 13.25/7.63 => ( ( C != zero_zero_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ A @ B )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ D )
% 13.25/7.63 => ( ( ( divide_divide_int @ B @ A )
% 13.25/7.63 = ( divide_divide_int @ D @ C ) )
% 13.25/7.63 = ( ( times_times_int @ B @ C )
% 13.25/7.63 = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4162_unit__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 13.25/7.63 = zero_z3403309356797280102nteger )
% 13.25/7.63 = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4163_unit__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( ( divide_divide_nat @ A @ B )
% 13.25/7.63 = zero_zero_nat )
% 13.25/7.63 = ( A = zero_zero_nat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4164_unit__div__eq__0__iff,axiom,
% 13.25/7.63 ! [B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( ( divide_divide_int @ A @ B )
% 13.25/7.63 = zero_zero_int )
% 13.25/7.63 = ( A = zero_zero_int ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4165_even__numeral,axiom,
% 13.25/7.63 ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4166_even__numeral,axiom,
% 13.25/7.63 ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4167_even__numeral,axiom,
% 13.25/7.63 ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4168_inf__period_I4_J,axiom,
% 13.25/7.63 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 13.25/7.63 => ! [X5: code_integer,K4: code_integer] :
% 13.25/7.63 ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4169_inf__period_I4_J,axiom,
% 13.25/7.63 ! [D: real,D4: real,T: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ D @ D4 )
% 13.25/7.63 => ! [X5: real,K4: real] :
% 13.25/7.63 ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4170_inf__period_I4_J,axiom,
% 13.25/7.63 ! [D: rat,D4: rat,T: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ D @ D4 )
% 13.25/7.63 => ! [X5: rat,K4: rat] :
% 13.25/7.63 ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4171_inf__period_I4_J,axiom,
% 13.25/7.63 ! [D: int,D4: int,T: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ D @ D4 )
% 13.25/7.63 => ! [X5: int,K4: int] :
% 13.25/7.63 ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4172_inf__period_I3_J,axiom,
% 13.25/7.63 ! [D: code_integer,D4: code_integer,T: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ D @ D4 )
% 13.25/7.63 => ! [X5: code_integer,K4: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X5 @ T ) )
% 13.25/7.63 = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X5 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4173_inf__period_I3_J,axiom,
% 13.25/7.63 ! [D: real,D4: real,T: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ D @ D4 )
% 13.25/7.63 => ! [X5: real,K4: real] :
% 13.25/7.63 ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
% 13.25/7.63 = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4174_inf__period_I3_J,axiom,
% 13.25/7.63 ! [D: rat,D4: rat,T: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ D @ D4 )
% 13.25/7.63 => ! [X5: rat,K4: rat] :
% 13.25/7.63 ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
% 13.25/7.63 = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4175_inf__period_I3_J,axiom,
% 13.25/7.63 ! [D: int,D4: int,T: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ D @ D4 )
% 13.25/7.63 => ! [X5: int,K4: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 13.25/7.63 = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D4 ) ) @ T ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4176_is__unit__div__mult2__eq,axiom,
% 13.25/7.63 ! [B: code_integer,C: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 13.25/7.63 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4177_is__unit__div__mult2__eq,axiom,
% 13.25/7.63 ! [B: nat,C: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 13.25/7.63 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 13.25/7.63 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4178_is__unit__div__mult2__eq,axiom,
% 13.25/7.63 ! [B: int,C: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ C @ one_one_int )
% 13.25/7.63 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 13.25/7.63 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4179_unit__div__mult__swap,axiom,
% 13.25/7.63 ! [C: code_integer,A: code_integer,B: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 13.25/7.63 => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4180_unit__div__mult__swap,axiom,
% 13.25/7.63 ! [C: nat,A: nat,B: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 13.25/7.63 => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 13.25/7.63 = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4181_unit__div__mult__swap,axiom,
% 13.25/7.63 ! [C: int,A: int,B: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ one_one_int )
% 13.25/7.63 => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 13.25/7.63 = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4182_unit__div__commute,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4183_unit__div__commute,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 13.25/7.63 = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4184_unit__div__commute,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 13.25/7.63 = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4185_div__mult__unit2,axiom,
% 13.25/7.63 ! [C: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 13.25/7.63 => ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.63 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 13.25/7.63 = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4186_div__mult__unit2,axiom,
% 13.25/7.63 ! [C: nat,B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ C @ one_one_nat )
% 13.25/7.63 => ( ( dvd_dvd_nat @ B @ A )
% 13.25/7.63 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 13.25/7.63 = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4187_div__mult__unit2,axiom,
% 13.25/7.63 ! [C: int,B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ C @ one_one_int )
% 13.25/7.63 => ( ( dvd_dvd_int @ B @ A )
% 13.25/7.63 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 13.25/7.63 = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4188_unit__eq__div2,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( A
% 13.25/7.63 = ( divide6298287555418463151nteger @ C @ B ) )
% 13.25/7.63 = ( ( times_3573771949741848930nteger @ A @ B )
% 13.25/7.63 = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4189_unit__eq__div2,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( A
% 13.25/7.63 = ( divide_divide_nat @ C @ B ) )
% 13.25/7.63 = ( ( times_times_nat @ A @ B )
% 13.25/7.63 = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4190_unit__eq__div2,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( A
% 13.25/7.63 = ( divide_divide_int @ C @ B ) )
% 13.25/7.63 = ( ( times_times_int @ A @ B )
% 13.25/7.63 = C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4191_unit__eq__div1,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer,C: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( ( divide6298287555418463151nteger @ A @ B )
% 13.25/7.63 = C )
% 13.25/7.63 = ( A
% 13.25/7.63 = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4192_unit__eq__div1,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( ( divide_divide_nat @ A @ B )
% 13.25/7.63 = C )
% 13.25/7.63 = ( A
% 13.25/7.63 = ( times_times_nat @ C @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4193_unit__eq__div1,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( ( divide_divide_int @ A @ B )
% 13.25/7.63 = C )
% 13.25/7.63 = ( A
% 13.25/7.63 = ( times_times_int @ C @ B ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4194_unit__imp__mod__eq__0,axiom,
% 13.25/7.63 ! [B: nat,A: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.63 => ( ( modulo_modulo_nat @ A @ B )
% 13.25/7.63 = zero_zero_nat ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4195_unit__imp__mod__eq__0,axiom,
% 13.25/7.63 ! [B: int,A: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.63 => ( ( modulo_modulo_int @ A @ B )
% 13.25/7.63 = zero_zero_int ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4196_unit__imp__mod__eq__0,axiom,
% 13.25/7.63 ! [B: code_integer,A: code_integer] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.63 => ( ( modulo364778990260209775nteger @ A @ B )
% 13.25/7.63 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4197_is__unit__power__iff,axiom,
% 13.25/7.63 ! [A: code_integer,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer )
% 13.25/7.63 = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.63 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4198_is__unit__power__iff,axiom,
% 13.25/7.63 ! [A: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
% 13.25/7.63 = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.63 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4199_is__unit__power__iff,axiom,
% 13.25/7.63 ! [A: int,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
% 13.25/7.63 = ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.63 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4200_dvd__imp__le,axiom,
% 13.25/7.63 ! [K: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ K @ N )
% 13.25/7.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.63 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4201_nat__mult__dvd__cancel1,axiom,
% 13.25/7.63 ! [K: nat,M: nat,N: nat] :
% 13.25/7.63 ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.25/7.63 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.25/7.63 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4202_dvd__mult__cancel,axiom,
% 13.25/7.63 ! [K: nat,M: nat,N: nat] :
% 13.25/7.63 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 13.25/7.63 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.25/7.63 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4203_set__update__memI,axiom,
% 13.25/7.63 ! [N: nat,Xs2: list_real,X: real] :
% 13.25/7.63 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
% 13.25/7.63 => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4204_set__update__memI,axiom,
% 13.25/7.63 ! [N: nat,Xs2: list_complex,X: complex] :
% 13.25/7.63 ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
% 13.25/7.63 => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4205_set__update__memI,axiom,
% 13.25/7.63 ! [N: nat,Xs2: list_P8469869581646625389at_nat,X: produc859450856879609959at_nat] :
% 13.25/7.63 ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs2 ) )
% 13.25/7.63 => ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( list_u5003261594476800725at_nat @ Xs2 @ N @ X ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4206_set__update__memI,axiom,
% 13.25/7.63 ! [N: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
% 13.25/7.63 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.63 => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4207_set__update__memI,axiom,
% 13.25/7.63 ! [N: nat,Xs2: list_o,X: $o] :
% 13.25/7.63 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.63 => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4208_set__update__memI,axiom,
% 13.25/7.63 ! [N: nat,Xs2: list_nat,X: nat] :
% 13.25/7.63 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.63 => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4209_set__update__memI,axiom,
% 13.25/7.63 ! [N: nat,Xs2: list_int,X: int] :
% 13.25/7.63 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.63 => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4210_zdvd__imp__le,axiom,
% 13.25/7.63 ! [Z: int,N: int] :
% 13.25/7.63 ( ( dvd_dvd_int @ Z @ N )
% 13.25/7.63 => ( ( ord_less_int @ zero_zero_int @ N )
% 13.25/7.63 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4211_nle__le,axiom,
% 13.25/7.63 ! [A: rat,B: rat] :
% 13.25/7.63 ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 13.25/7.63 = ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.63 & ( B != A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4212_nle__le,axiom,
% 13.25/7.63 ! [A: num,B: num] :
% 13.25/7.63 ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 13.25/7.63 = ( ( ord_less_eq_num @ B @ A )
% 13.25/7.63 & ( B != A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4213_nle__le,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 13.25/7.63 = ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.63 & ( B != A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4214_nle__le,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 13.25/7.63 = ( ( ord_less_eq_int @ B @ A )
% 13.25/7.63 & ( B != A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4215_le__cases3,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.63 ( ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.63 => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 13.25/7.63 => ( ( ( ord_less_eq_rat @ Y3 @ X )
% 13.25/7.63 => ~ ( ord_less_eq_rat @ X @ Z ) )
% 13.25/7.63 => ( ( ( ord_less_eq_rat @ X @ Z )
% 13.25/7.63 => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 13.25/7.63 => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 13.25/7.63 => ~ ( ord_less_eq_rat @ Y3 @ X ) )
% 13.25/7.63 => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 13.25/7.63 => ~ ( ord_less_eq_rat @ Z @ X ) )
% 13.25/7.63 => ~ ( ( ord_less_eq_rat @ Z @ X )
% 13.25/7.63 => ~ ( ord_less_eq_rat @ X @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4216_le__cases3,axiom,
% 13.25/7.63 ! [X: num,Y3: num,Z: num] :
% 13.25/7.63 ( ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.63 => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 13.25/7.63 => ( ( ( ord_less_eq_num @ Y3 @ X )
% 13.25/7.63 => ~ ( ord_less_eq_num @ X @ Z ) )
% 13.25/7.63 => ( ( ( ord_less_eq_num @ X @ Z )
% 13.25/7.63 => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 13.25/7.63 => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 13.25/7.63 => ~ ( ord_less_eq_num @ Y3 @ X ) )
% 13.25/7.63 => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 13.25/7.63 => ~ ( ord_less_eq_num @ Z @ X ) )
% 13.25/7.63 => ~ ( ( ord_less_eq_num @ Z @ X )
% 13.25/7.63 => ~ ( ord_less_eq_num @ X @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4217_le__cases3,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.63 ( ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.63 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 13.25/7.63 => ( ( ( ord_less_eq_nat @ Y3 @ X )
% 13.25/7.63 => ~ ( ord_less_eq_nat @ X @ Z ) )
% 13.25/7.63 => ( ( ( ord_less_eq_nat @ X @ Z )
% 13.25/7.63 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 13.25/7.63 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 13.25/7.63 => ~ ( ord_less_eq_nat @ Y3 @ X ) )
% 13.25/7.63 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 13.25/7.63 => ~ ( ord_less_eq_nat @ Z @ X ) )
% 13.25/7.63 => ~ ( ( ord_less_eq_nat @ Z @ X )
% 13.25/7.63 => ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4218_le__cases3,axiom,
% 13.25/7.63 ! [X: int,Y3: int,Z: int] :
% 13.25/7.63 ( ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.63 => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 13.25/7.63 => ( ( ( ord_less_eq_int @ Y3 @ X )
% 13.25/7.63 => ~ ( ord_less_eq_int @ X @ Z ) )
% 13.25/7.63 => ( ( ( ord_less_eq_int @ X @ Z )
% 13.25/7.63 => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 13.25/7.63 => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 13.25/7.63 => ~ ( ord_less_eq_int @ Y3 @ X ) )
% 13.25/7.63 => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 13.25/7.63 => ~ ( ord_less_eq_int @ Z @ X ) )
% 13.25/7.63 => ~ ( ( ord_less_eq_int @ Z @ X )
% 13.25/7.63 => ~ ( ord_less_eq_int @ X @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4219_order__class_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [X2: set_int,Y: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ X2 @ Y )
% 13.25/7.63 & ( ord_less_eq_set_int @ Y @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4220_order__class_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [X2: rat,Y: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X2 @ Y )
% 13.25/7.63 & ( ord_less_eq_rat @ Y @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4221_order__class_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [X2: num,Y: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X2 @ Y )
% 13.25/7.63 & ( ord_less_eq_num @ Y @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4222_order__class_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [X2: nat,Y: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X2 @ Y )
% 13.25/7.63 & ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4223_order__class_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [X2: int,Y: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ X2 @ Y )
% 13.25/7.63 & ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4224_ord__eq__le__trans,axiom,
% 13.25/7.63 ! [A: set_int,B: set_int,C: set_int] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_eq_set_int @ B @ C )
% 13.25/7.63 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4225_ord__eq__le__trans,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4226_ord__eq__le__trans,axiom,
% 13.25/7.63 ! [A: num,B: num,C: num] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4227_ord__eq__le__trans,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ B @ C )
% 13.25/7.63 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4228_ord__eq__le__trans,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_eq_int @ B @ C )
% 13.25/7.63 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4229_ord__le__eq__trans,axiom,
% 13.25/7.63 ! [A: set_int,B: set_int,C: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4230_ord__le__eq__trans,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4231_ord__le__eq__trans,axiom,
% 13.25/7.63 ! [A: num,B: num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4232_ord__le__eq__trans,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4233_ord__le__eq__trans,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4234_order__antisym,axiom,
% 13.25/7.63 ! [X: set_int,Y3: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_set_int @ Y3 @ X )
% 13.25/7.63 => ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4235_order__antisym,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_rat @ Y3 @ X )
% 13.25/7.63 => ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4236_order__antisym,axiom,
% 13.25/7.63 ! [X: num,Y3: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_num @ Y3 @ X )
% 13.25/7.63 => ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4237_order__antisym,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_nat @ Y3 @ X )
% 13.25/7.63 => ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4238_order__antisym,axiom,
% 13.25/7.63 ! [X: int,Y3: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_int @ Y3 @ X )
% 13.25/7.63 => ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4239_order_Otrans,axiom,
% 13.25/7.63 ! [A: set_int,B: set_int,C: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_set_int @ B @ C )
% 13.25/7.63 => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4240_order_Otrans,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4241_order_Otrans,axiom,
% 13.25/7.63 ! [A: num,B: num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ord_less_eq_num @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4242_order_Otrans,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ B @ C )
% 13.25/7.63 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4243_order_Otrans,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_int @ B @ C )
% 13.25/7.63 => ( ord_less_eq_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4244_order__trans,axiom,
% 13.25/7.63 ! [X: set_int,Y3: set_int,Z: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 13.25/7.63 => ( ord_less_eq_set_int @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4245_order__trans,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 13.25/7.63 => ( ord_less_eq_rat @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4246_order__trans,axiom,
% 13.25/7.63 ! [X: num,Y3: num,Z: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_num @ Y3 @ Z )
% 13.25/7.63 => ( ord_less_eq_num @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4247_order__trans,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 13.25/7.63 => ( ord_less_eq_nat @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4248_order__trans,axiom,
% 13.25/7.63 ! [X: int,Y3: int,Z: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.63 => ( ( ord_less_eq_int @ Y3 @ Z )
% 13.25/7.63 => ( ord_less_eq_int @ X @ Z ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4249_linorder__wlog,axiom,
% 13.25/7.63 ! [P: rat > rat > $o,A: rat,B: rat] :
% 13.25/7.63 ( ! [A5: rat,B3: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A5 @ B3 )
% 13.25/7.63 => ( P @ A5 @ B3 ) )
% 13.25/7.63 => ( ! [A5: rat,B3: rat] :
% 13.25/7.63 ( ( P @ B3 @ A5 )
% 13.25/7.63 => ( P @ A5 @ B3 ) )
% 13.25/7.63 => ( P @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4250_linorder__wlog,axiom,
% 13.25/7.63 ! [P: num > num > $o,A: num,B: num] :
% 13.25/7.63 ( ! [A5: num,B3: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ A5 @ B3 )
% 13.25/7.63 => ( P @ A5 @ B3 ) )
% 13.25/7.63 => ( ! [A5: num,B3: num] :
% 13.25/7.63 ( ( P @ B3 @ A5 )
% 13.25/7.63 => ( P @ A5 @ B3 ) )
% 13.25/7.63 => ( P @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4251_linorder__wlog,axiom,
% 13.25/7.63 ! [P: nat > nat > $o,A: nat,B: nat] :
% 13.25/7.63 ( ! [A5: nat,B3: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A5 @ B3 )
% 13.25/7.63 => ( P @ A5 @ B3 ) )
% 13.25/7.63 => ( ! [A5: nat,B3: nat] :
% 13.25/7.63 ( ( P @ B3 @ A5 )
% 13.25/7.63 => ( P @ A5 @ B3 ) )
% 13.25/7.63 => ( P @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4252_linorder__wlog,axiom,
% 13.25/7.63 ! [P: int > int > $o,A: int,B: int] :
% 13.25/7.63 ( ! [A5: int,B3: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ A5 @ B3 )
% 13.25/7.63 => ( P @ A5 @ B3 ) )
% 13.25/7.63 => ( ! [A5: int,B3: int] :
% 13.25/7.63 ( ( P @ B3 @ A5 )
% 13.25/7.63 => ( P @ A5 @ B3 ) )
% 13.25/7.63 => ( P @ A @ B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4253_dual__order_Oeq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: set_int,B4: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ B4 @ A3 )
% 13.25/7.63 & ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4254_dual__order_Oeq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: rat,B4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ B4 @ A3 )
% 13.25/7.63 & ( ord_less_eq_rat @ A3 @ B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4255_dual__order_Oeq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: num,B4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ B4 @ A3 )
% 13.25/7.63 & ( ord_less_eq_num @ A3 @ B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4256_dual__order_Oeq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ B4 @ A3 )
% 13.25/7.63 & ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4257_dual__order_Oeq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: int,B4: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ B4 @ A3 )
% 13.25/7.63 & ( ord_less_eq_int @ A3 @ B4 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4258_dual__order_Oantisym,axiom,
% 13.25/7.63 ! [B: set_int,A: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_set_int @ A @ B )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4259_dual__order_Oantisym,axiom,
% 13.25/7.63 ! [B: rat,A: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4260_dual__order_Oantisym,axiom,
% 13.25/7.63 ! [B: num,A: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4261_dual__order_Oantisym,axiom,
% 13.25/7.63 ! [B: nat,A: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4262_dual__order_Oantisym,axiom,
% 13.25/7.63 ! [B: int,A: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_int @ A @ B )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4263_dual__order_Otrans,axiom,
% 13.25/7.63 ! [B: set_int,A: set_int,C: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_set_int @ C @ B )
% 13.25/7.63 => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4264_dual__order_Otrans,axiom,
% 13.25/7.63 ! [B: rat,A: rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_rat @ C @ B )
% 13.25/7.63 => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4265_dual__order_Otrans,axiom,
% 13.25/7.63 ! [B: num,A: num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_num @ C @ B )
% 13.25/7.63 => ( ord_less_eq_num @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4266_dual__order_Otrans,axiom,
% 13.25/7.63 ! [B: nat,A: nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_nat @ C @ B )
% 13.25/7.63 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4267_dual__order_Otrans,axiom,
% 13.25/7.63 ! [B: int,A: int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ B @ A )
% 13.25/7.63 => ( ( ord_less_eq_int @ C @ B )
% 13.25/7.63 => ( ord_less_eq_int @ C @ A ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4268_antisym,axiom,
% 13.25/7.63 ! [A: set_int,B: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_set_int @ B @ A )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4269_antisym,axiom,
% 13.25/7.63 ! [A: rat,B: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4270_antisym,axiom,
% 13.25/7.63 ! [A: num,B: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ A )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4271_antisym,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4272_antisym,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_int @ B @ A )
% 13.25/7.63 => ( A = B ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4273_Orderings_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: set_int,Z4: set_int] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: set_int,B4: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ A3 @ B4 )
% 13.25/7.63 & ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4274_Orderings_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: rat,Z4: rat] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: rat,B4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A3 @ B4 )
% 13.25/7.63 & ( ord_less_eq_rat @ B4 @ A3 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4275_Orderings_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: num,B4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ A3 @ B4 )
% 13.25/7.63 & ( ord_less_eq_num @ B4 @ A3 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4276_Orderings_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A3 @ B4 )
% 13.25/7.63 & ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4277_Orderings_Oorder__eq__iff,axiom,
% 13.25/7.63 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.63 = ( ^ [A3: int,B4: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ A3 @ B4 )
% 13.25/7.63 & ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4278_order__subst1,axiom,
% 13.25/7.63 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4279_order__subst1,axiom,
% 13.25/7.63 ! [A: rat,F: num > rat,B: num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4280_order__subst1,axiom,
% 13.25/7.63 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_nat @ B @ C )
% 13.25/7.63 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4281_order__subst1,axiom,
% 13.25/7.63 ! [A: rat,F: int > rat,B: int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_int @ B @ C )
% 13.25/7.63 => ( ! [X3: int,Y4: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4282_order__subst1,axiom,
% 13.25/7.63 ! [A: num,F: rat > num,B: rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4283_order__subst1,axiom,
% 13.25/7.63 ! [A: num,F: num > num,B: num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4284_order__subst1,axiom,
% 13.25/7.63 ! [A: num,F: nat > num,B: nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_nat @ B @ C )
% 13.25/7.63 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4285_order__subst1,axiom,
% 13.25/7.63 ! [A: num,F: int > num,B: int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_int @ B @ C )
% 13.25/7.63 => ( ! [X3: int,Y4: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4286_order__subst1,axiom,
% 13.25/7.63 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4287_order__subst1,axiom,
% 13.25/7.63 ! [A: nat,F: num > nat,B: num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4288_order__subst2,axiom,
% 13.25/7.63 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4289_order__subst2,axiom,
% 13.25/7.63 ! [A: rat,B: rat,F: rat > num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4290_order__subst2,axiom,
% 13.25/7.63 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4291_order__subst2,axiom,
% 13.25/7.63 ! [A: rat,B: rat,F: rat > int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4292_order__subst2,axiom,
% 13.25/7.63 ! [A: num,B: num,F: num > rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4293_order__subst2,axiom,
% 13.25/7.63 ! [A: num,B: num,F: num > num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4294_order__subst2,axiom,
% 13.25/7.63 ! [A: num,B: num,F: num > nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4295_order__subst2,axiom,
% 13.25/7.63 ! [A: num,B: num,F: num > int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4296_order__subst2,axiom,
% 13.25/7.63 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4297_order__subst2,axiom,
% 13.25/7.63 ! [A: nat,B: nat,F: nat > num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.63 => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 13.25/7.63 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4298_order__eq__refl,axiom,
% 13.25/7.63 ! [X: set_int,Y3: set_int] :
% 13.25/7.63 ( ( X = Y3 )
% 13.25/7.63 => ( ord_less_eq_set_int @ X @ Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4299_order__eq__refl,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat] :
% 13.25/7.63 ( ( X = Y3 )
% 13.25/7.63 => ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4300_order__eq__refl,axiom,
% 13.25/7.63 ! [X: num,Y3: num] :
% 13.25/7.63 ( ( X = Y3 )
% 13.25/7.63 => ( ord_less_eq_num @ X @ Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4301_order__eq__refl,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat] :
% 13.25/7.63 ( ( X = Y3 )
% 13.25/7.63 => ( ord_less_eq_nat @ X @ Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4302_order__eq__refl,axiom,
% 13.25/7.63 ! [X: int,Y3: int] :
% 13.25/7.63 ( ( X = Y3 )
% 13.25/7.63 => ( ord_less_eq_int @ X @ Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4303_linorder__linear,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.63 | ( ord_less_eq_rat @ Y3 @ X ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4304_linorder__linear,axiom,
% 13.25/7.63 ! [X: num,Y3: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.63 | ( ord_less_eq_num @ Y3 @ X ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4305_linorder__linear,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.63 | ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4306_linorder__linear,axiom,
% 13.25/7.63 ! [X: int,Y3: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.63 | ( ord_less_eq_int @ Y3 @ X ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4307_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4308_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: num,F: rat > num,B: rat,C: rat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4309_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4310_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: int,F: rat > int,B: rat,C: rat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4311_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: rat,F: num > rat,B: num,C: num] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4312_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: num,F: num > num,B: num,C: num] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4313_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: nat,F: num > nat,B: num,C: num] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4314_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: int,F: num > int,B: num,C: num] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4315_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_nat @ B @ C )
% 13.25/7.63 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4316_ord__eq__le__subst,axiom,
% 13.25/7.63 ! [A: num,F: nat > num,B: nat,C: nat] :
% 13.25/7.63 ( ( A
% 13.25/7.63 = ( F @ B ) )
% 13.25/7.63 => ( ( ord_less_eq_nat @ B @ C )
% 13.25/7.63 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4317_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4318_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: rat,B: rat,F: rat > num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4319_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4320_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: rat,B: rat,F: rat > int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4321_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: num,B: num,F: num > rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4322_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: num,B: num,F: num > num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4323_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: num,B: num,F: num > nat,C: nat] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4324_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: num,B: num,F: num > int,C: int] :
% 13.25/7.63 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: num,Y4: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4325_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4326_ord__le__eq__subst,axiom,
% 13.25/7.63 ! [A: nat,B: nat,F: nat > num,C: num] :
% 13.25/7.63 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.63 => ( ( ( F @ B )
% 13.25/7.63 = C )
% 13.25/7.63 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ X3 @ Y4 )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.63 => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4327_linorder__le__cases,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat] :
% 13.25/7.63 ( ~ ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.63 => ( ord_less_eq_rat @ Y3 @ X ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4328_linorder__le__cases,axiom,
% 13.25/7.63 ! [X: num,Y3: num] :
% 13.25/7.63 ( ~ ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.63 => ( ord_less_eq_num @ Y3 @ X ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4329_linorder__le__cases,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat] :
% 13.25/7.63 ( ~ ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.63 => ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4330_linorder__le__cases,axiom,
% 13.25/7.63 ! [X: int,Y3: int] :
% 13.25/7.63 ( ~ ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.63 => ( ord_less_eq_int @ Y3 @ X ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4331_order__antisym__conv,axiom,
% 13.25/7.63 ! [Y3: set_int,X: set_int] :
% 13.25/7.63 ( ( ord_less_eq_set_int @ Y3 @ X )
% 13.25/7.63 => ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4332_order__antisym__conv,axiom,
% 13.25/7.63 ! [Y3: rat,X: rat] :
% 13.25/7.63 ( ( ord_less_eq_rat @ Y3 @ X )
% 13.25/7.63 => ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4333_order__antisym__conv,axiom,
% 13.25/7.63 ! [Y3: num,X: num] :
% 13.25/7.63 ( ( ord_less_eq_num @ Y3 @ X )
% 13.25/7.63 => ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4334_order__antisym__conv,axiom,
% 13.25/7.63 ! [Y3: nat,X: nat] :
% 13.25/7.63 ( ( ord_less_eq_nat @ Y3 @ X )
% 13.25/7.63 => ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4335_order__antisym__conv,axiom,
% 13.25/7.63 ! [Y3: int,X: int] :
% 13.25/7.63 ( ( ord_less_eq_int @ Y3 @ X )
% 13.25/7.63 => ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4336_mod__greater__zero__iff__not__dvd,axiom,
% 13.25/7.63 ! [M: nat,N: nat] :
% 13.25/7.63 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 13.25/7.63 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4337_list__update__same__conv,axiom,
% 13.25/7.63 ! [I: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
% 13.25/7.63 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.63 => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X )
% 13.25/7.63 = Xs2 )
% 13.25/7.63 = ( ( nth_VEBT_VEBT @ Xs2 @ I )
% 13.25/7.63 = X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4338_list__update__same__conv,axiom,
% 13.25/7.63 ! [I: nat,Xs2: list_o,X: $o] :
% 13.25/7.63 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.63 => ( ( ( list_update_o @ Xs2 @ I @ X )
% 13.25/7.63 = Xs2 )
% 13.25/7.63 = ( ( nth_o @ Xs2 @ I )
% 13.25/7.63 = X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4339_list__update__same__conv,axiom,
% 13.25/7.63 ! [I: nat,Xs2: list_nat,X: nat] :
% 13.25/7.63 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.63 => ( ( ( list_update_nat @ Xs2 @ I @ X )
% 13.25/7.63 = Xs2 )
% 13.25/7.63 = ( ( nth_nat @ Xs2 @ I )
% 13.25/7.63 = X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4340_list__update__same__conv,axiom,
% 13.25/7.63 ! [I: nat,Xs2: list_int,X: int] :
% 13.25/7.63 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.63 => ( ( ( list_update_int @ Xs2 @ I @ X )
% 13.25/7.63 = Xs2 )
% 13.25/7.63 = ( ( nth_int @ Xs2 @ I )
% 13.25/7.63 = X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4341_nth__list__update,axiom,
% 13.25/7.63 ! [I: nat,Xs2: list_VEBT_VEBT,J: nat,X: vEBT_VEBT] :
% 13.25/7.63 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 13.25/7.63 => ( ( ( I = J )
% 13.25/7.63 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
% 13.25/7.63 = X ) )
% 13.25/7.63 & ( ( I != J )
% 13.25/7.63 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X ) @ J )
% 13.25/7.63 = ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4342_nth__list__update,axiom,
% 13.25/7.63 ! [I: nat,Xs2: list_o,X: $o,J: nat] :
% 13.25/7.63 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
% 13.25/7.63 => ( ( nth_o @ ( list_update_o @ Xs2 @ I @ X ) @ J )
% 13.25/7.63 = ( ( ( I = J )
% 13.25/7.63 => X )
% 13.25/7.63 & ( ( I != J )
% 13.25/7.63 => ( nth_o @ Xs2 @ J ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4343_nth__list__update,axiom,
% 13.25/7.63 ! [I: nat,Xs2: list_nat,J: nat,X: nat] :
% 13.25/7.63 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
% 13.25/7.63 => ( ( ( I = J )
% 13.25/7.63 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
% 13.25/7.63 = X ) )
% 13.25/7.63 & ( ( I != J )
% 13.25/7.63 => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J )
% 13.25/7.63 = ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4344_nth__list__update,axiom,
% 13.25/7.63 ! [I: nat,Xs2: list_int,J: nat,X: int] :
% 13.25/7.63 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
% 13.25/7.63 => ( ( ( I = J )
% 13.25/7.63 => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
% 13.25/7.63 = X ) )
% 13.25/7.63 & ( ( I != J )
% 13.25/7.63 => ( ( nth_int @ ( list_update_int @ Xs2 @ I @ X ) @ J )
% 13.25/7.63 = ( nth_int @ Xs2 @ J ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4345_lt__ex,axiom,
% 13.25/7.63 ! [X: real] :
% 13.25/7.63 ? [Y4: real] : ( ord_less_real @ Y4 @ X ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4346_lt__ex,axiom,
% 13.25/7.63 ! [X: rat] :
% 13.25/7.63 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4347_lt__ex,axiom,
% 13.25/7.63 ! [X: int] :
% 13.25/7.63 ? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4348_gt__ex,axiom,
% 13.25/7.63 ! [X: real] :
% 13.25/7.63 ? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4349_gt__ex,axiom,
% 13.25/7.63 ! [X: rat] :
% 13.25/7.63 ? [X_1: rat] : ( ord_less_rat @ X @ X_1 ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4350_gt__ex,axiom,
% 13.25/7.63 ! [X: nat] :
% 13.25/7.63 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4351_gt__ex,axiom,
% 13.25/7.63 ! [X: int] :
% 13.25/7.63 ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4352_dense,axiom,
% 13.25/7.63 ! [X: real,Y3: real] :
% 13.25/7.63 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.63 => ? [Z5: real] :
% 13.25/7.63 ( ( ord_less_real @ X @ Z5 )
% 13.25/7.63 & ( ord_less_real @ Z5 @ Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4353_dense,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat] :
% 13.25/7.63 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.63 => ? [Z5: rat] :
% 13.25/7.63 ( ( ord_less_rat @ X @ Z5 )
% 13.25/7.63 & ( ord_less_rat @ Z5 @ Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4354_less__imp__neq,axiom,
% 13.25/7.63 ! [X: real,Y3: real] :
% 13.25/7.63 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.63 => ( X != Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4355_less__imp__neq,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat] :
% 13.25/7.63 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.63 => ( X != Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4356_less__imp__neq,axiom,
% 13.25/7.63 ! [X: num,Y3: num] :
% 13.25/7.63 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.63 => ( X != Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4357_less__imp__neq,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat] :
% 13.25/7.63 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.63 => ( X != Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4358_less__imp__neq,axiom,
% 13.25/7.63 ! [X: int,Y3: int] :
% 13.25/7.63 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.63 => ( X != Y3 ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4359_order_Oasym,axiom,
% 13.25/7.63 ! [A: real,B: real] :
% 13.25/7.63 ( ( ord_less_real @ A @ B )
% 13.25/7.63 => ~ ( ord_less_real @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4360_order_Oasym,axiom,
% 13.25/7.63 ! [A: rat,B: rat] :
% 13.25/7.63 ( ( ord_less_rat @ A @ B )
% 13.25/7.63 => ~ ( ord_less_rat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4361_order_Oasym,axiom,
% 13.25/7.63 ! [A: num,B: num] :
% 13.25/7.63 ( ( ord_less_num @ A @ B )
% 13.25/7.63 => ~ ( ord_less_num @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4362_order_Oasym,axiom,
% 13.25/7.63 ! [A: nat,B: nat] :
% 13.25/7.63 ( ( ord_less_nat @ A @ B )
% 13.25/7.63 => ~ ( ord_less_nat @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4363_order_Oasym,axiom,
% 13.25/7.63 ! [A: int,B: int] :
% 13.25/7.63 ( ( ord_less_int @ A @ B )
% 13.25/7.63 => ~ ( ord_less_int @ B @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4364_ord__eq__less__trans,axiom,
% 13.25/7.63 ! [A: real,B: real,C: real] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_real @ B @ C )
% 13.25/7.63 => ( ord_less_real @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4365_ord__eq__less__trans,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_rat @ B @ C )
% 13.25/7.63 => ( ord_less_rat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4366_ord__eq__less__trans,axiom,
% 13.25/7.63 ! [A: num,B: num,C: num] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_num @ B @ C )
% 13.25/7.63 => ( ord_less_num @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4367_ord__eq__less__trans,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_nat @ B @ C )
% 13.25/7.63 => ( ord_less_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4368_ord__eq__less__trans,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( A = B )
% 13.25/7.63 => ( ( ord_less_int @ B @ C )
% 13.25/7.63 => ( ord_less_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4369_ord__less__eq__trans,axiom,
% 13.25/7.63 ! [A: real,B: real,C: real] :
% 13.25/7.63 ( ( ord_less_real @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_real @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4370_ord__less__eq__trans,axiom,
% 13.25/7.63 ! [A: rat,B: rat,C: rat] :
% 13.25/7.63 ( ( ord_less_rat @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_rat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4371_ord__less__eq__trans,axiom,
% 13.25/7.63 ! [A: num,B: num,C: num] :
% 13.25/7.63 ( ( ord_less_num @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_num @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4372_ord__less__eq__trans,axiom,
% 13.25/7.63 ! [A: nat,B: nat,C: nat] :
% 13.25/7.63 ( ( ord_less_nat @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_nat @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4373_ord__less__eq__trans,axiom,
% 13.25/7.63 ! [A: int,B: int,C: int] :
% 13.25/7.63 ( ( ord_less_int @ A @ B )
% 13.25/7.63 => ( ( B = C )
% 13.25/7.63 => ( ord_less_int @ A @ C ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4374_less__induct,axiom,
% 13.25/7.63 ! [P: nat > $o,A: nat] :
% 13.25/7.63 ( ! [X3: nat] :
% 13.25/7.63 ( ! [Y5: nat] :
% 13.25/7.63 ( ( ord_less_nat @ Y5 @ X3 )
% 13.25/7.63 => ( P @ Y5 ) )
% 13.25/7.63 => ( P @ X3 ) )
% 13.25/7.63 => ( P @ A ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4375_antisym__conv3,axiom,
% 13.25/7.63 ! [Y3: real,X: real] :
% 13.25/7.63 ( ~ ( ord_less_real @ Y3 @ X )
% 13.25/7.63 => ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4376_antisym__conv3,axiom,
% 13.25/7.63 ! [Y3: rat,X: rat] :
% 13.25/7.63 ( ~ ( ord_less_rat @ Y3 @ X )
% 13.25/7.63 => ( ( ~ ( ord_less_rat @ X @ Y3 ) )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4377_antisym__conv3,axiom,
% 13.25/7.63 ! [Y3: num,X: num] :
% 13.25/7.63 ( ~ ( ord_less_num @ Y3 @ X )
% 13.25/7.63 => ( ( ~ ( ord_less_num @ X @ Y3 ) )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4378_antisym__conv3,axiom,
% 13.25/7.63 ! [Y3: nat,X: nat] :
% 13.25/7.63 ( ~ ( ord_less_nat @ Y3 @ X )
% 13.25/7.63 => ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4379_antisym__conv3,axiom,
% 13.25/7.63 ! [Y3: int,X: int] :
% 13.25/7.63 ( ~ ( ord_less_int @ Y3 @ X )
% 13.25/7.63 => ( ( ~ ( ord_less_int @ X @ Y3 ) )
% 13.25/7.63 = ( X = Y3 ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4380_linorder__cases,axiom,
% 13.25/7.63 ! [X: real,Y3: real] :
% 13.25/7.63 ( ~ ( ord_less_real @ X @ Y3 )
% 13.25/7.63 => ( ( X != Y3 )
% 13.25/7.63 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4381_linorder__cases,axiom,
% 13.25/7.63 ! [X: rat,Y3: rat] :
% 13.25/7.63 ( ~ ( ord_less_rat @ X @ Y3 )
% 13.25/7.63 => ( ( X != Y3 )
% 13.25/7.63 => ( ord_less_rat @ Y3 @ X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4382_linorder__cases,axiom,
% 13.25/7.63 ! [X: num,Y3: num] :
% 13.25/7.63 ( ~ ( ord_less_num @ X @ Y3 )
% 13.25/7.63 => ( ( X != Y3 )
% 13.25/7.63 => ( ord_less_num @ Y3 @ X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4383_linorder__cases,axiom,
% 13.25/7.63 ! [X: nat,Y3: nat] :
% 13.25/7.63 ( ~ ( ord_less_nat @ X @ Y3 )
% 13.25/7.63 => ( ( X != Y3 )
% 13.25/7.63 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4384_linorder__cases,axiom,
% 13.25/7.63 ! [X: int,Y3: int] :
% 13.25/7.63 ( ~ ( ord_less_int @ X @ Y3 )
% 13.25/7.63 => ( ( X != Y3 )
% 13.25/7.63 => ( ord_less_int @ Y3 @ X ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4385_dual__order_Oasym,axiom,
% 13.25/7.63 ! [B: real,A: real] :
% 13.25/7.63 ( ( ord_less_real @ B @ A )
% 13.25/7.63 => ~ ( ord_less_real @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4386_dual__order_Oasym,axiom,
% 13.25/7.63 ! [B: rat,A: rat] :
% 13.25/7.63 ( ( ord_less_rat @ B @ A )
% 13.25/7.63 => ~ ( ord_less_rat @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4387_dual__order_Oasym,axiom,
% 13.25/7.63 ! [B: num,A: num] :
% 13.25/7.63 ( ( ord_less_num @ B @ A )
% 13.25/7.63 => ~ ( ord_less_num @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4388_dual__order_Oasym,axiom,
% 13.25/7.63 ! [B: nat,A: nat] :
% 13.25/7.63 ( ( ord_less_nat @ B @ A )
% 13.25/7.63 => ~ ( ord_less_nat @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4389_dual__order_Oasym,axiom,
% 13.25/7.63 ! [B: int,A: int] :
% 13.25/7.63 ( ( ord_less_int @ B @ A )
% 13.25/7.63 => ~ ( ord_less_int @ A @ B ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4390_dual__order_Oirrefl,axiom,
% 13.25/7.63 ! [A: real] :
% 13.25/7.63 ~ ( ord_less_real @ A @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4391_dual__order_Oirrefl,axiom,
% 13.25/7.63 ! [A: rat] :
% 13.25/7.63 ~ ( ord_less_rat @ A @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4392_dual__order_Oirrefl,axiom,
% 13.25/7.63 ! [A: num] :
% 13.25/7.63 ~ ( ord_less_num @ A @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4393_dual__order_Oirrefl,axiom,
% 13.25/7.63 ! [A: nat] :
% 13.25/7.63 ~ ( ord_less_nat @ A @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4394_dual__order_Oirrefl,axiom,
% 13.25/7.63 ! [A: int] :
% 13.25/7.63 ~ ( ord_less_int @ A @ A ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4395_exists__least__iff,axiom,
% 13.25/7.63 ( ( ^ [P2: nat > $o] :
% 13.25/7.63 ? [X4: nat] : ( P2 @ X4 ) )
% 13.25/7.63 = ( ^ [P3: nat > $o] :
% 13.25/7.63 ? [N2: nat] :
% 13.25/7.63 ( ( P3 @ N2 )
% 13.25/7.63 & ! [M3: nat] :
% 13.25/7.63 ( ( ord_less_nat @ M3 @ N2 )
% 13.25/7.63 => ~ ( P3 @ M3 ) ) ) ) ) ).
% 13.25/7.63
% 13.25/7.63 thf(fact_4396_linorder__less__wlog,axiom,
% 13.25/7.63 ! [P: real > real > $o,A: real,B: real] :
% 13.25/7.63 ( ! [A5: real,B3: real] :
% 13.25/7.63 ( ( ord_less_real @ A5 @ B3 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( ! [A5: real] : ( P @ A5 @ A5 )
% 13.25/7.64 => ( ! [A5: real,B3: real] :
% 13.25/7.64 ( ( P @ B3 @ A5 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( P @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4397_linorder__less__wlog,axiom,
% 13.25/7.64 ! [P: rat > rat > $o,A: rat,B: rat] :
% 13.25/7.64 ( ! [A5: rat,B3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A5 @ B3 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( ! [A5: rat] : ( P @ A5 @ A5 )
% 13.25/7.64 => ( ! [A5: rat,B3: rat] :
% 13.25/7.64 ( ( P @ B3 @ A5 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( P @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4398_linorder__less__wlog,axiom,
% 13.25/7.64 ! [P: num > num > $o,A: num,B: num] :
% 13.25/7.64 ( ! [A5: num,B3: num] :
% 13.25/7.64 ( ( ord_less_num @ A5 @ B3 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( ! [A5: num] : ( P @ A5 @ A5 )
% 13.25/7.64 => ( ! [A5: num,B3: num] :
% 13.25/7.64 ( ( P @ B3 @ A5 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( P @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4399_linorder__less__wlog,axiom,
% 13.25/7.64 ! [P: nat > nat > $o,A: nat,B: nat] :
% 13.25/7.64 ( ! [A5: nat,B3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ A5 @ B3 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( ! [A5: nat] : ( P @ A5 @ A5 )
% 13.25/7.64 => ( ! [A5: nat,B3: nat] :
% 13.25/7.64 ( ( P @ B3 @ A5 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( P @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4400_linorder__less__wlog,axiom,
% 13.25/7.64 ! [P: int > int > $o,A: int,B: int] :
% 13.25/7.64 ( ! [A5: int,B3: int] :
% 13.25/7.64 ( ( ord_less_int @ A5 @ B3 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( ! [A5: int] : ( P @ A5 @ A5 )
% 13.25/7.64 => ( ! [A5: int,B3: int] :
% 13.25/7.64 ( ( P @ B3 @ A5 )
% 13.25/7.64 => ( P @ A5 @ B3 ) )
% 13.25/7.64 => ( P @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4401_order_Ostrict__trans,axiom,
% 13.25/7.64 ! [A: real,B: real,C: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ord_less_real @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4402_order_Ostrict__trans,axiom,
% 13.25/7.64 ! [A: rat,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ord_less_rat @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4403_order_Ostrict__trans,axiom,
% 13.25/7.64 ! [A: num,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_num @ B @ C )
% 13.25/7.64 => ( ord_less_num @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4404_order_Ostrict__trans,axiom,
% 13.25/7.64 ! [A: nat,B: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_nat @ A @ B )
% 13.25/7.64 => ( ( ord_less_nat @ B @ C )
% 13.25/7.64 => ( ord_less_nat @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4405_order_Ostrict__trans,axiom,
% 13.25/7.64 ! [A: int,B: int,C: int] :
% 13.25/7.64 ( ( ord_less_int @ A @ B )
% 13.25/7.64 => ( ( ord_less_int @ B @ C )
% 13.25/7.64 => ( ord_less_int @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4406_not__less__iff__gr__or__eq,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 13.25/7.64 = ( ( ord_less_real @ Y3 @ X )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4407_not__less__iff__gr__or__eq,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ~ ( ord_less_rat @ X @ Y3 ) )
% 13.25/7.64 = ( ( ord_less_rat @ Y3 @ X )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4408_not__less__iff__gr__or__eq,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ~ ( ord_less_num @ X @ Y3 ) )
% 13.25/7.64 = ( ( ord_less_num @ Y3 @ X )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4409_not__less__iff__gr__or__eq,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 13.25/7.64 = ( ( ord_less_nat @ Y3 @ X )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4410_not__less__iff__gr__or__eq,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ~ ( ord_less_int @ X @ Y3 ) )
% 13.25/7.64 = ( ( ord_less_int @ Y3 @ X )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4411_dual__order_Ostrict__trans,axiom,
% 13.25/7.64 ! [B: real,A: real,C: real] :
% 13.25/7.64 ( ( ord_less_real @ B @ A )
% 13.25/7.64 => ( ( ord_less_real @ C @ B )
% 13.25/7.64 => ( ord_less_real @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4412_dual__order_Ostrict__trans,axiom,
% 13.25/7.64 ! [B: rat,A: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ B @ A )
% 13.25/7.64 => ( ( ord_less_rat @ C @ B )
% 13.25/7.64 => ( ord_less_rat @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4413_dual__order_Ostrict__trans,axiom,
% 13.25/7.64 ! [B: num,A: num,C: num] :
% 13.25/7.64 ( ( ord_less_num @ B @ A )
% 13.25/7.64 => ( ( ord_less_num @ C @ B )
% 13.25/7.64 => ( ord_less_num @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4414_dual__order_Ostrict__trans,axiom,
% 13.25/7.64 ! [B: nat,A: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_nat @ B @ A )
% 13.25/7.64 => ( ( ord_less_nat @ C @ B )
% 13.25/7.64 => ( ord_less_nat @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4415_dual__order_Ostrict__trans,axiom,
% 13.25/7.64 ! [B: int,A: int,C: int] :
% 13.25/7.64 ( ( ord_less_int @ B @ A )
% 13.25/7.64 => ( ( ord_less_int @ C @ B )
% 13.25/7.64 => ( ord_less_int @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4416_order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [A: real,B: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4417_order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [A: rat,B: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4418_order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [A: num,B: num] :
% 13.25/7.64 ( ( ord_less_num @ A @ B )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4419_order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ( ord_less_nat @ A @ B )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4420_order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ( ord_less_int @ A @ B )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4421_dual__order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [B: real,A: real] :
% 13.25/7.64 ( ( ord_less_real @ B @ A )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4422_dual__order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [B: rat,A: rat] :
% 13.25/7.64 ( ( ord_less_rat @ B @ A )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4423_dual__order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [B: num,A: num] :
% 13.25/7.64 ( ( ord_less_num @ B @ A )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4424_dual__order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [B: nat,A: nat] :
% 13.25/7.64 ( ( ord_less_nat @ B @ A )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4425_dual__order_Ostrict__implies__not__eq,axiom,
% 13.25/7.64 ! [B: int,A: int] :
% 13.25/7.64 ( ( ord_less_int @ B @ A )
% 13.25/7.64 => ( A != B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4426_linorder__neqE,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 => ( ~ ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( ord_less_real @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4427_linorder__neqE,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 => ( ~ ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( ord_less_rat @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4428_linorder__neqE,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 => ( ~ ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( ord_less_num @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4429_linorder__neqE,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 => ( ~ ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( ord_less_nat @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4430_linorder__neqE,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 => ( ~ ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( ord_less_int @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4431_order__less__asym,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4432_order__less__asym,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_rat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4433_order__less__asym,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_num @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4434_order__less__asym,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4435_order__less__asym,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_int @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4436_linorder__neq__iff,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 = ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 | ( ord_less_real @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4437_linorder__neq__iff,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 = ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 | ( ord_less_rat @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4438_linorder__neq__iff,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 = ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 | ( ord_less_num @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4439_linorder__neq__iff,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 = ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 | ( ord_less_nat @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4440_linorder__neq__iff,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( X != Y3 )
% 13.25/7.64 = ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 | ( ord_less_int @ Y3 @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4441_order__less__asym_H,axiom,
% 13.25/7.64 ! [A: real,B: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ~ ( ord_less_real @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4442_order__less__asym_H,axiom,
% 13.25/7.64 ! [A: rat,B: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ~ ( ord_less_rat @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4443_order__less__asym_H,axiom,
% 13.25/7.64 ! [A: num,B: num] :
% 13.25/7.64 ( ( ord_less_num @ A @ B )
% 13.25/7.64 => ~ ( ord_less_num @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4444_order__less__asym_H,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ( ord_less_nat @ A @ B )
% 13.25/7.64 => ~ ( ord_less_nat @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4445_order__less__asym_H,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ( ord_less_int @ A @ B )
% 13.25/7.64 => ~ ( ord_less_int @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4446_order__less__trans,axiom,
% 13.25/7.64 ! [X: real,Y3: real,Z: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_real @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_real @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4447_order__less__trans,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_rat @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_rat @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4448_order__less__trans,axiom,
% 13.25/7.64 ! [X: num,Y3: num,Z: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_num @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_num @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4449_order__less__trans,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_nat @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_nat @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4450_order__less__trans,axiom,
% 13.25/7.64 ! [X: int,Y3: int,Z: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_int @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_int @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4451_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: real,F: real > real,B: real,C: real] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4452_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: rat,F: real > rat,B: real,C: real] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4453_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: num,F: real > num,B: real,C: real] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4454_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: nat,F: real > nat,B: real,C: real] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4455_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: int,F: real > int,B: real,C: real] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4456_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: real,F: rat > real,B: rat,C: rat] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4457_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4458_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: num,F: rat > num,B: rat,C: rat] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4459_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4460_ord__eq__less__subst,axiom,
% 13.25/7.64 ! [A: int,F: rat > int,B: rat,C: rat] :
% 13.25/7.64 ( ( A
% 13.25/7.64 = ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4461_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > real,C: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4462_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4463_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > num,C: num] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4464_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > nat,C: nat] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4465_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > int,C: int] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4466_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > real,C: real] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4467_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4468_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > num,C: num] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4469_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4470_ord__less__eq__subst,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > int,C: int] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ( F @ B )
% 13.25/7.64 = C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4471_order__less__irrefl,axiom,
% 13.25/7.64 ! [X: real] :
% 13.25/7.64 ~ ( ord_less_real @ X @ X ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4472_order__less__irrefl,axiom,
% 13.25/7.64 ! [X: rat] :
% 13.25/7.64 ~ ( ord_less_rat @ X @ X ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4473_order__less__irrefl,axiom,
% 13.25/7.64 ! [X: num] :
% 13.25/7.64 ~ ( ord_less_num @ X @ X ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4474_order__less__irrefl,axiom,
% 13.25/7.64 ! [X: nat] :
% 13.25/7.64 ~ ( ord_less_nat @ X @ X ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4475_order__less__irrefl,axiom,
% 13.25/7.64 ! [X: int] :
% 13.25/7.64 ~ ( ord_less_int @ X @ X ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4476_order__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: real > real,B: real,C: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4477_order__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: rat > real,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4478_order__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: num > real,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4479_order__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: nat > real,B: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_nat @ B @ C )
% 13.25/7.64 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4480_order__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: int > real,B: int,C: int] :
% 13.25/7.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_int @ B @ C )
% 13.25/7.64 => ( ! [X3: int,Y4: int] :
% 13.25/7.64 ( ( ord_less_int @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4481_order__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: real > rat,B: real,C: real] :
% 13.25/7.64 ( ( ord_less_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4482_order__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4483_order__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: num > rat,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4484_order__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_nat @ B @ C )
% 13.25/7.64 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4485_order__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: int > rat,B: int,C: int] :
% 13.25/7.64 ( ( ord_less_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_int @ B @ C )
% 13.25/7.64 => ( ! [X3: int,Y4: int] :
% 13.25/7.64 ( ( ord_less_int @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4486_order__less__subst2,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > real,C: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4487_order__less__subst2,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4488_order__less__subst2,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > num,C: num] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_num @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4489_order__less__subst2,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > nat,C: nat] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4490_order__less__subst2,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > int,C: int] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_int @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4491_order__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > real,C: real] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4492_order__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4493_order__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > num,C: num] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_num @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4494_order__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4495_order__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > int,C: int] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_int @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4496_order__less__not__sym,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4497_order__less__not__sym,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_rat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4498_order__less__not__sym,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_num @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4499_order__less__not__sym,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4500_order__less__not__sym,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_int @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4501_order__less__imp__triv,axiom,
% 13.25/7.64 ! [X: real,Y3: real,P: $o] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_real @ Y3 @ X )
% 13.25/7.64 => P ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4502_order__less__imp__triv,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat,P: $o] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_rat @ Y3 @ X )
% 13.25/7.64 => P ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4503_order__less__imp__triv,axiom,
% 13.25/7.64 ! [X: num,Y3: num,P: $o] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_num @ Y3 @ X )
% 13.25/7.64 => P ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4504_order__less__imp__triv,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat,P: $o] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_nat @ Y3 @ X )
% 13.25/7.64 => P ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4505_order__less__imp__triv,axiom,
% 13.25/7.64 ! [X: int,Y3: int,P: $o] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_int @ Y3 @ X )
% 13.25/7.64 => P ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4506_linorder__less__linear,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 )
% 13.25/7.64 | ( ord_less_real @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4507_linorder__less__linear,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 )
% 13.25/7.64 | ( ord_less_rat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4508_linorder__less__linear,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 )
% 13.25/7.64 | ( ord_less_num @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4509_linorder__less__linear,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 )
% 13.25/7.64 | ( ord_less_nat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4510_linorder__less__linear,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 )
% 13.25/7.64 | ( ord_less_int @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4511_order__less__imp__not__eq,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( X != Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4512_order__less__imp__not__eq,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( X != Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4513_order__less__imp__not__eq,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( X != Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4514_order__less__imp__not__eq,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( X != Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4515_order__less__imp__not__eq,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( X != Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4516_order__less__imp__not__eq2,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( Y3 != X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4517_order__less__imp__not__eq2,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( Y3 != X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4518_order__less__imp__not__eq2,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( Y3 != X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4519_order__less__imp__not__eq2,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( Y3 != X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4520_order__less__imp__not__eq2,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( Y3 != X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4521_order__less__imp__not__less,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_real @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4522_order__less__imp__not__less,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_rat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4523_order__less__imp__not__less,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_num @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4524_order__less__imp__not__less,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_nat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4525_order__less__imp__not__less,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ~ ( ord_less_int @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4526_mod__eq__dvd__iff__nat,axiom,
% 13.25/7.64 ! [N: nat,M: nat,Q2: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.64 => ( ( ( modulo_modulo_nat @ M @ Q2 )
% 13.25/7.64 = ( modulo_modulo_nat @ N @ Q2 ) )
% 13.25/7.64 = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4527_even__zero,axiom,
% 13.25/7.64 dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4528_even__zero,axiom,
% 13.25/7.64 dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4529_even__zero,axiom,
% 13.25/7.64 dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4530_is__unit__div__mult__cancel__right,axiom,
% 13.25/7.64 ! [A: code_integer,B: code_integer] :
% 13.25/7.64 ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.64 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.64 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 13.25/7.64 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4531_is__unit__div__mult__cancel__right,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ( A != zero_zero_nat )
% 13.25/7.64 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.64 => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 13.25/7.64 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4532_is__unit__div__mult__cancel__right,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ( A != zero_zero_int )
% 13.25/7.64 => ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.64 => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 13.25/7.64 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4533_is__unit__div__mult__cancel__left,axiom,
% 13.25/7.64 ! [A: code_integer,B: code_integer] :
% 13.25/7.64 ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.64 => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 13.25/7.64 => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 13.25/7.64 = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4534_is__unit__div__mult__cancel__left,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ( A != zero_zero_nat )
% 13.25/7.64 => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 13.25/7.64 => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 13.25/7.64 = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4535_is__unit__div__mult__cancel__left,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ( A != zero_zero_int )
% 13.25/7.64 => ( ( dvd_dvd_int @ B @ one_one_int )
% 13.25/7.64 => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 13.25/7.64 = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4536_is__unitE,axiom,
% 13.25/7.64 ! [A: code_integer,C: code_integer] :
% 13.25/7.64 ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 13.25/7.64 => ~ ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.64 => ! [B3: code_integer] :
% 13.25/7.64 ( ( B3 != zero_z3403309356797280102nteger )
% 13.25/7.64 => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
% 13.25/7.64 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 13.25/7.64 = B3 )
% 13.25/7.64 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
% 13.25/7.64 = A )
% 13.25/7.64 => ( ( ( times_3573771949741848930nteger @ A @ B3 )
% 13.25/7.64 = one_one_Code_integer )
% 13.25/7.64 => ( ( divide6298287555418463151nteger @ C @ A )
% 13.25/7.64 != ( times_3573771949741848930nteger @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4537_is__unitE,axiom,
% 13.25/7.64 ! [A: nat,C: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ A @ one_one_nat )
% 13.25/7.64 => ~ ( ( A != zero_zero_nat )
% 13.25/7.64 => ! [B3: nat] :
% 13.25/7.64 ( ( B3 != zero_zero_nat )
% 13.25/7.64 => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
% 13.25/7.64 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 13.25/7.64 = B3 )
% 13.25/7.64 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
% 13.25/7.64 = A )
% 13.25/7.64 => ( ( ( times_times_nat @ A @ B3 )
% 13.25/7.64 = one_one_nat )
% 13.25/7.64 => ( ( divide_divide_nat @ C @ A )
% 13.25/7.64 != ( times_times_nat @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4538_is__unitE,axiom,
% 13.25/7.64 ! [A: int,C: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ A @ one_one_int )
% 13.25/7.64 => ~ ( ( A != zero_zero_int )
% 13.25/7.64 => ! [B3: int] :
% 13.25/7.64 ( ( B3 != zero_zero_int )
% 13.25/7.64 => ( ( dvd_dvd_int @ B3 @ one_one_int )
% 13.25/7.64 => ( ( ( divide_divide_int @ one_one_int @ A )
% 13.25/7.64 = B3 )
% 13.25/7.64 => ( ( ( divide_divide_int @ one_one_int @ B3 )
% 13.25/7.64 = A )
% 13.25/7.64 => ( ( ( times_times_int @ A @ B3 )
% 13.25/7.64 = one_one_int )
% 13.25/7.64 => ( ( divide_divide_int @ C @ A )
% 13.25/7.64 != ( times_times_int @ C @ B3 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4539_evenE,axiom,
% 13.25/7.64 ! [A: code_integer] :
% 13.25/7.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ~ ! [B3: code_integer] :
% 13.25/7.64 ( A
% 13.25/7.64 != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4540_evenE,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ~ ! [B3: nat] :
% 13.25/7.64 ( A
% 13.25/7.64 != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4541_evenE,axiom,
% 13.25/7.64 ! [A: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ~ ! [B3: int] :
% 13.25/7.64 ( A
% 13.25/7.64 != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4542_odd__one,axiom,
% 13.25/7.64 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4543_odd__one,axiom,
% 13.25/7.64 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4544_odd__one,axiom,
% 13.25/7.64 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4545_odd__even__add,axiom,
% 13.25/7.64 ! [A: code_integer,B: code_integer] :
% 13.25/7.64 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 13.25/7.64 => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4546_odd__even__add,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 13.25/7.64 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4547_odd__even__add,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 13.25/7.64 => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4548_bit__eq__rec,axiom,
% 13.25/7.64 ( ( ^ [Y6: code_integer,Z4: code_integer] : ( Y6 = Z4 ) )
% 13.25/7.64 = ( ^ [A3: code_integer,B4: code_integer] :
% 13.25/7.64 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
% 13.25/7.64 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) )
% 13.25/7.64 & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.64 = ( divide6298287555418463151nteger @ B4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4549_bit__eq__rec,axiom,
% 13.25/7.64 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.64 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
% 13.25/7.64 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) )
% 13.25/7.64 & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.64 = ( divide_divide_nat @ B4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4550_bit__eq__rec,axiom,
% 13.25/7.64 ( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.64 = ( ^ [A3: int,B4: int] :
% 13.25/7.64 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
% 13.25/7.64 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) )
% 13.25/7.64 & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.64 = ( divide_divide_int @ B4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4551_dvd__power__iff,axiom,
% 13.25/7.64 ! [X: code_integer,M: nat,N: nat] :
% 13.25/7.64 ( ( X != zero_z3403309356797280102nteger )
% 13.25/7.64 => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M ) @ ( power_8256067586552552935nteger @ X @ N ) )
% 13.25/7.64 = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
% 13.25/7.64 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4552_dvd__power__iff,axiom,
% 13.25/7.64 ! [X: nat,M: nat,N: nat] :
% 13.25/7.64 ( ( X != zero_zero_nat )
% 13.25/7.64 => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
% 13.25/7.64 = ( ( dvd_dvd_nat @ X @ one_one_nat )
% 13.25/7.64 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4553_dvd__power__iff,axiom,
% 13.25/7.64 ! [X: int,M: nat,N: nat] :
% 13.25/7.64 ( ( X != zero_zero_int )
% 13.25/7.64 => ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
% 13.25/7.64 = ( ( dvd_dvd_int @ X @ one_one_int )
% 13.25/7.64 | ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4554_dvd__power,axiom,
% 13.25/7.64 ! [N: nat,X: code_integer] :
% 13.25/7.64 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 | ( X = one_one_Code_integer ) )
% 13.25/7.64 => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4555_dvd__power,axiom,
% 13.25/7.64 ! [N: nat,X: rat] :
% 13.25/7.64 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 | ( X = one_one_rat ) )
% 13.25/7.64 => ( dvd_dvd_rat @ X @ ( power_power_rat @ X @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4556_dvd__power,axiom,
% 13.25/7.64 ! [N: nat,X: nat] :
% 13.25/7.64 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 | ( X = one_one_nat ) )
% 13.25/7.64 => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4557_dvd__power,axiom,
% 13.25/7.64 ! [N: nat,X: real] :
% 13.25/7.64 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 | ( X = one_one_real ) )
% 13.25/7.64 => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4558_dvd__power,axiom,
% 13.25/7.64 ! [N: nat,X: int] :
% 13.25/7.64 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 | ( X = one_one_int ) )
% 13.25/7.64 => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4559_dvd__power,axiom,
% 13.25/7.64 ! [N: nat,X: complex] :
% 13.25/7.64 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 | ( X = one_one_complex ) )
% 13.25/7.64 => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4560_even__even__mod__4__iff,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4561_dvd__mult__cancel1,axiom,
% 13.25/7.64 ! [M: nat,N: nat] :
% 13.25/7.64 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.64 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 13.25/7.64 = ( N = one_one_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4562_dvd__mult__cancel2,axiom,
% 13.25/7.64 ! [M: nat,N: nat] :
% 13.25/7.64 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.64 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 13.25/7.64 = ( N = one_one_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4563_dvd__minus__add,axiom,
% 13.25/7.64 ! [Q2: nat,N: nat,R: nat,M: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ Q2 @ N )
% 13.25/7.64 => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R @ M ) )
% 13.25/7.64 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q2 ) )
% 13.25/7.64 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4564_power__dvd__imp__le,axiom,
% 13.25/7.64 ! [I: nat,M: nat,N: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 13.25/7.64 => ( ( ord_less_nat @ one_one_nat @ I )
% 13.25/7.64 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4565_mod__nat__eqI,axiom,
% 13.25/7.64 ! [R: nat,N: nat,M: nat] :
% 13.25/7.64 ( ( ord_less_nat @ R @ N )
% 13.25/7.64 => ( ( ord_less_eq_nat @ R @ M )
% 13.25/7.64 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R ) )
% 13.25/7.64 => ( ( modulo_modulo_nat @ M @ N )
% 13.25/7.64 = R ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4566_mod__int__pos__iff,axiom,
% 13.25/7.64 ! [K: int,L2: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L2 ) )
% 13.25/7.64 = ( ( dvd_dvd_int @ L2 @ K )
% 13.25/7.64 | ( ( L2 = zero_zero_int )
% 13.25/7.64 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 13.25/7.64 | ( ord_less_int @ zero_zero_int @ L2 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4567_bset_I9_J,axiom,
% 13.25/7.64 ! [D: int,D4: int,B5: set_int,T: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ D @ D4 )
% 13.25/7.64 => ! [X5: int] :
% 13.25/7.64 ( ! [Xa3: int] :
% 13.25/7.64 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.64 => ! [Xb3: int] :
% 13.25/7.64 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.64 => ( X5
% 13.25/7.64 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.64 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 13.25/7.64 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4568_bset_I10_J,axiom,
% 13.25/7.64 ! [D: int,D4: int,B5: set_int,T: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ D @ D4 )
% 13.25/7.64 => ! [X5: int] :
% 13.25/7.64 ( ! [Xa3: int] :
% 13.25/7.64 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.64 => ! [Xb3: int] :
% 13.25/7.64 ( ( member_int @ Xb3 @ B5 )
% 13.25/7.64 => ( X5
% 13.25/7.64 != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.64 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 13.25/7.64 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4569_aset_I9_J,axiom,
% 13.25/7.64 ! [D: int,D4: int,A2: set_int,T: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ D @ D4 )
% 13.25/7.64 => ! [X5: int] :
% 13.25/7.64 ( ! [Xa3: int] :
% 13.25/7.64 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.64 => ! [Xb3: int] :
% 13.25/7.64 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.64 => ( X5
% 13.25/7.64 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.64 => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 13.25/7.64 => ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4570_aset_I10_J,axiom,
% 13.25/7.64 ! [D: int,D4: int,A2: set_int,T: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ D @ D4 )
% 13.25/7.64 => ! [X5: int] :
% 13.25/7.64 ( ! [Xa3: int] :
% 13.25/7.64 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 13.25/7.64 => ! [Xb3: int] :
% 13.25/7.64 ( ( member_int @ Xb3 @ A2 )
% 13.25/7.64 => ( X5
% 13.25/7.64 != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
% 13.25/7.64 => ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
% 13.25/7.64 => ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( plus_plus_int @ X5 @ D4 ) @ T ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4571_even__two__times__div__two,axiom,
% 13.25/7.64 ! [A: code_integer] :
% 13.25/7.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 13.25/7.64 = A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4572_even__two__times__div__two,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.64 = A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4573_even__two__times__div__two,axiom,
% 13.25/7.64 ! [A: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 13.25/7.64 = A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4574_even__iff__mod__2__eq__zero,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_zero_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4575_even__iff__mod__2__eq__zero,axiom,
% 13.25/7.64 ! [A: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_zero_int ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4576_even__iff__mod__2__eq__zero,axiom,
% 13.25/7.64 ! [A: code_integer] :
% 13.25/7.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4577_odd__iff__mod__2__eq__one,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.64 = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.64 = one_one_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4578_odd__iff__mod__2__eq__one,axiom,
% 13.25/7.64 ! [A: int] :
% 13.25/7.64 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.64 = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.64 = one_one_int ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4579_odd__iff__mod__2__eq__one,axiom,
% 13.25/7.64 ! [A: code_integer] :
% 13.25/7.64 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.64 = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.64 = one_one_Code_integer ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4580_power__mono__odd,axiom,
% 13.25/7.64 ! [N: nat,A: real,B: real] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ( ord_less_eq_real @ A @ B )
% 13.25/7.64 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4581_power__mono__odd,axiom,
% 13.25/7.64 ! [N: nat,A: rat,B: rat] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4582_power__mono__odd,axiom,
% 13.25/7.64 ! [N: nat,A: int,B: int] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ( ord_less_eq_int @ A @ B )
% 13.25/7.64 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4583_odd__pos,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4584_dvd__power__iff__le,axiom,
% 13.25/7.64 ! [K: nat,M: nat,N: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 13.25/7.64 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 13.25/7.64 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4585_even__unset__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: code_integer] :
% 13.25/7.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 | ( M = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4586_even__unset__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 | ( M = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4587_even__unset__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 | ( M = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4588_even__set__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: code_integer] :
% 13.25/7.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 & ( M != zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4589_even__set__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 & ( M != zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4590_even__set__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 & ( M != zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4591_even__flip__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: code_integer] :
% 13.25/7.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 != ( M = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4592_even__flip__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 != ( M = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4593_even__flip__bit__iff,axiom,
% 13.25/7.64 ! [M: nat,A: nat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 13.25/7.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 != ( M = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4594_even__diff__iff,axiom,
% 13.25/7.64 ! [K: int,L2: int] :
% 13.25/7.64 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L2 ) )
% 13.25/7.64 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4595_oddE,axiom,
% 13.25/7.64 ! [A: code_integer] :
% 13.25/7.64 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ~ ! [B3: code_integer] :
% 13.25/7.64 ( A
% 13.25/7.64 != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) @ one_one_Code_integer ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4596_oddE,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ~ ! [B3: nat] :
% 13.25/7.64 ( A
% 13.25/7.64 != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) @ one_one_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4597_oddE,axiom,
% 13.25/7.64 ! [A: int] :
% 13.25/7.64 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ~ ! [B3: int] :
% 13.25/7.64 ( A
% 13.25/7.64 != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ one_one_int ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4598_parity__cases,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.64 != zero_zero_nat ) )
% 13.25/7.64 => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.64 != one_one_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4599_parity__cases,axiom,
% 13.25/7.64 ! [A: int] :
% 13.25/7.64 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.64 != zero_zero_int ) )
% 13.25/7.64 => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.64 != one_one_int ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4600_parity__cases,axiom,
% 13.25/7.64 ! [A: code_integer] :
% 13.25/7.64 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.64 != zero_z3403309356797280102nteger ) )
% 13.25/7.64 => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.64 != one_one_Code_integer ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4601_mod2__eq__if,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_zero_nat ) )
% 13.25/7.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.64 = one_one_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4602_mod2__eq__if,axiom,
% 13.25/7.64 ! [A: int] :
% 13.25/7.64 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_zero_int ) )
% 13.25/7.64 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.64 = one_one_int ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4603_mod2__eq__if,axiom,
% 13.25/7.64 ! [A: code_integer] :
% 13.25/7.64 ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_z3403309356797280102nteger ) )
% 13.25/7.64 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.64 => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.64 = one_one_Code_integer ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4604_zero__le__even__power,axiom,
% 13.25/7.64 ! [N: nat,A: real] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4605_zero__le__even__power,axiom,
% 13.25/7.64 ! [N: nat,A: rat] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4606_zero__le__even__power,axiom,
% 13.25/7.64 ! [N: nat,A: int] :
% 13.25/7.64 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4607_zero__le__odd__power,axiom,
% 13.25/7.64 ! [N: nat,A: real] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 13.25/7.64 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4608_zero__le__odd__power,axiom,
% 13.25/7.64 ! [N: nat,A: rat] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 13.25/7.64 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4609_zero__le__odd__power,axiom,
% 13.25/7.64 ! [N: nat,A: int] :
% 13.25/7.64 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 13.25/7.64 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4610_zero__le__power__eq,axiom,
% 13.25/7.64 ! [A: real,N: nat] :
% 13.25/7.64 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 13.25/7.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4611_zero__le__power__eq,axiom,
% 13.25/7.64 ! [A: rat,N: nat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 13.25/7.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4612_zero__le__power__eq,axiom,
% 13.25/7.64 ! [A: int,N: nat] :
% 13.25/7.64 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 13.25/7.64 = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4613_leD,axiom,
% 13.25/7.64 ! [Y3: real,X: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ Y3 @ X )
% 13.25/7.64 => ~ ( ord_less_real @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4614_leD,axiom,
% 13.25/7.64 ! [Y3: set_int,X: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ Y3 @ X )
% 13.25/7.64 => ~ ( ord_less_set_int @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4615_leD,axiom,
% 13.25/7.64 ! [Y3: rat,X: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ Y3 @ X )
% 13.25/7.64 => ~ ( ord_less_rat @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4616_leD,axiom,
% 13.25/7.64 ! [Y3: num,X: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ Y3 @ X )
% 13.25/7.64 => ~ ( ord_less_num @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4617_leD,axiom,
% 13.25/7.64 ! [Y3: nat,X: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ Y3 @ X )
% 13.25/7.64 => ~ ( ord_less_nat @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4618_leD,axiom,
% 13.25/7.64 ! [Y3: int,X: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ Y3 @ X )
% 13.25/7.64 => ~ ( ord_less_int @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4619_leI,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ~ ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_real @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4620_leI,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ~ ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_rat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4621_leI,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ~ ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_num @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4622_leI,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ~ ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4623_leI,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ~ ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_int @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4624_nless__le,axiom,
% 13.25/7.64 ! [A: real,B: real] :
% 13.25/7.64 ( ( ~ ( ord_less_real @ A @ B ) )
% 13.25/7.64 = ( ~ ( ord_less_eq_real @ A @ B )
% 13.25/7.64 | ( A = B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4625_nless__le,axiom,
% 13.25/7.64 ! [A: set_int,B: set_int] :
% 13.25/7.64 ( ( ~ ( ord_less_set_int @ A @ B ) )
% 13.25/7.64 = ( ~ ( ord_less_eq_set_int @ A @ B )
% 13.25/7.64 | ( A = B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4626_nless__le,axiom,
% 13.25/7.64 ! [A: rat,B: rat] :
% 13.25/7.64 ( ( ~ ( ord_less_rat @ A @ B ) )
% 13.25/7.64 = ( ~ ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 | ( A = B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4627_nless__le,axiom,
% 13.25/7.64 ! [A: num,B: num] :
% 13.25/7.64 ( ( ~ ( ord_less_num @ A @ B ) )
% 13.25/7.64 = ( ~ ( ord_less_eq_num @ A @ B )
% 13.25/7.64 | ( A = B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4628_nless__le,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ( ~ ( ord_less_nat @ A @ B ) )
% 13.25/7.64 = ( ~ ( ord_less_eq_nat @ A @ B )
% 13.25/7.64 | ( A = B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4629_nless__le,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ( ~ ( ord_less_int @ A @ B ) )
% 13.25/7.64 = ( ~ ( ord_less_eq_int @ A @ B )
% 13.25/7.64 | ( A = B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4630_antisym__conv1,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ~ ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4631_antisym__conv1,axiom,
% 13.25/7.64 ! [X: set_int,Y3: set_int] :
% 13.25/7.64 ( ~ ( ord_less_set_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4632_antisym__conv1,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ~ ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4633_antisym__conv1,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ~ ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4634_antisym__conv1,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ~ ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4635_antisym__conv1,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ~ ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4636_antisym__conv2,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.64 => ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4637_antisym__conv2,axiom,
% 13.25/7.64 ! [X: set_int,Y3: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.64 => ( ( ~ ( ord_less_set_int @ X @ Y3 ) )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4638_antisym__conv2,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.64 => ( ( ~ ( ord_less_rat @ X @ Y3 ) )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4639_antisym__conv2,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.64 => ( ( ~ ( ord_less_num @ X @ Y3 ) )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4640_antisym__conv2,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.64 => ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4641_antisym__conv2,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.64 => ( ( ~ ( ord_less_int @ X @ Y3 ) )
% 13.25/7.64 = ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4642_dense__ge,axiom,
% 13.25/7.64 ! [Z: real,Y3: real] :
% 13.25/7.64 ( ! [X3: real] :
% 13.25/7.64 ( ( ord_less_real @ Z @ X3 )
% 13.25/7.64 => ( ord_less_eq_real @ Y3 @ X3 ) )
% 13.25/7.64 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4643_dense__ge,axiom,
% 13.25/7.64 ! [Z: rat,Y3: rat] :
% 13.25/7.64 ( ! [X3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ Z @ X3 )
% 13.25/7.64 => ( ord_less_eq_rat @ Y3 @ X3 ) )
% 13.25/7.64 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4644_dense__le,axiom,
% 13.25/7.64 ! [Y3: real,Z: real] :
% 13.25/7.64 ( ! [X3: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y3 )
% 13.25/7.64 => ( ord_less_eq_real @ X3 @ Z ) )
% 13.25/7.64 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4645_dense__le,axiom,
% 13.25/7.64 ! [Y3: rat,Z: rat] :
% 13.25/7.64 ( ! [X3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y3 )
% 13.25/7.64 => ( ord_less_eq_rat @ X3 @ Z ) )
% 13.25/7.64 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4646_less__le__not__le,axiom,
% 13.25/7.64 ( ord_less_real
% 13.25/7.64 = ( ^ [X2: real,Y: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ X2 @ Y )
% 13.25/7.64 & ~ ( ord_less_eq_real @ Y @ X2 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4647_less__le__not__le,axiom,
% 13.25/7.64 ( ord_less_set_int
% 13.25/7.64 = ( ^ [X2: set_int,Y: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ X2 @ Y )
% 13.25/7.64 & ~ ( ord_less_eq_set_int @ Y @ X2 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4648_less__le__not__le,axiom,
% 13.25/7.64 ( ord_less_rat
% 13.25/7.64 = ( ^ [X2: rat,Y: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X2 @ Y )
% 13.25/7.64 & ~ ( ord_less_eq_rat @ Y @ X2 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4649_less__le__not__le,axiom,
% 13.25/7.64 ( ord_less_num
% 13.25/7.64 = ( ^ [X2: num,Y: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X2 @ Y )
% 13.25/7.64 & ~ ( ord_less_eq_num @ Y @ X2 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4650_less__le__not__le,axiom,
% 13.25/7.64 ( ord_less_nat
% 13.25/7.64 = ( ^ [X2: nat,Y: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 13.25/7.64 & ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4651_less__le__not__le,axiom,
% 13.25/7.64 ( ord_less_int
% 13.25/7.64 = ( ^ [X2: int,Y: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ X2 @ Y )
% 13.25/7.64 & ~ ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4652_not__le__imp__less,axiom,
% 13.25/7.64 ! [Y3: real,X: real] :
% 13.25/7.64 ( ~ ( ord_less_eq_real @ Y3 @ X )
% 13.25/7.64 => ( ord_less_real @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4653_not__le__imp__less,axiom,
% 13.25/7.64 ! [Y3: rat,X: rat] :
% 13.25/7.64 ( ~ ( ord_less_eq_rat @ Y3 @ X )
% 13.25/7.64 => ( ord_less_rat @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4654_not__le__imp__less,axiom,
% 13.25/7.64 ! [Y3: num,X: num] :
% 13.25/7.64 ( ~ ( ord_less_eq_num @ Y3 @ X )
% 13.25/7.64 => ( ord_less_num @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4655_not__le__imp__less,axiom,
% 13.25/7.64 ! [Y3: nat,X: nat] :
% 13.25/7.64 ( ~ ( ord_less_eq_nat @ Y3 @ X )
% 13.25/7.64 => ( ord_less_nat @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4656_not__le__imp__less,axiom,
% 13.25/7.64 ! [Y3: int,X: int] :
% 13.25/7.64 ( ~ ( ord_less_eq_int @ Y3 @ X )
% 13.25/7.64 => ( ord_less_int @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4657_order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_real
% 13.25/7.64 = ( ^ [A3: real,B4: real] :
% 13.25/7.64 ( ( ord_less_real @ A3 @ B4 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4658_order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_set_int
% 13.25/7.64 = ( ^ [A3: set_int,B4: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ A3 @ B4 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4659_order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_rat
% 13.25/7.64 = ( ^ [A3: rat,B4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A3 @ B4 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4660_order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_num
% 13.25/7.64 = ( ^ [A3: num,B4: num] :
% 13.25/7.64 ( ( ord_less_num @ A3 @ B4 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4661_order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_nat
% 13.25/7.64 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.64 ( ( ord_less_nat @ A3 @ B4 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4662_order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_int
% 13.25/7.64 = ( ^ [A3: int,B4: int] :
% 13.25/7.64 ( ( ord_less_int @ A3 @ B4 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4663_order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_real
% 13.25/7.64 = ( ^ [A3: real,B4: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ A3 @ B4 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4664_order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_set_int
% 13.25/7.64 = ( ^ [A3: set_int,B4: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ A3 @ B4 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4665_order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_rat
% 13.25/7.64 = ( ^ [A3: rat,B4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A3 @ B4 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4666_order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_num
% 13.25/7.64 = ( ^ [A3: num,B4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ A3 @ B4 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4667_order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_nat
% 13.25/7.64 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ A3 @ B4 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4668_order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_int
% 13.25/7.64 = ( ^ [A3: int,B4: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ A3 @ B4 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4669_order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [A: real,B: real,C: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ord_less_real @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4670_order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [A: set_int,B: set_int,C: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ A @ B )
% 13.25/7.64 => ( ( ord_less_set_int @ B @ C )
% 13.25/7.64 => ( ord_less_set_int @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4671_order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [A: rat,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ord_less_rat @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4672_order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [A: num,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_num @ B @ C )
% 13.25/7.64 => ( ord_less_num @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4673_order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [A: nat,B: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.64 => ( ( ord_less_nat @ B @ C )
% 13.25/7.64 => ( ord_less_nat @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4674_order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [A: int,B: int,C: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ A @ B )
% 13.25/7.64 => ( ( ord_less_int @ B @ C )
% 13.25/7.64 => ( ord_less_int @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4675_order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [A: real,B: real,C: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_real @ B @ C )
% 13.25/7.64 => ( ord_less_real @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4676_order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [A: set_int,B: set_int,C: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_set_int @ B @ C )
% 13.25/7.64 => ( ord_less_set_int @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4677_order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.64 => ( ord_less_rat @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4678_order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [A: num,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.64 => ( ord_less_num @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4679_order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [A: nat,B: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_nat @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_nat @ B @ C )
% 13.25/7.64 => ( ord_less_nat @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4680_order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [A: int,B: int,C: int] :
% 13.25/7.64 ( ( ord_less_int @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_int @ B @ C )
% 13.25/7.64 => ( ord_less_int @ A @ C ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4681_order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_real
% 13.25/7.64 = ( ^ [A3: real,B4: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ A3 @ B4 )
% 13.25/7.64 & ~ ( ord_less_eq_real @ B4 @ A3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4682_order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_set_int
% 13.25/7.64 = ( ^ [A3: set_int,B4: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ A3 @ B4 )
% 13.25/7.64 & ~ ( ord_less_eq_set_int @ B4 @ A3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4683_order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_rat
% 13.25/7.64 = ( ^ [A3: rat,B4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A3 @ B4 )
% 13.25/7.64 & ~ ( ord_less_eq_rat @ B4 @ A3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4684_order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_num
% 13.25/7.64 = ( ^ [A3: num,B4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ A3 @ B4 )
% 13.25/7.64 & ~ ( ord_less_eq_num @ B4 @ A3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4685_order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_nat
% 13.25/7.64 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ A3 @ B4 )
% 13.25/7.64 & ~ ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4686_order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_int
% 13.25/7.64 = ( ^ [A3: int,B4: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ A3 @ B4 )
% 13.25/7.64 & ~ ( ord_less_eq_int @ B4 @ A3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4687_dense__ge__bounded,axiom,
% 13.25/7.64 ! [Z: real,X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ Z @ X )
% 13.25/7.64 => ( ! [W2: real] :
% 13.25/7.64 ( ( ord_less_real @ Z @ W2 )
% 13.25/7.64 => ( ( ord_less_real @ W2 @ X )
% 13.25/7.64 => ( ord_less_eq_real @ Y3 @ W2 ) ) )
% 13.25/7.64 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4688_dense__ge__bounded,axiom,
% 13.25/7.64 ! [Z: rat,X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ Z @ X )
% 13.25/7.64 => ( ! [W2: rat] :
% 13.25/7.64 ( ( ord_less_rat @ Z @ W2 )
% 13.25/7.64 => ( ( ord_less_rat @ W2 @ X )
% 13.25/7.64 => ( ord_less_eq_rat @ Y3 @ W2 ) ) )
% 13.25/7.64 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4689_dense__le__bounded,axiom,
% 13.25/7.64 ! [X: real,Y3: real,Z: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( ! [W2: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ W2 )
% 13.25/7.64 => ( ( ord_less_real @ W2 @ Y3 )
% 13.25/7.64 => ( ord_less_eq_real @ W2 @ Z ) ) )
% 13.25/7.64 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4690_dense__le__bounded,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( ! [W2: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ W2 )
% 13.25/7.64 => ( ( ord_less_rat @ W2 @ Y3 )
% 13.25/7.64 => ( ord_less_eq_rat @ W2 @ Z ) ) )
% 13.25/7.64 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4691_dual__order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_real
% 13.25/7.64 = ( ^ [B4: real,A3: real] :
% 13.25/7.64 ( ( ord_less_real @ B4 @ A3 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4692_dual__order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_set_int
% 13.25/7.64 = ( ^ [B4: set_int,A3: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ B4 @ A3 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4693_dual__order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_rat
% 13.25/7.64 = ( ^ [B4: rat,A3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ B4 @ A3 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4694_dual__order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_num
% 13.25/7.64 = ( ^ [B4: num,A3: num] :
% 13.25/7.64 ( ( ord_less_num @ B4 @ A3 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4695_dual__order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_nat
% 13.25/7.64 = ( ^ [B4: nat,A3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ B4 @ A3 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4696_dual__order_Oorder__iff__strict,axiom,
% 13.25/7.64 ( ord_less_eq_int
% 13.25/7.64 = ( ^ [B4: int,A3: int] :
% 13.25/7.64 ( ( ord_less_int @ B4 @ A3 )
% 13.25/7.64 | ( A3 = B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4697_dual__order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_real
% 13.25/7.64 = ( ^ [B4: real,A3: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ B4 @ A3 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4698_dual__order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_set_int
% 13.25/7.64 = ( ^ [B4: set_int,A3: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ B4 @ A3 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4699_dual__order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_rat
% 13.25/7.64 = ( ^ [B4: rat,A3: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ B4 @ A3 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4700_dual__order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_num
% 13.25/7.64 = ( ^ [B4: num,A3: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ B4 @ A3 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4701_dual__order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_nat
% 13.25/7.64 = ( ^ [B4: nat,A3: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ B4 @ A3 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4702_dual__order_Ostrict__iff__order,axiom,
% 13.25/7.64 ( ord_less_int
% 13.25/7.64 = ( ^ [B4: int,A3: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ B4 @ A3 )
% 13.25/7.64 & ( A3 != B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4703_dual__order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [B: real,A: real,C: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ B @ A )
% 13.25/7.64 => ( ( ord_less_real @ C @ B )
% 13.25/7.64 => ( ord_less_real @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4704_dual__order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [B: set_int,A: set_int,C: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ B @ A )
% 13.25/7.64 => ( ( ord_less_set_int @ C @ B )
% 13.25/7.64 => ( ord_less_set_int @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4705_dual__order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [B: rat,A: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ B @ A )
% 13.25/7.64 => ( ( ord_less_rat @ C @ B )
% 13.25/7.64 => ( ord_less_rat @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4706_dual__order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [B: num,A: num,C: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ B @ A )
% 13.25/7.64 => ( ( ord_less_num @ C @ B )
% 13.25/7.64 => ( ord_less_num @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4707_dual__order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [B: nat,A: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ B @ A )
% 13.25/7.64 => ( ( ord_less_nat @ C @ B )
% 13.25/7.64 => ( ord_less_nat @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4708_dual__order_Ostrict__trans1,axiom,
% 13.25/7.64 ! [B: int,A: int,C: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ B @ A )
% 13.25/7.64 => ( ( ord_less_int @ C @ B )
% 13.25/7.64 => ( ord_less_int @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4709_dual__order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [B: real,A: real,C: real] :
% 13.25/7.64 ( ( ord_less_real @ B @ A )
% 13.25/7.64 => ( ( ord_less_eq_real @ C @ B )
% 13.25/7.64 => ( ord_less_real @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4710_dual__order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [B: set_int,A: set_int,C: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ B @ A )
% 13.25/7.64 => ( ( ord_less_eq_set_int @ C @ B )
% 13.25/7.64 => ( ord_less_set_int @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4711_dual__order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [B: rat,A: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ B @ A )
% 13.25/7.64 => ( ( ord_less_eq_rat @ C @ B )
% 13.25/7.64 => ( ord_less_rat @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4712_dual__order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [B: num,A: num,C: num] :
% 13.25/7.64 ( ( ord_less_num @ B @ A )
% 13.25/7.64 => ( ( ord_less_eq_num @ C @ B )
% 13.25/7.64 => ( ord_less_num @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4713_dual__order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [B: nat,A: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_nat @ B @ A )
% 13.25/7.64 => ( ( ord_less_eq_nat @ C @ B )
% 13.25/7.64 => ( ord_less_nat @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4714_dual__order_Ostrict__trans2,axiom,
% 13.25/7.64 ! [B: int,A: int,C: int] :
% 13.25/7.64 ( ( ord_less_int @ B @ A )
% 13.25/7.64 => ( ( ord_less_eq_int @ C @ B )
% 13.25/7.64 => ( ord_less_int @ C @ A ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4715_dual__order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_real
% 13.25/7.64 = ( ^ [B4: real,A3: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ B4 @ A3 )
% 13.25/7.64 & ~ ( ord_less_eq_real @ A3 @ B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4716_dual__order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_set_int
% 13.25/7.64 = ( ^ [B4: set_int,A3: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ B4 @ A3 )
% 13.25/7.64 & ~ ( ord_less_eq_set_int @ A3 @ B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4717_dual__order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_rat
% 13.25/7.64 = ( ^ [B4: rat,A3: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ B4 @ A3 )
% 13.25/7.64 & ~ ( ord_less_eq_rat @ A3 @ B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4718_dual__order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_num
% 13.25/7.64 = ( ^ [B4: num,A3: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ B4 @ A3 )
% 13.25/7.64 & ~ ( ord_less_eq_num @ A3 @ B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4719_dual__order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_nat
% 13.25/7.64 = ( ^ [B4: nat,A3: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ B4 @ A3 )
% 13.25/7.64 & ~ ( ord_less_eq_nat @ A3 @ B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4720_dual__order_Ostrict__iff__not,axiom,
% 13.25/7.64 ( ord_less_int
% 13.25/7.64 = ( ^ [B4: int,A3: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ B4 @ A3 )
% 13.25/7.64 & ~ ( ord_less_eq_int @ A3 @ B4 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4721_order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [A: real,B: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ord_less_eq_real @ A @ B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4722_order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [A: set_int,B: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ A @ B )
% 13.25/7.64 => ( ord_less_eq_set_int @ A @ B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4723_order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [A: rat,B: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ord_less_eq_rat @ A @ B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4724_order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [A: num,B: num] :
% 13.25/7.64 ( ( ord_less_num @ A @ B )
% 13.25/7.64 => ( ord_less_eq_num @ A @ B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4725_order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ( ord_less_nat @ A @ B )
% 13.25/7.64 => ( ord_less_eq_nat @ A @ B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4726_order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ( ord_less_int @ A @ B )
% 13.25/7.64 => ( ord_less_eq_int @ A @ B ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4727_dual__order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [B: real,A: real] :
% 13.25/7.64 ( ( ord_less_real @ B @ A )
% 13.25/7.64 => ( ord_less_eq_real @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4728_dual__order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [B: set_int,A: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ B @ A )
% 13.25/7.64 => ( ord_less_eq_set_int @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4729_dual__order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [B: rat,A: rat] :
% 13.25/7.64 ( ( ord_less_rat @ B @ A )
% 13.25/7.64 => ( ord_less_eq_rat @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4730_dual__order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [B: num,A: num] :
% 13.25/7.64 ( ( ord_less_num @ B @ A )
% 13.25/7.64 => ( ord_less_eq_num @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4731_dual__order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [B: nat,A: nat] :
% 13.25/7.64 ( ( ord_less_nat @ B @ A )
% 13.25/7.64 => ( ord_less_eq_nat @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4732_dual__order_Ostrict__implies__order,axiom,
% 13.25/7.64 ! [B: int,A: int] :
% 13.25/7.64 ( ( ord_less_int @ B @ A )
% 13.25/7.64 => ( ord_less_eq_int @ B @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4733_order__le__less,axiom,
% 13.25/7.64 ( ord_less_eq_real
% 13.25/7.64 = ( ^ [X2: real,Y: real] :
% 13.25/7.64 ( ( ord_less_real @ X2 @ Y )
% 13.25/7.64 | ( X2 = Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4734_order__le__less,axiom,
% 13.25/7.64 ( ord_less_eq_set_int
% 13.25/7.64 = ( ^ [X2: set_int,Y: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ X2 @ Y )
% 13.25/7.64 | ( X2 = Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4735_order__le__less,axiom,
% 13.25/7.64 ( ord_less_eq_rat
% 13.25/7.64 = ( ^ [X2: rat,Y: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X2 @ Y )
% 13.25/7.64 | ( X2 = Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4736_order__le__less,axiom,
% 13.25/7.64 ( ord_less_eq_num
% 13.25/7.64 = ( ^ [X2: num,Y: num] :
% 13.25/7.64 ( ( ord_less_num @ X2 @ Y )
% 13.25/7.64 | ( X2 = Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4737_order__le__less,axiom,
% 13.25/7.64 ( ord_less_eq_nat
% 13.25/7.64 = ( ^ [X2: nat,Y: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X2 @ Y )
% 13.25/7.64 | ( X2 = Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4738_order__le__less,axiom,
% 13.25/7.64 ( ord_less_eq_int
% 13.25/7.64 = ( ^ [X2: int,Y: int] :
% 13.25/7.64 ( ( ord_less_int @ X2 @ Y )
% 13.25/7.64 | ( X2 = Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4739_order__less__le,axiom,
% 13.25/7.64 ( ord_less_real
% 13.25/7.64 = ( ^ [X2: real,Y: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ X2 @ Y )
% 13.25/7.64 & ( X2 != Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4740_order__less__le,axiom,
% 13.25/7.64 ( ord_less_set_int
% 13.25/7.64 = ( ^ [X2: set_int,Y: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ X2 @ Y )
% 13.25/7.64 & ( X2 != Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4741_order__less__le,axiom,
% 13.25/7.64 ( ord_less_rat
% 13.25/7.64 = ( ^ [X2: rat,Y: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X2 @ Y )
% 13.25/7.64 & ( X2 != Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4742_order__less__le,axiom,
% 13.25/7.64 ( ord_less_num
% 13.25/7.64 = ( ^ [X2: num,Y: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X2 @ Y )
% 13.25/7.64 & ( X2 != Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4743_order__less__le,axiom,
% 13.25/7.64 ( ord_less_nat
% 13.25/7.64 = ( ^ [X2: nat,Y: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ X2 @ Y )
% 13.25/7.64 & ( X2 != Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4744_order__less__le,axiom,
% 13.25/7.64 ( ord_less_int
% 13.25/7.64 = ( ^ [X2: int,Y: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ X2 @ Y )
% 13.25/7.64 & ( X2 != Y ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4745_linorder__not__le,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ~ ( ord_less_eq_real @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_real @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4746_linorder__not__le,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ~ ( ord_less_eq_rat @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_rat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4747_linorder__not__le,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ~ ( ord_less_eq_num @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_num @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4748_linorder__not__le,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ~ ( ord_less_eq_nat @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_nat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4749_linorder__not__le,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ~ ( ord_less_eq_int @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_int @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4750_linorder__not__less,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ~ ( ord_less_real @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_eq_real @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4751_linorder__not__less,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ~ ( ord_less_rat @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_eq_rat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4752_linorder__not__less,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ~ ( ord_less_num @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_eq_num @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4753_linorder__not__less,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ~ ( ord_less_nat @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_eq_nat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4754_linorder__not__less,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ~ ( ord_less_int @ X @ Y3 ) )
% 13.25/7.64 = ( ord_less_eq_int @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4755_order__less__imp__le,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4756_order__less__imp__le,axiom,
% 13.25/7.64 ! [X: set_int,Y3: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_set_int @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4757_order__less__imp__le,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4758_order__less__imp__le,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_num @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4759_order__less__imp__le,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_nat @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4760_order__less__imp__le,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( ord_less_eq_int @ X @ Y3 ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4761_order__le__neq__trans,axiom,
% 13.25/7.64 ! [A: real,B: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.64 => ( ( A != B )
% 13.25/7.64 => ( ord_less_real @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4762_order__le__neq__trans,axiom,
% 13.25/7.64 ! [A: set_int,B: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ A @ B )
% 13.25/7.64 => ( ( A != B )
% 13.25/7.64 => ( ord_less_set_int @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4763_order__le__neq__trans,axiom,
% 13.25/7.64 ! [A: rat,B: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ( A != B )
% 13.25/7.64 => ( ord_less_rat @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4764_order__le__neq__trans,axiom,
% 13.25/7.64 ! [A: num,B: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.64 => ( ( A != B )
% 13.25/7.64 => ( ord_less_num @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4765_order__le__neq__trans,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.64 => ( ( A != B )
% 13.25/7.64 => ( ord_less_nat @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4766_order__le__neq__trans,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ A @ B )
% 13.25/7.64 => ( ( A != B )
% 13.25/7.64 => ( ord_less_int @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4767_order__neq__le__trans,axiom,
% 13.25/7.64 ! [A: real,B: real] :
% 13.25/7.64 ( ( A != B )
% 13.25/7.64 => ( ( ord_less_eq_real @ A @ B )
% 13.25/7.64 => ( ord_less_real @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4768_order__neq__le__trans,axiom,
% 13.25/7.64 ! [A: set_int,B: set_int] :
% 13.25/7.64 ( ( A != B )
% 13.25/7.64 => ( ( ord_less_eq_set_int @ A @ B )
% 13.25/7.64 => ( ord_less_set_int @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4769_order__neq__le__trans,axiom,
% 13.25/7.64 ! [A: rat,B: rat] :
% 13.25/7.64 ( ( A != B )
% 13.25/7.64 => ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ord_less_rat @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4770_order__neq__le__trans,axiom,
% 13.25/7.64 ! [A: num,B: num] :
% 13.25/7.64 ( ( A != B )
% 13.25/7.64 => ( ( ord_less_eq_num @ A @ B )
% 13.25/7.64 => ( ord_less_num @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4771_order__neq__le__trans,axiom,
% 13.25/7.64 ! [A: nat,B: nat] :
% 13.25/7.64 ( ( A != B )
% 13.25/7.64 => ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.64 => ( ord_less_nat @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4772_order__neq__le__trans,axiom,
% 13.25/7.64 ! [A: int,B: int] :
% 13.25/7.64 ( ( A != B )
% 13.25/7.64 => ( ( ord_less_eq_int @ A @ B )
% 13.25/7.64 => ( ord_less_int @ A @ B ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4773_order__le__less__trans,axiom,
% 13.25/7.64 ! [X: real,Y3: real,Z: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_real @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_real @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4774_order__le__less__trans,axiom,
% 13.25/7.64 ! [X: set_int,Y3: set_int,Z: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_set_int @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_set_int @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4775_order__le__less__trans,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_rat @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_rat @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4776_order__le__less__trans,axiom,
% 13.25/7.64 ! [X: num,Y3: num,Z: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_num @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_num @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4777_order__le__less__trans,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_nat @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_nat @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4778_order__le__less__trans,axiom,
% 13.25/7.64 ! [X: int,Y3: int,Z: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_int @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_int @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4779_order__less__le__trans,axiom,
% 13.25/7.64 ! [X: real,Y3: real,Z: real] :
% 13.25/7.64 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_real @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_real @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4780_order__less__le__trans,axiom,
% 13.25/7.64 ! [X: set_int,Y3: set_int,Z: set_int] :
% 13.25/7.64 ( ( ord_less_set_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_set_int @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4781_order__less__le__trans,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat,Z: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_rat @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4782_order__less__le__trans,axiom,
% 13.25/7.64 ! [X: num,Y3: num,Z: num] :
% 13.25/7.64 ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_num @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_num @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4783_order__less__le__trans,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_nat @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4784_order__less__le__trans,axiom,
% 13.25/7.64 ! [X: int,Y3: int,Z: int] :
% 13.25/7.64 ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_int @ Y3 @ Z )
% 13.25/7.64 => ( ord_less_int @ X @ Z ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4785_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: real > real,B: real,C: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4786_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: rat > real,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4787_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: num > real,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4788_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: nat > real,B: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_nat @ B @ C )
% 13.25/7.64 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4789_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: real,F: int > real,B: int,C: int] :
% 13.25/7.64 ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_int @ B @ C )
% 13.25/7.64 => ( ! [X3: int,Y4: int] :
% 13.25/7.64 ( ( ord_less_int @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4790_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: real > rat,B: real,C: real] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_real @ B @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4791_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4792_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: num > rat,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4793_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_nat @ B @ C )
% 13.25/7.64 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4794_order__le__less__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: int > rat,B: int,C: int] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_int @ B @ C )
% 13.25/7.64 => ( ! [X3: int,Y4: int] :
% 13.25/7.64 ( ( ord_less_int @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4795_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > real,C: real] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4796_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4797_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > num,C: num] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_num @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4798_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4799_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > int,C: int] :
% 13.25/7.64 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_int @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4800_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: num,B: num,F: num > real,C: real] :
% 13.25/7.64 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4801_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: num,B: num,F: num > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4802_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: num,B: num,F: num > num,C: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_num @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4803_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: num,B: num,F: num > nat,C: nat] :
% 13.25/7.64 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4804_order__le__less__subst2,axiom,
% 13.25/7.64 ! [A: num,B: num,F: num > int,C: int] :
% 13.25/7.64 ( ( ord_less_eq_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_int @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4805_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: real,F: rat > real,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4806_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4807_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: num,F: rat > num,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_num @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4808_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_nat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4809_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: int,F: rat > int,B: rat,C: rat] :
% 13.25/7.64 ( ( ord_less_int @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_rat @ B @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4810_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: real,F: num > real,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_real @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4811_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: rat,F: num > rat,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_rat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4812_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: num,F: num > num,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_num @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4813_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: nat,F: num > nat,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_nat @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4814_order__less__le__subst1,axiom,
% 13.25/7.64 ! [A: int,F: num > int,B: num,C: num] :
% 13.25/7.64 ( ( ord_less_int @ A @ ( F @ B ) )
% 13.25/7.64 => ( ( ord_less_eq_num @ B @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4815_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > real,C: real] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4816_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > real,C: real] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4817_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: num,B: num,F: num > real,C: real] :
% 13.25/7.64 ( ( ord_less_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4818_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: nat,B: nat,F: nat > real,C: real] :
% 13.25/7.64 ( ( ord_less_nat @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4819_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: int,B: int,F: int > real,C: real] :
% 13.25/7.64 ( ( ord_less_int @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: int,Y4: int] :
% 13.25/7.64 ( ( ord_less_int @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_real @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4820_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: real,B: real,F: real > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_real @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: real,Y4: real] :
% 13.25/7.64 ( ( ord_less_real @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4821_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_rat @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: rat,Y4: rat] :
% 13.25/7.64 ( ( ord_less_rat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4822_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: num,B: num,F: num > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_num @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: num,Y4: num] :
% 13.25/7.64 ( ( ord_less_num @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4823_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_nat @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: nat,Y4: nat] :
% 13.25/7.64 ( ( ord_less_nat @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4824_order__less__le__subst2,axiom,
% 13.25/7.64 ! [A: int,B: int,F: int > rat,C: rat] :
% 13.25/7.64 ( ( ord_less_int @ A @ B )
% 13.25/7.64 => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 13.25/7.64 => ( ! [X3: int,Y4: int] :
% 13.25/7.64 ( ( ord_less_int @ X3 @ Y4 )
% 13.25/7.64 => ( ord_less_rat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 13.25/7.64 => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4825_linorder__le__less__linear,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.64 | ( ord_less_real @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4826_linorder__le__less__linear,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.64 | ( ord_less_rat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4827_linorder__le__less__linear,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.64 | ( ord_less_num @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4828_linorder__le__less__linear,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.64 | ( ord_less_nat @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4829_linorder__le__less__linear,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.64 | ( ord_less_int @ Y3 @ X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4830_order__le__imp__less__or__eq,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_real @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4831_order__le__imp__less__or__eq,axiom,
% 13.25/7.64 ! [X: set_int,Y3: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_set_int @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4832_order__le__imp__less__or__eq,axiom,
% 13.25/7.64 ! [X: rat,Y3: rat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_rat @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4833_order__le__imp__less__or__eq,axiom,
% 13.25/7.64 ! [X: num,Y3: num] :
% 13.25/7.64 ( ( ord_less_eq_num @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_num @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4834_order__le__imp__less__or__eq,axiom,
% 13.25/7.64 ! [X: nat,Y3: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4835_order__le__imp__less__or__eq,axiom,
% 13.25/7.64 ! [X: int,Y3: int] :
% 13.25/7.64 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.64 => ( ( ord_less_int @ X @ Y3 )
% 13.25/7.64 | ( X = Y3 ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4836_bot_Oextremum,axiom,
% 13.25/7.64 ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4837_bot_Oextremum,axiom,
% 13.25/7.64 ! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4838_bot_Oextremum,axiom,
% 13.25/7.64 ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4839_bot_Oextremum,axiom,
% 13.25/7.64 ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4840_bot_Oextremum,axiom,
% 13.25/7.64 ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4841_bot_Oextremum__unique,axiom,
% 13.25/7.64 ! [A: set_nat] :
% 13.25/7.64 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 13.25/7.64 = ( A = bot_bot_set_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4842_bot_Oextremum__unique,axiom,
% 13.25/7.64 ! [A: extended_enat] :
% 13.25/7.64 ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 13.25/7.64 = ( A = bot_bo4199563552545308370d_enat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4843_bot_Oextremum__unique,axiom,
% 13.25/7.64 ! [A: set_real] :
% 13.25/7.64 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 13.25/7.64 = ( A = bot_bot_set_real ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4844_bot_Oextremum__unique,axiom,
% 13.25/7.64 ! [A: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 13.25/7.64 = ( A = bot_bot_set_int ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4845_bot_Oextremum__unique,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 13.25/7.64 = ( A = bot_bot_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4846_bot_Oextremum__uniqueI,axiom,
% 13.25/7.64 ! [A: set_nat] :
% 13.25/7.64 ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 13.25/7.64 => ( A = bot_bot_set_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4847_bot_Oextremum__uniqueI,axiom,
% 13.25/7.64 ! [A: extended_enat] :
% 13.25/7.64 ( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
% 13.25/7.64 => ( A = bot_bo4199563552545308370d_enat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4848_bot_Oextremum__uniqueI,axiom,
% 13.25/7.64 ! [A: set_real] :
% 13.25/7.64 ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 13.25/7.64 => ( A = bot_bot_set_real ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4849_bot_Oextremum__uniqueI,axiom,
% 13.25/7.64 ! [A: set_int] :
% 13.25/7.64 ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 13.25/7.64 => ( A = bot_bot_set_int ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4850_bot_Oextremum__uniqueI,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 13.25/7.64 => ( A = bot_bot_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4851_bot_Oextremum__strict,axiom,
% 13.25/7.64 ! [A: set_nat] :
% 13.25/7.64 ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4852_bot_Oextremum__strict,axiom,
% 13.25/7.64 ! [A: extended_enat] :
% 13.25/7.64 ~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4853_bot_Oextremum__strict,axiom,
% 13.25/7.64 ! [A: set_int] :
% 13.25/7.64 ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4854_bot_Oextremum__strict,axiom,
% 13.25/7.64 ! [A: set_real] :
% 13.25/7.64 ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4855_bot_Oextremum__strict,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4856_bot_Onot__eq__extremum,axiom,
% 13.25/7.64 ! [A: set_nat] :
% 13.25/7.64 ( ( A != bot_bot_set_nat )
% 13.25/7.64 = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4857_bot_Onot__eq__extremum,axiom,
% 13.25/7.64 ! [A: extended_enat] :
% 13.25/7.64 ( ( A != bot_bo4199563552545308370d_enat )
% 13.25/7.64 = ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4858_bot_Onot__eq__extremum,axiom,
% 13.25/7.64 ! [A: set_int] :
% 13.25/7.64 ( ( A != bot_bot_set_int )
% 13.25/7.64 = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4859_bot_Onot__eq__extremum,axiom,
% 13.25/7.64 ! [A: set_real] :
% 13.25/7.64 ( ( A != bot_bot_set_real )
% 13.25/7.64 = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4860_bot_Onot__eq__extremum,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( A != bot_bot_nat )
% 13.25/7.64 = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4861_zero__less__power__eq,axiom,
% 13.25/7.64 ! [A: real,N: nat] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) )
% 13.25/7.64 = ( ( N = zero_zero_nat )
% 13.25/7.64 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( A != zero_zero_real ) )
% 13.25/7.64 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4862_zero__less__power__eq,axiom,
% 13.25/7.64 ! [A: rat,N: nat] :
% 13.25/7.64 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N ) )
% 13.25/7.64 = ( ( N = zero_zero_nat )
% 13.25/7.64 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( A != zero_zero_rat ) )
% 13.25/7.64 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4863_zero__less__power__eq,axiom,
% 13.25/7.64 ! [A: int,N: nat] :
% 13.25/7.64 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) )
% 13.25/7.64 = ( ( N = zero_zero_nat )
% 13.25/7.64 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( A != zero_zero_int ) )
% 13.25/7.64 | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4864_Euclid__induct,axiom,
% 13.25/7.64 ! [P: nat > nat > $o,A: nat,B: nat] :
% 13.25/7.64 ( ! [A5: nat,B3: nat] :
% 13.25/7.64 ( ( P @ A5 @ B3 )
% 13.25/7.64 = ( P @ B3 @ A5 ) )
% 13.25/7.64 => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
% 13.25/7.64 => ( ! [A5: nat,B3: nat] :
% 13.25/7.64 ( ( P @ A5 @ B3 )
% 13.25/7.64 => ( P @ A5 @ ( plus_plus_nat @ A5 @ B3 ) ) )
% 13.25/7.64 => ( P @ A @ B ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4865_even__mask__div__iff_H,axiom,
% 13.25/7.64 ! [M: nat,N: nat] :
% 13.25/7.64 ( ( 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 ) ) )
% 13.25/7.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4866_even__mask__div__iff_H,axiom,
% 13.25/7.64 ! [M: nat,N: nat] :
% 13.25/7.64 ( ( 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 ) ) )
% 13.25/7.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4867_even__mask__div__iff_H,axiom,
% 13.25/7.64 ! [M: nat,N: nat] :
% 13.25/7.64 ( ( 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 ) ) )
% 13.25/7.64 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4868_vebt__insert_Osimps_I5_J,axiom,
% 13.25/7.64 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
% 13.25/7.64 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X )
% 13.25/7.64 = ( if_VEBT_VEBT
% 13.25/7.64 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 13.25/7.64 & ~ ( ( X = Mi )
% 13.25/7.64 | ( X = Ma ) ) )
% 13.25/7.64 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 13.25/7.64 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4869_power__le__zero__eq,axiom,
% 13.25/7.64 ! [A: real,N: nat] :
% 13.25/7.64 ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ zero_zero_real )
% 13.25/7.64 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 13.25/7.64 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( A = zero_zero_real ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4870_power__le__zero__eq,axiom,
% 13.25/7.64 ! [A: rat,N: nat] :
% 13.25/7.64 ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ zero_zero_rat )
% 13.25/7.64 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 13.25/7.64 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( A = zero_zero_rat ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4871_power__le__zero__eq,axiom,
% 13.25/7.64 ! [A: int,N: nat] :
% 13.25/7.64 ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ zero_zero_int )
% 13.25/7.64 = ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.64 & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 13.25/7.64 | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 & ( A = zero_zero_int ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4872_option_Osize__gen_I1_J,axiom,
% 13.25/7.64 ! [X: nat > nat] :
% 13.25/7.64 ( ( size_option_nat @ X @ none_nat )
% 13.25/7.64 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4873_option_Osize__gen_I1_J,axiom,
% 13.25/7.64 ! [X: product_prod_nat_nat > nat] :
% 13.25/7.64 ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
% 13.25/7.64 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4874_option_Osize__gen_I1_J,axiom,
% 13.25/7.64 ! [X: num > nat] :
% 13.25/7.64 ( ( size_option_num @ X @ none_num )
% 13.25/7.64 = ( suc @ zero_zero_nat ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4875_even__mod__4__div__2,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.64 = ( suc @ zero_zero_nat ) )
% 13.25/7.64 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4876_even__mask__div__iff,axiom,
% 13.25/7.64 ! [M: nat,N: nat] :
% 13.25/7.64 ( ( 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 ) ) )
% 13.25/7.64 = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 = zero_z3403309356797280102nteger )
% 13.25/7.64 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4877_even__mask__div__iff,axiom,
% 13.25/7.64 ! [M: nat,N: nat] :
% 13.25/7.64 ( ( 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 ) ) )
% 13.25/7.64 = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 = zero_zero_nat )
% 13.25/7.64 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4878_even__mask__div__iff,axiom,
% 13.25/7.64 ! [M: nat,N: nat] :
% 13.25/7.64 ( ( 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 ) ) )
% 13.25/7.64 = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 = zero_zero_int )
% 13.25/7.64 | ( ord_less_eq_nat @ M @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4879_even__mult__exp__div__exp__iff,axiom,
% 13.25/7.64 ! [A: code_integer,M: nat,N: nat] :
% 13.25/7.64 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.64 = ( ( ord_less_nat @ N @ M )
% 13.25/7.64 | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 = zero_z3403309356797280102nteger )
% 13.25/7.64 | ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.64 & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4880_even__mult__exp__div__exp__iff,axiom,
% 13.25/7.64 ! [A: nat,M: nat,N: nat] :
% 13.25/7.64 ( ( 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 ) ) )
% 13.25/7.64 = ( ( ord_less_nat @ N @ M )
% 13.25/7.64 | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 = zero_zero_nat )
% 13.25/7.64 | ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.64 & ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4881_even__mult__exp__div__exp__iff,axiom,
% 13.25/7.64 ! [A: int,M: nat,N: nat] :
% 13.25/7.64 ( ( 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 ) ) )
% 13.25/7.64 = ( ( ord_less_nat @ N @ M )
% 13.25/7.64 | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 13.25/7.64 = zero_zero_int )
% 13.25/7.64 | ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.64 & ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4882_vebt__insert_Oelims,axiom,
% 13.25/7.64 ! [X: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
% 13.25/7.64 ( ( ( vEBT_vebt_insert @ X @ Xa )
% 13.25/7.64 = Y3 )
% 13.25/7.64 => ( ! [A5: $o,B3: $o] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.64 => ~ ( ( ( Xa = zero_zero_nat )
% 13.25/7.64 => ( Y3
% 13.25/7.64 = ( vEBT_Leaf @ $true @ B3 ) ) )
% 13.25/7.64 & ( ( Xa != zero_zero_nat )
% 13.25/7.64 => ( ( ( Xa = one_one_nat )
% 13.25/7.64 => ( Y3
% 13.25/7.64 = ( vEBT_Leaf @ A5 @ $true ) ) )
% 13.25/7.64 & ( ( Xa != one_one_nat )
% 13.25/7.64 => ( Y3
% 13.25/7.64 = ( vEBT_Leaf @ A5 @ B3 ) ) ) ) ) ) )
% 13.25/7.64 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 13.25/7.64 => ( Y3
% 13.25/7.64 != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) ) )
% 13.25/7.64 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 13.25/7.64 => ( Y3
% 13.25/7.64 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) ) )
% 13.25/7.64 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.64 => ( Y3
% 13.25/7.64 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) ) )
% 13.25/7.64 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.64 => ( Y3
% 13.25/7.64 != ( if_VEBT_VEBT
% 13.25/7.64 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.64 & ~ ( ( Xa = Mi2 )
% 13.25/7.64 | ( Xa = Ma2 ) ) )
% 13.25/7.64 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 13.25/7.64 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4883_vebt__insert_Opelims,axiom,
% 13.25/7.64 ! [X: vEBT_VEBT,Xa: nat,Y3: vEBT_VEBT] :
% 13.25/7.64 ( ( ( vEBT_vebt_insert @ X @ Xa )
% 13.25/7.64 = Y3 )
% 13.25/7.64 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.64 => ( ! [A5: $o,B3: $o] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.64 => ( ( ( ( Xa = zero_zero_nat )
% 13.25/7.64 => ( Y3
% 13.25/7.64 = ( vEBT_Leaf @ $true @ B3 ) ) )
% 13.25/7.64 & ( ( Xa != zero_zero_nat )
% 13.25/7.64 => ( ( ( Xa = one_one_nat )
% 13.25/7.64 => ( Y3
% 13.25/7.64 = ( vEBT_Leaf @ A5 @ $true ) ) )
% 13.25/7.64 & ( ( Xa != one_one_nat )
% 13.25/7.64 => ( Y3
% 13.25/7.64 = ( vEBT_Leaf @ A5 @ B3 ) ) ) ) ) )
% 13.25/7.64 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B3 ) @ Xa ) ) ) )
% 13.25/7.64 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 13.25/7.64 => ( ( Y3
% 13.25/7.64 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) )
% 13.25/7.64 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S3 ) @ Xa ) ) ) )
% 13.25/7.64 => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 13.25/7.64 => ( ( Y3
% 13.25/7.64 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) )
% 13.25/7.64 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S3 ) @ Xa ) ) ) )
% 13.25/7.64 => ( ! [V2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.64 => ( ( Y3
% 13.25/7.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa @ Xa ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.64 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) )
% 13.25/7.64 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) )
% 13.25/7.64 => ( ( Y3
% 13.25/7.64 = ( if_VEBT_VEBT
% 13.25/7.64 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 13.25/7.64 & ~ ( ( Xa = Mi2 )
% 13.25/7.64 | ( Xa = Ma2 ) ) )
% 13.25/7.64 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Xa @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa @ Mi2 ) @ Mi2 @ Xa ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 13.25/7.64 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) ) )
% 13.25/7.64 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4884_triangle__def,axiom,
% 13.25/7.64 ( nat_triangle
% 13.25/7.64 = ( ^ [N2: nat] : ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4885_vebt__buildup_Oelims,axiom,
% 13.25/7.64 ! [X: nat,Y3: vEBT_VEBT] :
% 13.25/7.64 ( ( ( vEBT_vebt_buildup @ X )
% 13.25/7.64 = Y3 )
% 13.25/7.64 => ( ( ( X = zero_zero_nat )
% 13.25/7.64 => ( Y3
% 13.25/7.64 != ( vEBT_Leaf @ $false @ $false ) ) )
% 13.25/7.64 => ( ( ( X
% 13.25/7.64 = ( suc @ zero_zero_nat ) )
% 13.25/7.64 => ( Y3
% 13.25/7.64 != ( vEBT_Leaf @ $false @ $false ) ) )
% 13.25/7.64 => ~ ! [Va3: nat] :
% 13.25/7.64 ( ( X
% 13.25/7.64 = ( suc @ ( suc @ Va3 ) ) )
% 13.25/7.64 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 13.25/7.64 => ( Y3
% 13.25/7.64 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 13.25/7.64 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 13.25/7.64 => ( Y3
% 13.25/7.64 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4886_flip__bit__0,axiom,
% 13.25/7.64 ! [A: code_integer] :
% 13.25/7.64 ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 13.25/7.64 = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4887_flip__bit__0,axiom,
% 13.25/7.64 ! [A: int] :
% 13.25/7.64 ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 13.25/7.64 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4888_flip__bit__0,axiom,
% 13.25/7.64 ! [A: nat] :
% 13.25/7.64 ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 13.25/7.64 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4889_signed__take__bit__Suc,axiom,
% 13.25/7.64 ! [N: nat,A: code_integer] :
% 13.25/7.64 ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ A )
% 13.25/7.64 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4890_signed__take__bit__Suc,axiom,
% 13.25/7.64 ! [N: nat,A: int] :
% 13.25/7.64 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ A )
% 13.25/7.64 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4891_set__decode__Suc,axiom,
% 13.25/7.64 ! [N: nat,X: nat] :
% 13.25/7.64 ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
% 13.25/7.64 = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4892_add__scale__eq__noteq,axiom,
% 13.25/7.64 ! [R: complex,A: complex,B: complex,C: complex,D: complex] :
% 13.25/7.64 ( ( R != zero_zero_complex )
% 13.25/7.64 => ( ( ( A = B )
% 13.25/7.64 & ( C != D ) )
% 13.25/7.64 => ( ( plus_plus_complex @ A @ ( times_times_complex @ R @ C ) )
% 13.25/7.64 != ( plus_plus_complex @ B @ ( times_times_complex @ R @ D ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4893_add__scale__eq__noteq,axiom,
% 13.25/7.64 ! [R: real,A: real,B: real,C: real,D: real] :
% 13.25/7.64 ( ( R != zero_zero_real )
% 13.25/7.64 => ( ( ( A = B )
% 13.25/7.64 & ( C != D ) )
% 13.25/7.64 => ( ( plus_plus_real @ A @ ( times_times_real @ R @ C ) )
% 13.25/7.64 != ( plus_plus_real @ B @ ( times_times_real @ R @ D ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4894_add__scale__eq__noteq,axiom,
% 13.25/7.64 ! [R: rat,A: rat,B: rat,C: rat,D: rat] :
% 13.25/7.64 ( ( R != zero_zero_rat )
% 13.25/7.64 => ( ( ( A = B )
% 13.25/7.64 & ( C != D ) )
% 13.25/7.64 => ( ( plus_plus_rat @ A @ ( times_times_rat @ R @ C ) )
% 13.25/7.64 != ( plus_plus_rat @ B @ ( times_times_rat @ R @ D ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4895_add__scale__eq__noteq,axiom,
% 13.25/7.64 ! [R: nat,A: nat,B: nat,C: nat,D: nat] :
% 13.25/7.64 ( ( R != zero_zero_nat )
% 13.25/7.64 => ( ( ( A = B )
% 13.25/7.64 & ( C != D ) )
% 13.25/7.64 => ( ( plus_plus_nat @ A @ ( times_times_nat @ R @ C ) )
% 13.25/7.64 != ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4896_add__scale__eq__noteq,axiom,
% 13.25/7.64 ! [R: int,A: int,B: int,C: int,D: int] :
% 13.25/7.64 ( ( R != zero_zero_int )
% 13.25/7.64 => ( ( ( A = B )
% 13.25/7.64 & ( C != D ) )
% 13.25/7.64 => ( ( plus_plus_int @ A @ ( times_times_int @ R @ C ) )
% 13.25/7.64 != ( plus_plus_int @ B @ ( times_times_int @ R @ D ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4897_artanh__def,axiom,
% 13.25/7.64 ( artanh_real
% 13.25/7.64 = ( ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4898_intind,axiom,
% 13.25/7.64 ! [I: nat,N: nat,P: nat > $o,X: nat] :
% 13.25/7.64 ( ( ord_less_nat @ I @ N )
% 13.25/7.64 => ( ( P @ X )
% 13.25/7.64 => ( P @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4899_intind,axiom,
% 13.25/7.64 ! [I: nat,N: nat,P: int > $o,X: int] :
% 13.25/7.64 ( ( ord_less_nat @ I @ N )
% 13.25/7.64 => ( ( P @ X )
% 13.25/7.64 => ( P @ ( nth_int @ ( replicate_int @ N @ X ) @ I ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4900_intind,axiom,
% 13.25/7.64 ! [I: nat,N: nat,P: vEBT_VEBT > $o,X: vEBT_VEBT] :
% 13.25/7.64 ( ( ord_less_nat @ I @ N )
% 13.25/7.64 => ( ( P @ X )
% 13.25/7.64 => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4901_max__enat__simps_I2_J,axiom,
% 13.25/7.64 ! [Q2: extended_enat] :
% 13.25/7.64 ( ( ord_ma741700101516333627d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 13.25/7.64 = Q2 ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4902_max__enat__simps_I3_J,axiom,
% 13.25/7.64 ! [Q2: extended_enat] :
% 13.25/7.64 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 13.25/7.64 = Q2 ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4903_ln__inj__iff,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.64 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.64 => ( ( ( ln_ln_real @ X )
% 13.25/7.64 = ( ln_ln_real @ Y3 ) )
% 13.25/7.64 = ( X = Y3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4904_ln__less__cancel__iff,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.64 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.64 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) )
% 13.25/7.64 = ( ord_less_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4905_of__bool__less__eq__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 13.25/7.64 = ( P
% 13.25/7.64 => Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4906_of__bool__less__eq__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 13.25/7.64 = ( P
% 13.25/7.64 => Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4907_of__bool__less__eq__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 13.25/7.64 = ( P
% 13.25/7.64 => Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4908_of__bool__less__eq__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 13.25/7.64 = ( P
% 13.25/7.64 => Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4909_of__bool__eq_I1_J,axiom,
% 13.25/7.64 ( ( zero_n1201886186963655149omplex @ $false )
% 13.25/7.64 = zero_zero_complex ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4910_of__bool__eq_I1_J,axiom,
% 13.25/7.64 ( ( zero_n3304061248610475627l_real @ $false )
% 13.25/7.64 = zero_zero_real ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4911_of__bool__eq_I1_J,axiom,
% 13.25/7.64 ( ( zero_n2052037380579107095ol_rat @ $false )
% 13.25/7.64 = zero_zero_rat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4912_of__bool__eq_I1_J,axiom,
% 13.25/7.64 ( ( zero_n2687167440665602831ol_nat @ $false )
% 13.25/7.64 = zero_zero_nat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4913_of__bool__eq_I1_J,axiom,
% 13.25/7.64 ( ( zero_n2684676970156552555ol_int @ $false )
% 13.25/7.64 = zero_zero_int ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4914_of__bool__eq_I1_J,axiom,
% 13.25/7.64 ( ( zero_n356916108424825756nteger @ $false )
% 13.25/7.64 = zero_z3403309356797280102nteger ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4915_of__bool__eq__0__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n1201886186963655149omplex @ P )
% 13.25/7.64 = zero_zero_complex )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4916_of__bool__eq__0__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n3304061248610475627l_real @ P )
% 13.25/7.64 = zero_zero_real )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4917_of__bool__eq__0__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 13.25/7.64 = zero_zero_rat )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4918_of__bool__eq__0__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 13.25/7.64 = zero_zero_nat )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4919_of__bool__eq__0__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n2684676970156552555ol_int @ P )
% 13.25/7.64 = zero_zero_int )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4920_of__bool__eq__0__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n356916108424825756nteger @ P )
% 13.25/7.64 = zero_z3403309356797280102nteger )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4921_of__bool__less__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 13.25/7.64 = ( ~ P
% 13.25/7.64 & Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4922_of__bool__less__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 13.25/7.64 = ( ~ P
% 13.25/7.64 & Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4923_of__bool__less__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 13.25/7.64 = ( ~ P
% 13.25/7.64 & Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4924_of__bool__less__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 13.25/7.64 = ( ~ P
% 13.25/7.64 & Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4925_of__bool__less__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
% 13.25/7.64 = ( ~ P
% 13.25/7.64 & Q ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4926_of__bool__eq__1__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n1201886186963655149omplex @ P )
% 13.25/7.64 = one_one_complex )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4927_of__bool__eq__1__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n3304061248610475627l_real @ P )
% 13.25/7.64 = one_one_real )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4928_of__bool__eq__1__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n2052037380579107095ol_rat @ P )
% 13.25/7.64 = one_one_rat )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4929_of__bool__eq__1__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n2687167440665602831ol_nat @ P )
% 13.25/7.64 = one_one_nat )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4930_of__bool__eq__1__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n2684676970156552555ol_int @ P )
% 13.25/7.64 = one_one_int )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4931_of__bool__eq__1__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ( zero_n356916108424825756nteger @ P )
% 13.25/7.64 = one_one_Code_integer )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4932_of__bool__eq_I2_J,axiom,
% 13.25/7.64 ( ( zero_n1201886186963655149omplex @ $true )
% 13.25/7.64 = one_one_complex ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4933_of__bool__eq_I2_J,axiom,
% 13.25/7.64 ( ( zero_n3304061248610475627l_real @ $true )
% 13.25/7.64 = one_one_real ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4934_of__bool__eq_I2_J,axiom,
% 13.25/7.64 ( ( zero_n2052037380579107095ol_rat @ $true )
% 13.25/7.64 = one_one_rat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4935_of__bool__eq_I2_J,axiom,
% 13.25/7.64 ( ( zero_n2687167440665602831ol_nat @ $true )
% 13.25/7.64 = one_one_nat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4936_of__bool__eq_I2_J,axiom,
% 13.25/7.64 ( ( zero_n2684676970156552555ol_int @ $true )
% 13.25/7.64 = one_one_int ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4937_of__bool__eq_I2_J,axiom,
% 13.25/7.64 ( ( zero_n356916108424825756nteger @ $true )
% 13.25/7.64 = one_one_Code_integer ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4938_signed__take__bit__of__0,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 13.25/7.64 = zero_zero_int ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4939_of__bool__or__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( zero_n2687167440665602831ol_nat
% 13.25/7.64 @ ( P
% 13.25/7.64 | Q ) )
% 13.25/7.64 = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4940_of__bool__or__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( zero_n2684676970156552555ol_int
% 13.25/7.64 @ ( P
% 13.25/7.64 | Q ) )
% 13.25/7.64 = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4941_of__bool__or__iff,axiom,
% 13.25/7.64 ! [P: $o,Q: $o] :
% 13.25/7.64 ( ( zero_n356916108424825756nteger
% 13.25/7.64 @ ( P
% 13.25/7.64 | Q ) )
% 13.25/7.64 = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4942_ln__le__cancel__iff,axiom,
% 13.25/7.64 ! [X: real,Y3: real] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.64 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.64 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) )
% 13.25/7.64 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4943_zero__less__of__bool__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4944_zero__less__of__bool__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4945_zero__less__of__bool__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4946_zero__less__of__bool__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4947_zero__less__of__bool__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
% 13.25/7.64 = P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4948_ln__eq__zero__iff,axiom,
% 13.25/7.64 ! [X: real] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.64 => ( ( ( ln_ln_real @ X )
% 13.25/7.64 = zero_zero_real )
% 13.25/7.64 = ( X = one_one_real ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4949_ln__gt__zero__iff,axiom,
% 13.25/7.64 ! [X: real] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.64 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 13.25/7.64 = ( ord_less_real @ one_one_real @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4950_ln__less__zero__iff,axiom,
% 13.25/7.64 ! [X: real] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.64 => ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 13.25/7.64 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4951_of__bool__less__one__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4952_of__bool__less__one__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4953_of__bool__less__one__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4954_of__bool__less__one__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4955_of__bool__less__one__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
% 13.25/7.64 = ~ P ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4956_ln__one,axiom,
% 13.25/7.64 ( ( ln_ln_real @ one_one_real )
% 13.25/7.64 = zero_zero_real ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4957_of__bool__not__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( zero_n1201886186963655149omplex @ ~ P )
% 13.25/7.64 = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4958_of__bool__not__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( zero_n3304061248610475627l_real @ ~ P )
% 13.25/7.64 = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4959_of__bool__not__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 13.25/7.64 = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4960_of__bool__not__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( zero_n2684676970156552555ol_int @ ~ P )
% 13.25/7.64 = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4961_of__bool__not__iff,axiom,
% 13.25/7.64 ! [P: $o] :
% 13.25/7.64 ( ( zero_n356916108424825756nteger @ ~ P )
% 13.25/7.64 = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4962_Suc__0__mod__eq,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.64 = ( zero_n2687167440665602831ol_nat
% 13.25/7.64 @ ( N
% 13.25/7.64 != ( suc @ zero_zero_nat ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4963_signed__take__bit__Suc__1,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ one_one_int )
% 13.25/7.64 = one_one_int ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4964_signed__take__bit__numeral__of__1,axiom,
% 13.25/7.64 ! [K: num] :
% 13.25/7.64 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
% 13.25/7.64 = one_one_int ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4965_nth__replicate,axiom,
% 13.25/7.64 ! [I: nat,N: nat,X: nat] :
% 13.25/7.64 ( ( ord_less_nat @ I @ N )
% 13.25/7.64 => ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
% 13.25/7.64 = X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4966_nth__replicate,axiom,
% 13.25/7.64 ! [I: nat,N: nat,X: int] :
% 13.25/7.64 ( ( ord_less_nat @ I @ N )
% 13.25/7.64 => ( ( nth_int @ ( replicate_int @ N @ X ) @ I )
% 13.25/7.64 = X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4967_nth__replicate,axiom,
% 13.25/7.64 ! [I: nat,N: nat,X: vEBT_VEBT] :
% 13.25/7.64 ( ( ord_less_nat @ I @ N )
% 13.25/7.64 => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I )
% 13.25/7.64 = X ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4968_triangle__Suc,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( nat_triangle @ ( suc @ N ) )
% 13.25/7.64 = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4969_ln__ge__zero__iff,axiom,
% 13.25/7.64 ! [X: real] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.64 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 13.25/7.64 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4970_ln__le__zero__iff,axiom,
% 13.25/7.64 ! [X: real] :
% 13.25/7.64 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.64 => ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
% 13.25/7.64 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4971_signed__take__bit__Suc__bit0,axiom,
% 13.25/7.64 ! [N: nat,K: num] :
% 13.25/7.64 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 13.25/7.64 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4972_odd__of__bool__self,axiom,
% 13.25/7.64 ! [P6: $o] :
% 13.25/7.64 ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P6 ) ) )
% 13.25/7.64 = P6 ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4973_odd__of__bool__self,axiom,
% 13.25/7.64 ! [P6: $o] :
% 13.25/7.64 ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P6 ) ) )
% 13.25/7.64 = P6 ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4974_odd__of__bool__self,axiom,
% 13.25/7.64 ! [P6: $o] :
% 13.25/7.64 ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P6 ) ) )
% 13.25/7.64 = P6 ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4975_of__bool__half__eq__0,axiom,
% 13.25/7.64 ! [B: $o] :
% 13.25/7.64 ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_zero_nat ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4976_of__bool__half__eq__0,axiom,
% 13.25/7.64 ! [B: $o] :
% 13.25/7.64 ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_zero_int ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4977_of__bool__half__eq__0,axiom,
% 13.25/7.64 ! [B: $o] :
% 13.25/7.64 ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.64 = zero_z3403309356797280102nteger ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4978_set__decode__0,axiom,
% 13.25/7.64 ! [X: nat] :
% 13.25/7.64 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
% 13.25/7.64 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4979_one__div__2__pow__eq,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.64 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4980_one__div__2__pow__eq,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.64 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4981_one__div__2__pow__eq,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.64 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4982_bits__1__div__exp,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.64 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4983_bits__1__div__exp,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.64 = ( zero_n2684676970156552555ol_int @ ( N = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4984_bits__1__div__exp,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.64 = ( zero_n356916108424825756nteger @ ( N = zero_zero_nat ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4985_one__mod__2__pow__eq,axiom,
% 13.25/7.64 ! [N: nat] :
% 13.25/7.64 ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.64 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.64
% 13.25/7.64 thf(fact_4986_one__mod__2__pow__eq,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4987_one__mod__2__pow__eq,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4988_dvd__antisym,axiom,
% 13.25/7.65 ! [M: nat,N: nat] :
% 13.25/7.65 ( ( dvd_dvd_nat @ M @ N )
% 13.25/7.65 => ( ( dvd_dvd_nat @ N @ M )
% 13.25/7.65 => ( M = N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4989_of__bool__eq__iff,axiom,
% 13.25/7.65 ! [P6: $o,Q2: $o] :
% 13.25/7.65 ( ( ( zero_n2687167440665602831ol_nat @ P6 )
% 13.25/7.65 = ( zero_n2687167440665602831ol_nat @ Q2 ) )
% 13.25/7.65 = ( P6 = Q2 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4990_of__bool__eq__iff,axiom,
% 13.25/7.65 ! [P6: $o,Q2: $o] :
% 13.25/7.65 ( ( ( zero_n2684676970156552555ol_int @ P6 )
% 13.25/7.65 = ( zero_n2684676970156552555ol_int @ Q2 ) )
% 13.25/7.65 = ( P6 = Q2 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4991_of__bool__eq__iff,axiom,
% 13.25/7.65 ! [P6: $o,Q2: $o] :
% 13.25/7.65 ( ( ( zero_n356916108424825756nteger @ P6 )
% 13.25/7.65 = ( zero_n356916108424825756nteger @ Q2 ) )
% 13.25/7.65 = ( P6 = Q2 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4992_of__bool__conj,axiom,
% 13.25/7.65 ! [P: $o,Q: $o] :
% 13.25/7.65 ( ( zero_n3304061248610475627l_real
% 13.25/7.65 @ ( P
% 13.25/7.65 & Q ) )
% 13.25/7.65 = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4993_of__bool__conj,axiom,
% 13.25/7.65 ! [P: $o,Q: $o] :
% 13.25/7.65 ( ( zero_n2052037380579107095ol_rat
% 13.25/7.65 @ ( P
% 13.25/7.65 & Q ) )
% 13.25/7.65 = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4994_of__bool__conj,axiom,
% 13.25/7.65 ! [P: $o,Q: $o] :
% 13.25/7.65 ( ( zero_n2687167440665602831ol_nat
% 13.25/7.65 @ ( P
% 13.25/7.65 & Q ) )
% 13.25/7.65 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4995_of__bool__conj,axiom,
% 13.25/7.65 ! [P: $o,Q: $o] :
% 13.25/7.65 ( ( zero_n2684676970156552555ol_int
% 13.25/7.65 @ ( P
% 13.25/7.65 & Q ) )
% 13.25/7.65 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4996_of__bool__conj,axiom,
% 13.25/7.65 ! [P: $o,Q: $o] :
% 13.25/7.65 ( ( zero_n356916108424825756nteger
% 13.25/7.65 @ ( P
% 13.25/7.65 & Q ) )
% 13.25/7.65 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4997_signed__take__bit__mult,axiom,
% 13.25/7.65 ! [N: nat,K: int,L2: int] :
% 13.25/7.65 ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 13.25/7.65 = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4998_signed__take__bit__add,axiom,
% 13.25/7.65 ! [N: nat,K: int,L2: int] :
% 13.25/7.65 ( ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 13.25/7.65 = ( bit_ri631733984087533419it_int @ N @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_4999_signed__take__bit__diff,axiom,
% 13.25/7.65 ! [N: nat,K: int,L2: int] :
% 13.25/7.65 ( ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( bit_ri631733984087533419it_int @ N @ L2 ) ) )
% 13.25/7.65 = ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5000_ln__less__self,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5001_zero__less__eq__of__bool,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5002_zero__less__eq__of__bool,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5003_zero__less__eq__of__bool,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5004_zero__less__eq__of__bool,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5005_zero__less__eq__of__bool,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5006_of__bool__less__eq__one,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5007_of__bool__less__eq__one,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5008_of__bool__less__eq__one,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5009_of__bool__less__eq__one,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5010_of__bool__less__eq__one,axiom,
% 13.25/7.65 ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5011_of__bool__def,axiom,
% 13.25/7.65 ( zero_n1201886186963655149omplex
% 13.25/7.65 = ( ^ [P4: $o] : ( if_complex @ P4 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5012_of__bool__def,axiom,
% 13.25/7.65 ( zero_n3304061248610475627l_real
% 13.25/7.65 = ( ^ [P4: $o] : ( if_real @ P4 @ one_one_real @ zero_zero_real ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5013_of__bool__def,axiom,
% 13.25/7.65 ( zero_n2052037380579107095ol_rat
% 13.25/7.65 = ( ^ [P4: $o] : ( if_rat @ P4 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5014_of__bool__def,axiom,
% 13.25/7.65 ( zero_n2687167440665602831ol_nat
% 13.25/7.65 = ( ^ [P4: $o] : ( if_nat @ P4 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5015_of__bool__def,axiom,
% 13.25/7.65 ( zero_n2684676970156552555ol_int
% 13.25/7.65 = ( ^ [P4: $o] : ( if_int @ P4 @ one_one_int @ zero_zero_int ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5016_of__bool__def,axiom,
% 13.25/7.65 ( zero_n356916108424825756nteger
% 13.25/7.65 = ( ^ [P4: $o] : ( if_Code_integer @ P4 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5017_split__of__bool,axiom,
% 13.25/7.65 ! [P: complex > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
% 13.25/7.65 = ( ( P6
% 13.25/7.65 => ( P @ one_one_complex ) )
% 13.25/7.65 & ( ~ P6
% 13.25/7.65 => ( P @ zero_zero_complex ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5018_split__of__bool,axiom,
% 13.25/7.65 ! [P: real > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
% 13.25/7.65 = ( ( P6
% 13.25/7.65 => ( P @ one_one_real ) )
% 13.25/7.65 & ( ~ P6
% 13.25/7.65 => ( P @ zero_zero_real ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5019_split__of__bool,axiom,
% 13.25/7.65 ! [P: rat > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
% 13.25/7.65 = ( ( P6
% 13.25/7.65 => ( P @ one_one_rat ) )
% 13.25/7.65 & ( ~ P6
% 13.25/7.65 => ( P @ zero_zero_rat ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5020_split__of__bool,axiom,
% 13.25/7.65 ! [P: nat > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
% 13.25/7.65 = ( ( P6
% 13.25/7.65 => ( P @ one_one_nat ) )
% 13.25/7.65 & ( ~ P6
% 13.25/7.65 => ( P @ zero_zero_nat ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5021_split__of__bool,axiom,
% 13.25/7.65 ! [P: int > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
% 13.25/7.65 = ( ( P6
% 13.25/7.65 => ( P @ one_one_int ) )
% 13.25/7.65 & ( ~ P6
% 13.25/7.65 => ( P @ zero_zero_int ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5022_split__of__bool,axiom,
% 13.25/7.65 ! [P: code_integer > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
% 13.25/7.65 = ( ( P6
% 13.25/7.65 => ( P @ one_one_Code_integer ) )
% 13.25/7.65 & ( ~ P6
% 13.25/7.65 => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5023_split__of__bool__asm,axiom,
% 13.25/7.65 ! [P: complex > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n1201886186963655149omplex @ P6 ) )
% 13.25/7.65 = ( ~ ( ( P6
% 13.25/7.65 & ~ ( P @ one_one_complex ) )
% 13.25/7.65 | ( ~ P6
% 13.25/7.65 & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5024_split__of__bool__asm,axiom,
% 13.25/7.65 ! [P: real > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n3304061248610475627l_real @ P6 ) )
% 13.25/7.65 = ( ~ ( ( P6
% 13.25/7.65 & ~ ( P @ one_one_real ) )
% 13.25/7.65 | ( ~ P6
% 13.25/7.65 & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5025_split__of__bool__asm,axiom,
% 13.25/7.65 ! [P: rat > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n2052037380579107095ol_rat @ P6 ) )
% 13.25/7.65 = ( ~ ( ( P6
% 13.25/7.65 & ~ ( P @ one_one_rat ) )
% 13.25/7.65 | ( ~ P6
% 13.25/7.65 & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5026_split__of__bool__asm,axiom,
% 13.25/7.65 ! [P: nat > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n2687167440665602831ol_nat @ P6 ) )
% 13.25/7.65 = ( ~ ( ( P6
% 13.25/7.65 & ~ ( P @ one_one_nat ) )
% 13.25/7.65 | ( ~ P6
% 13.25/7.65 & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5027_split__of__bool__asm,axiom,
% 13.25/7.65 ! [P: int > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n2684676970156552555ol_int @ P6 ) )
% 13.25/7.65 = ( ~ ( ( P6
% 13.25/7.65 & ~ ( P @ one_one_int ) )
% 13.25/7.65 | ( ~ P6
% 13.25/7.65 & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5028_split__of__bool__asm,axiom,
% 13.25/7.65 ! [P: code_integer > $o,P6: $o] :
% 13.25/7.65 ( ( P @ ( zero_n356916108424825756nteger @ P6 ) )
% 13.25/7.65 = ( ~ ( ( P6
% 13.25/7.65 & ~ ( P @ one_one_Code_integer ) )
% 13.25/7.65 | ( ~ P6
% 13.25/7.65 & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5029_ln__bound,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5030_ln__gt__zero,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.65 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5031_ln__less__zero,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.65 => ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5032_ln__gt__zero__imp__gt__one,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 13.25/7.65 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ord_less_real @ one_one_real @ X ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5033_ln__ge__zero,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ one_one_real @ X )
% 13.25/7.65 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5034_ln__ge__zero__imp__ge__one,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
% 13.25/7.65 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5035_ln__add__one__self__le__self,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5036_ln__mult,axiom,
% 13.25/7.65 ! [X: real,Y3: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.65 => ( ( ln_ln_real @ ( times_times_real @ X @ Y3 ) )
% 13.25/7.65 = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5037_ln__eq__minus__one,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ( ( ln_ln_real @ X )
% 13.25/7.65 = ( minus_minus_real @ X @ one_one_real ) )
% 13.25/7.65 => ( X = one_one_real ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5038_ln__div,axiom,
% 13.25/7.65 ! [X: real,Y3: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.65 => ( ( ln_ln_real @ ( divide_divide_real @ X @ Y3 ) )
% 13.25/7.65 = ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5039_subset__decode__imp__le,axiom,
% 13.25/7.65 ! [M: nat,N: nat] :
% 13.25/7.65 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 13.25/7.65 => ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5040_ln__le__minus__one,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5041_ln__diff__le,axiom,
% 13.25/7.65 ! [X: real,Y3: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.65 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X @ Y3 ) @ Y3 ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5042_of__bool__odd__eq__mod__2,axiom,
% 13.25/7.65 ! [A: nat] :
% 13.25/7.65 ( ( zero_n2687167440665602831ol_nat
% 13.25/7.65 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.65 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5043_of__bool__odd__eq__mod__2,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( zero_n2684676970156552555ol_int
% 13.25/7.65 @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.65 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5044_of__bool__odd__eq__mod__2,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( zero_n356916108424825756nteger
% 13.25/7.65 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.65 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5045_signed__take__bit__int__less__exp,axiom,
% 13.25/7.65 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5046_even__signed__take__bit__iff,axiom,
% 13.25/7.65 ! [M: nat,A: code_integer] :
% 13.25/7.65 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
% 13.25/7.65 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5047_even__signed__take__bit__iff,axiom,
% 13.25/7.65 ! [M: nat,A: int] :
% 13.25/7.65 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
% 13.25/7.65 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5048_bits__induct,axiom,
% 13.25/7.65 ! [P: nat > $o,A: nat] :
% 13.25/7.65 ( ! [A5: nat] :
% 13.25/7.65 ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = A5 )
% 13.25/7.65 => ( P @ A5 ) )
% 13.25/7.65 => ( ! [A5: nat,B3: $o] :
% 13.25/7.65 ( ( P @ A5 )
% 13.25/7.65 => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = A5 )
% 13.25/7.65 => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 13.25/7.65 => ( P @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5049_bits__induct,axiom,
% 13.25/7.65 ! [P: int > $o,A: int] :
% 13.25/7.65 ( ! [A5: int] :
% 13.25/7.65 ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.65 = A5 )
% 13.25/7.65 => ( P @ A5 ) )
% 13.25/7.65 => ( ! [A5: int,B3: $o] :
% 13.25/7.65 ( ( P @ A5 )
% 13.25/7.65 => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.65 = A5 )
% 13.25/7.65 => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 13.25/7.65 => ( P @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5050_bits__induct,axiom,
% 13.25/7.65 ! [P: code_integer > $o,A: code_integer] :
% 13.25/7.65 ( ! [A5: code_integer] :
% 13.25/7.65 ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.65 = A5 )
% 13.25/7.65 => ( P @ A5 ) )
% 13.25/7.65 => ( ! [A5: code_integer,B3: $o] :
% 13.25/7.65 ( ( P @ A5 )
% 13.25/7.65 => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.65 = A5 )
% 13.25/7.65 => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
% 13.25/7.65 => ( P @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5051_signed__take__bit__int__less__self__iff,axiom,
% 13.25/7.65 ! [N: nat,K: int] :
% 13.25/7.65 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 13.25/7.65 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5052_signed__take__bit__int__greater__eq__self__iff,axiom,
% 13.25/7.65 ! [K: int,N: nat] :
% 13.25/7.65 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 13.25/7.65 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5053_add__0__iff,axiom,
% 13.25/7.65 ! [B: complex,A: complex] :
% 13.25/7.65 ( ( B
% 13.25/7.65 = ( plus_plus_complex @ B @ A ) )
% 13.25/7.65 = ( A = zero_zero_complex ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5054_add__0__iff,axiom,
% 13.25/7.65 ! [B: real,A: real] :
% 13.25/7.65 ( ( B
% 13.25/7.65 = ( plus_plus_real @ B @ A ) )
% 13.25/7.65 = ( A = zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5055_add__0__iff,axiom,
% 13.25/7.65 ! [B: rat,A: rat] :
% 13.25/7.65 ( ( B
% 13.25/7.65 = ( plus_plus_rat @ B @ A ) )
% 13.25/7.65 = ( A = zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5056_add__0__iff,axiom,
% 13.25/7.65 ! [B: nat,A: nat] :
% 13.25/7.65 ( ( B
% 13.25/7.65 = ( plus_plus_nat @ B @ A ) )
% 13.25/7.65 = ( A = zero_zero_nat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5057_add__0__iff,axiom,
% 13.25/7.65 ! [B: int,A: int] :
% 13.25/7.65 ( ( B
% 13.25/7.65 = ( plus_plus_int @ B @ A ) )
% 13.25/7.65 = ( A = zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5058_exp__mod__exp,axiom,
% 13.25/7.65 ! [M: nat,N: nat] :
% 13.25/7.65 ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5059_exp__mod__exp,axiom,
% 13.25/7.65 ! [M: nat,N: nat] :
% 13.25/7.65 ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5060_exp__mod__exp,axiom,
% 13.25/7.65 ! [M: nat,N: nat] :
% 13.25/7.65 ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5061_crossproduct__noteq,axiom,
% 13.25/7.65 ! [A: real,B: real,C: real,D: real] :
% 13.25/7.65 ( ( ( A != B )
% 13.25/7.65 & ( C != D ) )
% 13.25/7.65 = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 13.25/7.65 != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5062_crossproduct__noteq,axiom,
% 13.25/7.65 ! [A: rat,B: rat,C: rat,D: rat] :
% 13.25/7.65 ( ( ( A != B )
% 13.25/7.65 & ( C != D ) )
% 13.25/7.65 = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 13.25/7.65 != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5063_crossproduct__noteq,axiom,
% 13.25/7.65 ! [A: nat,B: nat,C: nat,D: nat] :
% 13.25/7.65 ( ( ( A != B )
% 13.25/7.65 & ( C != D ) )
% 13.25/7.65 = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 13.25/7.65 != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5064_crossproduct__noteq,axiom,
% 13.25/7.65 ! [A: int,B: int,C: int,D: int] :
% 13.25/7.65 ( ( ( A != B )
% 13.25/7.65 & ( C != D ) )
% 13.25/7.65 = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 13.25/7.65 != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5065_crossproduct__eq,axiom,
% 13.25/7.65 ! [W: real,Y3: real,X: real,Z: real] :
% 13.25/7.65 ( ( ( plus_plus_real @ ( times_times_real @ W @ Y3 ) @ ( times_times_real @ X @ Z ) )
% 13.25/7.65 = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X @ Y3 ) ) )
% 13.25/7.65 = ( ( W = X )
% 13.25/7.65 | ( Y3 = Z ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5066_crossproduct__eq,axiom,
% 13.25/7.65 ! [W: rat,Y3: rat,X: rat,Z: rat] :
% 13.25/7.65 ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y3 ) @ ( times_times_rat @ X @ Z ) )
% 13.25/7.65 = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X @ Y3 ) ) )
% 13.25/7.65 = ( ( W = X )
% 13.25/7.65 | ( Y3 = Z ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5067_crossproduct__eq,axiom,
% 13.25/7.65 ! [W: nat,Y3: nat,X: nat,Z: nat] :
% 13.25/7.65 ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X @ Z ) )
% 13.25/7.65 = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y3 ) ) )
% 13.25/7.65 = ( ( W = X )
% 13.25/7.65 | ( Y3 = Z ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5068_crossproduct__eq,axiom,
% 13.25/7.65 ! [W: int,Y3: int,X: int,Z: int] :
% 13.25/7.65 ( ( ( plus_plus_int @ ( times_times_int @ W @ Y3 ) @ ( times_times_int @ X @ Z ) )
% 13.25/7.65 = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y3 ) ) )
% 13.25/7.65 = ( ( W = X )
% 13.25/7.65 | ( Y3 = Z ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5069_signed__take__bit__int__less__eq,axiom,
% 13.25/7.65 ! [N: nat,K: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 13.25/7.65 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5070_ln__one__plus__pos__lower__bound,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.65 => ( 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 ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5071_set__decode__def,axiom,
% 13.25/7.65 ( nat_set_decode
% 13.25/7.65 = ( ^ [X2: nat] :
% 13.25/7.65 ( collect_nat
% 13.25/7.65 @ ^ [N2: nat] :
% 13.25/7.65 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5072_exp__div__exp__eq,axiom,
% 13.25/7.65 ! [M: nat,N: nat] :
% 13.25/7.65 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( times_times_nat
% 13.25/7.65 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.65 @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 13.25/7.65 != zero_zero_nat )
% 13.25/7.65 & ( ord_less_eq_nat @ N @ M ) ) )
% 13.25/7.65 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5073_exp__div__exp__eq,axiom,
% 13.25/7.65 ! [M: nat,N: nat] :
% 13.25/7.65 ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( times_times_int
% 13.25/7.65 @ ( zero_n2684676970156552555ol_int
% 13.25/7.65 @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 13.25/7.65 != zero_zero_int )
% 13.25/7.65 & ( ord_less_eq_nat @ N @ M ) ) )
% 13.25/7.65 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5074_exp__div__exp__eq,axiom,
% 13.25/7.65 ! [M: nat,N: nat] :
% 13.25/7.65 ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( times_3573771949741848930nteger
% 13.25/7.65 @ ( zero_n356916108424825756nteger
% 13.25/7.65 @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 13.25/7.65 != zero_z3403309356797280102nteger )
% 13.25/7.65 & ( ord_less_eq_nat @ N @ M ) ) )
% 13.25/7.65 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5075_vebt__buildup_Osimps_I3_J,axiom,
% 13.25/7.65 ! [Va: nat] :
% 13.25/7.65 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 13.25/7.65 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 13.25/7.65 = ( 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 ) ) ) ) ) ) )
% 13.25/7.65 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 13.25/7.65 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 13.25/7.65 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5076_ln__2__less__1,axiom,
% 13.25/7.65 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5077_tanh__ln__real,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ( tanh_real @ ( ln_ln_real @ X ) )
% 13.25/7.65 = ( 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 ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5078_Divides_Oadjust__div__eq,axiom,
% 13.25/7.65 ! [Q2: int,R: int] :
% 13.25/7.65 ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.65 = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5079_ln__one__minus__pos__lower__bound,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.65 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.65 => ( 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 ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5080_signed__take__bit__rec,axiom,
% 13.25/7.65 ( bit_ri6519982836138164636nteger
% 13.25/7.65 = ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = 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 @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5081_signed__take__bit__rec,axiom,
% 13.25/7.65 ( bit_ri631733984087533419it_int
% 13.25/7.65 = ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = 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 @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5082_abs__ln__one__plus__x__minus__x__bound,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.65 => ( 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 ) ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5083_vebt__buildup_Opelims,axiom,
% 13.25/7.65 ! [X: nat,Y3: vEBT_VEBT] :
% 13.25/7.65 ( ( ( vEBT_vebt_buildup @ X )
% 13.25/7.65 = Y3 )
% 13.25/7.65 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
% 13.25/7.65 => ( ( ( X = zero_zero_nat )
% 13.25/7.65 => ( ( Y3
% 13.25/7.65 = ( vEBT_Leaf @ $false @ $false ) )
% 13.25/7.65 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 13.25/7.65 => ( ( ( X
% 13.25/7.65 = ( suc @ zero_zero_nat ) )
% 13.25/7.65 => ( ( Y3
% 13.25/7.65 = ( vEBT_Leaf @ $false @ $false ) )
% 13.25/7.65 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 13.25/7.65 => ~ ! [Va3: nat] :
% 13.25/7.65 ( ( X
% 13.25/7.65 = ( suc @ ( suc @ Va3 ) ) )
% 13.25/7.65 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 13.25/7.65 => ( Y3
% 13.25/7.65 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 13.25/7.65 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
% 13.25/7.65 => ( Y3
% 13.25/7.65 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 13.25/7.65 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5084_verit__minus__simplify_I4_J,axiom,
% 13.25/7.65 ! [B: real] :
% 13.25/7.65 ( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5085_verit__minus__simplify_I4_J,axiom,
% 13.25/7.65 ! [B: int] :
% 13.25/7.65 ( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5086_verit__minus__simplify_I4_J,axiom,
% 13.25/7.65 ! [B: complex] :
% 13.25/7.65 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5087_verit__minus__simplify_I4_J,axiom,
% 13.25/7.65 ! [B: code_integer] :
% 13.25/7.65 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5088_verit__minus__simplify_I4_J,axiom,
% 13.25/7.65 ! [B: rat] :
% 13.25/7.65 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5089_neg__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ( uminus_uminus_real @ A )
% 13.25/7.65 = ( uminus_uminus_real @ B ) )
% 13.25/7.65 = ( A = B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5090_neg__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( ( uminus_uminus_int @ A )
% 13.25/7.65 = ( uminus_uminus_int @ B ) )
% 13.25/7.65 = ( A = B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5091_neg__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( ( uminus1482373934393186551omplex @ A )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.65 = ( A = B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5092_neg__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( ( uminus1351360451143612070nteger @ A )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.65 = ( A = B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5093_neg__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ( uminus_uminus_rat @ A )
% 13.25/7.65 = ( uminus_uminus_rat @ B ) )
% 13.25/7.65 = ( A = B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5094_add_Oinverse__inverse,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5095_add_Oinverse__inverse,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5096_add_Oinverse__inverse,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5097_add_Oinverse__inverse,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5098_add_Oinverse__inverse,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5099_Compl__anti__mono,axiom,
% 13.25/7.65 ! [A2: set_int,B5: set_int] :
% 13.25/7.65 ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.65 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B5 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5100_Compl__subset__Compl__iff,axiom,
% 13.25/7.65 ! [A2: set_int,B5: set_int] :
% 13.25/7.65 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B5 ) )
% 13.25/7.65 = ( ord_less_eq_set_int @ B5 @ A2 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5101_abs__abs,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 13.25/7.65 = ( abs_abs_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5102_abs__abs,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 13.25/7.65 = ( abs_abs_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5103_abs__abs,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 13.25/7.65 = ( abs_abs_Code_integer @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5104_abs__abs,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 13.25/7.65 = ( abs_abs_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5105_abs__idempotent,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 13.25/7.65 = ( abs_abs_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5106_abs__idempotent,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 13.25/7.65 = ( abs_abs_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5107_abs__idempotent,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 13.25/7.65 = ( abs_abs_Code_integer @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5108_abs__idempotent,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 13.25/7.65 = ( abs_abs_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5109_compl__le__compl__iff,axiom,
% 13.25/7.65 ! [X: set_int,Y3: set_int] :
% 13.25/7.65 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 13.25/7.65 = ( ord_less_eq_set_int @ Y3 @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5110_neg__le__iff__le,axiom,
% 13.25/7.65 ! [B: real,A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( ord_less_eq_real @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5111_neg__le__iff__le,axiom,
% 13.25/7.65 ! [B: code_integer,A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5112_neg__le__iff__le,axiom,
% 13.25/7.65 ! [B: rat,A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( ord_less_eq_rat @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5113_neg__le__iff__le,axiom,
% 13.25/7.65 ! [B: int,A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( ord_less_eq_int @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5114_neg__equal__zero,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ( uminus_uminus_real @ A )
% 13.25/7.65 = A )
% 13.25/7.65 = ( A = zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5115_neg__equal__zero,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ( uminus_uminus_int @ A )
% 13.25/7.65 = A )
% 13.25/7.65 = ( A = zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5116_neg__equal__zero,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ( uminus1351360451143612070nteger @ A )
% 13.25/7.65 = A )
% 13.25/7.65 = ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5117_neg__equal__zero,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ( uminus_uminus_rat @ A )
% 13.25/7.65 = A )
% 13.25/7.65 = ( A = zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5118_equal__neg__zero,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( A = zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5119_equal__neg__zero,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( A = zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5120_equal__neg__zero,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5121_equal__neg__zero,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( A = zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5122_neg__equal__0__iff__equal,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ( uminus_uminus_real @ A )
% 13.25/7.65 = zero_zero_real )
% 13.25/7.65 = ( A = zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5123_neg__equal__0__iff__equal,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ( uminus_uminus_int @ A )
% 13.25/7.65 = zero_zero_int )
% 13.25/7.65 = ( A = zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5124_neg__equal__0__iff__equal,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( ( uminus1482373934393186551omplex @ A )
% 13.25/7.65 = zero_zero_complex )
% 13.25/7.65 = ( A = zero_zero_complex ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5125_neg__equal__0__iff__equal,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ( uminus1351360451143612070nteger @ A )
% 13.25/7.65 = zero_z3403309356797280102nteger )
% 13.25/7.65 = ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5126_neg__equal__0__iff__equal,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ( uminus_uminus_rat @ A )
% 13.25/7.65 = zero_zero_rat )
% 13.25/7.65 = ( A = zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5127_neg__0__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( zero_zero_real
% 13.25/7.65 = ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( zero_zero_real = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5128_neg__0__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( zero_zero_int
% 13.25/7.65 = ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( zero_zero_int = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5129_neg__0__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( zero_zero_complex
% 13.25/7.65 = ( uminus1482373934393186551omplex @ A ) )
% 13.25/7.65 = ( zero_zero_complex = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5130_neg__0__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( zero_z3403309356797280102nteger
% 13.25/7.65 = ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( zero_z3403309356797280102nteger = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5131_neg__0__equal__iff__equal,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( zero_zero_rat
% 13.25/7.65 = ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( zero_zero_rat = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5132_add_Oinverse__neutral,axiom,
% 13.25/7.65 ( ( uminus_uminus_real @ zero_zero_real )
% 13.25/7.65 = zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5133_add_Oinverse__neutral,axiom,
% 13.25/7.65 ( ( uminus_uminus_int @ zero_zero_int )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5134_add_Oinverse__neutral,axiom,
% 13.25/7.65 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 13.25/7.65 = zero_zero_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5135_add_Oinverse__neutral,axiom,
% 13.25/7.65 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5136_add_Oinverse__neutral,axiom,
% 13.25/7.65 ( ( uminus_uminus_rat @ zero_zero_rat )
% 13.25/7.65 = zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5137_neg__less__iff__less,axiom,
% 13.25/7.65 ! [B: real,A: real] :
% 13.25/7.65 ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( ord_less_real @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5138_neg__less__iff__less,axiom,
% 13.25/7.65 ! [B: int,A: int] :
% 13.25/7.65 ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( ord_less_int @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5139_neg__less__iff__less,axiom,
% 13.25/7.65 ! [B: code_integer,A: code_integer] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5140_neg__less__iff__less,axiom,
% 13.25/7.65 ! [B: rat,A: rat] :
% 13.25/7.65 ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( ord_less_rat @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5141_neg__numeral__eq__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( M = N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5142_neg__numeral__eq__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( M = N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5143_neg__numeral__eq__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 13.25/7.65 = ( M = N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5144_neg__numeral__eq__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( M = N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5145_neg__numeral__eq__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( M = N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5146_mult__minus__left,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 13.25/7.65 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5147_mult__minus__left,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.65 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5148_mult__minus__left,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5149_mult__minus__left,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5150_mult__minus__left,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 13.25/7.65 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5151_minus__mult__minus,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 13.25/7.65 = ( times_times_real @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5152_minus__mult__minus,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 13.25/7.65 = ( times_times_int @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5153_minus__mult__minus,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.65 = ( times_times_complex @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5154_minus__mult__minus,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.65 = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5155_minus__mult__minus,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 13.25/7.65 = ( times_times_rat @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5156_mult__minus__right,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5157_mult__minus__right,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5158_mult__minus__right,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5159_mult__minus__right,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5160_mult__minus__right,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5161_add__minus__cancel,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5162_add__minus__cancel,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5163_add__minus__cancel,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5164_add__minus__cancel,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5165_add__minus__cancel,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5166_minus__add__cancel,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5167_minus__add__cancel,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5168_minus__add__cancel,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5169_minus__add__cancel,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5170_minus__add__cancel,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 13.25/7.65 = B ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5171_minus__add__distrib,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 13.25/7.65 = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5172_minus__add__distrib,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 13.25/7.65 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5173_minus__add__distrib,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 13.25/7.65 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5174_minus__add__distrib,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 13.25/7.65 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5175_minus__add__distrib,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 13.25/7.65 = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5176_minus__diff__eq,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 13.25/7.65 = ( minus_minus_real @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5177_minus__diff__eq,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 13.25/7.65 = ( minus_minus_int @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5178_minus__diff__eq,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
% 13.25/7.65 = ( minus_minus_complex @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5179_minus__diff__eq,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 13.25/7.65 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5180_minus__diff__eq,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 13.25/7.65 = ( minus_minus_rat @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5181_div__minus__minus,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 13.25/7.65 = ( divide_divide_int @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5182_div__minus__minus,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.65 = ( divide6298287555418463151nteger @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5183_abs__0,axiom,
% 13.25/7.65 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5184_abs__0,axiom,
% 13.25/7.65 ( ( abs_abs_complex @ zero_zero_complex )
% 13.25/7.65 = zero_zero_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5185_abs__0,axiom,
% 13.25/7.65 ( ( abs_abs_real @ zero_zero_real )
% 13.25/7.65 = zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5186_abs__0,axiom,
% 13.25/7.65 ( ( abs_abs_rat @ zero_zero_rat )
% 13.25/7.65 = zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5187_abs__0,axiom,
% 13.25/7.65 ( ( abs_abs_int @ zero_zero_int )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5188_abs__zero,axiom,
% 13.25/7.65 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5189_abs__zero,axiom,
% 13.25/7.65 ( ( abs_abs_real @ zero_zero_real )
% 13.25/7.65 = zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5190_abs__zero,axiom,
% 13.25/7.65 ( ( abs_abs_rat @ zero_zero_rat )
% 13.25/7.65 = zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5191_abs__zero,axiom,
% 13.25/7.65 ( ( abs_abs_int @ zero_zero_int )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5192_abs__eq__0,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ( abs_abs_Code_integer @ A )
% 13.25/7.65 = zero_z3403309356797280102nteger )
% 13.25/7.65 = ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5193_abs__eq__0,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ( abs_abs_real @ A )
% 13.25/7.65 = zero_zero_real )
% 13.25/7.65 = ( A = zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5194_abs__eq__0,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ( abs_abs_rat @ A )
% 13.25/7.65 = zero_zero_rat )
% 13.25/7.65 = ( A = zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5195_abs__eq__0,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ( abs_abs_int @ A )
% 13.25/7.65 = zero_zero_int )
% 13.25/7.65 = ( A = zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5196_abs__0__eq,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( zero_z3403309356797280102nteger
% 13.25/7.65 = ( abs_abs_Code_integer @ A ) )
% 13.25/7.65 = ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5197_abs__0__eq,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( zero_zero_real
% 13.25/7.65 = ( abs_abs_real @ A ) )
% 13.25/7.65 = ( A = zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5198_abs__0__eq,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( zero_zero_rat
% 13.25/7.65 = ( abs_abs_rat @ A ) )
% 13.25/7.65 = ( A = zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5199_abs__0__eq,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( zero_zero_int
% 13.25/7.65 = ( abs_abs_int @ A ) )
% 13.25/7.65 = ( A = zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5200_minus__dvd__iff,axiom,
% 13.25/7.65 ! [X: real,Y3: real] :
% 13.25/7.65 ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y3 )
% 13.25/7.65 = ( dvd_dvd_real @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5201_minus__dvd__iff,axiom,
% 13.25/7.65 ! [X: int,Y3: int] :
% 13.25/7.65 ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y3 )
% 13.25/7.65 = ( dvd_dvd_int @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5202_minus__dvd__iff,axiom,
% 13.25/7.65 ! [X: complex,Y3: complex] :
% 13.25/7.65 ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y3 )
% 13.25/7.65 = ( dvd_dvd_complex @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5203_minus__dvd__iff,axiom,
% 13.25/7.65 ! [X: code_integer,Y3: code_integer] :
% 13.25/7.65 ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y3 )
% 13.25/7.65 = ( dvd_dvd_Code_integer @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5204_minus__dvd__iff,axiom,
% 13.25/7.65 ! [X: rat,Y3: rat] :
% 13.25/7.65 ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y3 )
% 13.25/7.65 = ( dvd_dvd_rat @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5205_dvd__minus__iff,axiom,
% 13.25/7.65 ! [X: real,Y3: real] :
% 13.25/7.65 ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y3 ) )
% 13.25/7.65 = ( dvd_dvd_real @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5206_dvd__minus__iff,axiom,
% 13.25/7.65 ! [X: int,Y3: int] :
% 13.25/7.65 ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y3 ) )
% 13.25/7.65 = ( dvd_dvd_int @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5207_dvd__minus__iff,axiom,
% 13.25/7.65 ! [X: complex,Y3: complex] :
% 13.25/7.65 ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y3 ) )
% 13.25/7.65 = ( dvd_dvd_complex @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5208_dvd__minus__iff,axiom,
% 13.25/7.65 ! [X: code_integer,Y3: code_integer] :
% 13.25/7.65 ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y3 ) )
% 13.25/7.65 = ( dvd_dvd_Code_integer @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5209_dvd__minus__iff,axiom,
% 13.25/7.65 ! [X: rat,Y3: rat] :
% 13.25/7.65 ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y3 ) )
% 13.25/7.65 = ( dvd_dvd_rat @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5210_abs__numeral,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 13.25/7.65 = ( numera6620942414471956472nteger @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5211_abs__numeral,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
% 13.25/7.65 = ( numeral_numeral_real @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5212_abs__numeral,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
% 13.25/7.65 = ( numeral_numeral_rat @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5213_abs__numeral,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
% 13.25/7.65 = ( numeral_numeral_int @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5214_abs__mult__self__eq,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 13.25/7.65 = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5215_abs__mult__self__eq,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 13.25/7.65 = ( times_times_real @ A @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5216_abs__mult__self__eq,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 13.25/7.65 = ( times_times_rat @ A @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5217_abs__mult__self__eq,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 13.25/7.65 = ( times_times_int @ A @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5218_abs__1,axiom,
% 13.25/7.65 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 13.25/7.65 = one_one_Code_integer ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5219_abs__1,axiom,
% 13.25/7.65 ( ( abs_abs_complex @ one_one_complex )
% 13.25/7.65 = one_one_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5220_abs__1,axiom,
% 13.25/7.65 ( ( abs_abs_real @ one_one_real )
% 13.25/7.65 = one_one_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5221_abs__1,axiom,
% 13.25/7.65 ( ( abs_abs_rat @ one_one_rat )
% 13.25/7.65 = one_one_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5222_abs__1,axiom,
% 13.25/7.65 ( ( abs_abs_int @ one_one_int )
% 13.25/7.65 = one_one_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5223_abs__add__abs,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 13.25/7.65 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5224_abs__add__abs,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 13.25/7.65 = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5225_abs__add__abs,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 13.25/7.65 = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5226_abs__add__abs,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 13.25/7.65 = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5227_abs__divide,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.65 = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5228_abs__divide,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 13.25/7.65 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5229_abs__divide,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 13.25/7.65 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5230_abs__minus,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( abs_abs_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5231_abs__minus,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( abs_abs_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5232_abs__minus,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
% 13.25/7.65 = ( abs_abs_complex @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5233_abs__minus,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( abs_abs_Code_integer @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5234_abs__minus,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( abs_abs_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5235_abs__minus__cancel,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( abs_abs_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5236_abs__minus__cancel,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( abs_abs_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5237_abs__minus__cancel,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( abs_abs_Code_integer @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5238_abs__minus__cancel,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( abs_abs_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5239_mod__minus__minus,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5240_mod__minus__minus,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5241_dvd__abs__iff,axiom,
% 13.25/7.65 ! [M: real,K: real] :
% 13.25/7.65 ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 13.25/7.65 = ( dvd_dvd_real @ M @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5242_dvd__abs__iff,axiom,
% 13.25/7.65 ! [M: int,K: int] :
% 13.25/7.65 ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 13.25/7.65 = ( dvd_dvd_int @ M @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5243_dvd__abs__iff,axiom,
% 13.25/7.65 ! [M: code_integer,K: code_integer] :
% 13.25/7.65 ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 13.25/7.65 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5244_dvd__abs__iff,axiom,
% 13.25/7.65 ! [M: rat,K: rat] :
% 13.25/7.65 ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 13.25/7.65 = ( dvd_dvd_rat @ M @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5245_abs__dvd__iff,axiom,
% 13.25/7.65 ! [M: real,K: real] :
% 13.25/7.65 ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 13.25/7.65 = ( dvd_dvd_real @ M @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5246_abs__dvd__iff,axiom,
% 13.25/7.65 ! [M: int,K: int] :
% 13.25/7.65 ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 13.25/7.65 = ( dvd_dvd_int @ M @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5247_abs__dvd__iff,axiom,
% 13.25/7.65 ! [M: code_integer,K: code_integer] :
% 13.25/7.65 ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 13.25/7.65 = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5248_abs__dvd__iff,axiom,
% 13.25/7.65 ! [M: rat,K: rat] :
% 13.25/7.65 ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 13.25/7.65 = ( dvd_dvd_rat @ M @ K ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5249_abs__bool__eq,axiom,
% 13.25/7.65 ! [P: $o] :
% 13.25/7.65 ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 13.25/7.65 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5250_abs__bool__eq,axiom,
% 13.25/7.65 ! [P: $o] :
% 13.25/7.65 ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 13.25/7.65 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5251_abs__bool__eq,axiom,
% 13.25/7.65 ! [P: $o] :
% 13.25/7.65 ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 13.25/7.65 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5252_abs__bool__eq,axiom,
% 13.25/7.65 ! [P: $o] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 13.25/7.65 = ( zero_n356916108424825756nteger @ P ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5253_tanh__real__less__iff,axiom,
% 13.25/7.65 ! [X: real,Y3: real] :
% 13.25/7.65 ( ( ord_less_real @ ( tanh_real @ X ) @ ( tanh_real @ Y3 ) )
% 13.25/7.65 = ( ord_less_real @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5254_tanh__real__le__iff,axiom,
% 13.25/7.65 ! [X: real,Y3: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ ( tanh_real @ Y3 ) )
% 13.25/7.65 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5255_neg__less__eq__nonneg,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 13.25/7.65 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5256_neg__less__eq__nonneg,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 13.25/7.65 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5257_neg__less__eq__nonneg,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 13.25/7.65 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5258_neg__less__eq__nonneg,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 13.25/7.65 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5259_less__eq__neg__nonpos,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5260_less__eq__neg__nonpos,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5261_less__eq__neg__nonpos,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5262_less__eq__neg__nonpos,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5263_neg__le__0__iff__le,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 13.25/7.65 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5264_neg__le__0__iff__le,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 13.25/7.65 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5265_neg__le__0__iff__le,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 13.25/7.65 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5266_neg__le__0__iff__le,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 13.25/7.65 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5267_neg__0__le__iff__le,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5268_neg__0__le__iff__le,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5269_neg__0__le__iff__le,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5270_neg__0__le__iff__le,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5271_less__neg__neg,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5272_less__neg__neg,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5273_less__neg__neg,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5274_less__neg__neg,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5275_neg__less__pos,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 13.25/7.65 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5276_neg__less__pos,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 13.25/7.65 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5277_neg__less__pos,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 13.25/7.65 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5278_neg__less__pos,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 13.25/7.65 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5279_neg__0__less__iff__less,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5280_neg__0__less__iff__less,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5281_neg__0__less__iff__less,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5282_neg__0__less__iff__less,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5283_neg__less__0__iff__less,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 13.25/7.65 = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5284_neg__less__0__iff__less,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 13.25/7.65 = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5285_neg__less__0__iff__less,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 13.25/7.65 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5286_neg__less__0__iff__less,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 13.25/7.65 = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5287_add_Oright__inverse,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5288_add_Oright__inverse,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5289_add_Oright__inverse,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 13.25/7.65 = zero_zero_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5290_add_Oright__inverse,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5291_add_Oright__inverse,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 13.25/7.65 = zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5292_ab__left__minus,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 13.25/7.65 = zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5293_ab__left__minus,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5294_ab__left__minus,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 13.25/7.65 = zero_zero_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5295_ab__left__minus,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5296_ab__left__minus,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 13.25/7.65 = zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5297_diff__0,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( minus_minus_real @ zero_zero_real @ A )
% 13.25/7.65 = ( uminus_uminus_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5298_diff__0,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( minus_minus_int @ zero_zero_int @ A )
% 13.25/7.65 = ( uminus_uminus_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5299_diff__0,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( minus_minus_complex @ zero_zero_complex @ A )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5300_diff__0,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5301_diff__0,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( minus_minus_rat @ zero_zero_rat @ A )
% 13.25/7.65 = ( uminus_uminus_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5302_verit__minus__simplify_I3_J,axiom,
% 13.25/7.65 ! [B: real] :
% 13.25/7.65 ( ( minus_minus_real @ zero_zero_real @ B )
% 13.25/7.65 = ( uminus_uminus_real @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5303_verit__minus__simplify_I3_J,axiom,
% 13.25/7.65 ! [B: int] :
% 13.25/7.65 ( ( minus_minus_int @ zero_zero_int @ B )
% 13.25/7.65 = ( uminus_uminus_int @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5304_verit__minus__simplify_I3_J,axiom,
% 13.25/7.65 ! [B: complex] :
% 13.25/7.65 ( ( minus_minus_complex @ zero_zero_complex @ B )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5305_verit__minus__simplify_I3_J,axiom,
% 13.25/7.65 ! [B: code_integer] :
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5306_verit__minus__simplify_I3_J,axiom,
% 13.25/7.65 ! [B: rat] :
% 13.25/7.65 ( ( minus_minus_rat @ zero_zero_rat @ B )
% 13.25/7.65 = ( uminus_uminus_rat @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5307_add__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5308_add__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5309_add__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5310_add__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5311_add__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5312_mult__minus1__right,axiom,
% 13.25/7.65 ! [Z: real] :
% 13.25/7.65 ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = ( uminus_uminus_real @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5313_mult__minus1__right,axiom,
% 13.25/7.65 ! [Z: int] :
% 13.25/7.65 ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = ( uminus_uminus_int @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5314_mult__minus1__right,axiom,
% 13.25/7.65 ! [Z: complex] :
% 13.25/7.65 ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5315_mult__minus1__right,axiom,
% 13.25/7.65 ! [Z: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5316_mult__minus1__right,axiom,
% 13.25/7.65 ! [Z: rat] :
% 13.25/7.65 ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = ( uminus_uminus_rat @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5317_mult__minus1,axiom,
% 13.25/7.65 ! [Z: real] :
% 13.25/7.65 ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
% 13.25/7.65 = ( uminus_uminus_real @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5318_mult__minus1,axiom,
% 13.25/7.65 ! [Z: int] :
% 13.25/7.65 ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
% 13.25/7.65 = ( uminus_uminus_int @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5319_mult__minus1,axiom,
% 13.25/7.65 ! [Z: complex] :
% 13.25/7.65 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5320_mult__minus1,axiom,
% 13.25/7.65 ! [Z: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5321_mult__minus1,axiom,
% 13.25/7.65 ! [Z: rat] :
% 13.25/7.65 ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
% 13.25/7.65 = ( uminus_uminus_rat @ Z ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5322_abs__le__zero__iff,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 13.25/7.65 = ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5323_abs__le__zero__iff,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 13.25/7.65 = ( A = zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5324_abs__le__zero__iff,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 13.25/7.65 = ( A = zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5325_abs__le__zero__iff,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 13.25/7.65 = ( A = zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5326_abs__le__self__iff,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 13.25/7.65 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5327_abs__le__self__iff,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 13.25/7.65 = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5328_abs__le__self__iff,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 13.25/7.65 = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5329_abs__le__self__iff,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 13.25/7.65 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5330_abs__of__nonneg,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 13.25/7.65 => ( ( abs_abs_Code_integer @ A )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5331_abs__of__nonneg,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.65 => ( ( abs_abs_real @ A )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5332_abs__of__nonneg,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.25/7.65 => ( ( abs_abs_rat @ A )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5333_abs__of__nonneg,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.25/7.65 => ( ( abs_abs_int @ A )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5334_zero__less__abs__iff,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 13.25/7.65 = ( A != zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5335_zero__less__abs__iff,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 13.25/7.65 = ( A != zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5336_zero__less__abs__iff,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 13.25/7.65 = ( A != zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5337_zero__less__abs__iff,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 13.25/7.65 = ( A != zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5338_diff__minus__eq__add,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 13.25/7.65 = ( plus_plus_real @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5339_diff__minus__eq__add,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.65 = ( plus_plus_int @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5340_diff__minus__eq__add,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.65 = ( plus_plus_complex @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5341_diff__minus__eq__add,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.65 = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5342_diff__minus__eq__add,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 13.25/7.65 = ( plus_plus_rat @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5343_uminus__add__conv__diff,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 13.25/7.65 = ( minus_minus_real @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5344_uminus__add__conv__diff,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.65 = ( minus_minus_int @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5345_uminus__add__conv__diff,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 13.25/7.65 = ( minus_minus_complex @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5346_uminus__add__conv__diff,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 13.25/7.65 = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5347_uminus__add__conv__diff,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 13.25/7.65 = ( minus_minus_rat @ B @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5348_div__minus1__right,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = ( uminus_uminus_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5349_div__minus1__right,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5350_divide__minus1,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( divide_divide_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = ( uminus_uminus_real @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5351_divide__minus1,axiom,
% 13.25/7.65 ! [X: complex] :
% 13.25/7.65 ( ( divide1717551699836669952omplex @ X @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5352_divide__minus1,axiom,
% 13.25/7.65 ! [X: rat] :
% 13.25/7.65 ( ( divide_divide_rat @ X @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = ( uminus_uminus_rat @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5353_abs__neg__numeral,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_real @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5354_abs__neg__numeral,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_int @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5355_abs__neg__numeral,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( numera6620942414471956472nteger @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5356_abs__neg__numeral,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_rat @ N ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5357_abs__neg__one,axiom,
% 13.25/7.65 ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = one_one_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5358_abs__neg__one,axiom,
% 13.25/7.65 ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = one_one_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5359_abs__neg__one,axiom,
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = one_one_Code_integer ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5360_abs__neg__one,axiom,
% 13.25/7.65 ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = one_one_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5361_minus__mod__self1,axiom,
% 13.25/7.65 ! [B: int,A: int] :
% 13.25/7.65 ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
% 13.25/7.65 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5362_minus__mod__self1,axiom,
% 13.25/7.65 ! [B: code_integer,A: code_integer] :
% 13.25/7.65 ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
% 13.25/7.65 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5363_abs__power__minus,axiom,
% 13.25/7.65 ! [A: real,N: nat] :
% 13.25/7.65 ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 13.25/7.65 = ( abs_abs_real @ ( power_power_real @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5364_abs__power__minus,axiom,
% 13.25/7.65 ! [A: int,N: nat] :
% 13.25/7.65 ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 13.25/7.65 = ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5365_abs__power__minus,axiom,
% 13.25/7.65 ! [A: code_integer,N: nat] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 13.25/7.65 = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5366_abs__power__minus,axiom,
% 13.25/7.65 ! [A: rat,N: nat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 13.25/7.65 = ( abs_abs_rat @ ( power_power_rat @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5367_signed__take__bit__of__minus__1,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( bit_ri6519982836138164636nteger @ N @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5368_signed__take__bit__of__minus__1,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5369_real__add__minus__iff,axiom,
% 13.25/7.65 ! [X: real,A: real] :
% 13.25/7.65 ( ( ( plus_plus_real @ X @ ( uminus_uminus_real @ A ) )
% 13.25/7.65 = zero_zero_real )
% 13.25/7.65 = ( X = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5370_tanh__real__pos__iff,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X ) )
% 13.25/7.65 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5371_tanh__real__neg__iff,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ ( tanh_real @ X ) @ zero_zero_real )
% 13.25/7.65 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5372_tanh__real__nonneg__iff,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X ) )
% 13.25/7.65 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5373_tanh__real__nonpos__iff,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( tanh_real @ X ) @ zero_zero_real )
% 13.25/7.65 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5374_dbl__simps_I1_J,axiom,
% 13.25/7.65 ! [K: num] :
% 13.25/7.65 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5375_dbl__simps_I1_J,axiom,
% 13.25/7.65 ! [K: num] :
% 13.25/7.65 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5376_dbl__simps_I1_J,axiom,
% 13.25/7.65 ! [K: num] :
% 13.25/7.65 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5377_dbl__simps_I1_J,axiom,
% 13.25/7.65 ! [K: num] :
% 13.25/7.65 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5378_dbl__simps_I1_J,axiom,
% 13.25/7.65 ! [K: num] :
% 13.25/7.65 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5379_add__neg__numeral__special_I7_J,axiom,
% 13.25/7.65 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5380_add__neg__numeral__special_I7_J,axiom,
% 13.25/7.65 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5381_add__neg__numeral__special_I7_J,axiom,
% 13.25/7.65 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.65 = zero_zero_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5382_add__neg__numeral__special_I7_J,axiom,
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5383_add__neg__numeral__special_I7_J,axiom,
% 13.25/7.65 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5384_add__neg__numeral__special_I8_J,axiom,
% 13.25/7.65 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 13.25/7.65 = zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5385_add__neg__numeral__special_I8_J,axiom,
% 13.25/7.65 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5386_add__neg__numeral__special_I8_J,axiom,
% 13.25/7.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 13.25/7.65 = zero_zero_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5387_add__neg__numeral__special_I8_J,axiom,
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5388_add__neg__numeral__special_I8_J,axiom,
% 13.25/7.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 13.25/7.65 = zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5389_diff__numeral__special_I12_J,axiom,
% 13.25/7.65 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5390_diff__numeral__special_I12_J,axiom,
% 13.25/7.65 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5391_diff__numeral__special_I12_J,axiom,
% 13.25/7.65 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.65 = zero_zero_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5392_diff__numeral__special_I12_J,axiom,
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5393_diff__numeral__special_I12_J,axiom,
% 13.25/7.65 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5394_numeral__eq__neg__one__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
% 13.25/7.65 = ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5395_numeral__eq__neg__one__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
% 13.25/7.65 = ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5396_numeral__eq__neg__one__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5397_numeral__eq__neg__one__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5398_numeral__eq__neg__one__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) )
% 13.25/7.65 = ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5399_neg__one__eq__numeral__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_real @ one_one_real )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5400_neg__one__eq__numeral__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_int @ one_one_int )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5401_neg__one__eq__numeral__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus1482373934393186551omplex @ one_one_complex )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5402_neg__one__eq__numeral__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5403_neg__one__eq__numeral__iff,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( ( uminus_uminus_rat @ one_one_rat )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( N = one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5404_divide__le__0__abs__iff,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 13.25/7.65 = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.25/7.65 | ( B = zero_zero_real ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5405_divide__le__0__abs__iff,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 13.25/7.65 = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.25/7.65 | ( B = zero_zero_rat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5406_zero__le__divide__abs__iff,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 13.25/7.65 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.65 | ( B = zero_zero_real ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5407_zero__le__divide__abs__iff,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 13.25/7.65 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.25/7.65 | ( B = zero_zero_rat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5408_abs__of__nonpos,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.25/7.65 => ( ( abs_abs_real @ A )
% 13.25/7.65 = ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5409_abs__of__nonpos,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 13.25/7.65 => ( ( abs_abs_Code_integer @ A )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5410_abs__of__nonpos,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 13.25/7.65 => ( ( abs_abs_rat @ A )
% 13.25/7.65 = ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5411_abs__of__nonpos,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ A @ zero_zero_int )
% 13.25/7.65 => ( ( abs_abs_int @ A )
% 13.25/7.65 = ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5412_left__minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat,A: real] :
% 13.25/7.65 ( ( 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 ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5413_left__minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat,A: int] :
% 13.25/7.65 ( ( 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 ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5414_left__minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat,A: complex] :
% 13.25/7.65 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ A ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5415_left__minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat,A: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ A ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5416_left__minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat,A: rat] :
% 13.25/7.65 ( ( 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 ) )
% 13.25/7.65 = A ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5417_minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) )
% 13.25/7.65 = one_one_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5418_minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) )
% 13.25/7.65 = one_one_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5419_minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) )
% 13.25/7.65 = one_one_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5420_minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) )
% 13.25/7.65 = one_one_Code_integer ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5421_minus__one__mult__self,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) )
% 13.25/7.65 = one_one_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5422_mod__minus1__right,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5423_mod__minus1__right,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5424_max__number__of_I4_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5425_max__number__of_I4_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 13.25/7.65 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5426_max__number__of_I4_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5427_max__number__of_I4_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5428_max__number__of_I3_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 13.25/7.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 13.25/7.65 = ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 13.25/7.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5429_max__number__of_I3_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 13.25/7.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 13.25/7.65 = ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 13.25/7.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5430_max__number__of_I3_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 13.25/7.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 13.25/7.65 = ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 13.25/7.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5431_max__number__of_I3_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 13.25/7.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 13.25/7.65 = ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 13.25/7.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5432_max__number__of_I2_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 13.25/7.65 = ( numeral_numeral_real @ U ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5433_max__number__of_I2_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 13.25/7.65 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 13.25/7.65 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5434_max__number__of_I2_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 13.25/7.65 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5435_max__number__of_I2_J,axiom,
% 13.25/7.65 ! [U: num,V: num] :
% 13.25/7.65 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 13.25/7.65 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 13.25/7.65 = ( numeral_numeral_int @ U ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5436_semiring__norm_I168_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: real] :
% 13.25/7.65 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5437_semiring__norm_I168_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: int] :
% 13.25/7.65 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5438_semiring__norm_I168_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: complex] :
% 13.25/7.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5439_semiring__norm_I168_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: code_integer] :
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5440_semiring__norm_I168_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: rat] :
% 13.25/7.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5441_diff__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5442_diff__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5443_diff__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 13.25/7.65 = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5444_diff__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5445_diff__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5446_diff__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5447_diff__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5448_diff__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5449_diff__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5450_diff__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5451_semiring__norm_I172_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: real] :
% 13.25/7.65 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5452_semiring__norm_I172_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: int] :
% 13.25/7.65 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5453_semiring__norm_I172_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: complex] :
% 13.25/7.65 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5454_semiring__norm_I172_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5455_semiring__norm_I172_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: rat] :
% 13.25/7.65 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5456_semiring__norm_I171_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: real] :
% 13.25/7.65 ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5457_semiring__norm_I171_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: int] :
% 13.25/7.65 ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5458_semiring__norm_I171_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: complex] :
% 13.25/7.65 ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5459_semiring__norm_I171_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5460_semiring__norm_I171_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: rat] :
% 13.25/7.65 ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y3 ) )
% 13.25/7.65 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5461_semiring__norm_I170_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: real] :
% 13.25/7.65 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y3 ) )
% 13.25/7.65 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5462_semiring__norm_I170_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: int] :
% 13.25/7.65 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y3 ) )
% 13.25/7.65 = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5463_semiring__norm_I170_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: complex] :
% 13.25/7.65 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y3 ) )
% 13.25/7.65 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5464_semiring__norm_I170_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: code_integer] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y3 ) )
% 13.25/7.65 = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5465_semiring__norm_I170_J,axiom,
% 13.25/7.65 ! [V: num,W: num,Y3: rat] :
% 13.25/7.65 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y3 ) )
% 13.25/7.65 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y3 ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5466_mult__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5467_mult__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5468_mult__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5469_mult__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5470_mult__neg__numeral__simps_I3_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5471_mult__neg__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5472_mult__neg__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5473_mult__neg__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5474_mult__neg__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5475_mult__neg__numeral__simps_I2_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5476_mult__neg__numeral__simps_I1_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_real @ ( times_times_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5477_mult__neg__numeral__simps_I1_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_int @ ( times_times_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5478_mult__neg__numeral__simps_I1_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 13.25/7.65 = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5479_mult__neg__numeral__simps_I1_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5480_mult__neg__numeral__simps_I1_J,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_rat @ ( times_times_num @ M @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5481_artanh__minus__real,axiom,
% 13.25/7.65 ! [X: real] :
% 13.25/7.65 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.65 => ( ( artanh_real @ ( uminus_uminus_real @ X ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( artanh_real @ X ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5482_neg__numeral__le__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( ord_less_eq_num @ N @ M ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5483_neg__numeral__le__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( ord_less_eq_num @ N @ M ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5484_neg__numeral__le__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( ord_less_eq_num @ N @ M ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5485_neg__numeral__le__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( ord_less_eq_num @ N @ M ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5486_neg__numeral__less__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( ord_less_num @ N @ M ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5487_neg__numeral__less__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( ord_less_num @ N @ M ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5488_neg__numeral__less__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( ord_less_num @ N @ M ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5489_neg__numeral__less__iff,axiom,
% 13.25/7.65 ! [M: num,N: num] :
% 13.25/7.65 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( ord_less_num @ N @ M ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5490_not__neg__one__le__neg__numeral__iff,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 13.25/7.65 = ( M != one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5491_not__neg__one__le__neg__numeral__iff,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 13.25/7.65 = ( M != one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5492_not__neg__one__le__neg__numeral__iff,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 13.25/7.65 = ( M != one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5493_not__neg__one__le__neg__numeral__iff,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 13.25/7.65 = ( M != one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5494_le__divide__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [A: real,B: real,W: num] :
% 13.25/7.65 ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 13.25/7.65 = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5495_le__divide__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [A: rat,B: rat,W: num] :
% 13.25/7.65 ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 13.25/7.65 = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5496_divide__le__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [B: real,W: num,A: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 13.25/7.65 = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5497_divide__le__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [B: rat,W: num,A: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 13.25/7.65 = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5498_eq__divide__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [A: real,B: real,W: num] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 13.25/7.65 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 13.25/7.65 != zero_zero_real )
% 13.25/7.65 => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.65 = B ) )
% 13.25/7.65 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 13.25/7.65 = zero_zero_real )
% 13.25/7.65 => ( A = zero_zero_real ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5499_eq__divide__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [A: complex,B: complex,W: num] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
% 13.25/7.65 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.65 != zero_zero_complex )
% 13.25/7.65 => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.65 = B ) )
% 13.25/7.65 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.65 = zero_zero_complex )
% 13.25/7.65 => ( A = zero_zero_complex ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5500_eq__divide__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [A: rat,B: rat,W: num] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 13.25/7.65 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 13.25/7.65 != zero_zero_rat )
% 13.25/7.65 => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 13.25/7.65 = B ) )
% 13.25/7.65 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 13.25/7.65 = zero_zero_rat )
% 13.25/7.65 => ( A = zero_zero_rat ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5501_divide__eq__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [B: real,W: num,A: real] :
% 13.25/7.65 ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.65 = A )
% 13.25/7.65 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 13.25/7.65 != zero_zero_real )
% 13.25/7.65 => ( B
% 13.25/7.65 = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
% 13.25/7.65 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 13.25/7.65 = zero_zero_real )
% 13.25/7.65 => ( A = zero_zero_real ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5502_divide__eq__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [B: complex,W: num,A: complex] :
% 13.25/7.65 ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.65 = A )
% 13.25/7.65 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.65 != zero_zero_complex )
% 13.25/7.65 => ( B
% 13.25/7.65 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
% 13.25/7.65 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.65 = zero_zero_complex )
% 13.25/7.65 => ( A = zero_zero_complex ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5503_divide__eq__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [B: rat,W: num,A: rat] :
% 13.25/7.65 ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 13.25/7.65 = A )
% 13.25/7.65 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 13.25/7.65 != zero_zero_rat )
% 13.25/7.65 => ( B
% 13.25/7.65 = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
% 13.25/7.65 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 13.25/7.65 = zero_zero_rat )
% 13.25/7.65 => ( A = zero_zero_rat ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5504_neg__numeral__less__neg__one__iff,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = ( M != one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5505_neg__numeral__less__neg__one__iff,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = ( M != one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5506_neg__numeral__less__neg__one__iff,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = ( M != one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5507_neg__numeral__less__neg__one__iff,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = ( M != one ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5508_less__divide__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [A: real,B: real,W: num] :
% 13.25/7.65 ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
% 13.25/7.65 = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5509_less__divide__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [A: rat,B: rat,W: num] :
% 13.25/7.65 ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
% 13.25/7.65 = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5510_divide__less__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [B: real,W: num,A: real] :
% 13.25/7.65 ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
% 13.25/7.65 = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5511_divide__less__eq__numeral1_I2_J,axiom,
% 13.25/7.65 ! [B: rat,W: num,A: rat] :
% 13.25/7.65 ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
% 13.25/7.65 = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5512_power2__minus,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5513_power2__minus,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5514_power2__minus,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5515_power2__minus,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5516_power2__minus,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5517_zero__less__power__abs__iff,axiom,
% 13.25/7.65 ! [A: code_integer,N: nat] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) )
% 13.25/7.65 = ( ( A != zero_z3403309356797280102nteger )
% 13.25/7.65 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5518_zero__less__power__abs__iff,axiom,
% 13.25/7.65 ! [A: real,N: nat] :
% 13.25/7.65 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 13.25/7.65 = ( ( A != zero_zero_real )
% 13.25/7.65 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5519_zero__less__power__abs__iff,axiom,
% 13.25/7.65 ! [A: rat,N: nat] :
% 13.25/7.65 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) )
% 13.25/7.65 = ( ( A != zero_zero_rat )
% 13.25/7.65 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5520_zero__less__power__abs__iff,axiom,
% 13.25/7.65 ! [A: int,N: nat] :
% 13.25/7.65 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
% 13.25/7.65 = ( ( A != zero_zero_int )
% 13.25/7.65 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5521_power2__abs,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5522_power2__abs,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5523_power2__abs,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5524_power2__abs,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5525_abs__power2,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.65 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5526_abs__power2,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.65 = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5527_abs__power2,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.65 = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5528_abs__power2,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.65 = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5529_add__neg__numeral__special_I9_J,axiom,
% 13.25/7.65 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5530_add__neg__numeral__special_I9_J,axiom,
% 13.25/7.65 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5531_add__neg__numeral__special_I9_J,axiom,
% 13.25/7.65 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5532_add__neg__numeral__special_I9_J,axiom,
% 13.25/7.65 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5533_add__neg__numeral__special_I9_J,axiom,
% 13.25/7.65 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5534_diff__numeral__special_I10_J,axiom,
% 13.25/7.65 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5535_diff__numeral__special_I10_J,axiom,
% 13.25/7.65 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5536_diff__numeral__special_I10_J,axiom,
% 13.25/7.65 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5537_diff__numeral__special_I10_J,axiom,
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5538_diff__numeral__special_I10_J,axiom,
% 13.25/7.65 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5539_diff__numeral__special_I11_J,axiom,
% 13.25/7.65 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5540_diff__numeral__special_I11_J,axiom,
% 13.25/7.65 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5541_diff__numeral__special_I11_J,axiom,
% 13.25/7.65 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.65 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5542_diff__numeral__special_I11_J,axiom,
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5543_diff__numeral__special_I11_J,axiom,
% 13.25/7.65 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5544_minus__1__div__2__eq,axiom,
% 13.25/7.65 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5545_minus__1__div__2__eq,axiom,
% 13.25/7.65 ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5546_bits__minus__1__mod__2__eq,axiom,
% 13.25/7.65 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.65 = one_one_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5547_bits__minus__1__mod__2__eq,axiom,
% 13.25/7.65 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.65 = one_one_Code_integer ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5548_minus__1__mod__2__eq,axiom,
% 13.25/7.65 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.65 = one_one_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5549_minus__1__mod__2__eq,axiom,
% 13.25/7.65 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.65 = one_one_Code_integer ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5550_Power_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [A: real,N: nat] :
% 13.25/7.65 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5551_Power_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [A: int,N: nat] :
% 13.25/7.65 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5552_Power_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [A: complex,N: nat] :
% 13.25/7.65 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5553_Power_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [A: code_integer,N: nat] :
% 13.25/7.65 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5554_Power_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [A: rat,N: nat] :
% 13.25/7.65 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5555_Parity_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [N: nat,A: real] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 13.25/7.65 = ( power_power_real @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5556_Parity_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [N: nat,A: int] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 13.25/7.65 = ( power_power_int @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5557_Parity_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [N: nat,A: complex] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 13.25/7.65 = ( power_power_complex @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5558_Parity_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [N: nat,A: code_integer] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 13.25/7.65 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5559_Parity_Oring__1__class_Opower__minus__even,axiom,
% 13.25/7.65 ! [N: nat,A: rat] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 13.25/7.65 = ( power_power_rat @ A @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5560_power__minus__odd,axiom,
% 13.25/7.65 ! [N: nat,A: real] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 13.25/7.65 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5561_power__minus__odd,axiom,
% 13.25/7.65 ! [N: nat,A: int] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 13.25/7.65 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5562_power__minus__odd,axiom,
% 13.25/7.65 ! [N: nat,A: complex] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5563_power__minus__odd,axiom,
% 13.25/7.65 ! [N: nat,A: code_integer] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5564_power__minus__odd,axiom,
% 13.25/7.65 ! [N: nat,A: rat] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 13.25/7.65 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5565_power__even__abs__numeral,axiom,
% 13.25/7.65 ! [W: num,A: code_integer] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.65 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.65 = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5566_power__even__abs__numeral,axiom,
% 13.25/7.65 ! [W: num,A: rat] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.65 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.65 = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5567_power__even__abs__numeral,axiom,
% 13.25/7.65 ! [W: num,A: real] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.65 => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.65 = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5568_power__even__abs__numeral,axiom,
% 13.25/7.65 ! [W: num,A: int] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.65 => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
% 13.25/7.65 = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5569_diff__numeral__special_I4_J,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5570_diff__numeral__special_I4_J,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5571_diff__numeral__special_I4_J,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5572_diff__numeral__special_I4_J,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5573_diff__numeral__special_I4_J,axiom,
% 13.25/7.65 ! [M: num] :
% 13.25/7.65 ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5574_diff__numeral__special_I3_J,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5575_diff__numeral__special_I3_J,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5576_diff__numeral__special_I3_J,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 13.25/7.65 = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5577_diff__numeral__special_I3_J,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.65 = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5578_diff__numeral__special_I3_J,axiom,
% 13.25/7.65 ! [N: num] :
% 13.25/7.65 ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.65 = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5579_signed__take__bit__Suc__minus__bit0,axiom,
% 13.25/7.65 ! [N: nat,K: num] :
% 13.25/7.65 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 13.25/7.65 = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5580_dbl__simps_I4_J,axiom,
% 13.25/7.65 ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5581_dbl__simps_I4_J,axiom,
% 13.25/7.65 ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5582_dbl__simps_I4_J,axiom,
% 13.25/7.65 ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5583_dbl__simps_I4_J,axiom,
% 13.25/7.65 ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5584_dbl__simps_I4_J,axiom,
% 13.25/7.65 ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5585_power__minus1__even,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = one_one_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5586_power__minus1__even,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = one_one_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5587_power__minus1__even,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = one_one_complex ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5588_power__minus1__even,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = one_one_Code_integer ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5589_power__minus1__even,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.65 = one_one_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5590_neg__one__even__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 13.25/7.65 = one_one_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5591_neg__one__even__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 13.25/7.65 = one_one_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5592_neg__one__even__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 13.25/7.65 = one_one_complex ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5593_neg__one__even__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 13.25/7.65 = one_one_Code_integer ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5594_neg__one__even__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 13.25/7.65 = one_one_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5595_neg__one__odd__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 13.25/7.65 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5596_neg__one__odd__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 13.25/7.65 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5597_neg__one__odd__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 13.25/7.65 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5598_neg__one__odd__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5599_neg__one__odd__power,axiom,
% 13.25/7.65 ! [N: nat] :
% 13.25/7.65 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.65 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 13.25/7.65 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5600_signed__take__bit__0,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5601_signed__take__bit__0,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 13.25/7.65 = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5602_abs__leI,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.65 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5603_abs__leI,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ A @ B )
% 13.25/7.65 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 13.25/7.65 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5604_abs__leI,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.65 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5605_abs__leI,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ A @ B )
% 13.25/7.65 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5606_abs__le__D2,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5607_abs__le__D2,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 13.25/7.65 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5608_abs__le__D2,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5609_abs__le__D2,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5610_abs__le__iff,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 13.25/7.65 = ( ( ord_less_eq_real @ A @ B )
% 13.25/7.65 & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5611_abs__le__iff,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 13.25/7.65 = ( ( ord_le3102999989581377725nteger @ A @ B )
% 13.25/7.65 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5612_abs__le__iff,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 13.25/7.65 = ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.65 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5613_abs__le__iff,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 13.25/7.65 = ( ( ord_less_eq_int @ A @ B )
% 13.25/7.65 & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5614_abs__ge__minus__self,axiom,
% 13.25/7.65 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5615_abs__ge__minus__self,axiom,
% 13.25/7.65 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5616_abs__ge__minus__self,axiom,
% 13.25/7.65 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5617_abs__ge__minus__self,axiom,
% 13.25/7.65 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5618_abs__less__iff,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 13.25/7.65 = ( ( ord_less_real @ A @ B )
% 13.25/7.65 & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5619_abs__less__iff,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 13.25/7.65 = ( ( ord_less_int @ A @ B )
% 13.25/7.65 & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5620_abs__less__iff,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 13.25/7.65 = ( ( ord_le6747313008572928689nteger @ A @ B )
% 13.25/7.65 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5621_abs__less__iff,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 13.25/7.65 = ( ( ord_less_rat @ A @ B )
% 13.25/7.65 & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5622_signed__take__bit__minus,axiom,
% 13.25/7.65 ! [N: nat,K: int] :
% 13.25/7.65 ( ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N @ K ) ) )
% 13.25/7.65 = ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5623_verit__negate__coefficient_I3_J,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( A = B )
% 13.25/7.65 => ( ( uminus_uminus_real @ A )
% 13.25/7.65 = ( uminus_uminus_real @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5624_verit__negate__coefficient_I3_J,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( A = B )
% 13.25/7.65 => ( ( uminus_uminus_int @ A )
% 13.25/7.65 = ( uminus_uminus_int @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5625_verit__negate__coefficient_I3_J,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( A = B )
% 13.25/7.65 => ( ( uminus1351360451143612070nteger @ A )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5626_verit__negate__coefficient_I3_J,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( A = B )
% 13.25/7.65 => ( ( uminus_uminus_rat @ A )
% 13.25/7.65 = ( uminus_uminus_rat @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5627_abs__eq__iff,axiom,
% 13.25/7.65 ! [X: real,Y3: real] :
% 13.25/7.65 ( ( ( abs_abs_real @ X )
% 13.25/7.65 = ( abs_abs_real @ Y3 ) )
% 13.25/7.65 = ( ( X = Y3 )
% 13.25/7.65 | ( X
% 13.25/7.65 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5628_abs__eq__iff,axiom,
% 13.25/7.65 ! [X: int,Y3: int] :
% 13.25/7.65 ( ( ( abs_abs_int @ X )
% 13.25/7.65 = ( abs_abs_int @ Y3 ) )
% 13.25/7.65 = ( ( X = Y3 )
% 13.25/7.65 | ( X
% 13.25/7.65 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5629_abs__eq__iff,axiom,
% 13.25/7.65 ! [X: code_integer,Y3: code_integer] :
% 13.25/7.65 ( ( ( abs_abs_Code_integer @ X )
% 13.25/7.65 = ( abs_abs_Code_integer @ Y3 ) )
% 13.25/7.65 = ( ( X = Y3 )
% 13.25/7.65 | ( X
% 13.25/7.65 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5630_abs__eq__iff,axiom,
% 13.25/7.65 ! [X: rat,Y3: rat] :
% 13.25/7.65 ( ( ( abs_abs_rat @ X )
% 13.25/7.65 = ( abs_abs_rat @ Y3 ) )
% 13.25/7.65 = ( ( X = Y3 )
% 13.25/7.65 | ( X
% 13.25/7.65 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5631_minus__equation__iff,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ( uminus_uminus_real @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( uminus_uminus_real @ B )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5632_minus__equation__iff,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( ( uminus_uminus_int @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( uminus_uminus_int @ B )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5633_minus__equation__iff,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( ( uminus1482373934393186551omplex @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( uminus1482373934393186551omplex @ B )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5634_minus__equation__iff,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( ( uminus1351360451143612070nteger @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( uminus1351360451143612070nteger @ B )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5635_minus__equation__iff,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ( uminus_uminus_rat @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( uminus_uminus_rat @ B )
% 13.25/7.65 = A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5636_equation__minus__iff,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus_uminus_real @ B ) )
% 13.25/7.65 = ( B
% 13.25/7.65 = ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5637_equation__minus__iff,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus_uminus_int @ B ) )
% 13.25/7.65 = ( B
% 13.25/7.65 = ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5638_equation__minus__iff,axiom,
% 13.25/7.65 ! [A: complex,B: complex] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.65 = ( B
% 13.25/7.65 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5639_equation__minus__iff,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.65 = ( B
% 13.25/7.65 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5640_equation__minus__iff,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( uminus_uminus_rat @ B ) )
% 13.25/7.65 = ( B
% 13.25/7.65 = ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5641_abs__minus__le__zero,axiom,
% 13.25/7.65 ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5642_abs__minus__le__zero,axiom,
% 13.25/7.65 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5643_abs__minus__le__zero,axiom,
% 13.25/7.65 ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5644_abs__minus__le__zero,axiom,
% 13.25/7.65 ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5645_eq__abs__iff_H,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( abs_abs_real @ B ) )
% 13.25/7.65 = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.65 & ( ( B = A )
% 13.25/7.65 | ( B
% 13.25/7.65 = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5646_eq__abs__iff_H,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( abs_abs_Code_integer @ B ) )
% 13.25/7.65 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 13.25/7.65 & ( ( B = A )
% 13.25/7.65 | ( B
% 13.25/7.65 = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5647_eq__abs__iff_H,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( abs_abs_rat @ B ) )
% 13.25/7.65 = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.25/7.65 & ( ( B = A )
% 13.25/7.65 | ( B
% 13.25/7.65 = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5648_eq__abs__iff_H,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( A
% 13.25/7.65 = ( abs_abs_int @ B ) )
% 13.25/7.65 = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.25/7.65 & ( ( B = A )
% 13.25/7.65 | ( B
% 13.25/7.65 = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5649_abs__eq__iff_H,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ( abs_abs_real @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.25/7.65 & ( ( A = B )
% 13.25/7.65 | ( A
% 13.25/7.65 = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5650_abs__eq__iff_H,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( ( abs_abs_Code_integer @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 13.25/7.65 & ( ( A = B )
% 13.25/7.65 | ( A
% 13.25/7.65 = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5651_abs__eq__iff_H,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ( abs_abs_rat @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.25/7.65 & ( ( A = B )
% 13.25/7.65 | ( A
% 13.25/7.65 = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5652_abs__eq__iff_H,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( ( abs_abs_int @ A )
% 13.25/7.65 = B )
% 13.25/7.65 = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.25/7.65 & ( ( A = B )
% 13.25/7.65 | ( A
% 13.25/7.65 = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5653_abs__if__raw,axiom,
% 13.25/7.65 ( abs_abs_real
% 13.25/7.65 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5654_abs__if__raw,axiom,
% 13.25/7.65 ( abs_abs_int
% 13.25/7.65 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5655_abs__if__raw,axiom,
% 13.25/7.65 ( abs_abs_Code_integer
% 13.25/7.65 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5656_abs__if__raw,axiom,
% 13.25/7.65 ( abs_abs_rat
% 13.25/7.65 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5657_abs__of__neg,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ord_less_real @ A @ zero_zero_real )
% 13.25/7.65 => ( ( abs_abs_real @ A )
% 13.25/7.65 = ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5658_abs__of__neg,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ord_less_int @ A @ zero_zero_int )
% 13.25/7.65 => ( ( abs_abs_int @ A )
% 13.25/7.65 = ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5659_abs__of__neg,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 13.25/7.65 => ( ( abs_abs_Code_integer @ A )
% 13.25/7.65 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5660_abs__of__neg,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ord_less_rat @ A @ zero_zero_rat )
% 13.25/7.65 => ( ( abs_abs_rat @ A )
% 13.25/7.65 = ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5661_abs__if,axiom,
% 13.25/7.65 ( abs_abs_real
% 13.25/7.65 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5662_abs__if,axiom,
% 13.25/7.65 ( abs_abs_int
% 13.25/7.65 = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5663_abs__if,axiom,
% 13.25/7.65 ( abs_abs_Code_integer
% 13.25/7.65 = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5664_abs__if,axiom,
% 13.25/7.65 ( abs_abs_rat
% 13.25/7.65 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5665_abs__real__def,axiom,
% 13.25/7.65 ( abs_abs_real
% 13.25/7.65 = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5666_abs__le__D1,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_real @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5667_abs__le__D1,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 13.25/7.65 => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5668_abs__le__D1,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_rat @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5669_abs__le__D1,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 13.25/7.65 => ( ord_less_eq_int @ A @ B ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5670_abs__ge__self,axiom,
% 13.25/7.65 ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5671_abs__ge__self,axiom,
% 13.25/7.65 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5672_abs__ge__self,axiom,
% 13.25/7.65 ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5673_abs__ge__self,axiom,
% 13.25/7.65 ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5674_abs__eq__0__iff,axiom,
% 13.25/7.65 ! [A: code_integer] :
% 13.25/7.65 ( ( ( abs_abs_Code_integer @ A )
% 13.25/7.65 = zero_z3403309356797280102nteger )
% 13.25/7.65 = ( A = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5675_abs__eq__0__iff,axiom,
% 13.25/7.65 ! [A: complex] :
% 13.25/7.65 ( ( ( abs_abs_complex @ A )
% 13.25/7.65 = zero_zero_complex )
% 13.25/7.65 = ( A = zero_zero_complex ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5676_abs__eq__0__iff,axiom,
% 13.25/7.65 ! [A: real] :
% 13.25/7.65 ( ( ( abs_abs_real @ A )
% 13.25/7.65 = zero_zero_real )
% 13.25/7.65 = ( A = zero_zero_real ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5677_abs__eq__0__iff,axiom,
% 13.25/7.65 ! [A: rat] :
% 13.25/7.65 ( ( ( abs_abs_rat @ A )
% 13.25/7.65 = zero_zero_rat )
% 13.25/7.65 = ( A = zero_zero_rat ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5678_abs__eq__0__iff,axiom,
% 13.25/7.65 ! [A: int] :
% 13.25/7.65 ( ( ( abs_abs_int @ A )
% 13.25/7.65 = zero_zero_int )
% 13.25/7.65 = ( A = zero_zero_int ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5679_abs__mult,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 13.25/7.65 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5680_abs__mult,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 13.25/7.65 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5681_abs__mult,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 13.25/7.65 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5682_abs__mult,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 13.25/7.65 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5683_abs__one,axiom,
% 13.25/7.65 ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 13.25/7.65 = one_one_Code_integer ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5684_abs__one,axiom,
% 13.25/7.65 ( ( abs_abs_real @ one_one_real )
% 13.25/7.65 = one_one_real ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5685_abs__one,axiom,
% 13.25/7.65 ( ( abs_abs_rat @ one_one_rat )
% 13.25/7.65 = one_one_rat ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5686_abs__one,axiom,
% 13.25/7.65 ( ( abs_abs_int @ one_one_int )
% 13.25/7.65 = one_one_int ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5687_abs__minus__commute,axiom,
% 13.25/7.65 ! [A: code_integer,B: code_integer] :
% 13.25/7.65 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 13.25/7.65 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5688_abs__minus__commute,axiom,
% 13.25/7.65 ! [A: real,B: real] :
% 13.25/7.65 ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 13.25/7.65 = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5689_abs__minus__commute,axiom,
% 13.25/7.65 ! [A: rat,B: rat] :
% 13.25/7.65 ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 13.25/7.65 = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5690_abs__minus__commute,axiom,
% 13.25/7.65 ! [A: int,B: int] :
% 13.25/7.65 ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 13.25/7.65 = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 13.25/7.65
% 13.25/7.65 thf(fact_5691_tanh__real__gt__neg1,axiom,
% 13.25/7.65 ! [X: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5692_power__abs,axiom,
% 13.25/7.66 ! [A: code_integer,N: nat] :
% 13.25/7.66 ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) )
% 13.25/7.66 = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5693_power__abs,axiom,
% 13.25/7.66 ! [A: rat,N: nat] :
% 13.25/7.66 ( ( abs_abs_rat @ ( power_power_rat @ A @ N ) )
% 13.25/7.66 = ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5694_power__abs,axiom,
% 13.25/7.66 ! [A: real,N: nat] :
% 13.25/7.66 ( ( abs_abs_real @ ( power_power_real @ A @ N ) )
% 13.25/7.66 = ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5695_power__abs,axiom,
% 13.25/7.66 ! [A: int,N: nat] :
% 13.25/7.66 ( ( abs_abs_int @ ( power_power_int @ A @ N ) )
% 13.25/7.66 = ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5696_dvd__if__abs__eq,axiom,
% 13.25/7.66 ! [L2: real,K: real] :
% 13.25/7.66 ( ( ( abs_abs_real @ L2 )
% 13.25/7.66 = ( abs_abs_real @ K ) )
% 13.25/7.66 => ( dvd_dvd_real @ L2 @ K ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5697_dvd__if__abs__eq,axiom,
% 13.25/7.66 ! [L2: int,K: int] :
% 13.25/7.66 ( ( ( abs_abs_int @ L2 )
% 13.25/7.66 = ( abs_abs_int @ K ) )
% 13.25/7.66 => ( dvd_dvd_int @ L2 @ K ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5698_dvd__if__abs__eq,axiom,
% 13.25/7.66 ! [L2: code_integer,K: code_integer] :
% 13.25/7.66 ( ( ( abs_abs_Code_integer @ L2 )
% 13.25/7.66 = ( abs_abs_Code_integer @ K ) )
% 13.25/7.66 => ( dvd_dvd_Code_integer @ L2 @ K ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5699_dvd__if__abs__eq,axiom,
% 13.25/7.66 ! [L2: rat,K: rat] :
% 13.25/7.66 ( ( ( abs_abs_rat @ L2 )
% 13.25/7.66 = ( abs_abs_rat @ K ) )
% 13.25/7.66 => ( dvd_dvd_rat @ L2 @ K ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5700_compl__mono,axiom,
% 13.25/7.66 ! [X: set_int,Y3: set_int] :
% 13.25/7.66 ( ( ord_less_eq_set_int @ X @ Y3 )
% 13.25/7.66 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5701_compl__le__swap1,axiom,
% 13.25/7.66 ! [Y3: set_int,X: set_int] :
% 13.25/7.66 ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X ) )
% 13.25/7.66 => ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5702_compl__le__swap2,axiom,
% 13.25/7.66 ! [Y3: set_int,X: set_int] :
% 13.25/7.66 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X )
% 13.25/7.66 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X ) @ Y3 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5703_le__minus__iff,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 13.25/7.66 = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5704_le__minus__iff,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.66 = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5705_le__minus__iff,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 13.25/7.66 = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5706_le__minus__iff,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5707_minus__le__iff,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 13.25/7.66 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5708_minus__le__iff,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 13.25/7.66 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5709_minus__le__iff,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 13.25/7.66 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5710_minus__le__iff,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5711_le__imp__neg__le,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.66 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5712_le__imp__neg__le,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ A @ B )
% 13.25/7.66 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5713_le__imp__neg__le,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ A @ B )
% 13.25/7.66 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5714_le__imp__neg__le,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ A @ B )
% 13.25/7.66 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5715_less__minus__iff,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 13.25/7.66 = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5716_less__minus__iff,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5717_less__minus__iff,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.66 = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5718_less__minus__iff,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 13.25/7.66 = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5719_minus__less__iff,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 13.25/7.66 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5720_minus__less__iff,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5721_minus__less__iff,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 13.25/7.66 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5722_minus__less__iff,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 13.25/7.66 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5723_verit__negate__coefficient_I2_J,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ A @ B )
% 13.25/7.66 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5724_verit__negate__coefficient_I2_J,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ord_less_int @ A @ B )
% 13.25/7.66 => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5725_verit__negate__coefficient_I2_J,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ord_le6747313008572928689nteger @ A @ B )
% 13.25/7.66 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5726_verit__negate__coefficient_I2_J,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_rat @ A @ B )
% 13.25/7.66 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5727_numeral__neq__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( numeral_numeral_real @ M )
% 13.25/7.66 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5728_numeral__neq__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( numeral_numeral_int @ M )
% 13.25/7.66 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5729_numeral__neq__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( numera6690914467698888265omplex @ M )
% 13.25/7.66 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5730_numeral__neq__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( numera6620942414471956472nteger @ M )
% 13.25/7.66 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5731_numeral__neq__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( numeral_numeral_rat @ M )
% 13.25/7.66 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5732_neg__numeral__neq__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
% 13.25/7.66 != ( numeral_numeral_real @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5733_neg__numeral__neq__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
% 13.25/7.66 != ( numeral_numeral_int @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5734_neg__numeral__neq__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
% 13.25/7.66 != ( numera6690914467698888265omplex @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5735_neg__numeral__neq__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
% 13.25/7.66 != ( numera6620942414471956472nteger @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5736_neg__numeral__neq__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
% 13.25/7.66 != ( numeral_numeral_rat @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5737_square__eq__iff,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ( times_times_real @ A @ A )
% 13.25/7.66 = ( times_times_real @ B @ B ) )
% 13.25/7.66 = ( ( A = B )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus_uminus_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5738_square__eq__iff,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ( times_times_int @ A @ A )
% 13.25/7.66 = ( times_times_int @ B @ B ) )
% 13.25/7.66 = ( ( A = B )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus_uminus_int @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5739_square__eq__iff,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( ( times_times_complex @ A @ A )
% 13.25/7.66 = ( times_times_complex @ B @ B ) )
% 13.25/7.66 = ( ( A = B )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5740_square__eq__iff,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ( times_3573771949741848930nteger @ A @ A )
% 13.25/7.66 = ( times_3573771949741848930nteger @ B @ B ) )
% 13.25/7.66 = ( ( A = B )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5741_square__eq__iff,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ( times_times_rat @ A @ A )
% 13.25/7.66 = ( times_times_rat @ B @ B ) )
% 13.25/7.66 = ( ( A = B )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5742_minus__mult__commute,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 13.25/7.66 = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5743_minus__mult__commute,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5744_minus__mult__commute,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 13.25/7.66 = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5745_minus__mult__commute,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 13.25/7.66 = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5746_minus__mult__commute,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 13.25/7.66 = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5747_one__neq__neg__one,axiom,
% 13.25/7.66 ( one_one_real
% 13.25/7.66 != ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5748_one__neq__neg__one,axiom,
% 13.25/7.66 ( one_one_int
% 13.25/7.66 != ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5749_one__neq__neg__one,axiom,
% 13.25/7.66 ( one_one_complex
% 13.25/7.66 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5750_one__neq__neg__one,axiom,
% 13.25/7.66 ( one_one_Code_integer
% 13.25/7.66 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5751_one__neq__neg__one,axiom,
% 13.25/7.66 ( one_one_rat
% 13.25/7.66 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5752_group__cancel_Oneg1,axiom,
% 13.25/7.66 ! [A2: real,K: real,A: real] :
% 13.25/7.66 ( ( A2
% 13.25/7.66 = ( plus_plus_real @ K @ A ) )
% 13.25/7.66 => ( ( uminus_uminus_real @ A2 )
% 13.25/7.66 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5753_group__cancel_Oneg1,axiom,
% 13.25/7.66 ! [A2: int,K: int,A: int] :
% 13.25/7.66 ( ( A2
% 13.25/7.66 = ( plus_plus_int @ K @ A ) )
% 13.25/7.66 => ( ( uminus_uminus_int @ A2 )
% 13.25/7.66 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5754_group__cancel_Oneg1,axiom,
% 13.25/7.66 ! [A2: complex,K: complex,A: complex] :
% 13.25/7.66 ( ( A2
% 13.25/7.66 = ( plus_plus_complex @ K @ A ) )
% 13.25/7.66 => ( ( uminus1482373934393186551omplex @ A2 )
% 13.25/7.66 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5755_group__cancel_Oneg1,axiom,
% 13.25/7.66 ! [A2: code_integer,K: code_integer,A: code_integer] :
% 13.25/7.66 ( ( A2
% 13.25/7.66 = ( plus_p5714425477246183910nteger @ K @ A ) )
% 13.25/7.66 => ( ( uminus1351360451143612070nteger @ A2 )
% 13.25/7.66 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5756_group__cancel_Oneg1,axiom,
% 13.25/7.66 ! [A2: rat,K: rat,A: rat] :
% 13.25/7.66 ( ( A2
% 13.25/7.66 = ( plus_plus_rat @ K @ A ) )
% 13.25/7.66 => ( ( uminus_uminus_rat @ A2 )
% 13.25/7.66 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5757_add_Oinverse__distrib__swap,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 13.25/7.66 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5758_add_Oinverse__distrib__swap,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 13.25/7.66 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5759_add_Oinverse__distrib__swap,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 13.25/7.66 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5760_add_Oinverse__distrib__swap,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 13.25/7.66 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5761_add_Oinverse__distrib__swap,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 13.25/7.66 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5762_is__num__normalize_I8_J,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 13.25/7.66 = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5763_is__num__normalize_I8_J,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 13.25/7.66 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5764_is__num__normalize_I8_J,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
% 13.25/7.66 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5765_is__num__normalize_I8_J,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 13.25/7.66 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5766_is__num__normalize_I8_J,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 13.25/7.66 = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5767_minus__diff__commute,axiom,
% 13.25/7.66 ! [B: real,A: real] :
% 13.25/7.66 ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 13.25/7.66 = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5768_minus__diff__commute,axiom,
% 13.25/7.66 ! [B: int,A: int] :
% 13.25/7.66 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 13.25/7.66 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5769_minus__diff__commute,axiom,
% 13.25/7.66 ! [B: complex,A: complex] :
% 13.25/7.66 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
% 13.25/7.66 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5770_minus__diff__commute,axiom,
% 13.25/7.66 ! [B: code_integer,A: code_integer] :
% 13.25/7.66 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 13.25/7.66 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5771_minus__diff__commute,axiom,
% 13.25/7.66 ! [B: rat,A: rat] :
% 13.25/7.66 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 13.25/7.66 = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5772_minus__diff__minus,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 13.25/7.66 = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5773_minus__diff__minus,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5774_minus__diff__minus,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5775_minus__diff__minus,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5776_minus__diff__minus,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 13.25/7.66 = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5777_div__minus__right,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5778_div__minus__right,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.66 = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5779_minus__divide__left,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 13.25/7.66 = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5780_minus__divide__left,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.66 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5781_minus__divide__left,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 13.25/7.66 = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5782_minus__divide__divide,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 13.25/7.66 = ( divide_divide_real @ A @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5783_minus__divide__divide,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.66 = ( divide1717551699836669952omplex @ A @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5784_minus__divide__divide,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 13.25/7.66 = ( divide_divide_rat @ A @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5785_minus__divide__right,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 13.25/7.66 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5786_minus__divide__right,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.66 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5787_minus__divide__right,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 13.25/7.66 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5788_mod__minus__right,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5789_mod__minus__right,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5790_mod__minus__cong,axiom,
% 13.25/7.66 ! [A: int,B: int,A4: int] :
% 13.25/7.66 ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 = ( modulo_modulo_int @ A4 @ B ) )
% 13.25/7.66 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5791_mod__minus__cong,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer,A4: code_integer] :
% 13.25/7.66 ( ( ( modulo364778990260209775nteger @ A @ B )
% 13.25/7.66 = ( modulo364778990260209775nteger @ A4 @ B ) )
% 13.25/7.66 => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 13.25/7.66 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5792_mod__minus__eq,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
% 13.25/7.66 = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5793_mod__minus__eq,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
% 13.25/7.66 = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5794_abs__ge__zero,axiom,
% 13.25/7.66 ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5795_abs__ge__zero,axiom,
% 13.25/7.66 ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5796_abs__ge__zero,axiom,
% 13.25/7.66 ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5797_abs__ge__zero,axiom,
% 13.25/7.66 ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5798_abs__of__pos,axiom,
% 13.25/7.66 ! [A: code_integer] :
% 13.25/7.66 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 13.25/7.66 => ( ( abs_abs_Code_integer @ A )
% 13.25/7.66 = A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5799_abs__of__pos,axiom,
% 13.25/7.66 ! [A: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.66 => ( ( abs_abs_real @ A )
% 13.25/7.66 = A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5800_abs__of__pos,axiom,
% 13.25/7.66 ! [A: rat] :
% 13.25/7.66 ( ( ord_less_rat @ zero_zero_rat @ A )
% 13.25/7.66 => ( ( abs_abs_rat @ A )
% 13.25/7.66 = A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5801_abs__of__pos,axiom,
% 13.25/7.66 ! [A: int] :
% 13.25/7.66 ( ( ord_less_int @ zero_zero_int @ A )
% 13.25/7.66 => ( ( abs_abs_int @ A )
% 13.25/7.66 = A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5802_abs__not__less__zero,axiom,
% 13.25/7.66 ! [A: code_integer] :
% 13.25/7.66 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5803_abs__not__less__zero,axiom,
% 13.25/7.66 ! [A: real] :
% 13.25/7.66 ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5804_abs__not__less__zero,axiom,
% 13.25/7.66 ! [A: rat] :
% 13.25/7.66 ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5805_abs__not__less__zero,axiom,
% 13.25/7.66 ! [A: int] :
% 13.25/7.66 ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5806_abs__triangle__ineq,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5807_abs__triangle__ineq,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5808_abs__triangle__ineq,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5809_abs__triangle__ineq,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5810_abs__mult__less,axiom,
% 13.25/7.66 ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 13.25/7.66 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 13.25/7.66 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 13.25/7.66 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5811_abs__mult__less,axiom,
% 13.25/7.66 ! [A: real,C: real,B: real,D: real] :
% 13.25/7.66 ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 13.25/7.66 => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 13.25/7.66 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5812_abs__mult__less,axiom,
% 13.25/7.66 ! [A: rat,C: rat,B: rat,D: rat] :
% 13.25/7.66 ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 13.25/7.66 => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 13.25/7.66 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5813_abs__mult__less,axiom,
% 13.25/7.66 ! [A: int,C: int,B: int,D: int] :
% 13.25/7.66 ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 13.25/7.66 => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 13.25/7.66 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5814_abs__triangle__ineq2__sym,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5815_abs__triangle__ineq2__sym,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5816_abs__triangle__ineq2__sym,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5817_abs__triangle__ineq2__sym,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5818_abs__triangle__ineq3,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5819_abs__triangle__ineq3,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5820_abs__triangle__ineq3,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5821_abs__triangle__ineq3,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5822_abs__triangle__ineq2,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5823_abs__triangle__ineq2,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5824_abs__triangle__ineq2,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5825_abs__triangle__ineq2,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5826_nonzero__abs__divide,axiom,
% 13.25/7.66 ! [B: real,A: real] :
% 13.25/7.66 ( ( B != zero_zero_real )
% 13.25/7.66 => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 13.25/7.66 = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5827_nonzero__abs__divide,axiom,
% 13.25/7.66 ! [B: rat,A: rat] :
% 13.25/7.66 ( ( B != zero_zero_rat )
% 13.25/7.66 => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 13.25/7.66 = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5828_neg__numeral__le__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5829_neg__numeral__le__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5830_neg__numeral__le__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5831_neg__numeral__le__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5832_not__numeral__le__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5833_not__numeral__le__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5834_not__numeral__le__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5835_not__numeral__le__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5836_zero__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( zero_zero_real
% 13.25/7.66 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5837_zero__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( zero_zero_int
% 13.25/7.66 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5838_zero__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( zero_zero_complex
% 13.25/7.66 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5839_zero__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( zero_z3403309356797280102nteger
% 13.25/7.66 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5840_zero__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( zero_zero_rat
% 13.25/7.66 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5841_neg__numeral__less__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5842_neg__numeral__less__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5843_neg__numeral__less__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5844_neg__numeral__less__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5845_not__numeral__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5846_not__numeral__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5847_not__numeral__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5848_not__numeral__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5849_le__minus__one__simps_I4_J,axiom,
% 13.25/7.66 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5850_le__minus__one__simps_I4_J,axiom,
% 13.25/7.66 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5851_le__minus__one__simps_I4_J,axiom,
% 13.25/7.66 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5852_le__minus__one__simps_I4_J,axiom,
% 13.25/7.66 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5853_le__minus__one__simps_I2_J,axiom,
% 13.25/7.66 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5854_le__minus__one__simps_I2_J,axiom,
% 13.25/7.66 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5855_le__minus__one__simps_I2_J,axiom,
% 13.25/7.66 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5856_le__minus__one__simps_I2_J,axiom,
% 13.25/7.66 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5857_zero__neq__neg__one,axiom,
% 13.25/7.66 ( zero_zero_real
% 13.25/7.66 != ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5858_zero__neq__neg__one,axiom,
% 13.25/7.66 ( zero_zero_int
% 13.25/7.66 != ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5859_zero__neq__neg__one,axiom,
% 13.25/7.66 ( zero_zero_complex
% 13.25/7.66 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5860_zero__neq__neg__one,axiom,
% 13.25/7.66 ( zero_z3403309356797280102nteger
% 13.25/7.66 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5861_zero__neq__neg__one,axiom,
% 13.25/7.66 ( zero_zero_rat
% 13.25/7.66 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5862_add__eq__0__iff,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ( plus_plus_real @ A @ B )
% 13.25/7.66 = zero_zero_real )
% 13.25/7.66 = ( B
% 13.25/7.66 = ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5863_add__eq__0__iff,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ( plus_plus_int @ A @ B )
% 13.25/7.66 = zero_zero_int )
% 13.25/7.66 = ( B
% 13.25/7.66 = ( uminus_uminus_int @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5864_add__eq__0__iff,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( ( plus_plus_complex @ A @ B )
% 13.25/7.66 = zero_zero_complex )
% 13.25/7.66 = ( B
% 13.25/7.66 = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5865_add__eq__0__iff,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 13.25/7.66 = zero_z3403309356797280102nteger )
% 13.25/7.66 = ( B
% 13.25/7.66 = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5866_add__eq__0__iff,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ( plus_plus_rat @ A @ B )
% 13.25/7.66 = zero_zero_rat )
% 13.25/7.66 = ( B
% 13.25/7.66 = ( uminus_uminus_rat @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5867_ab__group__add__class_Oab__left__minus,axiom,
% 13.25/7.66 ! [A: real] :
% 13.25/7.66 ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 13.25/7.66 = zero_zero_real ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5868_ab__group__add__class_Oab__left__minus,axiom,
% 13.25/7.66 ! [A: int] :
% 13.25/7.66 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 13.25/7.66 = zero_zero_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5869_ab__group__add__class_Oab__left__minus,axiom,
% 13.25/7.66 ! [A: complex] :
% 13.25/7.66 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 13.25/7.66 = zero_zero_complex ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5870_ab__group__add__class_Oab__left__minus,axiom,
% 13.25/7.66 ! [A: code_integer] :
% 13.25/7.66 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 13.25/7.66 = zero_z3403309356797280102nteger ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5871_ab__group__add__class_Oab__left__minus,axiom,
% 13.25/7.66 ! [A: rat] :
% 13.25/7.66 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 13.25/7.66 = zero_zero_rat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5872_add_Oinverse__unique,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ( plus_plus_real @ A @ B )
% 13.25/7.66 = zero_zero_real )
% 13.25/7.66 => ( ( uminus_uminus_real @ A )
% 13.25/7.66 = B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5873_add_Oinverse__unique,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ( plus_plus_int @ A @ B )
% 13.25/7.66 = zero_zero_int )
% 13.25/7.66 => ( ( uminus_uminus_int @ A )
% 13.25/7.66 = B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5874_add_Oinverse__unique,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( ( plus_plus_complex @ A @ B )
% 13.25/7.66 = zero_zero_complex )
% 13.25/7.66 => ( ( uminus1482373934393186551omplex @ A )
% 13.25/7.66 = B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5875_add_Oinverse__unique,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 13.25/7.66 = zero_z3403309356797280102nteger )
% 13.25/7.66 => ( ( uminus1351360451143612070nteger @ A )
% 13.25/7.66 = B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5876_add_Oinverse__unique,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ( plus_plus_rat @ A @ B )
% 13.25/7.66 = zero_zero_rat )
% 13.25/7.66 => ( ( uminus_uminus_rat @ A )
% 13.25/7.66 = B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5877_eq__neg__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( A
% 13.25/7.66 = ( uminus_uminus_real @ B ) )
% 13.25/7.66 = ( ( plus_plus_real @ A @ B )
% 13.25/7.66 = zero_zero_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5878_eq__neg__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( A
% 13.25/7.66 = ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( ( plus_plus_int @ A @ B )
% 13.25/7.66 = zero_zero_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5879_eq__neg__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( A
% 13.25/7.66 = ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.66 = ( ( plus_plus_complex @ A @ B )
% 13.25/7.66 = zero_zero_complex ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5880_eq__neg__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( A
% 13.25/7.66 = ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.66 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 13.25/7.66 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5881_eq__neg__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( A
% 13.25/7.66 = ( uminus_uminus_rat @ B ) )
% 13.25/7.66 = ( ( plus_plus_rat @ A @ B )
% 13.25/7.66 = zero_zero_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5882_neg__eq__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ( uminus_uminus_real @ A )
% 13.25/7.66 = B )
% 13.25/7.66 = ( ( plus_plus_real @ A @ B )
% 13.25/7.66 = zero_zero_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5883_neg__eq__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ( uminus_uminus_int @ A )
% 13.25/7.66 = B )
% 13.25/7.66 = ( ( plus_plus_int @ A @ B )
% 13.25/7.66 = zero_zero_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5884_neg__eq__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( ( uminus1482373934393186551omplex @ A )
% 13.25/7.66 = B )
% 13.25/7.66 = ( ( plus_plus_complex @ A @ B )
% 13.25/7.66 = zero_zero_complex ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5885_neg__eq__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ( uminus1351360451143612070nteger @ A )
% 13.25/7.66 = B )
% 13.25/7.66 = ( ( plus_p5714425477246183910nteger @ A @ B )
% 13.25/7.66 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5886_neg__eq__iff__add__eq__0,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ( uminus_uminus_rat @ A )
% 13.25/7.66 = B )
% 13.25/7.66 = ( ( plus_plus_rat @ A @ B )
% 13.25/7.66 = zero_zero_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5887_less__minus__one__simps_I2_J,axiom,
% 13.25/7.66 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5888_less__minus__one__simps_I2_J,axiom,
% 13.25/7.66 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5889_less__minus__one__simps_I2_J,axiom,
% 13.25/7.66 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5890_less__minus__one__simps_I2_J,axiom,
% 13.25/7.66 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5891_less__minus__one__simps_I4_J,axiom,
% 13.25/7.66 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5892_less__minus__one__simps_I4_J,axiom,
% 13.25/7.66 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5893_less__minus__one__simps_I4_J,axiom,
% 13.25/7.66 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5894_less__minus__one__simps_I4_J,axiom,
% 13.25/7.66 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5895_numeral__times__minus__swap,axiom,
% 13.25/7.66 ! [W: num,X: real] :
% 13.25/7.66 ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X ) )
% 13.25/7.66 = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5896_numeral__times__minus__swap,axiom,
% 13.25/7.66 ! [W: num,X: int] :
% 13.25/7.66 ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X ) )
% 13.25/7.66 = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5897_numeral__times__minus__swap,axiom,
% 13.25/7.66 ! [W: num,X: complex] :
% 13.25/7.66 ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X ) )
% 13.25/7.66 = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5898_numeral__times__minus__swap,axiom,
% 13.25/7.66 ! [W: num,X: code_integer] :
% 13.25/7.66 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X ) )
% 13.25/7.66 = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5899_numeral__times__minus__swap,axiom,
% 13.25/7.66 ! [W: num,X: rat] :
% 13.25/7.66 ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X ) )
% 13.25/7.66 = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5900_nonzero__minus__divide__divide,axiom,
% 13.25/7.66 ! [B: real,A: real] :
% 13.25/7.66 ( ( B != zero_zero_real )
% 13.25/7.66 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 13.25/7.66 = ( divide_divide_real @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5901_nonzero__minus__divide__divide,axiom,
% 13.25/7.66 ! [B: complex,A: complex] :
% 13.25/7.66 ( ( B != zero_zero_complex )
% 13.25/7.66 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.66 = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5902_nonzero__minus__divide__divide,axiom,
% 13.25/7.66 ! [B: rat,A: rat] :
% 13.25/7.66 ( ( B != zero_zero_rat )
% 13.25/7.66 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 13.25/7.66 = ( divide_divide_rat @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5903_nonzero__minus__divide__right,axiom,
% 13.25/7.66 ! [B: real,A: real] :
% 13.25/7.66 ( ( B != zero_zero_real )
% 13.25/7.66 => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 13.25/7.66 = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5904_nonzero__minus__divide__right,axiom,
% 13.25/7.66 ! [B: complex,A: complex] :
% 13.25/7.66 ( ( B != zero_zero_complex )
% 13.25/7.66 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.66 = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5905_nonzero__minus__divide__right,axiom,
% 13.25/7.66 ! [B: rat,A: rat] :
% 13.25/7.66 ( ( B != zero_zero_rat )
% 13.25/7.66 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 13.25/7.66 = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5906_numeral__neq__neg__one,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( ( numeral_numeral_real @ N )
% 13.25/7.66 != ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5907_numeral__neq__neg__one,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( ( numeral_numeral_int @ N )
% 13.25/7.66 != ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5908_numeral__neq__neg__one,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( ( numera6690914467698888265omplex @ N )
% 13.25/7.66 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5909_numeral__neq__neg__one,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( ( numera6620942414471956472nteger @ N )
% 13.25/7.66 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5910_numeral__neq__neg__one,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( ( numeral_numeral_rat @ N )
% 13.25/7.66 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5911_one__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( one_one_real
% 13.25/7.66 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5912_one__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( one_one_int
% 13.25/7.66 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5913_one__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( one_one_complex
% 13.25/7.66 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5914_one__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( one_one_Code_integer
% 13.25/7.66 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5915_one__neq__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( one_one_rat
% 13.25/7.66 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5916_square__eq__1__iff,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ( times_times_real @ X @ X )
% 13.25/7.66 = one_one_real )
% 13.25/7.66 = ( ( X = one_one_real )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5917_square__eq__1__iff,axiom,
% 13.25/7.66 ! [X: int] :
% 13.25/7.66 ( ( ( times_times_int @ X @ X )
% 13.25/7.66 = one_one_int )
% 13.25/7.66 = ( ( X = one_one_int )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5918_square__eq__1__iff,axiom,
% 13.25/7.66 ! [X: complex] :
% 13.25/7.66 ( ( ( times_times_complex @ X @ X )
% 13.25/7.66 = one_one_complex )
% 13.25/7.66 = ( ( X = one_one_complex )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5919_square__eq__1__iff,axiom,
% 13.25/7.66 ! [X: code_integer] :
% 13.25/7.66 ( ( ( times_3573771949741848930nteger @ X @ X )
% 13.25/7.66 = one_one_Code_integer )
% 13.25/7.66 = ( ( X = one_one_Code_integer )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5920_square__eq__1__iff,axiom,
% 13.25/7.66 ! [X: rat] :
% 13.25/7.66 ( ( ( times_times_rat @ X @ X )
% 13.25/7.66 = one_one_rat )
% 13.25/7.66 = ( ( X = one_one_rat )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5921_group__cancel_Osub2,axiom,
% 13.25/7.66 ! [B5: real,K: real,B: real,A: real] :
% 13.25/7.66 ( ( B5
% 13.25/7.66 = ( plus_plus_real @ K @ B ) )
% 13.25/7.66 => ( ( minus_minus_real @ A @ B5 )
% 13.25/7.66 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5922_group__cancel_Osub2,axiom,
% 13.25/7.66 ! [B5: int,K: int,B: int,A: int] :
% 13.25/7.66 ( ( B5
% 13.25/7.66 = ( plus_plus_int @ K @ B ) )
% 13.25/7.66 => ( ( minus_minus_int @ A @ B5 )
% 13.25/7.66 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5923_group__cancel_Osub2,axiom,
% 13.25/7.66 ! [B5: complex,K: complex,B: complex,A: complex] :
% 13.25/7.66 ( ( B5
% 13.25/7.66 = ( plus_plus_complex @ K @ B ) )
% 13.25/7.66 => ( ( minus_minus_complex @ A @ B5 )
% 13.25/7.66 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5924_group__cancel_Osub2,axiom,
% 13.25/7.66 ! [B5: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 13.25/7.66 ( ( B5
% 13.25/7.66 = ( plus_p5714425477246183910nteger @ K @ B ) )
% 13.25/7.66 => ( ( minus_8373710615458151222nteger @ A @ B5 )
% 13.25/7.66 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5925_group__cancel_Osub2,axiom,
% 13.25/7.66 ! [B5: rat,K: rat,B: rat,A: rat] :
% 13.25/7.66 ( ( B5
% 13.25/7.66 = ( plus_plus_rat @ K @ B ) )
% 13.25/7.66 => ( ( minus_minus_rat @ A @ B5 )
% 13.25/7.66 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5926_diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_minus_real
% 13.25/7.66 = ( ^ [A3: real,B4: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5927_diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_minus_int
% 13.25/7.66 = ( ^ [A3: int,B4: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5928_diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_minus_complex
% 13.25/7.66 = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5929_diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_8373710615458151222nteger
% 13.25/7.66 = ( ^ [A3: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5930_diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_minus_rat
% 13.25/7.66 = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5931_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_minus_real
% 13.25/7.66 = ( ^ [A3: real,B4: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5932_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_minus_int
% 13.25/7.66 = ( ^ [A3: int,B4: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5933_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_minus_complex
% 13.25/7.66 = ( ^ [A3: complex,B4: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5934_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_8373710615458151222nteger
% 13.25/7.66 = ( ^ [A3: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5935_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 13.25/7.66 ( minus_minus_rat
% 13.25/7.66 = ( ^ [A3: rat,B4: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5936_dvd__neg__div,axiom,
% 13.25/7.66 ! [B: real,A: real] :
% 13.25/7.66 ( ( dvd_dvd_real @ B @ A )
% 13.25/7.66 => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 13.25/7.66 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5937_dvd__neg__div,axiom,
% 13.25/7.66 ! [B: int,A: int] :
% 13.25/7.66 ( ( dvd_dvd_int @ B @ A )
% 13.25/7.66 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5938_dvd__neg__div,axiom,
% 13.25/7.66 ! [B: complex,A: complex] :
% 13.25/7.66 ( ( dvd_dvd_complex @ B @ A )
% 13.25/7.66 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5939_dvd__neg__div,axiom,
% 13.25/7.66 ! [B: code_integer,A: code_integer] :
% 13.25/7.66 ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.66 => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5940_dvd__neg__div,axiom,
% 13.25/7.66 ! [B: rat,A: rat] :
% 13.25/7.66 ( ( dvd_dvd_rat @ B @ A )
% 13.25/7.66 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 13.25/7.66 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5941_dvd__div__neg,axiom,
% 13.25/7.66 ! [B: real,A: real] :
% 13.25/7.66 ( ( dvd_dvd_real @ B @ A )
% 13.25/7.66 => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 13.25/7.66 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5942_dvd__div__neg,axiom,
% 13.25/7.66 ! [B: int,A: int] :
% 13.25/7.66 ( ( dvd_dvd_int @ B @ A )
% 13.25/7.66 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5943_dvd__div__neg,axiom,
% 13.25/7.66 ! [B: complex,A: complex] :
% 13.25/7.66 ( ( dvd_dvd_complex @ B @ A )
% 13.25/7.66 => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5944_dvd__div__neg,axiom,
% 13.25/7.66 ! [B: code_integer,A: code_integer] :
% 13.25/7.66 ( ( dvd_dvd_Code_integer @ B @ A )
% 13.25/7.66 => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5945_dvd__div__neg,axiom,
% 13.25/7.66 ! [B: rat,A: rat] :
% 13.25/7.66 ( ( dvd_dvd_rat @ B @ A )
% 13.25/7.66 => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 13.25/7.66 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5946_subset__Compl__self__eq,axiom,
% 13.25/7.66 ! [A2: set_nat] :
% 13.25/7.66 ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 13.25/7.66 = ( A2 = bot_bot_set_nat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5947_subset__Compl__self__eq,axiom,
% 13.25/7.66 ! [A2: set_real] :
% 13.25/7.66 ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 13.25/7.66 = ( A2 = bot_bot_set_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5948_subset__Compl__self__eq,axiom,
% 13.25/7.66 ! [A2: set_int] :
% 13.25/7.66 ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 13.25/7.66 = ( A2 = bot_bot_set_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5949_sin__bound__lemma,axiom,
% 13.25/7.66 ! [X: real,Y3: real,U: real,V: real] :
% 13.25/7.66 ( ( X = Y3 )
% 13.25/7.66 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 13.25/7.66 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X @ U ) @ Y3 ) ) @ V ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5950_real__minus__mult__self__le,axiom,
% 13.25/7.66 ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5951_zmult__eq__1__iff,axiom,
% 13.25/7.66 ! [M: int,N: int] :
% 13.25/7.66 ( ( ( times_times_int @ M @ N )
% 13.25/7.66 = one_one_int )
% 13.25/7.66 = ( ( ( M = one_one_int )
% 13.25/7.66 & ( N = one_one_int ) )
% 13.25/7.66 | ( ( M
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.66 & ( N
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5952_pos__zmult__eq__1__iff__lemma,axiom,
% 13.25/7.66 ! [M: int,N: int] :
% 13.25/7.66 ( ( ( times_times_int @ M @ N )
% 13.25/7.66 = one_one_int )
% 13.25/7.66 => ( ( M = one_one_int )
% 13.25/7.66 | ( M
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5953_minus__int__code_I2_J,axiom,
% 13.25/7.66 ! [L2: int] :
% 13.25/7.66 ( ( minus_minus_int @ zero_zero_int @ L2 )
% 13.25/7.66 = ( uminus_uminus_int @ L2 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5954_zmod__zminus2__not__zero,axiom,
% 13.25/7.66 ! [K: int,L2: int] :
% 13.25/7.66 ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L2 ) )
% 13.25/7.66 != zero_zero_int )
% 13.25/7.66 => ( ( modulo_modulo_int @ K @ L2 )
% 13.25/7.66 != zero_zero_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5955_zmod__zminus1__not__zero,axiom,
% 13.25/7.66 ! [K: int,L2: int] :
% 13.25/7.66 ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 13.25/7.66 != zero_zero_int )
% 13.25/7.66 => ( ( modulo_modulo_int @ K @ L2 )
% 13.25/7.66 != zero_zero_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5956_minus__real__def,axiom,
% 13.25/7.66 ( minus_minus_real
% 13.25/7.66 = ( ^ [X2: real,Y: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5957_tanh__real__lt__1,axiom,
% 13.25/7.66 ! [X: real] : ( ord_less_real @ ( tanh_real @ X ) @ one_one_real ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5958_dense__eq0__I,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ! [E2: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ E2 )
% 13.25/7.66 => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
% 13.25/7.66 => ( X = zero_zero_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5959_dense__eq0__I,axiom,
% 13.25/7.66 ! [X: rat] :
% 13.25/7.66 ( ! [E2: rat] :
% 13.25/7.66 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 13.25/7.66 => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
% 13.25/7.66 => ( X = zero_zero_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5960_abs__mult__pos,axiom,
% 13.25/7.66 ! [X: code_integer,Y3: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
% 13.25/7.66 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X )
% 13.25/7.66 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5961_abs__mult__pos,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.66 => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X )
% 13.25/7.66 = ( abs_abs_real @ ( times_times_real @ Y3 @ X ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5962_abs__mult__pos,axiom,
% 13.25/7.66 ! [X: rat,Y3: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ zero_zero_rat @ X )
% 13.25/7.66 => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X )
% 13.25/7.66 = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5963_abs__mult__pos,axiom,
% 13.25/7.66 ! [X: int,Y3: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.66 => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X )
% 13.25/7.66 = ( abs_abs_int @ ( times_times_int @ Y3 @ X ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5964_abs__eq__mult,axiom,
% 13.25/7.66 ! [A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 13.25/7.66 | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 13.25/7.66 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 13.25/7.66 | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 13.25/7.66 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 13.25/7.66 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5965_abs__eq__mult,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.66 | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 13.25/7.66 & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 13.25/7.66 | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 13.25/7.66 => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 13.25/7.66 = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5966_abs__eq__mult,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 13.25/7.66 | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 13.25/7.66 & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 13.25/7.66 | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 13.25/7.66 => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 13.25/7.66 = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5967_abs__eq__mult,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 13.25/7.66 | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 13.25/7.66 & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.25/7.66 | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 13.25/7.66 => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 13.25/7.66 = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5968_abs__div__pos,axiom,
% 13.25/7.66 ! [Y3: real,X: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.66 => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y3 )
% 13.25/7.66 = ( abs_abs_real @ ( divide_divide_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5969_abs__div__pos,axiom,
% 13.25/7.66 ! [Y3: rat,X: rat] :
% 13.25/7.66 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 13.25/7.66 => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y3 )
% 13.25/7.66 = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y3 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5970_zero__le__power__abs,axiom,
% 13.25/7.66 ! [A: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5971_zero__le__power__abs,axiom,
% 13.25/7.66 ! [A: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5972_zero__le__power__abs,axiom,
% 13.25/7.66 ! [A: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5973_zero__le__power__abs,axiom,
% 13.25/7.66 ! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5974_abs__diff__triangle__ineq,axiom,
% 13.25/7.66 ! [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 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5975_abs__diff__triangle__ineq,axiom,
% 13.25/7.66 ! [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 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5976_abs__diff__triangle__ineq,axiom,
% 13.25/7.66 ! [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 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5977_abs__diff__triangle__ineq,axiom,
% 13.25/7.66 ! [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 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5978_abs__triangle__ineq4,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5979_abs__triangle__ineq4,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5980_abs__triangle__ineq4,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5981_abs__triangle__ineq4,axiom,
% 13.25/7.66 ! [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 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5982_abs__diff__le__iff,axiom,
% 13.25/7.66 ! [X: code_integer,A: code_integer,R: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 13.25/7.66 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 13.25/7.66 & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5983_abs__diff__le__iff,axiom,
% 13.25/7.66 ! [X: real,A: real,R: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 13.25/7.66 = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R ) @ X )
% 13.25/7.66 & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5984_abs__diff__le__iff,axiom,
% 13.25/7.66 ! [X: rat,A: rat,R: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 13.25/7.66 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 13.25/7.66 & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5985_abs__diff__le__iff,axiom,
% 13.25/7.66 ! [X: int,A: int,R: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 13.25/7.66 = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
% 13.25/7.66 & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5986_abs__diff__less__iff,axiom,
% 13.25/7.66 ! [X: code_integer,A: code_integer,R: code_integer] :
% 13.25/7.66 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R )
% 13.25/7.66 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R ) @ X )
% 13.25/7.66 & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5987_abs__diff__less__iff,axiom,
% 13.25/7.66 ! [X: real,A: real,R: real] :
% 13.25/7.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R )
% 13.25/7.66 = ( ( ord_less_real @ ( minus_minus_real @ A @ R ) @ X )
% 13.25/7.66 & ( ord_less_real @ X @ ( plus_plus_real @ A @ R ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5988_abs__diff__less__iff,axiom,
% 13.25/7.66 ! [X: rat,A: rat,R: rat] :
% 13.25/7.66 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R )
% 13.25/7.66 = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R ) @ X )
% 13.25/7.66 & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5989_abs__diff__less__iff,axiom,
% 13.25/7.66 ! [X: int,A: int,R: int] :
% 13.25/7.66 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
% 13.25/7.66 = ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
% 13.25/7.66 & ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5990_neg__numeral__le__zero,axiom,
% 13.25/7.66 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5991_neg__numeral__le__zero,axiom,
% 13.25/7.66 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5992_neg__numeral__le__zero,axiom,
% 13.25/7.66 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5993_neg__numeral__le__zero,axiom,
% 13.25/7.66 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5994_not__zero__le__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5995_not__zero__le__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5996_not__zero__le__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5997_not__zero__le__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5998_not__zero__less__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_5999_not__zero__less__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6000_not__zero__less__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6001_not__zero__less__neg__numeral,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6002_neg__numeral__less__zero,axiom,
% 13.25/7.66 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6003_neg__numeral__less__zero,axiom,
% 13.25/7.66 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6004_neg__numeral__less__zero,axiom,
% 13.25/7.66 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6005_neg__numeral__less__zero,axiom,
% 13.25/7.66 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6006_le__minus__one__simps_I1_J,axiom,
% 13.25/7.66 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6007_le__minus__one__simps_I1_J,axiom,
% 13.25/7.66 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6008_le__minus__one__simps_I1_J,axiom,
% 13.25/7.66 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6009_le__minus__one__simps_I1_J,axiom,
% 13.25/7.66 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6010_le__minus__one__simps_I3_J,axiom,
% 13.25/7.66 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6011_le__minus__one__simps_I3_J,axiom,
% 13.25/7.66 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6012_le__minus__one__simps_I3_J,axiom,
% 13.25/7.66 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6013_le__minus__one__simps_I3_J,axiom,
% 13.25/7.66 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6014_less__minus__one__simps_I1_J,axiom,
% 13.25/7.66 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6015_less__minus__one__simps_I1_J,axiom,
% 13.25/7.66 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6016_less__minus__one__simps_I1_J,axiom,
% 13.25/7.66 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6017_less__minus__one__simps_I1_J,axiom,
% 13.25/7.66 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6018_less__minus__one__simps_I3_J,axiom,
% 13.25/7.66 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6019_less__minus__one__simps_I3_J,axiom,
% 13.25/7.66 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6020_less__minus__one__simps_I3_J,axiom,
% 13.25/7.66 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6021_less__minus__one__simps_I3_J,axiom,
% 13.25/7.66 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6022_neg__numeral__le__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6023_neg__numeral__le__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6024_neg__numeral__le__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6025_neg__numeral__le__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6026_neg__one__le__numeral,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6027_neg__one__le__numeral,axiom,
% 13.25/7.66 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6028_neg__one__le__numeral,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6029_neg__one__le__numeral,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6030_neg__numeral__le__neg__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6031_neg__numeral__le__neg__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6032_neg__numeral__le__neg__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6033_neg__numeral__le__neg__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6034_not__numeral__le__neg__one,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6035_not__numeral__le__neg__one,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6036_not__numeral__le__neg__one,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6037_not__numeral__le__neg__one,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6038_not__one__le__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6039_not__one__le__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6040_not__one__le__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6041_not__one__le__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6042_not__neg__one__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6043_not__neg__one__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6044_not__neg__one__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6045_not__neg__one__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6046_not__one__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6047_not__one__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6048_not__one__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6049_not__one__less__neg__numeral,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6050_not__numeral__less__neg__one,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6051_not__numeral__less__neg__one,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6052_not__numeral__less__neg__one,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6053_not__numeral__less__neg__one,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6054_neg__one__less__numeral,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6055_neg__one__less__numeral,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6056_neg__one__less__numeral,axiom,
% 13.25/7.66 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6057_neg__one__less__numeral,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6058_neg__numeral__less__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6059_neg__numeral__less__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6060_neg__numeral__less__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6061_neg__numeral__less__one,axiom,
% 13.25/7.66 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6062_eq__minus__divide__eq,axiom,
% 13.25/7.66 ! [A: real,B: real,C: real] :
% 13.25/7.66 ( ( A
% 13.25/7.66 = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 13.25/7.66 = ( ( ( C != zero_zero_real )
% 13.25/7.66 => ( ( times_times_real @ A @ C )
% 13.25/7.66 = ( uminus_uminus_real @ B ) ) )
% 13.25/7.66 & ( ( C = zero_zero_real )
% 13.25/7.66 => ( A = zero_zero_real ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6063_eq__minus__divide__eq,axiom,
% 13.25/7.66 ! [A: complex,B: complex,C: complex] :
% 13.25/7.66 ( ( A
% 13.25/7.66 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 13.25/7.66 = ( ( ( C != zero_zero_complex )
% 13.25/7.66 => ( ( times_times_complex @ A @ C )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ B ) ) )
% 13.25/7.66 & ( ( C = zero_zero_complex )
% 13.25/7.66 => ( A = zero_zero_complex ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6064_eq__minus__divide__eq,axiom,
% 13.25/7.66 ! [A: rat,B: rat,C: rat] :
% 13.25/7.66 ( ( A
% 13.25/7.66 = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 13.25/7.66 = ( ( ( C != zero_zero_rat )
% 13.25/7.66 => ( ( times_times_rat @ A @ C )
% 13.25/7.66 = ( uminus_uminus_rat @ B ) ) )
% 13.25/7.66 & ( ( C = zero_zero_rat )
% 13.25/7.66 => ( A = zero_zero_rat ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6065_minus__divide__eq__eq,axiom,
% 13.25/7.66 ! [B: real,C: real,A: real] :
% 13.25/7.66 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 13.25/7.66 = A )
% 13.25/7.66 = ( ( ( C != zero_zero_real )
% 13.25/7.66 => ( ( uminus_uminus_real @ B )
% 13.25/7.66 = ( times_times_real @ A @ C ) ) )
% 13.25/7.66 & ( ( C = zero_zero_real )
% 13.25/7.66 => ( A = zero_zero_real ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6066_minus__divide__eq__eq,axiom,
% 13.25/7.66 ! [B: complex,C: complex,A: complex] :
% 13.25/7.66 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 13.25/7.66 = A )
% 13.25/7.66 = ( ( ( C != zero_zero_complex )
% 13.25/7.66 => ( ( uminus1482373934393186551omplex @ B )
% 13.25/7.66 = ( times_times_complex @ A @ C ) ) )
% 13.25/7.66 & ( ( C = zero_zero_complex )
% 13.25/7.66 => ( A = zero_zero_complex ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6067_minus__divide__eq__eq,axiom,
% 13.25/7.66 ! [B: rat,C: rat,A: rat] :
% 13.25/7.66 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 13.25/7.66 = A )
% 13.25/7.66 = ( ( ( C != zero_zero_rat )
% 13.25/7.66 => ( ( uminus_uminus_rat @ B )
% 13.25/7.66 = ( times_times_rat @ A @ C ) ) )
% 13.25/7.66 & ( ( C = zero_zero_rat )
% 13.25/7.66 => ( A = zero_zero_rat ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6068_nonzero__neg__divide__eq__eq,axiom,
% 13.25/7.66 ! [B: real,A: real,C: real] :
% 13.25/7.66 ( ( B != zero_zero_real )
% 13.25/7.66 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 13.25/7.66 = C )
% 13.25/7.66 = ( ( uminus_uminus_real @ A )
% 13.25/7.66 = ( times_times_real @ C @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6069_nonzero__neg__divide__eq__eq,axiom,
% 13.25/7.66 ! [B: complex,A: complex,C: complex] :
% 13.25/7.66 ( ( B != zero_zero_complex )
% 13.25/7.66 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.66 = C )
% 13.25/7.66 = ( ( uminus1482373934393186551omplex @ A )
% 13.25/7.66 = ( times_times_complex @ C @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6070_nonzero__neg__divide__eq__eq,axiom,
% 13.25/7.66 ! [B: rat,A: rat,C: rat] :
% 13.25/7.66 ( ( B != zero_zero_rat )
% 13.25/7.66 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 13.25/7.66 = C )
% 13.25/7.66 = ( ( uminus_uminus_rat @ A )
% 13.25/7.66 = ( times_times_rat @ C @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6071_nonzero__neg__divide__eq__eq2,axiom,
% 13.25/7.66 ! [B: real,C: real,A: real] :
% 13.25/7.66 ( ( B != zero_zero_real )
% 13.25/7.66 => ( ( C
% 13.25/7.66 = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 13.25/7.66 = ( ( times_times_real @ C @ B )
% 13.25/7.66 = ( uminus_uminus_real @ A ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6072_nonzero__neg__divide__eq__eq2,axiom,
% 13.25/7.66 ! [B: complex,C: complex,A: complex] :
% 13.25/7.66 ( ( B != zero_zero_complex )
% 13.25/7.66 => ( ( C
% 13.25/7.66 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 13.25/7.66 = ( ( times_times_complex @ C @ B )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6073_nonzero__neg__divide__eq__eq2,axiom,
% 13.25/7.66 ! [B: rat,C: rat,A: rat] :
% 13.25/7.66 ( ( B != zero_zero_rat )
% 13.25/7.66 => ( ( C
% 13.25/7.66 = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 13.25/7.66 = ( ( times_times_rat @ C @ B )
% 13.25/7.66 = ( uminus_uminus_rat @ A ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6074_mult__1s__ring__1_I1_J,axiom,
% 13.25/7.66 ! [B: real] :
% 13.25/7.66 ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
% 13.25/7.66 = ( uminus_uminus_real @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6075_mult__1s__ring__1_I1_J,axiom,
% 13.25/7.66 ! [B: int] :
% 13.25/7.66 ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
% 13.25/7.66 = ( uminus_uminus_int @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6076_mult__1s__ring__1_I1_J,axiom,
% 13.25/7.66 ! [B: complex] :
% 13.25/7.66 ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6077_mult__1s__ring__1_I1_J,axiom,
% 13.25/7.66 ! [B: code_integer] :
% 13.25/7.66 ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6078_mult__1s__ring__1_I1_J,axiom,
% 13.25/7.66 ! [B: rat] :
% 13.25/7.66 ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
% 13.25/7.66 = ( uminus_uminus_rat @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6079_mult__1s__ring__1_I2_J,axiom,
% 13.25/7.66 ! [B: real] :
% 13.25/7.66 ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
% 13.25/7.66 = ( uminus_uminus_real @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6080_mult__1s__ring__1_I2_J,axiom,
% 13.25/7.66 ! [B: int] :
% 13.25/7.66 ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
% 13.25/7.66 = ( uminus_uminus_int @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6081_mult__1s__ring__1_I2_J,axiom,
% 13.25/7.66 ! [B: complex] :
% 13.25/7.66 ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6082_mult__1s__ring__1_I2_J,axiom,
% 13.25/7.66 ! [B: code_integer] :
% 13.25/7.66 ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6083_mult__1s__ring__1_I2_J,axiom,
% 13.25/7.66 ! [B: rat] :
% 13.25/7.66 ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
% 13.25/7.66 = ( uminus_uminus_rat @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6084_divide__eq__minus__1__iff,axiom,
% 13.25/7.66 ! [A: real,B: real] :
% 13.25/7.66 ( ( ( divide_divide_real @ A @ B )
% 13.25/7.66 = ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.66 = ( ( B != zero_zero_real )
% 13.25/7.66 & ( A
% 13.25/7.66 = ( uminus_uminus_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6085_divide__eq__minus__1__iff,axiom,
% 13.25/7.66 ! [A: complex,B: complex] :
% 13.25/7.66 ( ( ( divide1717551699836669952omplex @ A @ B )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.66 = ( ( B != zero_zero_complex )
% 13.25/7.66 & ( A
% 13.25/7.66 = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6086_divide__eq__minus__1__iff,axiom,
% 13.25/7.66 ! [A: rat,B: rat] :
% 13.25/7.66 ( ( ( divide_divide_rat @ A @ B )
% 13.25/7.66 = ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.66 = ( ( B != zero_zero_rat )
% 13.25/7.66 & ( A
% 13.25/7.66 = ( uminus_uminus_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6087_uminus__numeral__One,axiom,
% 13.25/7.66 ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
% 13.25/7.66 = ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6088_uminus__numeral__One,axiom,
% 13.25/7.66 ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6089_uminus__numeral__One,axiom,
% 13.25/7.66 ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6090_uminus__numeral__One,axiom,
% 13.25/7.66 ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6091_uminus__numeral__One,axiom,
% 13.25/7.66 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
% 13.25/7.66 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6092_power__minus,axiom,
% 13.25/7.66 ! [A: real,N: nat] :
% 13.25/7.66 ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 13.25/7.66 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( power_power_real @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6093_power__minus,axiom,
% 13.25/7.66 ! [A: int,N: nat] :
% 13.25/7.66 ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 13.25/7.66 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N ) @ ( power_power_int @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6094_power__minus,axiom,
% 13.25/7.66 ! [A: complex,N: nat] :
% 13.25/7.66 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 13.25/7.66 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( power_power_complex @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6095_power__minus,axiom,
% 13.25/7.66 ! [A: code_integer,N: nat] :
% 13.25/7.66 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 13.25/7.66 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6096_power__minus,axiom,
% 13.25/7.66 ! [A: rat,N: nat] :
% 13.25/7.66 ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 13.25/7.66 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( power_power_rat @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6097_power__minus__Bit0,axiom,
% 13.25/7.66 ! [X: real,K: num] :
% 13.25/7.66 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 13.25/7.66 = ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6098_power__minus__Bit0,axiom,
% 13.25/7.66 ! [X: int,K: num] :
% 13.25/7.66 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 13.25/7.66 = ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6099_power__minus__Bit0,axiom,
% 13.25/7.66 ! [X: complex,K: num] :
% 13.25/7.66 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 13.25/7.66 = ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6100_power__minus__Bit0,axiom,
% 13.25/7.66 ! [X: code_integer,K: num] :
% 13.25/7.66 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 13.25/7.66 = ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6101_power__minus__Bit0,axiom,
% 13.25/7.66 ! [X: rat,K: num] :
% 13.25/7.66 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 13.25/7.66 = ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6102_lemma__interval__lt,axiom,
% 13.25/7.66 ! [A: real,X: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ A @ X )
% 13.25/7.66 => ( ( ord_less_real @ X @ B )
% 13.25/7.66 => ? [D3: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.66 & ! [Y5: real] :
% 13.25/7.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
% 13.25/7.66 => ( ( ord_less_real @ A @ Y5 )
% 13.25/7.66 & ( ord_less_real @ Y5 @ B ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6103_real__0__less__add__iff,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X @ Y3 ) )
% 13.25/7.66 = ( ord_less_real @ ( uminus_uminus_real @ X ) @ Y3 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6104_real__add__less__0__iff,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
% 13.25/7.66 = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6105_real__add__le__0__iff,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( plus_plus_real @ X @ Y3 ) @ zero_zero_real )
% 13.25/7.66 = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6106_real__0__le__add__iff,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X @ Y3 ) )
% 13.25/7.66 = ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ Y3 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6107_zmod__zminus2__eq__if,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 = zero_zero_int )
% 13.25/7.66 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = zero_zero_int ) )
% 13.25/7.66 & ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 != zero_zero_int )
% 13.25/7.66 => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6108_zmod__zminus1__eq__if,axiom,
% 13.25/7.66 ! [A: int,B: int] :
% 13.25/7.66 ( ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 = zero_zero_int )
% 13.25/7.66 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = zero_zero_int ) )
% 13.25/7.66 & ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 != zero_zero_int )
% 13.25/7.66 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6109_abs__add__one__gt__zero,axiom,
% 13.25/7.66 ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6110_abs__add__one__gt__zero,axiom,
% 13.25/7.66 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6111_abs__add__one__gt__zero,axiom,
% 13.25/7.66 ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6112_abs__add__one__gt__zero,axiom,
% 13.25/7.66 ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6113_pos__minus__divide__less__eq,axiom,
% 13.25/7.66 ! [C: real,B: real,A: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 13.25/7.66 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6114_pos__minus__divide__less__eq,axiom,
% 13.25/7.66 ! [C: rat,B: rat,A: rat] :
% 13.25/7.66 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 13.25/7.66 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6115_pos__less__minus__divide__eq,axiom,
% 13.25/7.66 ! [C: real,A: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 13.25/7.66 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6116_pos__less__minus__divide__eq,axiom,
% 13.25/7.66 ! [C: rat,A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 13.25/7.66 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6117_neg__minus__divide__less__eq,axiom,
% 13.25/7.66 ! [C: real,B: real,A: real] :
% 13.25/7.66 ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 13.25/7.66 = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6118_neg__minus__divide__less__eq,axiom,
% 13.25/7.66 ! [C: rat,B: rat,A: rat] :
% 13.25/7.66 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 13.25/7.66 = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6119_neg__less__minus__divide__eq,axiom,
% 13.25/7.66 ! [C: real,A: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 13.25/7.66 = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6120_neg__less__minus__divide__eq,axiom,
% 13.25/7.66 ! [C: rat,A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 13.25/7.66 = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6121_minus__divide__less__eq,axiom,
% 13.25/7.66 ! [B: real,C: real,A: real] :
% 13.25/7.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 13.25/7.66 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6122_minus__divide__less__eq,axiom,
% 13.25/7.66 ! [B: rat,C: rat,A: rat] :
% 13.25/7.66 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 13.25/7.66 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6123_less__minus__divide__eq,axiom,
% 13.25/7.66 ! [A: real,B: real,C: real] :
% 13.25/7.66 ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 13.25/7.66 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6124_less__minus__divide__eq,axiom,
% 13.25/7.66 ! [A: rat,B: rat,C: rat] :
% 13.25/7.66 ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 13.25/7.66 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6125_eq__divide__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [W: num,B: real,C: real] :
% 13.25/7.66 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 13.25/7.66 = ( divide_divide_real @ B @ C ) )
% 13.25/7.66 = ( ( ( C != zero_zero_real )
% 13.25/7.66 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
% 13.25/7.66 = B ) )
% 13.25/7.66 & ( ( C = zero_zero_real )
% 13.25/7.66 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 13.25/7.66 = zero_zero_real ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6126_eq__divide__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [W: num,B: complex,C: complex] :
% 13.25/7.66 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.66 = ( divide1717551699836669952omplex @ B @ C ) )
% 13.25/7.66 = ( ( ( C != zero_zero_complex )
% 13.25/7.66 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
% 13.25/7.66 = B ) )
% 13.25/7.66 & ( ( C = zero_zero_complex )
% 13.25/7.66 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.66 = zero_zero_complex ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6127_eq__divide__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [W: num,B: rat,C: rat] :
% 13.25/7.66 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 13.25/7.66 = ( divide_divide_rat @ B @ C ) )
% 13.25/7.66 = ( ( ( C != zero_zero_rat )
% 13.25/7.66 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
% 13.25/7.66 = B ) )
% 13.25/7.66 & ( ( C = zero_zero_rat )
% 13.25/7.66 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 13.25/7.66 = zero_zero_rat ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6128_divide__eq__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [B: real,C: real,W: num] :
% 13.25/7.66 ( ( ( divide_divide_real @ B @ C )
% 13.25/7.66 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.66 = ( ( ( C != zero_zero_real )
% 13.25/7.66 => ( B
% 13.25/7.66 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ( C = zero_zero_real )
% 13.25/7.66 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
% 13.25/7.66 = zero_zero_real ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6129_divide__eq__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [B: complex,C: complex,W: num] :
% 13.25/7.66 ( ( ( divide1717551699836669952omplex @ B @ C )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.66 = ( ( ( C != zero_zero_complex )
% 13.25/7.66 => ( B
% 13.25/7.66 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ( C = zero_zero_complex )
% 13.25/7.66 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.66 = zero_zero_complex ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6130_divide__eq__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [B: rat,C: rat,W: num] :
% 13.25/7.66 ( ( ( divide_divide_rat @ B @ C )
% 13.25/7.66 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 13.25/7.66 = ( ( ( C != zero_zero_rat )
% 13.25/7.66 => ( B
% 13.25/7.66 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ( C = zero_zero_rat )
% 13.25/7.66 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
% 13.25/7.66 = zero_zero_rat ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6131_minus__divide__add__eq__iff,axiom,
% 13.25/7.66 ! [Z: real,X: real,Y3: real] :
% 13.25/7.66 ( ( Z != zero_zero_real )
% 13.25/7.66 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y3 )
% 13.25/7.66 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6132_minus__divide__add__eq__iff,axiom,
% 13.25/7.66 ! [Z: complex,X: complex,Y3: complex] :
% 13.25/7.66 ( ( Z != zero_zero_complex )
% 13.25/7.66 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y3 )
% 13.25/7.66 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6133_minus__divide__add__eq__iff,axiom,
% 13.25/7.66 ! [Z: rat,X: rat,Y3: rat] :
% 13.25/7.66 ( ( Z != zero_zero_rat )
% 13.25/7.66 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y3 )
% 13.25/7.66 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6134_add__divide__eq__if__simps_I3_J,axiom,
% 13.25/7.66 ! [Z: real,A: real,B: real] :
% 13.25/7.66 ( ( ( Z = zero_zero_real )
% 13.25/7.66 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 13.25/7.66 = B ) )
% 13.25/7.66 & ( ( Z != zero_zero_real )
% 13.25/7.66 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 13.25/7.66 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6135_add__divide__eq__if__simps_I3_J,axiom,
% 13.25/7.66 ! [Z: complex,A: complex,B: complex] :
% 13.25/7.66 ( ( ( Z = zero_zero_complex )
% 13.25/7.66 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 13.25/7.66 = B ) )
% 13.25/7.66 & ( ( Z != zero_zero_complex )
% 13.25/7.66 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 13.25/7.66 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6136_add__divide__eq__if__simps_I3_J,axiom,
% 13.25/7.66 ! [Z: rat,A: rat,B: rat] :
% 13.25/7.66 ( ( ( Z = zero_zero_rat )
% 13.25/7.66 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 13.25/7.66 = B ) )
% 13.25/7.66 & ( ( Z != zero_zero_rat )
% 13.25/7.66 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 13.25/7.66 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6137_minus__divide__diff__eq__iff,axiom,
% 13.25/7.66 ! [Z: real,X: real,Y3: real] :
% 13.25/7.66 ( ( Z != zero_zero_real )
% 13.25/7.66 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z ) ) @ Y3 )
% 13.25/7.66 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6138_minus__divide__diff__eq__iff,axiom,
% 13.25/7.66 ! [Z: complex,X: complex,Y3: complex] :
% 13.25/7.66 ( ( Z != zero_zero_complex )
% 13.25/7.66 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z ) ) @ Y3 )
% 13.25/7.66 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6139_minus__divide__diff__eq__iff,axiom,
% 13.25/7.66 ! [Z: rat,X: rat,Y3: rat] :
% 13.25/7.66 ( ( Z != zero_zero_rat )
% 13.25/7.66 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z ) ) @ Y3 )
% 13.25/7.66 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6140_add__divide__eq__if__simps_I5_J,axiom,
% 13.25/7.66 ! [Z: real,A: real,B: real] :
% 13.25/7.66 ( ( ( Z = zero_zero_real )
% 13.25/7.66 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 13.25/7.66 = ( uminus_uminus_real @ B ) ) )
% 13.25/7.66 & ( ( Z != zero_zero_real )
% 13.25/7.66 => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
% 13.25/7.66 = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6141_add__divide__eq__if__simps_I5_J,axiom,
% 13.25/7.66 ! [Z: complex,A: complex,B: complex] :
% 13.25/7.66 ( ( ( Z = zero_zero_complex )
% 13.25/7.66 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ B ) ) )
% 13.25/7.66 & ( ( Z != zero_zero_complex )
% 13.25/7.66 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
% 13.25/7.66 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6142_add__divide__eq__if__simps_I5_J,axiom,
% 13.25/7.66 ! [Z: rat,A: rat,B: rat] :
% 13.25/7.66 ( ( ( Z = zero_zero_rat )
% 13.25/7.66 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 13.25/7.66 = ( uminus_uminus_rat @ B ) ) )
% 13.25/7.66 & ( ( Z != zero_zero_rat )
% 13.25/7.66 => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
% 13.25/7.66 = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6143_add__divide__eq__if__simps_I6_J,axiom,
% 13.25/7.66 ! [Z: real,A: real,B: real] :
% 13.25/7.66 ( ( ( Z = zero_zero_real )
% 13.25/7.66 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 13.25/7.66 = ( uminus_uminus_real @ B ) ) )
% 13.25/7.66 & ( ( Z != zero_zero_real )
% 13.25/7.66 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
% 13.25/7.66 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6144_add__divide__eq__if__simps_I6_J,axiom,
% 13.25/7.66 ! [Z: complex,A: complex,B: complex] :
% 13.25/7.66 ( ( ( Z = zero_zero_complex )
% 13.25/7.66 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ B ) ) )
% 13.25/7.66 & ( ( Z != zero_zero_complex )
% 13.25/7.66 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
% 13.25/7.66 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6145_add__divide__eq__if__simps_I6_J,axiom,
% 13.25/7.66 ! [Z: rat,A: rat,B: rat] :
% 13.25/7.66 ( ( ( Z = zero_zero_rat )
% 13.25/7.66 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 13.25/7.66 = ( uminus_uminus_rat @ B ) ) )
% 13.25/7.66 & ( ( Z != zero_zero_rat )
% 13.25/7.66 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
% 13.25/7.66 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6146_even__minus,axiom,
% 13.25/7.66 ! [A: int] :
% 13.25/7.66 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
% 13.25/7.66 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6147_even__minus,axiom,
% 13.25/7.66 ! [A: code_integer] :
% 13.25/7.66 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
% 13.25/7.66 = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6148_lemma__interval,axiom,
% 13.25/7.66 ! [A: real,X: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ A @ X )
% 13.25/7.66 => ( ( ord_less_real @ X @ B )
% 13.25/7.66 => ? [D3: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.66 & ! [Y5: real] :
% 13.25/7.66 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y5 ) ) @ D3 )
% 13.25/7.66 => ( ( ord_less_eq_real @ A @ Y5 )
% 13.25/7.66 & ( ord_less_eq_real @ Y5 @ B ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6149_power2__eq__iff,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ( X = Y3 )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6150_power2__eq__iff,axiom,
% 13.25/7.66 ! [X: int,Y3: int] :
% 13.25/7.66 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ( X = Y3 )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6151_power2__eq__iff,axiom,
% 13.25/7.66 ! [X: complex,Y3: complex] :
% 13.25/7.66 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ( X = Y3 )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus1482373934393186551omplex @ Y3 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6152_power2__eq__iff,axiom,
% 13.25/7.66 ! [X: code_integer,Y3: code_integer] :
% 13.25/7.66 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ( X = Y3 )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6153_power2__eq__iff,axiom,
% 13.25/7.66 ! [X: rat,Y3: rat] :
% 13.25/7.66 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ( X = Y3 )
% 13.25/7.66 | ( X
% 13.25/7.66 = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6154_ln__add__one__self__le__self2,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.66 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6155_verit__less__mono__div__int2,axiom,
% 13.25/7.66 ! [A2: int,B5: int,N: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ A2 @ B5 )
% 13.25/7.66 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 13.25/7.66 => ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A2 @ N ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6156_div__eq__minus1,axiom,
% 13.25/7.66 ! [B: int] :
% 13.25/7.66 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.66 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6157_abs__le__square__iff,axiom,
% 13.25/7.66 ! [X: code_integer,Y3: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y3 ) )
% 13.25/7.66 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6158_abs__le__square__iff,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y3 ) )
% 13.25/7.66 = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6159_abs__le__square__iff,axiom,
% 13.25/7.66 ! [X: rat,Y3: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y3 ) )
% 13.25/7.66 = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6160_abs__le__square__iff,axiom,
% 13.25/7.66 ! [X: int,Y3: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y3 ) )
% 13.25/7.66 = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6161_abs__square__eq__1,axiom,
% 13.25/7.66 ! [X: code_integer] :
% 13.25/7.66 ( ( ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_Code_integer )
% 13.25/7.66 = ( ( abs_abs_Code_integer @ X )
% 13.25/7.66 = one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6162_abs__square__eq__1,axiom,
% 13.25/7.66 ! [X: rat] :
% 13.25/7.66 ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_rat )
% 13.25/7.66 = ( ( abs_abs_rat @ X )
% 13.25/7.66 = one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6163_abs__square__eq__1,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_real )
% 13.25/7.66 = ( ( abs_abs_real @ X )
% 13.25/7.66 = one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6164_abs__square__eq__1,axiom,
% 13.25/7.66 ! [X: int] :
% 13.25/7.66 ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_int )
% 13.25/7.66 = ( ( abs_abs_int @ X )
% 13.25/7.66 = one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6165_power__even__abs,axiom,
% 13.25/7.66 ! [N: nat,A: code_integer] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N )
% 13.25/7.66 = ( power_8256067586552552935nteger @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6166_power__even__abs,axiom,
% 13.25/7.66 ! [N: nat,A: rat] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N )
% 13.25/7.66 = ( power_power_rat @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6167_power__even__abs,axiom,
% 13.25/7.66 ! [N: nat,A: real] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_real @ ( abs_abs_real @ A ) @ N )
% 13.25/7.66 = ( power_power_real @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6168_power__even__abs,axiom,
% 13.25/7.66 ! [N: nat,A: int] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_int @ ( abs_abs_int @ A ) @ N )
% 13.25/7.66 = ( power_power_int @ A @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6169_pos__minus__divide__le__eq,axiom,
% 13.25/7.66 ! [C: real,B: real,A: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 13.25/7.66 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6170_pos__minus__divide__le__eq,axiom,
% 13.25/7.66 ! [C: rat,B: rat,A: rat] :
% 13.25/7.66 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 13.25/7.66 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6171_pos__le__minus__divide__eq,axiom,
% 13.25/7.66 ! [C: real,A: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 13.25/7.66 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6172_pos__le__minus__divide__eq,axiom,
% 13.25/7.66 ! [C: rat,A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 13.25/7.66 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6173_neg__minus__divide__le__eq,axiom,
% 13.25/7.66 ! [C: real,B: real,A: real] :
% 13.25/7.66 ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 13.25/7.66 = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6174_neg__minus__divide__le__eq,axiom,
% 13.25/7.66 ! [C: rat,B: rat,A: rat] :
% 13.25/7.66 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 13.25/7.66 = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6175_neg__le__minus__divide__eq,axiom,
% 13.25/7.66 ! [C: real,A: real,B: real] :
% 13.25/7.66 ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 13.25/7.66 = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6176_neg__le__minus__divide__eq,axiom,
% 13.25/7.66 ! [C: rat,A: rat,B: rat] :
% 13.25/7.66 ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 13.25/7.66 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6177_minus__divide__le__eq,axiom,
% 13.25/7.66 ! [B: real,C: real,A: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 13.25/7.66 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6178_minus__divide__le__eq,axiom,
% 13.25/7.66 ! [B: rat,C: rat,A: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 13.25/7.66 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6179_le__minus__divide__eq,axiom,
% 13.25/7.66 ! [A: real,B: real,C: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 13.25/7.66 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6180_le__minus__divide__eq,axiom,
% 13.25/7.66 ! [A: rat,B: rat,C: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 13.25/7.66 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6181_divide__less__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [B: real,C: real,W: num] :
% 13.25/7.66 ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.66 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6182_divide__less__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [B: rat,C: rat,W: num] :
% 13.25/7.66 ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 13.25/7.66 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6183_less__divide__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [W: num,B: real,C: real] :
% 13.25/7.66 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 13.25/7.66 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6184_less__divide__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [W: num,B: rat,C: rat] :
% 13.25/7.66 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 13.25/7.66 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6185_power2__eq__1__iff,axiom,
% 13.25/7.66 ! [A: real] :
% 13.25/7.66 ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_real )
% 13.25/7.66 = ( ( A = one_one_real )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6186_power2__eq__1__iff,axiom,
% 13.25/7.66 ! [A: int] :
% 13.25/7.66 ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_int )
% 13.25/7.66 = ( ( A = one_one_int )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6187_power2__eq__1__iff,axiom,
% 13.25/7.66 ! [A: complex] :
% 13.25/7.66 ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_complex )
% 13.25/7.66 = ( ( A = one_one_complex )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6188_power2__eq__1__iff,axiom,
% 13.25/7.66 ! [A: code_integer] :
% 13.25/7.66 ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_Code_integer )
% 13.25/7.66 = ( ( A = one_one_Code_integer )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6189_power2__eq__1__iff,axiom,
% 13.25/7.66 ! [A: rat] :
% 13.25/7.66 ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = one_one_rat )
% 13.25/7.66 = ( ( A = one_one_rat )
% 13.25/7.66 | ( A
% 13.25/7.66 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6190_uminus__power__if,axiom,
% 13.25/7.66 ! [N: nat,A: real] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 13.25/7.66 = ( power_power_real @ A @ N ) ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N )
% 13.25/7.66 = ( uminus_uminus_real @ ( power_power_real @ A @ N ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6191_uminus__power__if,axiom,
% 13.25/7.66 ! [N: nat,A: int] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 13.25/7.66 = ( power_power_int @ A @ N ) ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N )
% 13.25/7.66 = ( uminus_uminus_int @ ( power_power_int @ A @ N ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6192_uminus__power__if,axiom,
% 13.25/7.66 ! [N: nat,A: complex] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 13.25/7.66 = ( power_power_complex @ A @ N ) ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6193_uminus__power__if,axiom,
% 13.25/7.66 ! [N: nat,A: code_integer] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 13.25/7.66 = ( power_8256067586552552935nteger @ A @ N ) ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6194_uminus__power__if,axiom,
% 13.25/7.66 ! [N: nat,A: rat] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 13.25/7.66 = ( power_power_rat @ A @ N ) ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N )
% 13.25/7.66 = ( uminus_uminus_rat @ ( power_power_rat @ A @ N ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6195_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 13.25/7.66 ! [K: nat,N: nat] :
% 13.25/7.66 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.66 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 13.25/7.66 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6196_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 13.25/7.66 ! [K: nat,N: nat] :
% 13.25/7.66 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.66 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 13.25/7.66 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6197_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 13.25/7.66 ! [K: nat,N: nat] :
% 13.25/7.66 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.66 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 13.25/7.66 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6198_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 13.25/7.66 ! [K: nat,N: nat] :
% 13.25/7.66 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.66 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 13.25/7.66 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6199_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 13.25/7.66 ! [K: nat,N: nat] :
% 13.25/7.66 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.66 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 13.25/7.66 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6200_realpow__square__minus__le,axiom,
% 13.25/7.66 ! [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 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6201_ln__one__minus__pos__upper__bound,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.66 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.66 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) @ ( uminus_uminus_real @ X ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6202_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 13.25/7.66 ! [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 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6203_signed__take__bit__int__less__eq__self__iff,axiom,
% 13.25/7.66 ! [N: nat,K: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 13.25/7.66 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6204_signed__take__bit__int__greater__self__iff,axiom,
% 13.25/7.66 ! [K: int,N: nat] :
% 13.25/7.66 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 13.25/7.66 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6205_minus__mod__int__eq,axiom,
% 13.25/7.66 ! [L2: int,K: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 13.25/7.66 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L2 )
% 13.25/7.66 = ( minus_minus_int @ ( minus_minus_int @ L2 @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L2 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6206_zmod__minus1,axiom,
% 13.25/7.66 ! [B: int] :
% 13.25/7.66 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.66 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 13.25/7.66 = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6207_zdiv__zminus1__eq__if,axiom,
% 13.25/7.66 ! [B: int,A: int] :
% 13.25/7.66 ( ( B != zero_zero_int )
% 13.25/7.66 => ( ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 = zero_zero_int )
% 13.25/7.66 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 13.25/7.66 & ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 != zero_zero_int )
% 13.25/7.66 => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 13.25/7.66 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6208_zdiv__zminus2__eq__if,axiom,
% 13.25/7.66 ! [B: int,A: int] :
% 13.25/7.66 ( ( B != zero_zero_int )
% 13.25/7.66 => ( ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 = zero_zero_int )
% 13.25/7.66 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
% 13.25/7.66 & ( ( ( modulo_modulo_int @ A @ B )
% 13.25/7.66 != zero_zero_int )
% 13.25/7.66 => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 13.25/7.66 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6209_abs__sqrt__wlog,axiom,
% 13.25/7.66 ! [P: code_integer > code_integer > $o,X: code_integer] :
% 13.25/7.66 ( ! [X3: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 13.25/7.66 => ( P @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.66 => ( P @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6210_abs__sqrt__wlog,axiom,
% 13.25/7.66 ! [P: real > real > $o,X: real] :
% 13.25/7.66 ( ! [X3: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 13.25/7.66 => ( P @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.66 => ( P @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6211_abs__sqrt__wlog,axiom,
% 13.25/7.66 ! [P: rat > rat > $o,X: rat] :
% 13.25/7.66 ( ! [X3: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 13.25/7.66 => ( P @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.66 => ( P @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6212_abs__sqrt__wlog,axiom,
% 13.25/7.66 ! [P: int > int > $o,X: int] :
% 13.25/7.66 ( ! [X3: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 13.25/7.66 => ( P @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.66 => ( P @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6213_power2__le__iff__abs__le,axiom,
% 13.25/7.66 ! [Y3: code_integer,X: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 13.25/7.66 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y3 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6214_power2__le__iff__abs__le,axiom,
% 13.25/7.66 ! [Y3: real,X: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.66 => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y3 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6215_power2__le__iff__abs__le,axiom,
% 13.25/7.66 ! [Y3: rat,X: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 13.25/7.66 => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y3 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6216_power2__le__iff__abs__le,axiom,
% 13.25/7.66 ! [Y3: int,X: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.66 => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.66 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y3 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6217_abs__square__le__1,axiom,
% 13.25/7.66 ! [X: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 13.25/7.66 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6218_abs__square__le__1,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 13.25/7.66 = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6219_abs__square__le__1,axiom,
% 13.25/7.66 ! [X: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 13.25/7.66 = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6220_abs__square__le__1,axiom,
% 13.25/7.66 ! [X: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 13.25/7.66 = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6221_abs__square__less__1,axiom,
% 13.25/7.66 ! [X: code_integer] :
% 13.25/7.66 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 13.25/7.66 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6222_abs__square__less__1,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 13.25/7.66 = ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6223_abs__square__less__1,axiom,
% 13.25/7.66 ! [X: rat] :
% 13.25/7.66 ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 13.25/7.66 = ( ord_less_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6224_abs__square__less__1,axiom,
% 13.25/7.66 ! [X: int] :
% 13.25/7.66 ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 13.25/7.66 = ( ord_less_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6225_power__mono__even,axiom,
% 13.25/7.66 ! [N: nat,A: code_integer,B: code_integer] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 13.25/7.66 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6226_power__mono__even,axiom,
% 13.25/7.66 ! [N: nat,A: real,B: real] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 13.25/7.66 => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6227_power__mono__even,axiom,
% 13.25/7.66 ! [N: nat,A: rat,B: rat] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 13.25/7.66 => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6228_power__mono__even,axiom,
% 13.25/7.66 ! [N: nat,A: int,B: int] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 13.25/7.66 => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6229_zminus1__lemma,axiom,
% 13.25/7.66 ! [A: int,B: int,Q2: int,R: int] :
% 13.25/7.66 ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R ) )
% 13.25/7.66 => ( ( B != zero_zero_int )
% 13.25/7.66 => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6230_divide__le__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [B: real,C: real,W: num] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.66 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6231_divide__le__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [B: rat,C: rat,W: num] :
% 13.25/7.66 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
% 13.25/7.66 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6232_le__divide__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [W: num,B: real,C: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
% 13.25/7.66 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.66 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 13.25/7.66 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6233_le__divide__eq__numeral_I2_J,axiom,
% 13.25/7.66 ! [W: num,B: rat,C: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
% 13.25/7.66 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 13.25/7.66 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
% 13.25/7.66 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6234_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.66 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.25/7.66 => ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6235_square__le__1,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.66 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.66 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6236_square__le__1,axiom,
% 13.25/7.66 ! [X: code_integer] :
% 13.25/7.66 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
% 13.25/7.66 => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
% 13.25/7.66 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6237_square__le__1,axiom,
% 13.25/7.66 ! [X: rat] :
% 13.25/7.66 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
% 13.25/7.66 => ( ( ord_less_eq_rat @ X @ one_one_rat )
% 13.25/7.66 => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6238_square__le__1,axiom,
% 13.25/7.66 ! [X: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
% 13.25/7.66 => ( ( ord_less_eq_int @ X @ one_one_int )
% 13.25/7.66 => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6239_minus__power__mult__self,axiom,
% 13.25/7.66 ! [A: real,N: nat] :
% 13.25/7.66 ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N ) )
% 13.25/7.66 = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6240_minus__power__mult__self,axiom,
% 13.25/7.66 ! [A: int,N: nat] :
% 13.25/7.66 ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
% 13.25/7.66 = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6241_minus__power__mult__self,axiom,
% 13.25/7.66 ! [A: complex,N: nat] :
% 13.25/7.66 ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N ) )
% 13.25/7.66 = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6242_minus__power__mult__self,axiom,
% 13.25/7.66 ! [A: code_integer,N: nat] :
% 13.25/7.66 ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N ) )
% 13.25/7.66 = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6243_minus__power__mult__self,axiom,
% 13.25/7.66 ! [A: rat,N: nat] :
% 13.25/7.66 ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N ) )
% 13.25/7.66 = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6244_minus__one__power__iff,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 13.25/7.66 = one_one_real ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N )
% 13.25/7.66 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6245_minus__one__power__iff,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 13.25/7.66 = one_one_int ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N )
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6246_minus__one__power__iff,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 13.25/7.66 = one_one_complex ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6247_minus__one__power__iff,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 13.25/7.66 = one_one_Code_integer ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6248_minus__one__power__iff,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 13.25/7.66 = one_one_rat ) )
% 13.25/7.66 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.66 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N )
% 13.25/7.66 = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6249_signed__take__bit__int__eq__self__iff,axiom,
% 13.25/7.66 ! [N: nat,K: int] :
% 13.25/7.66 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 13.25/7.66 = K )
% 13.25/7.66 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 13.25/7.66 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6250_signed__take__bit__int__eq__self,axiom,
% 13.25/7.66 ! [N: nat,K: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 13.25/7.66 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.66 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 13.25/7.66 = K ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6251_minus__1__div__exp__eq__int,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6252_div__pos__neg__trivial,axiom,
% 13.25/7.66 ! [K: int,L2: int] :
% 13.25/7.66 ( ( ord_less_int @ zero_zero_int @ K )
% 13.25/7.66 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L2 ) @ zero_zero_int )
% 13.25/7.66 => ( ( divide_divide_int @ K @ L2 )
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6253_power__minus1__odd,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.66 = ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6254_power__minus1__odd,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.66 = ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6255_power__minus1__odd,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.66 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6256_power__minus1__odd,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.66 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6257_power__minus1__odd,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.66 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6258_int__bit__induct,axiom,
% 13.25/7.66 ! [P: int > $o,K: int] :
% 13.25/7.66 ( ( P @ zero_zero_int )
% 13.25/7.66 => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.66 => ( ! [K3: int] :
% 13.25/7.66 ( ( P @ K3 )
% 13.25/7.66 => ( ( K3 != zero_zero_int )
% 13.25/7.66 => ( P @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.66 => ( ! [K3: int] :
% 13.25/7.66 ( ( P @ K3 )
% 13.25/7.66 => ( ( K3
% 13.25/7.66 != ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.66 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
% 13.25/7.66 => ( P @ K ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6259_eq__diff__eq_H,axiom,
% 13.25/7.66 ! [X: real,Y3: real,Z: real] :
% 13.25/7.66 ( ( X
% 13.25/7.66 = ( minus_minus_real @ Y3 @ Z ) )
% 13.25/7.66 = ( Y3
% 13.25/7.66 = ( plus_plus_real @ X @ Z ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6260_signed__take__bit__int__greater__eq,axiom,
% 13.25/7.66 ! [K: int,N: nat] :
% 13.25/7.66 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.66 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6261_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.66 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.66 => ( 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 ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6262_arctan__double,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.66 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
% 13.25/7.66 = ( 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 ) ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6263_signed__take__bit__Suc__minus__bit1,axiom,
% 13.25/7.66 ! [N: nat,K: num] :
% 13.25/7.66 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 13.25/7.66 = ( 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 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6264_Sum__Icc__int,axiom,
% 13.25/7.66 ! [M: int,N: int] :
% 13.25/7.66 ( ( ord_less_eq_int @ M @ N )
% 13.25/7.66 => ( ( groups4538972089207619220nt_int
% 13.25/7.66 @ ^ [X2: int] : X2
% 13.25/7.66 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 13.25/7.66 = ( 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 ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6265_divmod__step__def,axiom,
% 13.25/7.66 ( unique5026877609467782581ep_nat
% 13.25/7.66 = ( ^ [L: num] :
% 13.25/7.66 ( produc2626176000494625587at_nat
% 13.25/7.66 @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6266_divmod__step__def,axiom,
% 13.25/7.66 ( unique5024387138958732305ep_int
% 13.25/7.66 = ( ^ [L: num] :
% 13.25/7.66 ( produc4245557441103728435nt_int
% 13.25/7.66 @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6267_divmod__step__def,axiom,
% 13.25/7.66 ( unique4921790084139445826nteger
% 13.25/7.66 = ( ^ [L: num] :
% 13.25/7.66 ( produc6916734918728496179nteger
% 13.25/7.66 @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6268_even__set__encode__iff,axiom,
% 13.25/7.66 ! [A2: set_nat] :
% 13.25/7.66 ( ( finite_finite_nat @ A2 )
% 13.25/7.66 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 13.25/7.66 = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6269_signed__take__bit__Suc__bit1,axiom,
% 13.25/7.66 ! [N: nat,K: num] :
% 13.25/7.66 ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 13.25/7.66 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6270_take__bit__rec,axiom,
% 13.25/7.66 ( bit_se1745604003318907178nteger
% 13.25/7.66 = ( ^ [N2: nat,A3: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6271_take__bit__rec,axiom,
% 13.25/7.66 ( bit_se2923211474154528505it_int
% 13.25/7.66 = ( ^ [N2: nat,A3: int] : ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6272_take__bit__rec,axiom,
% 13.25/7.66 ( bit_se2925701944663578781it_nat
% 13.25/7.66 = ( ^ [N2: nat,A3: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6273_verit__eq__simplify_I9_J,axiom,
% 13.25/7.66 ! [X32: num,Y32: num] :
% 13.25/7.66 ( ( ( bit1 @ X32 )
% 13.25/7.66 = ( bit1 @ Y32 ) )
% 13.25/7.66 = ( X32 = Y32 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6274_semiring__norm_I90_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( ( bit1 @ M )
% 13.25/7.66 = ( bit1 @ N ) )
% 13.25/7.66 = ( M = N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6275_semiring__norm_I89_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( bit1 @ M )
% 13.25/7.66 != ( bit0 @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6276_semiring__norm_I88_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( bit0 @ M )
% 13.25/7.66 != ( bit1 @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6277_semiring__norm_I86_J,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ( ( bit1 @ M )
% 13.25/7.66 != one ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6278_semiring__norm_I84_J,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( one
% 13.25/7.66 != ( bit1 @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6279_take__bit__of__0,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ N @ zero_zero_int )
% 13.25/7.66 = zero_zero_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6280_take__bit__of__0,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ N @ zero_zero_nat )
% 13.25/7.66 = zero_zero_nat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6281_concat__bit__of__zero__2,axiom,
% 13.25/7.66 ! [N: nat,K: int] :
% 13.25/7.66 ( ( bit_concat_bit @ N @ K @ zero_zero_int )
% 13.25/7.66 = ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6282_semiring__norm_I73_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.66 = ( ord_less_eq_num @ M @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6283_semiring__norm_I80_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.66 = ( ord_less_num @ M @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6284_take__bit__0,axiom,
% 13.25/7.66 ! [A: int] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 13.25/7.66 = zero_zero_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6285_take__bit__0,axiom,
% 13.25/7.66 ! [A: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 13.25/7.66 = zero_zero_nat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6286_take__bit__Suc__1,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
% 13.25/7.66 = one_one_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6287_take__bit__Suc__1,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
% 13.25/7.66 = one_one_nat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6288_take__bit__numeral__1,axiom,
% 13.25/7.66 ! [L2: num] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ one_one_int )
% 13.25/7.66 = one_one_int ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6289_take__bit__numeral__1,axiom,
% 13.25/7.66 ! [L2: num] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ one_one_nat )
% 13.25/7.66 = one_one_nat ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6290_zabs__less__one__iff,axiom,
% 13.25/7.66 ! [Z: int] :
% 13.25/7.66 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 13.25/7.66 = ( Z = zero_zero_int ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6291_semiring__norm_I7_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.66 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6292_semiring__norm_I9_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.66 = ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6293_arctan__less__zero__iff,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_real @ ( arctan @ X ) @ zero_zero_real )
% 13.25/7.66 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6294_zero__less__arctan__iff,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X ) )
% 13.25/7.66 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6295_zero__le__arctan__iff,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X ) )
% 13.25/7.66 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6296_arctan__le__zero__iff,axiom,
% 13.25/7.66 ! [X: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( arctan @ X ) @ zero_zero_real )
% 13.25/7.66 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6297_semiring__norm_I14_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.66 = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6298_semiring__norm_I15_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.66 = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6299_semiring__norm_I72_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.66 = ( ord_less_eq_num @ M @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6300_semiring__norm_I81_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.66 = ( ord_less_num @ M @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6301_semiring__norm_I70_J,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6302_semiring__norm_I77_J,axiom,
% 13.25/7.66 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6303_sum__abs,axiom,
% 13.25/7.66 ! [F: int > int,A2: set_int] :
% 13.25/7.66 ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 13.25/7.66 @ ( groups4538972089207619220nt_int
% 13.25/7.66 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 13.25/7.66 @ A2 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6304_sum__abs,axiom,
% 13.25/7.66 ! [F: nat > real,A2: set_nat] :
% 13.25/7.66 ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 13.25/7.66 @ ( groups6591440286371151544t_real
% 13.25/7.66 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 13.25/7.66 @ A2 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6305_take__bit__of__1__eq__0__iff,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 13.25/7.66 = zero_zero_int )
% 13.25/7.66 = ( N = zero_zero_nat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6306_take__bit__of__1__eq__0__iff,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 13.25/7.66 = zero_zero_nat )
% 13.25/7.66 = ( N = zero_zero_nat ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6307_zdiv__numeral__Bit1,axiom,
% 13.25/7.66 ! [V: num,W: num] :
% 13.25/7.66 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 13.25/7.66 = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6308_semiring__norm_I10_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.66 = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6309_semiring__norm_I8_J,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ( ( plus_plus_num @ ( bit1 @ M ) @ one )
% 13.25/7.66 = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6310_semiring__norm_I5_J,axiom,
% 13.25/7.66 ! [M: num] :
% 13.25/7.66 ( ( plus_plus_num @ ( bit0 @ M ) @ one )
% 13.25/7.66 = ( bit1 @ M ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6311_semiring__norm_I4_J,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( ( plus_plus_num @ one @ ( bit1 @ N ) )
% 13.25/7.66 = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6312_semiring__norm_I3_J,axiom,
% 13.25/7.66 ! [N: num] :
% 13.25/7.66 ( ( plus_plus_num @ one @ ( bit0 @ N ) )
% 13.25/7.66 = ( bit1 @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6313_take__bit__of__Suc__0,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 13.25/7.66 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6314_sum__abs__ge__zero,axiom,
% 13.25/7.66 ! [F: int > int,A2: set_int] :
% 13.25/7.66 ( ord_less_eq_int @ zero_zero_int
% 13.25/7.66 @ ( groups4538972089207619220nt_int
% 13.25/7.66 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 13.25/7.66 @ A2 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6315_sum__abs__ge__zero,axiom,
% 13.25/7.66 ! [F: nat > real,A2: set_nat] :
% 13.25/7.66 ( ord_less_eq_real @ zero_zero_real
% 13.25/7.66 @ ( groups6591440286371151544t_real
% 13.25/7.66 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 13.25/7.66 @ A2 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6316_semiring__norm_I16_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.66 = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6317_semiring__norm_I79_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.66 = ( ord_less_eq_num @ M @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6318_semiring__norm_I74_J,axiom,
% 13.25/7.66 ! [M: num,N: num] :
% 13.25/7.66 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.66 = ( ord_less_num @ M @ N ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6319_take__bit__of__1,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se1745604003318907178nteger @ N @ one_one_Code_integer )
% 13.25/7.66 = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6320_take__bit__of__1,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ N @ one_one_int )
% 13.25/7.66 = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6321_take__bit__of__1,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ N @ one_one_nat )
% 13.25/7.66 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6322_even__take__bit__eq,axiom,
% 13.25/7.66 ! [N: nat,A: code_integer] :
% 13.25/7.66 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
% 13.25/7.66 = ( ( N = zero_zero_nat )
% 13.25/7.66 | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6323_even__take__bit__eq,axiom,
% 13.25/7.66 ! [N: nat,A: int] :
% 13.25/7.66 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 13.25/7.66 = ( ( N = zero_zero_nat )
% 13.25/7.66 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6324_even__take__bit__eq,axiom,
% 13.25/7.66 ! [N: nat,A: nat] :
% 13.25/7.66 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
% 13.25/7.66 = ( ( N = zero_zero_nat )
% 13.25/7.66 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6325_div__Suc__eq__div__add3,axiom,
% 13.25/7.66 ! [M: nat,N: nat] :
% 13.25/7.66 ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 13.25/7.66 = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6326_Suc__div__eq__add3__div__numeral,axiom,
% 13.25/7.66 ! [M: nat,V: num] :
% 13.25/7.66 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 13.25/7.66 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6327_Suc__mod__eq__add3__mod__numeral,axiom,
% 13.25/7.66 ! [M: nat,V: num] :
% 13.25/7.66 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 13.25/7.66 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6328_mod__Suc__eq__mod__add3,axiom,
% 13.25/7.66 ! [M: nat,N: nat] :
% 13.25/7.66 ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 13.25/7.66 = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6329_take__bit__Suc__0,axiom,
% 13.25/7.66 ! [A: code_integer] :
% 13.25/7.66 ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 13.25/7.66 = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6330_take__bit__Suc__0,axiom,
% 13.25/7.66 ! [A: int] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 13.25/7.66 = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6331_take__bit__Suc__0,axiom,
% 13.25/7.66 ! [A: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 13.25/7.66 = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6332_take__bit__of__exp,axiom,
% 13.25/7.66 ! [M: nat,N: nat] :
% 13.25/7.66 ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.66 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6333_take__bit__of__exp,axiom,
% 13.25/7.66 ! [M: nat,N: nat] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.66 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6334_take__bit__of__exp,axiom,
% 13.25/7.66 ! [M: nat,N: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.66 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6335_take__bit__of__2,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.66 = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6336_take__bit__of__2,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.66 = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6337_take__bit__of__2,axiom,
% 13.25/7.66 ! [N: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.66 = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6338_zmod__numeral__Bit1,axiom,
% 13.25/7.66 ! [V: num,W: num] :
% 13.25/7.66 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
% 13.25/7.66 = ( 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 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6339_take__bit__minus,axiom,
% 13.25/7.66 ! [N: nat,K: int] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) )
% 13.25/7.66 = ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6340_take__bit__add,axiom,
% 13.25/7.66 ! [N: nat,A: int,B: int] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
% 13.25/7.66 = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6341_take__bit__add,axiom,
% 13.25/7.66 ! [N: nat,A: nat,B: nat] :
% 13.25/7.66 ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
% 13.25/7.66 = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6342_take__bit__tightened,axiom,
% 13.25/7.66 ! [N: nat,A: int,B: int,M: nat] :
% 13.25/7.66 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 13.25/7.66 = ( bit_se2923211474154528505it_int @ N @ B ) )
% 13.25/7.66 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.66 => ( ( bit_se2923211474154528505it_int @ M @ A )
% 13.25/7.66 = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6343_take__bit__tightened,axiom,
% 13.25/7.66 ! [N: nat,A: nat,B: nat,M: nat] :
% 13.25/7.66 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 13.25/7.66 = ( bit_se2925701944663578781it_nat @ N @ B ) )
% 13.25/7.66 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.66 => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 13.25/7.66 = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6344_take__bit__tightened__less__eq__nat,axiom,
% 13.25/7.66 ! [M: nat,N: nat,Q2: nat] :
% 13.25/7.66 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.66 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6345_take__bit__nat__less__eq__self,axiom,
% 13.25/7.66 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6346_take__bit__mult,axiom,
% 13.25/7.66 ! [N: nat,K: int,L2: int] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
% 13.25/7.66 = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K @ L2 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6347_verit__eq__simplify_I14_J,axiom,
% 13.25/7.66 ! [X22: num,X32: num] :
% 13.25/7.66 ( ( bit0 @ X22 )
% 13.25/7.66 != ( bit1 @ X32 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6348_verit__eq__simplify_I12_J,axiom,
% 13.25/7.66 ! [X32: num] :
% 13.25/7.66 ( one
% 13.25/7.66 != ( bit1 @ X32 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6349_take__bit__diff,axiom,
% 13.25/7.66 ! [N: nat,K: int,L2: int] :
% 13.25/7.66 ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( bit_se2923211474154528505it_int @ N @ L2 ) ) )
% 13.25/7.66 = ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ L2 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6350_arctan__monotone,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.66 => ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y3 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6351_arctan__less__iff,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_real @ ( arctan @ X ) @ ( arctan @ Y3 ) )
% 13.25/7.66 = ( ord_less_real @ X @ Y3 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6352_arctan__monotone_H,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.66 => ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y3 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6353_arctan__le__iff,axiom,
% 13.25/7.66 ! [X: real,Y3: real] :
% 13.25/7.66 ( ( ord_less_eq_real @ ( arctan @ X ) @ ( arctan @ Y3 ) )
% 13.25/7.66 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6354_concat__bit__eq__iff,axiom,
% 13.25/7.66 ! [N: nat,K: int,L2: int,R: int,S: int] :
% 13.25/7.66 ( ( ( bit_concat_bit @ N @ K @ L2 )
% 13.25/7.66 = ( bit_concat_bit @ N @ R @ S ) )
% 13.25/7.66 = ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 13.25/7.66 = ( bit_se2923211474154528505it_int @ N @ R ) )
% 13.25/7.66 & ( L2 = S ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6355_concat__bit__take__bit__eq,axiom,
% 13.25/7.66 ! [N: nat,B: int] :
% 13.25/7.66 ( ( bit_concat_bit @ N @ ( bit_se2923211474154528505it_int @ N @ B ) )
% 13.25/7.66 = ( bit_concat_bit @ N @ B ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6356_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 13.25/7.66 ( ! [I2: nat] :
% 13.25/7.66 ( ( member_nat @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6357_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_real,F: real > rat,G: real > rat] :
% 13.25/7.66 ( ! [I2: real] :
% 13.25/7.66 ( ( member_real @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6358_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_int,F: int > rat,G: int > rat] :
% 13.25/7.66 ( ! [I2: int] :
% 13.25/7.66 ( ( member_int @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6359_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_complex,F: complex > rat,G: complex > rat] :
% 13.25/7.66 ( ! [I2: complex] :
% 13.25/7.66 ( ( member_complex @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6360_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_real,F: real > nat,G: real > nat] :
% 13.25/7.66 ( ! [I2: real] :
% 13.25/7.66 ( ( member_real @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6361_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_int,F: int > nat,G: int > nat] :
% 13.25/7.66 ( ! [I2: int] :
% 13.25/7.66 ( ( member_int @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6362_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_complex,F: complex > nat,G: complex > nat] :
% 13.25/7.66 ( ! [I2: complex] :
% 13.25/7.66 ( ( member_complex @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6363_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_nat,F: nat > int,G: nat > int] :
% 13.25/7.66 ( ! [I2: nat] :
% 13.25/7.66 ( ( member_nat @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6364_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_real,F: real > int,G: real > int] :
% 13.25/7.66 ( ! [I2: real] :
% 13.25/7.66 ( ( member_real @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6365_sum__mono,axiom,
% 13.25/7.66 ! [K5: set_complex,F: complex > int,G: complex > int] :
% 13.25/7.66 ( ! [I2: complex] :
% 13.25/7.66 ( ( member_complex @ I2 @ K5 )
% 13.25/7.66 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.66 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6366_sum__product,axiom,
% 13.25/7.66 ! [F: int > int,A2: set_int,G: int > int,B5: set_int] :
% 13.25/7.66 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ B5 ) )
% 13.25/7.66 = ( groups4538972089207619220nt_int
% 13.25/7.66 @ ^ [I4: int] :
% 13.25/7.66 ( groups4538972089207619220nt_int
% 13.25/7.66 @ ^ [J3: int] : ( times_times_int @ ( F @ I4 ) @ ( G @ J3 ) )
% 13.25/7.66 @ B5 )
% 13.25/7.66 @ A2 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6367_sum__product,axiom,
% 13.25/7.66 ! [F: complex > complex,A2: set_complex,G: complex > complex,B5: set_complex] :
% 13.25/7.66 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ B5 ) )
% 13.25/7.66 = ( groups7754918857620584856omplex
% 13.25/7.66 @ ^ [I4: complex] :
% 13.25/7.66 ( groups7754918857620584856omplex
% 13.25/7.66 @ ^ [J3: complex] : ( times_times_complex @ ( F @ I4 ) @ ( G @ J3 ) )
% 13.25/7.66 @ B5 )
% 13.25/7.66 @ A2 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6368_sum__product,axiom,
% 13.25/7.66 ! [F: nat > nat,A2: set_nat,G: nat > nat,B5: set_nat] :
% 13.25/7.66 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ B5 ) )
% 13.25/7.66 = ( groups3542108847815614940at_nat
% 13.25/7.66 @ ^ [I4: nat] :
% 13.25/7.66 ( groups3542108847815614940at_nat
% 13.25/7.66 @ ^ [J3: nat] : ( times_times_nat @ ( F @ I4 ) @ ( G @ J3 ) )
% 13.25/7.66 @ B5 )
% 13.25/7.66 @ A2 ) ) ).
% 13.25/7.66
% 13.25/7.66 thf(fact_6369_sum__product,axiom,
% 13.25/7.66 ! [F: nat > real,A2: set_nat,G: nat > real,B5: set_nat] :
% 13.25/7.66 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ B5 ) )
% 13.25/7.66 = ( groups6591440286371151544t_real
% 13.25/7.66 @ ^ [I4: nat] :
% 13.25/7.66 ( groups6591440286371151544t_real
% 13.25/7.66 @ ^ [J3: nat] : ( times_times_real @ ( F @ I4 ) @ ( G @ J3 ) )
% 13.25/7.66 @ B5 )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6370_sum__distrib__right,axiom,
% 13.25/7.67 ! [F: int > int,A2: set_int,R: int] :
% 13.25/7.67 ( ( times_times_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ R )
% 13.25/7.67 = ( groups4538972089207619220nt_int
% 13.25/7.67 @ ^ [N2: int] : ( times_times_int @ ( F @ N2 ) @ R )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6371_sum__distrib__right,axiom,
% 13.25/7.67 ! [F: complex > complex,A2: set_complex,R: complex] :
% 13.25/7.67 ( ( times_times_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
% 13.25/7.67 = ( groups7754918857620584856omplex
% 13.25/7.67 @ ^ [N2: complex] : ( times_times_complex @ ( F @ N2 ) @ R )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6372_sum__distrib__right,axiom,
% 13.25/7.67 ! [F: nat > nat,A2: set_nat,R: nat] :
% 13.25/7.67 ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ R )
% 13.25/7.67 = ( groups3542108847815614940at_nat
% 13.25/7.67 @ ^ [N2: nat] : ( times_times_nat @ ( F @ N2 ) @ R )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6373_sum__distrib__right,axiom,
% 13.25/7.67 ! [F: nat > real,A2: set_nat,R: real] :
% 13.25/7.67 ( ( times_times_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
% 13.25/7.67 = ( groups6591440286371151544t_real
% 13.25/7.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ R )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6374_sum__distrib__left,axiom,
% 13.25/7.67 ! [R: int,F: int > int,A2: set_int] :
% 13.25/7.67 ( ( times_times_int @ R @ ( groups4538972089207619220nt_int @ F @ A2 ) )
% 13.25/7.67 = ( groups4538972089207619220nt_int
% 13.25/7.67 @ ^ [N2: int] : ( times_times_int @ R @ ( F @ N2 ) )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6375_sum__distrib__left,axiom,
% 13.25/7.67 ! [R: complex,F: complex > complex,A2: set_complex] :
% 13.25/7.67 ( ( times_times_complex @ R @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 13.25/7.67 = ( groups7754918857620584856omplex
% 13.25/7.67 @ ^ [N2: complex] : ( times_times_complex @ R @ ( F @ N2 ) )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6376_sum__distrib__left,axiom,
% 13.25/7.67 ! [R: nat,F: nat > nat,A2: set_nat] :
% 13.25/7.67 ( ( times_times_nat @ R @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 13.25/7.67 = ( groups3542108847815614940at_nat
% 13.25/7.67 @ ^ [N2: nat] : ( times_times_nat @ R @ ( F @ N2 ) )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6377_sum__distrib__left,axiom,
% 13.25/7.67 ! [R: real,F: nat > real,A2: set_nat] :
% 13.25/7.67 ( ( times_times_real @ R @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 13.25/7.67 = ( groups6591440286371151544t_real
% 13.25/7.67 @ ^ [N2: nat] : ( times_times_real @ R @ ( F @ N2 ) )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6378_sum_Odistrib,axiom,
% 13.25/7.67 ! [G: int > int,H2: int > int,A2: set_int] :
% 13.25/7.67 ( ( groups4538972089207619220nt_int
% 13.25/7.67 @ ^ [X2: int] : ( plus_plus_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 13.25/7.67 @ A2 )
% 13.25/7.67 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H2 @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6379_sum_Odistrib,axiom,
% 13.25/7.67 ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
% 13.25/7.67 ( ( groups7754918857620584856omplex
% 13.25/7.67 @ ^ [X2: complex] : ( plus_plus_complex @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 13.25/7.67 @ A2 )
% 13.25/7.67 = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6380_sum_Odistrib,axiom,
% 13.25/7.67 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 13.25/7.67 ( ( groups3542108847815614940at_nat
% 13.25/7.67 @ ^ [X2: nat] : ( plus_plus_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 13.25/7.67 @ A2 )
% 13.25/7.67 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6381_sum_Odistrib,axiom,
% 13.25/7.67 ! [G: nat > real,H2: nat > real,A2: set_nat] :
% 13.25/7.67 ( ( groups6591440286371151544t_real
% 13.25/7.67 @ ^ [X2: nat] : ( plus_plus_real @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 13.25/7.67 @ A2 )
% 13.25/7.67 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6382_sum__subtractf,axiom,
% 13.25/7.67 ! [F: int > int,G: int > int,A2: set_int] :
% 13.25/7.67 ( ( groups4538972089207619220nt_int
% 13.25/7.67 @ ^ [X2: int] : ( minus_minus_int @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.67 @ A2 )
% 13.25/7.67 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6383_sum__subtractf,axiom,
% 13.25/7.67 ! [F: complex > complex,G: complex > complex,A2: set_complex] :
% 13.25/7.67 ( ( groups7754918857620584856omplex
% 13.25/7.67 @ ^ [X2: complex] : ( minus_minus_complex @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.67 @ A2 )
% 13.25/7.67 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6384_sum__subtractf,axiom,
% 13.25/7.67 ! [F: nat > real,G: nat > real,A2: set_nat] :
% 13.25/7.67 ( ( groups6591440286371151544t_real
% 13.25/7.67 @ ^ [X2: nat] : ( minus_minus_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.67 @ A2 )
% 13.25/7.67 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6385_sum__divide__distrib,axiom,
% 13.25/7.67 ! [F: complex > complex,A2: set_complex,R: complex] :
% 13.25/7.67 ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R )
% 13.25/7.67 = ( groups7754918857620584856omplex
% 13.25/7.67 @ ^ [N2: complex] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ R )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6386_sum__divide__distrib,axiom,
% 13.25/7.67 ! [F: nat > real,A2: set_nat,R: real] :
% 13.25/7.67 ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R )
% 13.25/7.67 = ( groups6591440286371151544t_real
% 13.25/7.67 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ R )
% 13.25/7.67 @ A2 ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6387_mod__sum__eq,axiom,
% 13.25/7.67 ! [F: int > int,A: int,A2: set_int] :
% 13.25/7.67 ( ( modulo_modulo_int
% 13.25/7.67 @ ( groups4538972089207619220nt_int
% 13.25/7.67 @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 13.25/7.67 @ A2 )
% 13.25/7.67 @ A )
% 13.25/7.67 = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6388_mod__sum__eq,axiom,
% 13.25/7.67 ! [F: nat > nat,A: nat,A2: set_nat] :
% 13.25/7.67 ( ( modulo_modulo_nat
% 13.25/7.67 @ ( groups3542108847815614940at_nat
% 13.25/7.67 @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 13.25/7.67 @ A2 )
% 13.25/7.67 @ A )
% 13.25/7.67 = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6389_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > real] :
% 13.25/7.67 ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6390_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > real] :
% 13.25/7.67 ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6391_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > real] :
% 13.25/7.67 ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ ( F @ X3 ) @ zero_zero_real ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6392_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_nat,F: nat > rat] :
% 13.25/7.67 ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6393_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > rat] :
% 13.25/7.67 ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6394_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > rat] :
% 13.25/7.67 ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6395_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > rat] :
% 13.25/7.67 ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ X3 ) @ zero_zero_rat ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6396_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > nat] :
% 13.25/7.67 ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
% 13.25/7.67 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6397_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > nat] :
% 13.25/7.67 ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
% 13.25/7.67 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6398_sum__nonpos,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > nat] :
% 13.25/7.67 ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
% 13.25/7.67 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6399_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > real] :
% 13.25/7.67 ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6400_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > real] :
% 13.25/7.67 ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6401_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > real] :
% 13.25/7.67 ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6402_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_nat,F: nat > rat] :
% 13.25/7.67 ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6403_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > rat] :
% 13.25/7.67 ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6404_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > rat] :
% 13.25/7.67 ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6405_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > rat] :
% 13.25/7.67 ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6406_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > nat] :
% 13.25/7.67 ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6407_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > nat] :
% 13.25/7.67 ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6408_sum__nonneg,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > nat] :
% 13.25/7.67 ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6409_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
% 13.25/7.67 ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 13.25/7.67 = ( groups1300246762558778688al_rat @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_real @ I @ I5 )
% 13.25/7.67 => ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6410_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
% 13.25/7.67 ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 13.25/7.67 = ( groups2906978787729119204at_rat @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: nat] :
% 13.25/7.67 ( ( member_nat @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_nat @ I @ I5 )
% 13.25/7.67 => ( ( finite_finite_nat @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6411_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
% 13.25/7.67 ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 13.25/7.67 = ( groups3906332499630173760nt_rat @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_int @ I @ I5 )
% 13.25/7.67 => ( ( finite_finite_int @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6412_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
% 13.25/7.67 ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_complex @ I @ I5 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6413_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
% 13.25/7.67 ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 13.25/7.67 = ( groups1935376822645274424al_nat @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_real @ I @ I5 )
% 13.25/7.67 => ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6414_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
% 13.25/7.67 ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 13.25/7.67 = ( groups4541462559716669496nt_nat @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_int @ I @ I5 )
% 13.25/7.67 => ( ( finite_finite_int @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6415_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
% 13.25/7.67 ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_complex @ I @ I5 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6416_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: real > int,I5: set_real,G: real > int,I: real] :
% 13.25/7.67 ( ( ( groups1932886352136224148al_int @ F @ I5 )
% 13.25/7.67 = ( groups1932886352136224148al_int @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_real @ I @ I5 )
% 13.25/7.67 => ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6417_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
% 13.25/7.67 ( ( ( groups3539618377306564664at_int @ F @ I5 )
% 13.25/7.67 = ( groups3539618377306564664at_int @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: nat] :
% 13.25/7.67 ( ( member_nat @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_nat @ I @ I5 )
% 13.25/7.67 => ( ( finite_finite_nat @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6418_sum__mono__inv,axiom,
% 13.25/7.67 ! [F: complex > int,I5: set_complex,G: complex > int,I: complex] :
% 13.25/7.67 ( ( ( groups5690904116761175830ex_int @ F @ I5 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ G @ I5 ) )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 13.25/7.67 => ( ( member_complex @ I @ I5 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = ( G @ I ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6419_take__bit__tightened__less__eq__int,axiom,
% 13.25/7.67 ! [M: nat,N: nat,K: int] :
% 13.25/7.67 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.67 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6420_take__bit__int__less__eq__self__iff,axiom,
% 13.25/7.67 ! [N: nat,K: int] :
% 13.25/7.67 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 13.25/7.67 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6421_take__bit__nonnegative,axiom,
% 13.25/7.67 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6422_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 13.25/7.67 ! [N: nat,A: int,B: int] :
% 13.25/7.67 ( ( ( bit_ri631733984087533419it_int @ N @ A )
% 13.25/7.67 = ( bit_ri631733984087533419it_int @ N @ B ) )
% 13.25/7.67 = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 13.25/7.67 = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6423_take__bit__int__greater__self__iff,axiom,
% 13.25/7.67 ! [K: int,N: nat] :
% 13.25/7.67 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 13.25/7.67 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6424_not__take__bit__negative,axiom,
% 13.25/7.67 ! [N: nat,K: int] :
% 13.25/7.67 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6425_signed__take__bit__take__bit,axiom,
% 13.25/7.67 ! [M: nat,N: nat,A: int] :
% 13.25/7.67 ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) )
% 13.25/7.67 = ( if_int_int @ ( ord_less_eq_nat @ N @ M ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6426_xor__num_Ocases,axiom,
% 13.25/7.67 ! [X: product_prod_num_num] :
% 13.25/7.67 ( ( X
% 13.25/7.67 != ( product_Pair_num_num @ one @ one ) )
% 13.25/7.67 => ( ! [N4: num] :
% 13.25/7.67 ( X
% 13.25/7.67 != ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) )
% 13.25/7.67 => ( ! [N4: num] :
% 13.25/7.67 ( X
% 13.25/7.67 != ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) )
% 13.25/7.67 => ( ! [M4: num] :
% 13.25/7.67 ( X
% 13.25/7.67 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
% 13.25/7.67 => ( ! [M4: num,N4: num] :
% 13.25/7.67 ( X
% 13.25/7.67 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) )
% 13.25/7.67 => ( ! [M4: num,N4: num] :
% 13.25/7.67 ( X
% 13.25/7.67 != ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) )
% 13.25/7.67 => ( ! [M4: num] :
% 13.25/7.67 ( X
% 13.25/7.67 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
% 13.25/7.67 => ( ! [M4: num,N4: num] :
% 13.25/7.67 ( X
% 13.25/7.67 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) )
% 13.25/7.67 => ~ ! [M4: num,N4: num] :
% 13.25/7.67 ( X
% 13.25/7.67 != ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6427_num_Oexhaust,axiom,
% 13.25/7.67 ! [Y3: num] :
% 13.25/7.67 ( ( Y3 != one )
% 13.25/7.67 => ( ! [X23: num] :
% 13.25/7.67 ( Y3
% 13.25/7.67 != ( bit0 @ X23 ) )
% 13.25/7.67 => ~ ! [X33: num] :
% 13.25/7.67 ( Y3
% 13.25/7.67 != ( bit1 @ X33 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6428_zabs__def,axiom,
% 13.25/7.67 ( abs_abs_int
% 13.25/7.67 = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6429_take__bit__unset__bit__eq,axiom,
% 13.25/7.67 ! [N: nat,M: nat,A: int] :
% 13.25/7.67 ( ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 13.25/7.67 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 13.25/7.67 & ( ~ ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 13.25/7.67 = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6430_take__bit__unset__bit__eq,axiom,
% 13.25/7.67 ! [N: nat,M: nat,A: nat] :
% 13.25/7.67 ( ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 13.25/7.67 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 13.25/7.67 & ( ~ ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 13.25/7.67 = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6431_take__bit__set__bit__eq,axiom,
% 13.25/7.67 ! [N: nat,M: nat,A: int] :
% 13.25/7.67 ( ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 13.25/7.67 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 13.25/7.67 & ( ~ ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 13.25/7.67 = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6432_take__bit__set__bit__eq,axiom,
% 13.25/7.67 ! [N: nat,M: nat,A: nat] :
% 13.25/7.67 ( ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 13.25/7.67 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 13.25/7.67 & ( ~ ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 13.25/7.67 = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6433_take__bit__flip__bit__eq,axiom,
% 13.25/7.67 ! [N: nat,M: nat,A: int] :
% 13.25/7.67 ( ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 13.25/7.67 = ( bit_se2923211474154528505it_int @ N @ A ) ) )
% 13.25/7.67 & ( ~ ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 13.25/7.67 = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6434_take__bit__flip__bit__eq,axiom,
% 13.25/7.67 ! [N: nat,M: nat,A: nat] :
% 13.25/7.67 ( ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 13.25/7.67 = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
% 13.25/7.67 & ( ~ ( ord_less_eq_nat @ N @ M )
% 13.25/7.67 => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 13.25/7.67 = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6435_abs__zmult__eq__1,axiom,
% 13.25/7.67 ! [M: int,N: int] :
% 13.25/7.67 ( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
% 13.25/7.67 = one_one_int )
% 13.25/7.67 => ( ( abs_abs_int @ M )
% 13.25/7.67 = one_one_int ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6436_abs__div,axiom,
% 13.25/7.67 ! [Y3: int,X: int] :
% 13.25/7.67 ( ( dvd_dvd_int @ Y3 @ X )
% 13.25/7.67 => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y3 ) )
% 13.25/7.67 = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y3 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6437_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 13.25/7.67 = zero_zero_real )
% 13.25/7.67 = ( ! [X2: real] :
% 13.25/7.67 ( ( member_real @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6438_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 13.25/7.67 = zero_zero_real )
% 13.25/7.67 = ( ! [X2: int] :
% 13.25/7.67 ( ( member_int @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6439_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 13.25/7.67 = zero_zero_real )
% 13.25/7.67 = ( ! [X2: complex] :
% 13.25/7.67 ( ( member_complex @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_real ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6440_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 13.25/7.67 = zero_zero_rat )
% 13.25/7.67 = ( ! [X2: real] :
% 13.25/7.67 ( ( member_real @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6441_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 13.25/7.67 = zero_zero_rat )
% 13.25/7.67 = ( ! [X2: nat] :
% 13.25/7.67 ( ( member_nat @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6442_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 13.25/7.67 = zero_zero_rat )
% 13.25/7.67 = ( ! [X2: int] :
% 13.25/7.67 ( ( member_int @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6443_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 13.25/7.67 = zero_zero_rat )
% 13.25/7.67 = ( ! [X2: complex] :
% 13.25/7.67 ( ( member_complex @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_rat ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6444_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 13.25/7.67 = zero_zero_nat )
% 13.25/7.67 = ( ! [X2: real] :
% 13.25/7.67 ( ( member_real @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_nat ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6445_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > nat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 13.25/7.67 = zero_zero_nat )
% 13.25/7.67 = ( ! [X2: int] :
% 13.25/7.67 ( ( member_int @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_nat ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6446_sum__nonneg__eq__0__iff,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 13.25/7.67 = zero_zero_nat )
% 13.25/7.67 = ( ! [X2: complex] :
% 13.25/7.67 ( ( member_complex @ X2 @ A2 )
% 13.25/7.67 => ( ( F @ X2 )
% 13.25/7.67 = zero_zero_nat ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6447_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ( finite_finite_int @ T )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S )
% 13.25/7.67 => ? [Xa2: int] :
% 13.25/7.67 ( ( member_int @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6448_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ T )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S )
% 13.25/7.67 => ? [Xa2: complex] :
% 13.25/7.67 ( ( member_complex @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6449_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ S )
% 13.25/7.67 => ( ( finite_finite_int @ T )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S )
% 13.25/7.67 => ? [Xa2: int] :
% 13.25/7.67 ( ( member_int @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6450_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ S )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ T )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S )
% 13.25/7.67 => ? [Xa2: complex] :
% 13.25/7.67 ( ( member_complex @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6451_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ S )
% 13.25/7.67 => ( ( finite_finite_nat @ T )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ S )
% 13.25/7.67 => ? [Xa2: nat] :
% 13.25/7.67 ( ( member_nat @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6452_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ S )
% 13.25/7.67 => ( ( finite_finite_int @ T )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ S )
% 13.25/7.67 => ? [Xa2: int] :
% 13.25/7.67 ( ( member_int @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6453_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ S )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ T )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ S )
% 13.25/7.67 => ? [Xa2: complex] :
% 13.25/7.67 ( ( member_complex @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6454_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ( finite_finite_nat @ T )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S )
% 13.25/7.67 => ? [Xa2: nat] :
% 13.25/7.67 ( ( member_nat @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6455_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ( finite_finite_int @ T )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S )
% 13.25/7.67 => ? [Xa2: int] :
% 13.25/7.67 ( ( member_int @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6456_sum__le__included,axiom,
% 13.25/7.67 ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ T )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ T )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S )
% 13.25/7.67 => ? [Xa2: complex] :
% 13.25/7.67 ( ( member_complex @ Xa2 @ T )
% 13.25/7.67 & ( ( I @ Xa2 )
% 13.25/7.67 = X3 )
% 13.25/7.67 & ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6457_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > real,G: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: int] :
% 13.25/7.67 ( ( member_int @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6458_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: complex] :
% 13.25/7.67 ( ( member_complex @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6459_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: nat] :
% 13.25/7.67 ( ( member_nat @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6460_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > rat,G: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: int] :
% 13.25/7.67 ( ( member_int @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6461_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: complex] :
% 13.25/7.67 ( ( member_complex @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6462_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > nat,G: int > nat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: int] :
% 13.25/7.67 ( ( member_int @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6463_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: complex] :
% 13.25/7.67 ( ( member_complex @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6464_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_nat,F: nat > int,G: nat > int] :
% 13.25/7.67 ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: nat] :
% 13.25/7.67 ( ( member_nat @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6465_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > int,G: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: complex] :
% 13.25/7.67 ( ( member_complex @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6466_sum__strict__mono__ex1,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > int,G: int > int] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_eq_int @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ? [X5: int] :
% 13.25/7.67 ( ( member_int @ X5 @ A2 )
% 13.25/7.67 & ( ord_less_int @ ( F @ X5 ) @ ( G @ X5 ) ) )
% 13.25/7.67 => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6467_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 13.25/7.67 ( ( R4 @ zero_zero_complex @ zero_zero_complex )
% 13.25/7.67 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite_finite_nat @ S2 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups2073611262835488442omplex @ H2 @ S2 ) @ ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6468_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 13.25/7.67 ( ( R4 @ zero_zero_complex @ zero_zero_complex )
% 13.25/7.67 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_complex @ X1 @ Y1 ) @ ( plus_plus_complex @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite_finite_int @ S2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups3049146728041665814omplex @ H2 @ S2 ) @ ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6469_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 13.25/7.67 ( ( R4 @ zero_zero_real @ zero_zero_real )
% 13.25/7.67 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite_finite_int @ S2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups8778361861064173332t_real @ H2 @ S2 ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6470_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 13.25/7.67 ( ( R4 @ zero_zero_real @ zero_zero_real )
% 13.25/7.67 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups5808333547571424918x_real @ H2 @ S2 ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6471_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 13.25/7.67 ( ( R4 @ zero_zero_rat @ zero_zero_rat )
% 13.25/7.67 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite_finite_nat @ S2 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups2906978787729119204at_rat @ H2 @ S2 ) @ ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6472_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 13.25/7.67 ( ( R4 @ zero_zero_rat @ zero_zero_rat )
% 13.25/7.67 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite_finite_int @ S2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups3906332499630173760nt_rat @ H2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6473_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 13.25/7.67 ( ( R4 @ zero_zero_rat @ zero_zero_rat )
% 13.25/7.67 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups5058264527183730370ex_rat @ H2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6474_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 13.25/7.67 ( ( R4 @ zero_zero_nat @ zero_zero_nat )
% 13.25/7.67 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite_finite_int @ S2 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups4541462559716669496nt_nat @ H2 @ S2 ) @ ( groups4541462559716669496nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6475_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: nat > nat > $o,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 13.25/7.67 ( ( R4 @ zero_zero_nat @ zero_zero_nat )
% 13.25/7.67 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups5693394587270226106ex_nat @ H2 @ S2 ) @ ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6476_sum_Orelated,axiom,
% 13.25/7.67 ! [R4: int > int > $o,S2: set_nat,H2: nat > int,G: nat > int] :
% 13.25/7.67 ( ( R4 @ zero_zero_int @ zero_zero_int )
% 13.25/7.67 => ( ! [X1: int,Y1: int,X23: int,Y22: int] :
% 13.25/7.67 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.67 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.67 => ( R4 @ ( plus_plus_int @ X1 @ Y1 ) @ ( plus_plus_int @ X23 @ Y22 ) ) )
% 13.25/7.67 => ( ( finite_finite_nat @ S2 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ S2 )
% 13.25/7.67 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( R4 @ ( groups3539618377306564664at_int @ H2 @ S2 ) @ ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6477_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_complex )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6478_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > real,G: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_int )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6479_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > real,G: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_real )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_real @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6480_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_complex )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6481_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_nat )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6482_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > rat,G: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_int )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6483_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > rat,G: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_real )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_rat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6484_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_complex )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6485_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_int,F: int > nat,G: int > nat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_int )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6486_sum__strict__mono,axiom,
% 13.25/7.67 ! [A2: set_real,F: real > nat,G: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ( A2 != bot_bot_set_real )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ A2 )
% 13.25/7.67 => ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.67 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6487_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 13.25/7.67 ( ( finite_finite_real @ S4 )
% 13.25/7.67 => ( ( finite_finite_real @ T4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups5754745047067104278omplex @ G @ S2 )
% 13.25/7.67 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6488_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 13.25/7.67 ( ( finite_finite_real @ S4 )
% 13.25/7.67 => ( ( finite_finite_int @ T4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups5754745047067104278omplex @ G @ S2 )
% 13.25/7.67 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6489_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 13.25/7.67 ( ( finite_finite_int @ S4 )
% 13.25/7.67 => ( ( finite_finite_real @ T4 )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.67 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups3049146728041665814omplex @ G @ S2 )
% 13.25/7.67 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6490_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 13.25/7.67 ( ( finite_finite_int @ S4 )
% 13.25/7.67 => ( ( finite_finite_int @ T4 )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.67 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups3049146728041665814omplex @ G @ S2 )
% 13.25/7.67 = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6491_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ S4 )
% 13.25/7.67 => ( ( finite_finite_real @ T4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups8097168146408367636l_real @ G @ S2 )
% 13.25/7.67 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6492_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 13.25/7.67 ( ( finite_finite_real @ S4 )
% 13.25/7.67 => ( ( finite_finite_int @ T4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups8097168146408367636l_real @ G @ S2 )
% 13.25/7.67 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6493_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 13.25/7.67 ( ( finite_finite_real @ S4 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ T4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.67 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.67 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups8097168146408367636l_real @ G @ S2 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6494_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 13.25/7.67 ( ( finite_finite_int @ S4 )
% 13.25/7.67 => ( ( finite_finite_real @ T4 )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.67 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups8778361861064173332t_real @ G @ S2 )
% 13.25/7.67 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6495_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ S4 )
% 13.25/7.67 => ( ( finite_finite_int @ T4 )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.67 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: int] :
% 13.25/7.67 ( ( member_int @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups8778361861064173332t_real @ G @ S2 )
% 13.25/7.67 = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6496_sum_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.67 ! [S4: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 13.25/7.67 ( ( finite_finite_int @ S4 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ T4 )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( ( I @ ( J @ A5 ) )
% 13.25/7.67 = A5 ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.67 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.67 => ( ( J @ ( I @ B3 ) )
% 13.25/7.67 = B3 ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.67 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S4 )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ T4 )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [A5: int] :
% 13.25/7.67 ( ( member_int @ A5 @ S2 )
% 13.25/7.67 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.67 = ( G @ A5 ) ) )
% 13.25/7.67 => ( ( groups8778361861064173332t_real @ G @ S2 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6497_take__bit__signed__take__bit,axiom,
% 13.25/7.67 ! [M: nat,N: nat,A: int] :
% 13.25/7.67 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.67 => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N @ A ) )
% 13.25/7.67 = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6498_numeral__Bit1,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6499_numeral__Bit1,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6500_numeral__Bit1,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6501_numeral__Bit1,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6502_numeral__Bit1,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6503_take__bit__decr__eq,axiom,
% 13.25/7.67 ! [N: nat,K: int] :
% 13.25/7.67 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 13.25/7.67 != zero_zero_int )
% 13.25/7.67 => ( ( bit_se2923211474154528505it_int @ N @ ( minus_minus_int @ K @ one_one_int ) )
% 13.25/7.67 = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ one_one_int ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6504_eval__nat__numeral_I3_J,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 13.25/7.67 = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6505_cong__exp__iff__simps_I13_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 13.25/7.67 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6506_cong__exp__iff__simps_I13_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 13.25/7.67 = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6507_cong__exp__iff__simps_I13_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 13.25/7.67 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6508_cong__exp__iff__simps_I12_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6509_cong__exp__iff__simps_I12_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6510_cong__exp__iff__simps_I12_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6511_cong__exp__iff__simps_I10_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6512_cong__exp__iff__simps_I10_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6513_cong__exp__iff__simps_I10_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num,N: num] :
% 13.25/7.67 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6514_power__minus__Bit1,axiom,
% 13.25/7.67 ! [X: real,K: num] :
% 13.25/7.67 ( ( power_power_real @ ( uminus_uminus_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 13.25/7.67 = ( uminus_uminus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6515_power__minus__Bit1,axiom,
% 13.25/7.67 ! [X: int,K: num] :
% 13.25/7.67 ( ( power_power_int @ ( uminus_uminus_int @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 13.25/7.67 = ( uminus_uminus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6516_power__minus__Bit1,axiom,
% 13.25/7.67 ! [X: complex,K: num] :
% 13.25/7.67 ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 13.25/7.67 = ( uminus1482373934393186551omplex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6517_power__minus__Bit1,axiom,
% 13.25/7.67 ! [X: code_integer,K: num] :
% 13.25/7.67 ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 13.25/7.67 = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6518_power__minus__Bit1,axiom,
% 13.25/7.67 ! [X: rat,K: num] :
% 13.25/7.67 ( ( power_power_rat @ ( uminus_uminus_rat @ X ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 13.25/7.67 = ( uminus_uminus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6519_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_real,F: real > real,I: real] :
% 13.25/7.67 ( ( finite_finite_real @ S )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups8097168146408367636l_real @ F @ S )
% 13.25/7.67 = zero_zero_real )
% 13.25/7.67 => ( ( member_real @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_real ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6520_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_int,F: int > real,I: int] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups8778361861064173332t_real @ F @ S )
% 13.25/7.67 = zero_zero_real )
% 13.25/7.67 => ( ( member_int @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_real ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6521_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_complex,F: complex > real,I: complex] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ S )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups5808333547571424918x_real @ F @ S )
% 13.25/7.67 = zero_zero_real )
% 13.25/7.67 => ( ( member_complex @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_real ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6522_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_real,F: real > rat,I: real] :
% 13.25/7.67 ( ( finite_finite_real @ S )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 13.25/7.67 = zero_zero_rat )
% 13.25/7.67 => ( ( member_real @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_rat ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6523_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_nat,F: nat > rat,I: nat] :
% 13.25/7.67 ( ( finite_finite_nat @ S )
% 13.25/7.67 => ( ! [I2: nat] :
% 13.25/7.67 ( ( member_nat @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 13.25/7.67 = zero_zero_rat )
% 13.25/7.67 => ( ( member_nat @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_rat ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6524_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_int,F: int > rat,I: int] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 13.25/7.67 = zero_zero_rat )
% 13.25/7.67 => ( ( member_int @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_rat ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6525_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_complex,F: complex > rat,I: complex] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ S )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 13.25/7.67 = zero_zero_rat )
% 13.25/7.67 => ( ( member_complex @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_rat ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6526_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_real,F: real > nat,I: real] :
% 13.25/7.67 ( ( finite_finite_real @ S )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 13.25/7.67 = zero_zero_nat )
% 13.25/7.67 => ( ( member_real @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_nat ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6527_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_int,F: int > nat,I: int] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 13.25/7.67 = zero_zero_nat )
% 13.25/7.67 => ( ( member_int @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_nat ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6528_sum__nonneg__0,axiom,
% 13.25/7.67 ! [S: set_complex,F: complex > nat,I: complex] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ S )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 13.25/7.67 = zero_zero_nat )
% 13.25/7.67 => ( ( member_complex @ I @ S )
% 13.25/7.67 => ( ( F @ I )
% 13.25/7.67 = zero_zero_nat ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6529_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_real,F: real > real,B5: real,I: real] :
% 13.25/7.67 ( ( finite_finite_real @ S )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups8097168146408367636l_real @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_real @ I @ S )
% 13.25/7.67 => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6530_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_int,F: int > real,B5: real,I: int] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups8778361861064173332t_real @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_int @ I @ S )
% 13.25/7.67 => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6531_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_complex,F: complex > real,B5: real,I: complex] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ S )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups5808333547571424918x_real @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_complex @ I @ S )
% 13.25/7.67 => ( ord_less_eq_real @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6532_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_real,F: real > rat,B5: rat,I: real] :
% 13.25/7.67 ( ( finite_finite_real @ S )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_real @ I @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6533_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_nat,F: nat > rat,B5: rat,I: nat] :
% 13.25/7.67 ( ( finite_finite_nat @ S )
% 13.25/7.67 => ( ! [I2: nat] :
% 13.25/7.67 ( ( member_nat @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_nat @ I @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6534_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_int,F: int > rat,B5: rat,I: int] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_int @ I @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6535_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_complex,F: complex > rat,B5: rat,I: complex] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ S )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_complex @ I @ S )
% 13.25/7.67 => ( ord_less_eq_rat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6536_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_real,F: real > nat,B5: nat,I: real] :
% 13.25/7.67 ( ( finite_finite_real @ S )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_real @ I @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6537_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_int,F: int > nat,B5: nat,I: int] :
% 13.25/7.67 ( ( finite_finite_int @ S )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_int @ I @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6538_sum__nonneg__leq__bound,axiom,
% 13.25/7.67 ! [S: set_complex,F: complex > nat,B5: nat,I: complex] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ S )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 13.25/7.67 = B5 )
% 13.25/7.67 => ( ( member_complex @ I @ S )
% 13.25/7.67 => ( ord_less_eq_nat @ ( F @ I ) @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6539_dvd__imp__le__int,axiom,
% 13.25/7.67 ! [I: int,D: int] :
% 13.25/7.67 ( ( I != zero_zero_int )
% 13.25/7.67 => ( ( dvd_dvd_int @ D @ I )
% 13.25/7.67 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6540_take__bit__Suc__bit1,axiom,
% 13.25/7.67 ! [N: nat,K: num] :
% 13.25/7.67 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 13.25/7.67 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6541_take__bit__Suc__bit1,axiom,
% 13.25/7.67 ! [N: nat,K: num] :
% 13.25/7.67 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 13.25/7.67 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6542_abs__mod__less,axiom,
% 13.25/7.67 ! [L2: int,K: int] :
% 13.25/7.67 ( ( L2 != zero_zero_int )
% 13.25/7.67 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L2 ) ) @ ( abs_abs_int @ L2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6543_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_real,G: real > complex] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ( groups5754745047067104278omplex @ G
% 13.25/7.67 @ ( minus_minus_set_real @ A2
% 13.25/7.67 @ ( collect_real
% 13.25/7.67 @ ^ [X2: real] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_complex ) ) ) )
% 13.25/7.67 = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6544_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_int,G: int > complex] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( groups3049146728041665814omplex @ G
% 13.25/7.67 @ ( minus_minus_set_int @ A2
% 13.25/7.67 @ ( collect_int
% 13.25/7.67 @ ^ [X2: int] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_complex ) ) ) )
% 13.25/7.67 = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6545_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_real,G: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ( groups8097168146408367636l_real @ G
% 13.25/7.67 @ ( minus_minus_set_real @ A2
% 13.25/7.67 @ ( collect_real
% 13.25/7.67 @ ^ [X2: real] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_real ) ) ) )
% 13.25/7.67 = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6546_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_int,G: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( groups8778361861064173332t_real @ G
% 13.25/7.67 @ ( minus_minus_set_int @ A2
% 13.25/7.67 @ ( collect_int
% 13.25/7.67 @ ^ [X2: int] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_real ) ) ) )
% 13.25/7.67 = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6547_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_complex,G: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( groups5808333547571424918x_real @ G
% 13.25/7.67 @ ( minus_811609699411566653omplex @ A2
% 13.25/7.67 @ ( collect_complex
% 13.25/7.67 @ ^ [X2: complex] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_real ) ) ) )
% 13.25/7.67 = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6548_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_real,G: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ( groups1300246762558778688al_rat @ G
% 13.25/7.67 @ ( minus_minus_set_real @ A2
% 13.25/7.67 @ ( collect_real
% 13.25/7.67 @ ^ [X2: real] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_rat ) ) ) )
% 13.25/7.67 = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6549_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_int,G: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( groups3906332499630173760nt_rat @ G
% 13.25/7.67 @ ( minus_minus_set_int @ A2
% 13.25/7.67 @ ( collect_int
% 13.25/7.67 @ ^ [X2: int] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_rat ) ) ) )
% 13.25/7.67 = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6550_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_complex,G: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( groups5058264527183730370ex_rat @ G
% 13.25/7.67 @ ( minus_811609699411566653omplex @ A2
% 13.25/7.67 @ ( collect_complex
% 13.25/7.67 @ ^ [X2: complex] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_rat ) ) ) )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6551_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_real,G: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ A2 )
% 13.25/7.67 => ( ( groups1935376822645274424al_nat @ G
% 13.25/7.67 @ ( minus_minus_set_real @ A2
% 13.25/7.67 @ ( collect_real
% 13.25/7.67 @ ^ [X2: real] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_nat ) ) ) )
% 13.25/7.67 = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6552_sum_Osetdiff__irrelevant,axiom,
% 13.25/7.67 ! [A2: set_int,G: int > nat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( groups4541462559716669496nt_nat @ G
% 13.25/7.67 @ ( minus_minus_set_int @ A2
% 13.25/7.67 @ ( collect_int
% 13.25/7.67 @ ^ [X2: int] :
% 13.25/7.67 ( ( G @ X2 )
% 13.25/7.67 = zero_zero_nat ) ) ) )
% 13.25/7.67 = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6553_numeral__code_I3_J,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numera6690914467698888265omplex @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) @ one_one_complex ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6554_numeral__code_I3_J,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_real @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) @ one_one_real ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6555_numeral__code_I3_J,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_rat @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N ) @ ( numeral_numeral_rat @ N ) ) @ one_one_rat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6556_numeral__code_I3_J,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_nat @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6557_numeral__code_I3_J,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( numeral_numeral_int @ ( bit1 @ N ) )
% 13.25/7.67 = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) @ one_one_int ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6558_power__numeral__odd,axiom,
% 13.25/7.67 ! [Z: complex,W: num] :
% 13.25/7.67 ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 13.25/7.67 = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6559_power__numeral__odd,axiom,
% 13.25/7.67 ! [Z: real,W: num] :
% 13.25/7.67 ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 13.25/7.67 = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6560_power__numeral__odd,axiom,
% 13.25/7.67 ! [Z: rat,W: num] :
% 13.25/7.67 ( ( power_power_rat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 13.25/7.67 = ( times_times_rat @ ( times_times_rat @ Z @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6561_power__numeral__odd,axiom,
% 13.25/7.67 ! [Z: nat,W: num] :
% 13.25/7.67 ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 13.25/7.67 = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6562_power__numeral__odd,axiom,
% 13.25/7.67 ! [Z: int,W: num] :
% 13.25/7.67 ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
% 13.25/7.67 = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6563_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_real,I: real,F: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( member_real @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6564_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_int,I: int,F: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ I5 )
% 13.25/7.67 => ( ( member_int @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6565_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_complex,I: complex,F: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( member_complex @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6566_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_real,I: real,F: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( member_real @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6567_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_nat,I: nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ I5 )
% 13.25/7.67 => ( ( member_nat @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: nat] :
% 13.25/7.67 ( ( member_nat @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6568_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_int,I: int,F: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ I5 )
% 13.25/7.67 => ( ( member_int @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6569_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_complex,I: complex,F: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( member_complex @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6570_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_real,I: real,F: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( member_real @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6571_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_int,I: int,F: int > nat] :
% 13.25/7.67 ( ( finite_finite_int @ I5 )
% 13.25/7.67 => ( ( member_int @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6572_sum__pos2,axiom,
% 13.25/7.67 ! [I5: set_complex,I: complex,F: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( member_complex @ I @ I5 )
% 13.25/7.67 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6573_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_complex,F: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_complex )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6574_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_int,F: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_int )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6575_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_real,F: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_real )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6576_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_complex,F: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_complex )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6577_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_nat )
% 13.25/7.67 => ( ! [I2: nat] :
% 13.25/7.67 ( ( member_nat @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6578_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_int,F: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_int )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6579_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_real,F: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_real )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6580_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_complex,F: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_complex )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6581_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_int,F: int > nat] :
% 13.25/7.67 ( ( finite_finite_int @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_int )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6582_sum__pos,axiom,
% 13.25/7.67 ! [I5: set_real,F: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ I5 )
% 13.25/7.67 => ( ( I5 != bot_bot_set_real )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) )
% 13.25/7.67 => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6583_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,G: real > complex,H2: real > complex] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5754745047067104278omplex @ G @ T3 )
% 13.25/7.67 = ( groups5754745047067104278omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6584_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,G: real > real,H2: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups8097168146408367636l_real @ G @ T3 )
% 13.25/7.67 = ( groups8097168146408367636l_real @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6585_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > real,H2: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6586_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,G: real > rat,H2: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 13.25/7.67 = ( groups1300246762558778688al_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6587_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > rat,H2: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6588_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,G: real > nat,H2: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 13.25/7.67 = ( groups1935376822645274424al_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6589_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > nat,H2: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6590_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,G: real > int,H2: real > int] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups1932886352136224148al_int @ G @ T3 )
% 13.25/7.67 = ( groups1932886352136224148al_int @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6591_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > int,H2: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6592_sum_Omono__neutral__cong__right,axiom,
% 13.25/7.67 ! [T3: set_nat,S2: set_nat,G: nat > complex,H2: nat > complex] :
% 13.25/7.67 ( ( finite_finite_nat @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 13.25/7.67 = ( groups2073611262835488442omplex @ H2 @ S2 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6593_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,H2: real > complex,G: real > complex] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5754745047067104278omplex @ G @ S2 )
% 13.25/7.67 = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6594_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,H2: real > real,G: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups8097168146408367636l_real @ G @ S2 )
% 13.25/7.67 = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6595_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,H2: complex > real,G: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5808333547571424918x_real @ G @ S2 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6596_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,H2: real > rat,G: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups1300246762558778688al_rat @ G @ S2 )
% 13.25/7.67 = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6597_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6598_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,H2: real > nat,G: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups1935376822645274424al_nat @ G @ S2 )
% 13.25/7.67 = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6599_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,H2: complex > nat,G: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6600_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_real,S2: set_real,H2: real > int,G: real > int] :
% 13.25/7.67 ( ( finite_finite_real @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ ( minus_minus_set_real @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups1932886352136224148al_int @ G @ S2 )
% 13.25/7.67 = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6601_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,H2: complex > int,G: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6602_sum_Omono__neutral__cong__left,axiom,
% 13.25/7.67 ! [T3: set_nat,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 13.25/7.67 ( ( finite_finite_nat @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 13.25/7.67 => ( ( H2 @ X3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ S2 )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = ( H2 @ X3 ) ) )
% 13.25/7.67 => ( ( groups2073611262835488442omplex @ G @ S2 )
% 13.25/7.67 = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6603_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ( groups5808333547571424918x_real @ G @ T3 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6604_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6605_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6606_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6607_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 13.25/7.67 ( ( finite_finite_nat @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ( groups2073611262835488442omplex @ G @ T3 )
% 13.25/7.67 = ( groups2073611262835488442omplex @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6608_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 13.25/7.67 = ( groups2906978787729119204at_rat @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6609_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 13.25/7.67 ( ( finite_finite_nat @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ( groups3539618377306564664at_int @ G @ T3 )
% 13.25/7.67 = ( groups3539618377306564664at_int @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6610_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_int,S2: set_int,G: int > complex] :
% 13.25/7.67 ( ( finite_finite_int @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ( groups3049146728041665814omplex @ G @ T3 )
% 13.25/7.67 = ( groups3049146728041665814omplex @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6611_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_int,S2: set_int,G: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ( groups8778361861064173332t_real @ G @ T3 )
% 13.25/7.67 = ( groups8778361861064173332t_real @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6612_sum_Omono__neutral__right,axiom,
% 13.25/7.67 ! [T3: set_int,S2: set_int,G: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 13.25/7.67 = ( groups3906332499630173760nt_rat @ G @ S2 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6613_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ( groups5808333547571424918x_real @ G @ S2 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6614_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( groups5058264527183730370ex_rat @ G @ S2 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6615_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ( groups5693394587270226106ex_nat @ G @ S2 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6616_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_complex,S2: set_complex,G: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ T3 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ( groups5690904116761175830ex_int @ G @ S2 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6617_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_nat,S2: set_nat,G: nat > complex] :
% 13.25/7.67 ( ( finite_finite_nat @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ( groups2073611262835488442omplex @ G @ S2 )
% 13.25/7.67 = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6618_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_nat,S2: set_nat,G: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( groups2906978787729119204at_rat @ G @ S2 )
% 13.25/7.67 = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6619_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_nat,S2: set_nat,G: nat > int] :
% 13.25/7.67 ( ( finite_finite_nat @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ( groups3539618377306564664at_int @ G @ S2 )
% 13.25/7.67 = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6620_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_int,S2: set_int,G: int > complex] :
% 13.25/7.67 ( ( finite_finite_int @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ( groups3049146728041665814omplex @ G @ S2 )
% 13.25/7.67 = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6621_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_int,S2: set_int,G: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ( groups8778361861064173332t_real @ G @ S2 )
% 13.25/7.67 = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6622_sum_Omono__neutral__left,axiom,
% 13.25/7.67 ! [T3: set_int,S2: set_int,G: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ T3 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ S2 @ T3 )
% 13.25/7.67 => ( ! [X3: int] :
% 13.25/7.67 ( ( member_int @ X3 @ ( minus_minus_set_int @ T3 @ S2 ) )
% 13.25/7.67 => ( ( G @ X3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( groups3906332499630173760nt_rat @ G @ S2 )
% 13.25/7.67 = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6623_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ( ( groups5754745047067104278omplex @ G @ C4 )
% 13.25/7.67 = ( groups5754745047067104278omplex @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 13.25/7.67 = ( groups5754745047067104278omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6624_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ( ( groups8097168146408367636l_real @ G @ C4 )
% 13.25/7.67 = ( groups8097168146408367636l_real @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 13.25/7.67 = ( groups8097168146408367636l_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6625_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: complex] :
% 13.25/7.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ( ( groups5808333547571424918x_real @ G @ C4 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6626_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
% 13.25/7.67 = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 13.25/7.67 = ( groups1300246762558778688al_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6627_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: complex] :
% 13.25/7.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6628_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
% 13.25/7.67 = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 13.25/7.67 = ( groups1935376822645274424al_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6629_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: complex] :
% 13.25/7.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6630_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > int,H2: real > int] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ( ( groups1932886352136224148al_int @ G @ C4 )
% 13.25/7.67 = ( groups1932886352136224148al_int @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups1932886352136224148al_int @ G @ A2 )
% 13.25/7.67 = ( groups1932886352136224148al_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6631_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: complex] :
% 13.25/7.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6632_sum_Osame__carrierI,axiom,
% 13.25/7.67 ! [C4: set_nat,A2: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 13.25/7.67 ( ( finite_finite_nat @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: nat] :
% 13.25/7.67 ( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [B3: nat] :
% 13.25/7.67 ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ( ( groups2073611262835488442omplex @ G @ C4 )
% 13.25/7.67 = ( groups2073611262835488442omplex @ H2 @ C4 ) )
% 13.25/7.67 => ( ( groups2073611262835488442omplex @ G @ A2 )
% 13.25/7.67 = ( groups2073611262835488442omplex @ H2 @ B5 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6633_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > complex,H2: real > complex] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 13.25/7.67 = ( groups5754745047067104278omplex @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups5754745047067104278omplex @ G @ C4 )
% 13.25/7.67 = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6634_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > real,H2: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 13.25/7.67 = ( groups8097168146408367636l_real @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups8097168146408367636l_real @ G @ C4 )
% 13.25/7.67 = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6635_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > real,H2: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: complex] :
% 13.25/7.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_real ) )
% 13.25/7.67 => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups5808333547571424918x_real @ G @ C4 )
% 13.25/7.67 = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6636_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > rat,H2: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 13.25/7.67 = ( groups1300246762558778688al_rat @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups1300246762558778688al_rat @ G @ C4 )
% 13.25/7.67 = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6637_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > rat,H2: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: complex] :
% 13.25/7.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_rat ) )
% 13.25/7.67 => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 13.25/7.67 = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6638_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > nat,H2: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 13.25/7.67 = ( groups1935376822645274424al_nat @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups1935376822645274424al_nat @ G @ C4 )
% 13.25/7.67 = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6639_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > nat,H2: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: complex] :
% 13.25/7.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_nat ) )
% 13.25/7.67 => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 13.25/7.67 = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6640_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_real,A2: set_real,B5: set_real,G: real > int,H2: real > int] :
% 13.25/7.67 ( ( finite_finite_real @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: real] :
% 13.25/7.67 ( ( member_real @ A5 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ( ( groups1932886352136224148al_int @ G @ A2 )
% 13.25/7.67 = ( groups1932886352136224148al_int @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups1932886352136224148al_int @ G @ C4 )
% 13.25/7.67 = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6641_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_complex,A2: set_complex,B5: set_complex,G: complex > int,H2: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: complex] :
% 13.25/7.67 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_int ) )
% 13.25/7.67 => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups5690904116761175830ex_int @ G @ C4 )
% 13.25/7.67 = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6642_sum_Osame__carrier,axiom,
% 13.25/7.67 ! [C4: set_nat,A2: set_nat,B5: set_nat,G: nat > complex,H2: nat > complex] :
% 13.25/7.67 ( ( finite_finite_nat @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ B5 @ C4 )
% 13.25/7.67 => ( ! [A5: nat] :
% 13.25/7.67 ( ( member_nat @ A5 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 13.25/7.67 => ( ( G @ A5 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ! [B3: nat] :
% 13.25/7.67 ( ( member_nat @ B3 @ ( minus_minus_set_nat @ C4 @ B5 ) )
% 13.25/7.67 => ( ( H2 @ B3 )
% 13.25/7.67 = zero_zero_complex ) )
% 13.25/7.67 => ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 13.25/7.67 = ( groups2073611262835488442omplex @ H2 @ B5 ) )
% 13.25/7.67 = ( ( groups2073611262835488442omplex @ G @ C4 )
% 13.25/7.67 = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6643_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,G: complex > real] :
% 13.25/7.67 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 13.25/7.67 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5808333547571424918x_real @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6644_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,G: complex > rat] :
% 13.25/7.67 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 13.25/7.67 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5058264527183730370ex_rat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6645_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,G: complex > nat] :
% 13.25/7.67 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 13.25/7.67 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5693394587270226106ex_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6646_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,G: complex > int] :
% 13.25/7.67 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.67 => ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 13.25/7.67 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups5690904116761175830ex_int @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6647_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_nat,A2: set_nat,G: nat > rat] :
% 13.25/7.67 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.67 => ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 13.25/7.67 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups2906978787729119204at_rat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6648_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_nat,A2: set_nat,G: nat > int] :
% 13.25/7.67 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.67 => ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ( groups3539618377306564664at_int @ G @ A2 )
% 13.25/7.67 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups3539618377306564664at_int @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6649_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_int,A2: set_int,G: int > real] :
% 13.25/7.67 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.67 => ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( groups8778361861064173332t_real @ G @ A2 )
% 13.25/7.67 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups8778361861064173332t_real @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6650_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_int,A2: set_int,G: int > rat] :
% 13.25/7.67 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.67 => ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 13.25/7.67 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups3906332499630173760nt_rat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6651_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_int,A2: set_int,G: int > nat] :
% 13.25/7.67 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.67 => ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 13.25/7.67 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups4541462559716669496nt_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6652_sum_Osubset__diff,axiom,
% 13.25/7.67 ! [B5: set_int,A2: set_int,G: int > int] :
% 13.25/7.67 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.67 => ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 13.25/7.67 = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups4538972089207619220nt_int @ G @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6653_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_complex,B5: set_complex,F: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.67 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6654_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_complex,B5: set_complex,F: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.67 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6655_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_complex,B5: set_complex,F: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.67 => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6656_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_nat,B5: set_nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.67 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6657_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_nat,B5: set_nat,F: nat > int] :
% 13.25/7.67 ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.67 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6658_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_int,B5: set_int,F: int > real] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.67 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6659_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_int,B5: set_int,F: int > rat] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.67 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6660_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_int,B5: set_int,F: int > int] :
% 13.25/7.67 ( ( finite_finite_int @ A2 )
% 13.25/7.67 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.67 => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6661_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_complex,B5: set_complex,F: complex > complex] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.67 => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6662_sum__diff,axiom,
% 13.25/7.67 ! [A2: set_nat,B5: set_nat,F: nat > real] :
% 13.25/7.67 ( ( finite_finite_nat @ A2 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.67 => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.67 = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6663_numeral__Bit1__div__2,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.67 = ( numeral_numeral_nat @ N ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6664_numeral__Bit1__div__2,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.67 = ( numeral_numeral_int @ N ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6665_odd__numeral,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6666_odd__numeral,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6667_odd__numeral,axiom,
% 13.25/7.67 ! [N: num] :
% 13.25/7.67 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6668_cong__exp__iff__simps_I3_J,axiom,
% 13.25/7.67 ! [N: num,Q2: num] :
% 13.25/7.67 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != zero_zero_nat ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6669_cong__exp__iff__simps_I3_J,axiom,
% 13.25/7.67 ! [N: num,Q2: num] :
% 13.25/7.67 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != zero_zero_int ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6670_cong__exp__iff__simps_I3_J,axiom,
% 13.25/7.67 ! [N: num,Q2: num] :
% 13.25/7.67 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 != zero_z3403309356797280102nteger ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6671_power3__eq__cube,axiom,
% 13.25/7.67 ! [A: complex] :
% 13.25/7.67 ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 13.25/7.67 = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6672_power3__eq__cube,axiom,
% 13.25/7.67 ! [A: real] :
% 13.25/7.67 ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 13.25/7.67 = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6673_power3__eq__cube,axiom,
% 13.25/7.67 ! [A: rat] :
% 13.25/7.67 ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 13.25/7.67 = ( times_times_rat @ ( times_times_rat @ A @ A ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6674_power3__eq__cube,axiom,
% 13.25/7.67 ! [A: nat] :
% 13.25/7.67 ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 13.25/7.67 = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6675_power3__eq__cube,axiom,
% 13.25/7.67 ! [A: int] :
% 13.25/7.67 ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 13.25/7.67 = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6676_numeral__3__eq__3,axiom,
% 13.25/7.67 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 13.25/7.67 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6677_Suc3__eq__add__3,axiom,
% 13.25/7.67 ! [N: nat] :
% 13.25/7.67 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 13.25/7.67 = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6678_zdvd__mult__cancel1,axiom,
% 13.25/7.67 ! [M: int,N: int] :
% 13.25/7.67 ( ( M != zero_zero_int )
% 13.25/7.67 => ( ( dvd_dvd_int @ ( times_times_int @ M @ N ) @ M )
% 13.25/7.67 = ( ( abs_abs_int @ N )
% 13.25/7.67 = one_one_int ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6679_take__bit__Suc__bit0,axiom,
% 13.25/7.67 ! [N: nat,K: num] :
% 13.25/7.67 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 13.25/7.67 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6680_take__bit__Suc__bit0,axiom,
% 13.25/7.67 ! [N: nat,K: num] :
% 13.25/7.67 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 13.25/7.67 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6681_take__bit__eq__mod,axiom,
% 13.25/7.67 ( bit_se1745604003318907178nteger
% 13.25/7.67 = ( ^ [N2: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6682_take__bit__eq__mod,axiom,
% 13.25/7.67 ( bit_se2923211474154528505it_int
% 13.25/7.67 = ( ^ [N2: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6683_take__bit__eq__mod,axiom,
% 13.25/7.67 ( bit_se2925701944663578781it_nat
% 13.25/7.67 = ( ^ [N2: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6684_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_real,A2: set_real,F: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6685_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,F: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6686_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_real,A2: set_real,F: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6687_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,F: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6688_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_nat,A2: set_nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: nat] :
% 13.25/7.67 ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6689_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_real,A2: set_real,F: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6690_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,F: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6691_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_real,A2: set_real,F: real > int] :
% 13.25/7.67 ( ( finite_finite_real @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: real] :
% 13.25/7.67 ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6692_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,F: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: complex] :
% 13.25/7.67 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6693_sum__mono2,axiom,
% 13.25/7.67 ! [B5: set_nat,A2: set_nat,F: nat > int] :
% 13.25/7.67 ( ( finite_finite_nat @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.67 => ( ! [B3: nat] :
% 13.25/7.67 ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A2 ) )
% 13.25/7.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ B3 ) ) )
% 13.25/7.67 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6694_take__bit__nat__eq__self,axiom,
% 13.25/7.67 ! [M: nat,N: nat] :
% 13.25/7.67 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.67 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 13.25/7.67 = M ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6695_take__bit__nat__less__exp,axiom,
% 13.25/7.67 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6696_take__bit__nat__eq__self__iff,axiom,
% 13.25/7.67 ! [N: nat,M: nat] :
% 13.25/7.67 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 13.25/7.67 = M )
% 13.25/7.67 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6697_num_Osize_I6_J,axiom,
% 13.25/7.67 ! [X32: num] :
% 13.25/7.67 ( ( size_size_num @ ( bit1 @ X32 ) )
% 13.25/7.67 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6698_take__bit__nat__def,axiom,
% 13.25/7.67 ( bit_se2925701944663578781it_nat
% 13.25/7.67 = ( ^ [N2: nat,M3: nat] : ( modulo_modulo_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6699_take__bit__Suc__minus__bit0,axiom,
% 13.25/7.67 ! [N: nat,K: num] :
% 13.25/7.67 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 13.25/7.67 = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6700_take__bit__int__less__exp,axiom,
% 13.25/7.67 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6701_take__bit__int__def,axiom,
% 13.25/7.67 ( bit_se2923211474154528505it_int
% 13.25/7.67 = ( ^ [N2: nat,K2: int] : ( modulo_modulo_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6702_cong__exp__iff__simps_I11_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num] :
% 13.25/7.67 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q2 ) )
% 13.25/7.67 = zero_zero_nat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6703_cong__exp__iff__simps_I11_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num] :
% 13.25/7.67 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q2 ) )
% 13.25/7.67 = zero_zero_int ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6704_cong__exp__iff__simps_I11_J,axiom,
% 13.25/7.67 ! [M: num,Q2: num] :
% 13.25/7.67 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 13.25/7.67 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6705_cong__exp__iff__simps_I7_J,axiom,
% 13.25/7.67 ! [Q2: num,N: num] :
% 13.25/7.67 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
% 13.25/7.67 = zero_zero_nat ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6706_cong__exp__iff__simps_I7_J,axiom,
% 13.25/7.67 ! [Q2: num,N: num] :
% 13.25/7.67 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
% 13.25/7.67 = zero_zero_int ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6707_cong__exp__iff__simps_I7_J,axiom,
% 13.25/7.67 ! [Q2: num,N: num] :
% 13.25/7.67 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) )
% 13.25/7.67 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q2 ) ) ) )
% 13.25/7.67 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q2 ) )
% 13.25/7.67 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6708_even__add__abs__iff,axiom,
% 13.25/7.67 ! [K: int,L2: int] :
% 13.25/7.67 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L2 ) ) )
% 13.25/7.67 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6709_even__abs__add__iff,axiom,
% 13.25/7.67 ! [K: int,L2: int] :
% 13.25/7.67 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L2 ) )
% 13.25/7.67 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L2 ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6710_Suc__div__eq__add3__div,axiom,
% 13.25/7.67 ! [M: nat,N: nat] :
% 13.25/7.67 ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 13.25/7.67 = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6711_Suc__mod__eq__add3__mod,axiom,
% 13.25/7.67 ! [M: nat,N: nat] :
% 13.25/7.67 ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
% 13.25/7.67 = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6712_take__bit__eq__0__iff,axiom,
% 13.25/7.67 ! [N: nat,A: code_integer] :
% 13.25/7.67 ( ( ( bit_se1745604003318907178nteger @ N @ A )
% 13.25/7.67 = zero_z3403309356797280102nteger )
% 13.25/7.67 = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6713_take__bit__eq__0__iff,axiom,
% 13.25/7.67 ! [N: nat,A: int] :
% 13.25/7.67 ( ( ( bit_se2923211474154528505it_int @ N @ A )
% 13.25/7.67 = zero_zero_int )
% 13.25/7.67 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6714_take__bit__eq__0__iff,axiom,
% 13.25/7.67 ! [N: nat,A: nat] :
% 13.25/7.67 ( ( ( bit_se2925701944663578781it_nat @ N @ A )
% 13.25/7.67 = zero_zero_nat )
% 13.25/7.67 = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6715_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_real,A2: set_real,B: real,F: real > real] :
% 13.25/7.67 ( ( finite_finite_real @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.67 => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6716_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.67 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6717_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_real,A2: set_real,B: real,F: real > rat] :
% 13.25/7.67 ( ( finite_finite_real @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.67 => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6718_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.67 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6719_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_nat,A2: set_nat,B: nat,F: nat > rat] :
% 13.25/7.67 ( ( finite_finite_nat @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.67 => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6720_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_real,A2: set_real,B: real,F: real > nat] :
% 13.25/7.67 ( ( finite_finite_real @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.67 => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6721_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.67 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6722_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_real,A2: set_real,B: real,F: real > int] :
% 13.25/7.67 ( ( finite_finite_real @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.67 => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: real] :
% 13.25/7.67 ( ( member_real @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6723_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_complex,A2: set_complex,B: complex,F: complex > int] :
% 13.25/7.67 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.67 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.67 => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: complex] :
% 13.25/7.67 ( ( member_complex @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6724_sum__strict__mono2,axiom,
% 13.25/7.67 ! [B5: set_nat,A2: set_nat,B: nat,F: nat > int] :
% 13.25/7.67 ( ( finite_finite_nat @ B5 )
% 13.25/7.67 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.67 => ( ( member_nat @ B @ ( minus_minus_set_nat @ B5 @ A2 ) )
% 13.25/7.67 => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 13.25/7.67 => ( ! [X3: nat] :
% 13.25/7.67 ( ( member_nat @ X3 @ B5 )
% 13.25/7.67 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 13.25/7.67 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B5 ) ) ) ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6725_take__bit__nat__less__self__iff,axiom,
% 13.25/7.67 ! [N: nat,M: nat] :
% 13.25/7.67 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 13.25/7.67 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6726_convex__sum__bound__le,axiom,
% 13.25/7.67 ! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 13.25/7.67 ( ! [I2: nat] :
% 13.25/7.67 ( ( member_nat @ I2 @ I5 )
% 13.25/7.67 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups7501900531339628137nteger @ X @ I5 )
% 13.25/7.67 = one_one_Code_integer )
% 13.25/7.67 => ( ! [I2: nat] :
% 13.25/7.67 ( ( member_nat @ I2 @ I5 )
% 13.25/7.67 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.67 => ( ord_le3102999989581377725nteger
% 13.25/7.67 @ ( abs_abs_Code_integer
% 13.25/7.67 @ ( minus_8373710615458151222nteger
% 13.25/7.67 @ ( groups7501900531339628137nteger
% 13.25/7.67 @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.67 @ I5 )
% 13.25/7.67 @ B ) )
% 13.25/7.67 @ Delta ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6727_convex__sum__bound__le,axiom,
% 13.25/7.67 ! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 13.25/7.67 ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups7713935264441627589nteger @ X @ I5 )
% 13.25/7.67 = one_one_Code_integer )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.67 => ( ord_le3102999989581377725nteger
% 13.25/7.67 @ ( abs_abs_Code_integer
% 13.25/7.67 @ ( minus_8373710615458151222nteger
% 13.25/7.67 @ ( groups7713935264441627589nteger
% 13.25/7.67 @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.67 @ I5 )
% 13.25/7.67 @ B ) )
% 13.25/7.67 @ Delta ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6728_convex__sum__bound__le,axiom,
% 13.25/7.67 ! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 13.25/7.67 ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups7873554091576472773nteger @ X @ I5 )
% 13.25/7.67 = one_one_Code_integer )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.67 => ( ord_le3102999989581377725nteger
% 13.25/7.67 @ ( abs_abs_Code_integer
% 13.25/7.67 @ ( minus_8373710615458151222nteger
% 13.25/7.67 @ ( groups7873554091576472773nteger
% 13.25/7.67 @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.67 @ I5 )
% 13.25/7.67 @ B ) )
% 13.25/7.67 @ Delta ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6729_convex__sum__bound__le,axiom,
% 13.25/7.67 ! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
% 13.25/7.67 ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups6621422865394947399nteger @ X @ I5 )
% 13.25/7.67 = one_one_Code_integer )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.67 => ( ord_le3102999989581377725nteger
% 13.25/7.67 @ ( abs_abs_Code_integer
% 13.25/7.67 @ ( minus_8373710615458151222nteger
% 13.25/7.67 @ ( groups6621422865394947399nteger
% 13.25/7.67 @ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.67 @ I5 )
% 13.25/7.67 @ B ) )
% 13.25/7.67 @ Delta ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6730_convex__sum__bound__le,axiom,
% 13.25/7.67 ! [I5: set_real,X: real > real,A: real > real,B: real,Delta: real] :
% 13.25/7.67 ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups8097168146408367636l_real @ X @ I5 )
% 13.25/7.67 = one_one_real )
% 13.25/7.67 => ( ! [I2: real] :
% 13.25/7.67 ( ( member_real @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.67 => ( ord_less_eq_real
% 13.25/7.67 @ ( abs_abs_real
% 13.25/7.67 @ ( minus_minus_real
% 13.25/7.67 @ ( groups8097168146408367636l_real
% 13.25/7.67 @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.67 @ I5 )
% 13.25/7.67 @ B ) )
% 13.25/7.67 @ Delta ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6731_convex__sum__bound__le,axiom,
% 13.25/7.67 ! [I5: set_int,X: int > real,A: int > real,B: real,Delta: real] :
% 13.25/7.67 ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups8778361861064173332t_real @ X @ I5 )
% 13.25/7.67 = one_one_real )
% 13.25/7.67 => ( ! [I2: int] :
% 13.25/7.67 ( ( member_int @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.67 => ( ord_less_eq_real
% 13.25/7.67 @ ( abs_abs_real
% 13.25/7.67 @ ( minus_minus_real
% 13.25/7.67 @ ( groups8778361861064173332t_real
% 13.25/7.67 @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.67 @ I5 )
% 13.25/7.67 @ B ) )
% 13.25/7.67 @ Delta ) ) ) ) ).
% 13.25/7.67
% 13.25/7.67 thf(fact_6732_convex__sum__bound__le,axiom,
% 13.25/7.67 ! [I5: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
% 13.25/7.67 ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ zero_zero_real @ ( X @ I2 ) ) )
% 13.25/7.67 => ( ( ( groups5808333547571424918x_real @ X @ I5 )
% 13.25/7.67 = one_one_real )
% 13.25/7.67 => ( ! [I2: complex] :
% 13.25/7.67 ( ( member_complex @ I2 @ I5 )
% 13.25/7.67 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.67 => ( ord_less_eq_real
% 13.25/7.67 @ ( abs_abs_real
% 13.25/7.67 @ ( minus_minus_real
% 13.25/7.67 @ ( groups5808333547571424918x_real
% 13.25/7.68 @ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.68 @ I5 )
% 13.25/7.68 @ B ) )
% 13.25/7.68 @ Delta ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6733_convex__sum__bound__le,axiom,
% 13.25/7.68 ! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 13.25/7.68 ( ! [I2: nat] :
% 13.25/7.68 ( ( member_nat @ I2 @ I5 )
% 13.25/7.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 13.25/7.68 => ( ( ( groups2906978787729119204at_rat @ X @ I5 )
% 13.25/7.68 = one_one_rat )
% 13.25/7.68 => ( ! [I2: nat] :
% 13.25/7.68 ( ( member_nat @ I2 @ I5 )
% 13.25/7.68 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.68 => ( ord_less_eq_rat
% 13.25/7.68 @ ( abs_abs_rat
% 13.25/7.68 @ ( minus_minus_rat
% 13.25/7.68 @ ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.68 @ I5 )
% 13.25/7.68 @ B ) )
% 13.25/7.68 @ Delta ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6734_convex__sum__bound__le,axiom,
% 13.25/7.68 ! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
% 13.25/7.68 ( ! [I2: real] :
% 13.25/7.68 ( ( member_real @ I2 @ I5 )
% 13.25/7.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 13.25/7.68 => ( ( ( groups1300246762558778688al_rat @ X @ I5 )
% 13.25/7.68 = one_one_rat )
% 13.25/7.68 => ( ! [I2: real] :
% 13.25/7.68 ( ( member_real @ I2 @ I5 )
% 13.25/7.68 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.68 => ( ord_less_eq_rat
% 13.25/7.68 @ ( abs_abs_rat
% 13.25/7.68 @ ( minus_minus_rat
% 13.25/7.68 @ ( groups1300246762558778688al_rat
% 13.25/7.68 @ ^ [I4: real] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.68 @ I5 )
% 13.25/7.68 @ B ) )
% 13.25/7.68 @ Delta ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6735_convex__sum__bound__le,axiom,
% 13.25/7.68 ! [I5: set_int,X: int > rat,A: int > rat,B: rat,Delta: rat] :
% 13.25/7.68 ( ! [I2: int] :
% 13.25/7.68 ( ( member_int @ I2 @ I5 )
% 13.25/7.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I2 ) ) )
% 13.25/7.68 => ( ( ( groups3906332499630173760nt_rat @ X @ I5 )
% 13.25/7.68 = one_one_rat )
% 13.25/7.68 => ( ! [I2: int] :
% 13.25/7.68 ( ( member_int @ I2 @ I5 )
% 13.25/7.68 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I2 ) @ B ) ) @ Delta ) )
% 13.25/7.68 => ( ord_less_eq_rat
% 13.25/7.68 @ ( abs_abs_rat
% 13.25/7.68 @ ( minus_minus_rat
% 13.25/7.68 @ ( groups3906332499630173760nt_rat
% 13.25/7.68 @ ^ [I4: int] : ( times_times_rat @ ( A @ I4 ) @ ( X @ I4 ) )
% 13.25/7.68 @ I5 )
% 13.25/7.68 @ B ) )
% 13.25/7.68 @ Delta ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6736_take__bit__int__less__self__iff,axiom,
% 13.25/7.68 ! [N: nat,K: int] :
% 13.25/7.68 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6737_take__bit__int__greater__eq__self__iff,axiom,
% 13.25/7.68 ! [K: int,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 13.25/7.68 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6738_nat__intermed__int__val,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > int,K: int] :
% 13.25/7.68 ( ! [I2: nat] :
% 13.25/7.68 ( ( ( ord_less_eq_nat @ M @ I2 )
% 13.25/7.68 & ( ord_less_nat @ I2 @ N ) )
% 13.25/7.68 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 13.25/7.68 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 13.25/7.68 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 13.25/7.68 => ? [I2: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ I2 )
% 13.25/7.68 & ( ord_less_eq_nat @ I2 @ N )
% 13.25/7.68 & ( ( F @ I2 )
% 13.25/7.68 = K ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6739_mod__exhaust__less__4,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.68 = zero_zero_nat )
% 13.25/7.68 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.68 = one_one_nat )
% 13.25/7.68 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.68 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.68 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.68 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6740_incr__lemma,axiom,
% 13.25/7.68 ! [D: int,Z: int,X: int] :
% 13.25/7.68 ( ( ord_less_int @ zero_zero_int @ D )
% 13.25/7.68 => ( 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 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6741_decr__lemma,axiom,
% 13.25/7.68 ! [D: int,X: int,Z: int] :
% 13.25/7.68 ( ( ord_less_int @ zero_zero_int @ D )
% 13.25/7.68 => ( 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 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6742_take__bit__int__eq__self,axiom,
% 13.25/7.68 ! [K: int,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.68 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.68 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 13.25/7.68 = K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6743_take__bit__int__eq__self__iff,axiom,
% 13.25/7.68 ! [N: nat,K: int] :
% 13.25/7.68 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 13.25/7.68 = K )
% 13.25/7.68 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.68 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6744_take__bit__incr__eq,axiom,
% 13.25/7.68 ! [N: nat,K: int] :
% 13.25/7.68 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 13.25/7.68 != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
% 13.25/7.68 => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 13.25/7.68 = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6745_nat__ivt__aux,axiom,
% 13.25/7.68 ! [N: nat,F: nat > int,K: int] :
% 13.25/7.68 ( ! [I2: nat] :
% 13.25/7.68 ( ( ord_less_nat @ I2 @ N )
% 13.25/7.68 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 13.25/7.68 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 13.25/7.68 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 13.25/7.68 => ? [I2: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ I2 @ N )
% 13.25/7.68 & ( ( F @ I2 )
% 13.25/7.68 = K ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6746_take__bit__Suc__minus__1__eq,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.68 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6747_take__bit__Suc__minus__1__eq,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.68 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6748_take__bit__numeral__minus__1__eq,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.68 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6749_take__bit__numeral__minus__1__eq,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.68 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6750_take__bit__Suc,axiom,
% 13.25/7.68 ! [N: nat,A: code_integer] :
% 13.25/7.68 ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
% 13.25/7.68 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6751_take__bit__Suc,axiom,
% 13.25/7.68 ! [N: nat,A: int] :
% 13.25/7.68 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
% 13.25/7.68 = ( 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 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6752_take__bit__Suc,axiom,
% 13.25/7.68 ! [N: nat,A: nat] :
% 13.25/7.68 ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
% 13.25/7.68 = ( 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 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6753_take__bit__int__less__eq,axiom,
% 13.25/7.68 ! [N: nat,K: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 13.25/7.68 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.68 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6754_take__bit__int__greater__eq,axiom,
% 13.25/7.68 ! [K: int,N: nat] :
% 13.25/7.68 ( ( ord_less_int @ K @ zero_zero_int )
% 13.25/7.68 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6755_take__bit__minus__small__eq,axiom,
% 13.25/7.68 ! [K: int,N: nat] :
% 13.25/7.68 ( ( ord_less_int @ zero_zero_int @ K )
% 13.25/7.68 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.68 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 13.25/7.68 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6756_signed__take__bit__eq__take__bit__shift,axiom,
% 13.25/7.68 ( bit_ri631733984087533419it_int
% 13.25/7.68 = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6757_nat0__intermed__int__val,axiom,
% 13.25/7.68 ! [N: nat,F: nat > int,K: int] :
% 13.25/7.68 ( ! [I2: nat] :
% 13.25/7.68 ( ( ord_less_nat @ I2 @ N )
% 13.25/7.68 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 13.25/7.68 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 13.25/7.68 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 13.25/7.68 => ? [I2: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ I2 @ N )
% 13.25/7.68 & ( ( F @ I2 )
% 13.25/7.68 = K ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6758_stable__imp__take__bit__eq,axiom,
% 13.25/7.68 ! [A: code_integer,N: nat] :
% 13.25/7.68 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.68 = A )
% 13.25/7.68 => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.68 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 13.25/7.68 = zero_z3403309356797280102nteger ) )
% 13.25/7.68 & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.68 => ( ( bit_se1745604003318907178nteger @ N @ A )
% 13.25/7.68 = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6759_stable__imp__take__bit__eq,axiom,
% 13.25/7.68 ! [A: int,N: nat] :
% 13.25/7.68 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.68 = A )
% 13.25/7.68 => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.68 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 13.25/7.68 = zero_zero_int ) )
% 13.25/7.68 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.68 => ( ( bit_se2923211474154528505it_int @ N @ A )
% 13.25/7.68 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6760_stable__imp__take__bit__eq,axiom,
% 13.25/7.68 ! [A: nat,N: nat] :
% 13.25/7.68 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.68 = A )
% 13.25/7.68 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.68 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 13.25/7.68 = zero_zero_nat ) )
% 13.25/7.68 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.68 => ( ( bit_se2925701944663578781it_nat @ N @ A )
% 13.25/7.68 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6761_divmod__step__nat__def,axiom,
% 13.25/7.68 ( unique5026877609467782581ep_nat
% 13.25/7.68 = ( ^ [L: num] :
% 13.25/7.68 ( produc2626176000494625587at_nat
% 13.25/7.68 @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6762_arctan__add,axiom,
% 13.25/7.68 ! [X: real,Y3: real] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.68 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 13.25/7.68 => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y3 ) )
% 13.25/7.68 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6763_divmod__step__int__def,axiom,
% 13.25/7.68 ( unique5024387138958732305ep_int
% 13.25/7.68 = ( ^ [L: num] :
% 13.25/7.68 ( produc4245557441103728435nt_int
% 13.25/7.68 @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6764_odd__mod__4__div__2,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.68 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 13.25/7.68 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6765_divmod__algorithm__code_I6_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.68 = ( produc4245557441103728435nt_int
% 13.25/7.68 @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 13.25/7.68 @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6766_divmod__algorithm__code_I6_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.68 = ( produc2626176000494625587at_nat
% 13.25/7.68 @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 13.25/7.68 @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6767_divmod__algorithm__code_I6_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.68 = ( produc6916734918728496179nteger
% 13.25/7.68 @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 13.25/7.68 @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6768_signed__take__bit__numeral__minus__bit1,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6769_dbl__dec__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.68 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6770_dbl__dec__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6771_dbl__dec__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.68 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6772_dbl__dec__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.68 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6773_dbl__dec__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.68 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6774_divmod__algorithm__code_I7_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( ( ord_less_eq_num @ M @ N )
% 13.25/7.68 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_eq_num @ M @ N )
% 13.25/7.68 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6775_divmod__algorithm__code_I7_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( ( ord_less_eq_num @ M @ N )
% 13.25/7.68 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_eq_num @ M @ N )
% 13.25/7.68 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6776_divmod__algorithm__code_I7_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( ( ord_less_eq_num @ M @ N )
% 13.25/7.68 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_eq_num @ M @ N )
% 13.25/7.68 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6777_divmod__algorithm__code_I8_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( ( ord_less_num @ M @ N )
% 13.25/7.68 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_num @ M @ N )
% 13.25/7.68 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6778_divmod__algorithm__code_I8_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( ( ord_less_num @ M @ N )
% 13.25/7.68 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_num @ M @ N )
% 13.25/7.68 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6779_divmod__algorithm__code_I8_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( ( ord_less_num @ M @ N )
% 13.25/7.68 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_num @ M @ N )
% 13.25/7.68 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.68 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6780_signed__take__bit__numeral__bit1,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6781_dbl__dec__simps_I3_J,axiom,
% 13.25/7.68 ( ( neg_nu6511756317524482435omplex @ one_one_complex )
% 13.25/7.68 = one_one_complex ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6782_dbl__dec__simps_I3_J,axiom,
% 13.25/7.68 ( ( neg_nu6075765906172075777c_real @ one_one_real )
% 13.25/7.68 = one_one_real ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6783_dbl__dec__simps_I3_J,axiom,
% 13.25/7.68 ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
% 13.25/7.68 = one_one_rat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6784_dbl__dec__simps_I3_J,axiom,
% 13.25/7.68 ( ( neg_nu3811975205180677377ec_int @ one_one_int )
% 13.25/7.68 = one_one_int ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6785_pred__numeral__simps_I1_J,axiom,
% 13.25/7.68 ( ( pred_numeral @ one )
% 13.25/7.68 = zero_zero_nat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6786_eq__numeral__Suc,axiom,
% 13.25/7.68 ! [K: num,N: nat] :
% 13.25/7.68 ( ( ( numeral_numeral_nat @ K )
% 13.25/7.68 = ( suc @ N ) )
% 13.25/7.68 = ( ( pred_numeral @ K )
% 13.25/7.68 = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6787_Suc__eq__numeral,axiom,
% 13.25/7.68 ! [N: nat,K: num] :
% 13.25/7.68 ( ( ( suc @ N )
% 13.25/7.68 = ( numeral_numeral_nat @ K ) )
% 13.25/7.68 = ( N
% 13.25/7.68 = ( pred_numeral @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6788_less__numeral__Suc,axiom,
% 13.25/7.68 ! [K: num,N: nat] :
% 13.25/7.68 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 13.25/7.68 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6789_less__Suc__numeral,axiom,
% 13.25/7.68 ! [N: nat,K: num] :
% 13.25/7.68 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 13.25/7.68 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6790_pred__numeral__simps_I3_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( pred_numeral @ ( bit1 @ K ) )
% 13.25/7.68 = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6791_le__Suc__numeral,axiom,
% 13.25/7.68 ! [N: nat,K: num] :
% 13.25/7.68 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 13.25/7.68 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6792_le__numeral__Suc,axiom,
% 13.25/7.68 ! [K: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6793_diff__numeral__Suc,axiom,
% 13.25/7.68 ! [K: num,N: nat] :
% 13.25/7.68 ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 13.25/7.68 = ( minus_minus_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6794_diff__Suc__numeral,axiom,
% 13.25/7.68 ! [N: nat,K: num] :
% 13.25/7.68 ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 13.25/7.68 = ( minus_minus_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6795_max__Suc__numeral,axiom,
% 13.25/7.68 ! [N: nat,K: num] :
% 13.25/7.68 ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 13.25/7.68 = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6796_max__numeral__Suc,axiom,
% 13.25/7.68 ! [K: num,N: nat] :
% 13.25/7.68 ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 13.25/7.68 = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6797_dbl__dec__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 13.25/7.68 = ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6798_dbl__dec__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 13.25/7.68 = ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6799_dbl__dec__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 13.25/7.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6800_dbl__dec__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 13.25/7.68 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6801_dbl__dec__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 13.25/7.68 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6802_numeral__div__minus__numeral,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6803_minus__numeral__div__numeral,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6804_sum_Ocl__ivl__Suc,axiom,
% 13.25/7.68 ! [N: nat,M: nat,G: nat > complex] :
% 13.25/7.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = zero_zero_complex ) )
% 13.25/7.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6805_sum_Ocl__ivl__Suc,axiom,
% 13.25/7.68 ! [N: nat,M: nat,G: nat > rat] :
% 13.25/7.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = zero_zero_rat ) )
% 13.25/7.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6806_sum_Ocl__ivl__Suc,axiom,
% 13.25/7.68 ! [N: nat,M: nat,G: nat > int] :
% 13.25/7.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = zero_zero_int ) )
% 13.25/7.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6807_sum_Ocl__ivl__Suc,axiom,
% 13.25/7.68 ! [N: nat,M: nat,G: nat > nat] :
% 13.25/7.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = zero_zero_nat ) )
% 13.25/7.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6808_sum_Ocl__ivl__Suc,axiom,
% 13.25/7.68 ! [N: nat,M: nat,G: nat > real] :
% 13.25/7.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = zero_zero_real ) )
% 13.25/7.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.68 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6809_dvd__numeral__simp,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.68 = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6810_dvd__numeral__simp,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6811_dvd__numeral__simp,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 13.25/7.68 = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6812_divmod__algorithm__code_I2_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( unique5052692396658037445od_int @ M @ one )
% 13.25/7.68 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6813_divmod__algorithm__code_I2_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( unique5055182867167087721od_nat @ M @ one )
% 13.25/7.68 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6814_divmod__algorithm__code_I2_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( unique3479559517661332726nteger @ M @ one )
% 13.25/7.68 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6815_sum__zero__power,axiom,
% 13.25/7.68 ! [A2: set_nat,C: nat > complex] :
% 13.25/7.68 ( ( ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups2073611262835488442omplex
% 13.25/7.68 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( C @ zero_zero_nat ) ) )
% 13.25/7.68 & ( ~ ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups2073611262835488442omplex
% 13.25/7.68 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = zero_zero_complex ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6816_sum__zero__power,axiom,
% 13.25/7.68 ! [A2: set_nat,C: nat > rat] :
% 13.25/7.68 ( ( ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( C @ zero_zero_nat ) ) )
% 13.25/7.68 & ( ~ ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = zero_zero_rat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6817_sum__zero__power,axiom,
% 13.25/7.68 ! [A2: set_nat,C: nat > real] :
% 13.25/7.68 ( ( ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( C @ zero_zero_nat ) ) )
% 13.25/7.68 & ( ~ ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = zero_zero_real ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6818_divmod__algorithm__code_I3_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 13.25/7.68 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6819_divmod__algorithm__code_I3_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 13.25/7.68 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6820_divmod__algorithm__code_I3_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 13.25/7.68 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6821_divmod__algorithm__code_I4_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 13.25/7.68 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6822_divmod__algorithm__code_I4_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 13.25/7.68 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6823_divmod__algorithm__code_I4_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 13.25/7.68 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6824_sum__zero__power_H,axiom,
% 13.25/7.68 ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 13.25/7.68 ( ( ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups2073611262835488442omplex
% 13.25/7.68 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 13.25/7.68 & ( ~ ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups2073611262835488442omplex
% 13.25/7.68 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = zero_zero_complex ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6825_sum__zero__power_H,axiom,
% 13.25/7.68 ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 13.25/7.68 ( ( ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 13.25/7.68 & ( ~ ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = zero_zero_rat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6826_sum__zero__power_H,axiom,
% 13.25/7.68 ! [A2: set_nat,C: nat > real,D: nat > real] :
% 13.25/7.68 ( ( ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 13.25/7.68 & ( ~ ( ( finite_finite_nat @ A2 )
% 13.25/7.68 & ( member_nat @ zero_zero_nat @ A2 ) )
% 13.25/7.68 => ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = zero_zero_real ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6827_one__div__minus__numeral,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6828_minus__one__div__numeral,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6829_signed__take__bit__numeral__bit0,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 13.25/7.68 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6830_divmod__algorithm__code_I5_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.25/7.68 = ( produc4245557441103728435nt_int
% 13.25/7.68 @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 13.25/7.68 @ ( unique5052692396658037445od_int @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6831_divmod__algorithm__code_I5_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.25/7.68 = ( produc2626176000494625587at_nat
% 13.25/7.68 @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 13.25/7.68 @ ( unique5055182867167087721od_nat @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6832_divmod__algorithm__code_I5_J,axiom,
% 13.25/7.68 ! [M: num,N: num] :
% 13.25/7.68 ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.25/7.68 = ( produc6916734918728496179nteger
% 13.25/7.68 @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 13.25/7.68 @ ( unique3479559517661332726nteger @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6833_signed__take__bit__numeral__minus__bit0,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 13.25/7.68 = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6834_numeral__eq__Suc,axiom,
% 13.25/7.68 ( numeral_numeral_nat
% 13.25/7.68 = ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6835_sum__subtractf__nat,axiom,
% 13.25/7.68 ! [A2: set_real,G: real > nat,F: real > nat] :
% 13.25/7.68 ( ! [X3: real] :
% 13.25/7.68 ( ( member_real @ X3 @ A2 )
% 13.25/7.68 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 13.25/7.68 => ( ( groups1935376822645274424al_nat
% 13.25/7.68 @ ^ [X2: real] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6836_sum__subtractf__nat,axiom,
% 13.25/7.68 ! [A2: set_int,G: int > nat,F: int > nat] :
% 13.25/7.68 ( ! [X3: int] :
% 13.25/7.68 ( ( member_int @ X3 @ A2 )
% 13.25/7.68 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 13.25/7.68 => ( ( groups4541462559716669496nt_nat
% 13.25/7.68 @ ^ [X2: int] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6837_sum__subtractf__nat,axiom,
% 13.25/7.68 ! [A2: set_complex,G: complex > nat,F: complex > nat] :
% 13.25/7.68 ( ! [X3: complex] :
% 13.25/7.68 ( ( member_complex @ X3 @ A2 )
% 13.25/7.68 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 13.25/7.68 => ( ( groups5693394587270226106ex_nat
% 13.25/7.68 @ ^ [X2: complex] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6838_sum__subtractf__nat,axiom,
% 13.25/7.68 ! [A2: set_Pr8693737435421807431at_nat,G: produc859450856879609959at_nat > nat,F: produc859450856879609959at_nat > nat] :
% 13.25/7.68 ( ! [X3: produc859450856879609959at_nat] :
% 13.25/7.68 ( ( member8206827879206165904at_nat @ X3 @ A2 )
% 13.25/7.68 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 13.25/7.68 => ( ( groups1900718384385340925at_nat
% 13.25/7.68 @ ^ [X2: produc859450856879609959at_nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( minus_minus_nat @ ( groups1900718384385340925at_nat @ F @ A2 ) @ ( groups1900718384385340925at_nat @ G @ A2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6839_sum__subtractf__nat,axiom,
% 13.25/7.68 ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 13.25/7.68 ( ! [X3: nat] :
% 13.25/7.68 ( ( member_nat @ X3 @ A2 )
% 13.25/7.68 => ( ord_less_eq_nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
% 13.25/7.68 => ( ( groups3542108847815614940at_nat
% 13.25/7.68 @ ^ [X2: nat] : ( minus_minus_nat @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.68 @ A2 )
% 13.25/7.68 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6840_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 13.25/7.68 ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 13.25/7.68 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 13.25/7.68 = ( groups3542108847815614940at_nat
% 13.25/7.68 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6841_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 13.25/7.68 ! [G: nat > real,M: nat,K: nat,N: nat] :
% 13.25/7.68 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 13.25/7.68 = ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6842_sum__SucD,axiom,
% 13.25/7.68 ! [F: nat > nat,A2: set_nat,N: nat] :
% 13.25/7.68 ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 13.25/7.68 = ( suc @ N ) )
% 13.25/7.68 => ? [X3: nat] :
% 13.25/7.68 ( ( member_nat @ X3 @ A2 )
% 13.25/7.68 & ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6843_sum__eq__1__iff,axiom,
% 13.25/7.68 ! [A2: set_int,F: int > nat] :
% 13.25/7.68 ( ( finite_finite_int @ A2 )
% 13.25/7.68 => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 13.25/7.68 = one_one_nat )
% 13.25/7.68 = ( ? [X2: int] :
% 13.25/7.68 ( ( member_int @ X2 @ A2 )
% 13.25/7.68 & ( ( F @ X2 )
% 13.25/7.68 = one_one_nat )
% 13.25/7.68 & ! [Y: int] :
% 13.25/7.68 ( ( member_int @ Y @ A2 )
% 13.25/7.68 => ( ( X2 != Y )
% 13.25/7.68 => ( ( F @ Y )
% 13.25/7.68 = zero_zero_nat ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6844_sum__eq__1__iff,axiom,
% 13.25/7.68 ! [A2: set_complex,F: complex > nat] :
% 13.25/7.68 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.68 => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 13.25/7.68 = one_one_nat )
% 13.25/7.68 = ( ? [X2: complex] :
% 13.25/7.68 ( ( member_complex @ X2 @ A2 )
% 13.25/7.68 & ( ( F @ X2 )
% 13.25/7.68 = one_one_nat )
% 13.25/7.68 & ! [Y: complex] :
% 13.25/7.68 ( ( member_complex @ Y @ A2 )
% 13.25/7.68 => ( ( X2 != Y )
% 13.25/7.68 => ( ( F @ Y )
% 13.25/7.68 = zero_zero_nat ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6845_sum__eq__1__iff,axiom,
% 13.25/7.68 ! [A2: set_nat,F: nat > nat] :
% 13.25/7.68 ( ( finite_finite_nat @ A2 )
% 13.25/7.68 => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 13.25/7.68 = one_one_nat )
% 13.25/7.68 = ( ? [X2: nat] :
% 13.25/7.68 ( ( member_nat @ X2 @ A2 )
% 13.25/7.68 & ( ( F @ X2 )
% 13.25/7.68 = one_one_nat )
% 13.25/7.68 & ! [Y: nat] :
% 13.25/7.68 ( ( member_nat @ Y @ A2 )
% 13.25/7.68 => ( ( X2 != Y )
% 13.25/7.68 => ( ( F @ Y )
% 13.25/7.68 = zero_zero_nat ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6846_sum__power__add,axiom,
% 13.25/7.68 ! [X: complex,M: nat,I5: set_nat] :
% 13.25/7.68 ( ( groups2073611262835488442omplex
% 13.25/7.68 @ ^ [I4: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M @ I4 ) )
% 13.25/7.68 @ I5 )
% 13.25/7.68 = ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6847_sum__power__add,axiom,
% 13.25/7.68 ! [X: rat,M: nat,I5: set_nat] :
% 13.25/7.68 ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M @ I4 ) )
% 13.25/7.68 @ I5 )
% 13.25/7.68 = ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6848_sum__power__add,axiom,
% 13.25/7.68 ! [X: int,M: nat,I5: set_nat] :
% 13.25/7.68 ( ( groups3539618377306564664at_int
% 13.25/7.68 @ ^ [I4: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M @ I4 ) )
% 13.25/7.68 @ I5 )
% 13.25/7.68 = ( times_times_int @ ( power_power_int @ X @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6849_sum__power__add,axiom,
% 13.25/7.68 ! [X: real,M: nat,I5: set_nat] :
% 13.25/7.68 ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M @ I4 ) )
% 13.25/7.68 @ I5 )
% 13.25/7.68 = ( times_times_real @ ( power_power_real @ X @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6850_sum_OatLeastAtMost__rev,axiom,
% 13.25/7.68 ! [G: nat > nat,N: nat,M: nat] :
% 13.25/7.68 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 13.25/7.68 = ( groups3542108847815614940at_nat
% 13.25/7.68 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6851_sum_OatLeastAtMost__rev,axiom,
% 13.25/7.68 ! [G: nat > real,N: nat,M: nat] :
% 13.25/7.68 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 13.25/7.68 = ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6852_pred__numeral__def,axiom,
% 13.25/7.68 ( pred_numeral
% 13.25/7.68 = ( ^ [K2: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ one_one_nat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6853_sum__nth__roots,axiom,
% 13.25/7.68 ! [N: nat,C: complex] :
% 13.25/7.68 ( ( ord_less_nat @ one_one_nat @ N )
% 13.25/7.68 => ( ( groups7754918857620584856omplex
% 13.25/7.68 @ ^ [X2: complex] : X2
% 13.25/7.68 @ ( collect_complex
% 13.25/7.68 @ ^ [Z2: complex] :
% 13.25/7.68 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.68 = C ) ) )
% 13.25/7.68 = zero_zero_complex ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6854_sum__roots__unity,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ord_less_nat @ one_one_nat @ N )
% 13.25/7.68 => ( ( groups7754918857620584856omplex
% 13.25/7.68 @ ^ [X2: complex] : X2
% 13.25/7.68 @ ( collect_complex
% 13.25/7.68 @ ^ [Z2: complex] :
% 13.25/7.68 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.68 = one_one_complex ) ) )
% 13.25/7.68 = zero_zero_complex ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6855_sum__diff__nat,axiom,
% 13.25/7.68 ! [B5: set_complex,A2: set_complex,F: complex > nat] :
% 13.25/7.68 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.68 => ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.68 => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B5 ) )
% 13.25/7.68 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6856_sum__diff__nat,axiom,
% 13.25/7.68 ! [B5: set_int,A2: set_int,F: int > nat] :
% 13.25/7.68 ( ( finite_finite_int @ B5 )
% 13.25/7.68 => ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.68 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B5 ) )
% 13.25/7.68 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6857_sum__diff__nat,axiom,
% 13.25/7.68 ! [B5: set_nat,A2: set_nat,F: nat > nat] :
% 13.25/7.68 ( ( finite_finite_nat @ B5 )
% 13.25/7.68 => ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.68 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B5 ) )
% 13.25/7.68 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B5 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6858_sum_OatLeast0__atMost__Suc,axiom,
% 13.25/7.68 ! [G: nat > rat,N: nat] :
% 13.25/7.68 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6859_sum_OatLeast0__atMost__Suc,axiom,
% 13.25/7.68 ! [G: nat > int,N: nat] :
% 13.25/7.68 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6860_sum_OatLeast0__atMost__Suc,axiom,
% 13.25/7.68 ! [G: nat > nat,N: nat] :
% 13.25/7.68 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6861_sum_OatLeast0__atMost__Suc,axiom,
% 13.25/7.68 ! [G: nat > real,N: nat] :
% 13.25/7.68 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6862_sum_OatLeast__Suc__atMost,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > rat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6863_sum_OatLeast__Suc__atMost,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > int] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6864_sum_OatLeast__Suc__atMost,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6865_sum_OatLeast__Suc__atMost,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > real] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6866_sum_Onat__ivl__Suc_H,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > rat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6867_sum_Onat__ivl__Suc_H,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > int] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.68 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6868_sum_Onat__ivl__Suc_H,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.68 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6869_sum_Onat__ivl__Suc_H,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > real] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.68 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6870_sum_OSuc__reindex__ivl,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > rat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_rat @ ( G @ M )
% 13.25/7.68 @ ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6871_sum_OSuc__reindex__ivl,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > int] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( G @ M )
% 13.25/7.68 @ ( groups3539618377306564664at_int
% 13.25/7.68 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6872_sum_OSuc__reindex__ivl,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_nat @ ( G @ M )
% 13.25/7.68 @ ( groups3542108847815614940at_nat
% 13.25/7.68 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6873_sum_OSuc__reindex__ivl,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > real] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
% 13.25/7.68 = ( plus_plus_real @ ( G @ M )
% 13.25/7.68 @ ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6874_sum__Suc__diff,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > rat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6875_sum__Suc__diff,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > int] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.68 => ( ( groups3539618377306564664at_int
% 13.25/7.68 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6876_sum__Suc__diff,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > real] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.68 => ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6877_sum_Oub__add__nat,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > rat,P6: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 13.25/7.68 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6878_sum_Oub__add__nat,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > int,P6: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 13.25/7.68 => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6879_sum_Oub__add__nat,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > nat,P6: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 13.25/7.68 => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 13.25/7.68 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6880_sum_Oub__add__nat,axiom,
% 13.25/7.68 ! [M: nat,N: nat,G: nat > real,P6: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 13.25/7.68 => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 13.25/7.68 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6881_set__encode__def,axiom,
% 13.25/7.68 ( nat_set_encode
% 13.25/7.68 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6882_divmod__int__def,axiom,
% 13.25/7.68 ( unique5052692396658037445od_int
% 13.25/7.68 = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6883_divmod__def,axiom,
% 13.25/7.68 ( unique5052692396658037445od_int
% 13.25/7.68 = ( ^ [M3: num,N2: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6884_divmod__def,axiom,
% 13.25/7.68 ( unique5055182867167087721od_nat
% 13.25/7.68 = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6885_divmod__def,axiom,
% 13.25/7.68 ( unique3479559517661332726nteger
% 13.25/7.68 = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6886_divmod_H__nat__def,axiom,
% 13.25/7.68 ( unique5055182867167087721od_nat
% 13.25/7.68 = ( ^ [M3: num,N2: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6887_dbl__dec__def,axiom,
% 13.25/7.68 ( neg_nu6511756317524482435omplex
% 13.25/7.68 = ( ^ [X2: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6888_dbl__dec__def,axiom,
% 13.25/7.68 ( neg_nu6075765906172075777c_real
% 13.25/7.68 = ( ^ [X2: real] : ( minus_minus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6889_dbl__dec__def,axiom,
% 13.25/7.68 ( neg_nu3179335615603231917ec_rat
% 13.25/7.68 = ( ^ [X2: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6890_dbl__dec__def,axiom,
% 13.25/7.68 ( neg_nu3811975205180677377ec_int
% 13.25/7.68 = ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6891_sum__natinterval__diff,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > complex] :
% 13.25/7.68 ( ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups2073611262835488442omplex
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups2073611262835488442omplex
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_complex @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = zero_zero_complex ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6892_sum__natinterval__diff,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > rat] :
% 13.25/7.68 ( ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = zero_zero_rat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6893_sum__natinterval__diff,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > int] :
% 13.25/7.68 ( ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups3539618377306564664at_int
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups3539618377306564664at_int
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = zero_zero_int ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6894_sum__natinterval__diff,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > real] :
% 13.25/7.68 ( ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 13.25/7.68 & ( ~ ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( plus_plus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = zero_zero_real ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6895_sum__telescope_H_H,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > rat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_rat @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 13.25/7.68 = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6896_sum__telescope_H_H,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > int] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups3539618377306564664at_int
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_int @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 13.25/7.68 = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6897_sum__telescope_H_H,axiom,
% 13.25/7.68 ! [M: nat,N: nat,F: nat > real] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [K2: nat] : ( minus_minus_real @ ( F @ K2 ) @ ( F @ ( minus_minus_nat @ K2 @ one_one_nat ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 13.25/7.68 = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6898_take__bit__numeral__bit0,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 13.25/7.68 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6899_take__bit__numeral__bit0,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 13.25/7.68 = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6900_take__bit__numeral__minus__bit0,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 13.25/7.68 = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6901_mask__eq__sum__exp,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 13.25/7.68 = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.68 @ ( collect_nat
% 13.25/7.68 @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6902_mask__eq__sum__exp,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 13.25/7.68 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.68 @ ( collect_nat
% 13.25/7.68 @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6903_sum__gp__multiplied,axiom,
% 13.25/7.68 ! [M: nat,N: nat,X: complex] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 13.25/7.68 = ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6904_sum__gp__multiplied,axiom,
% 13.25/7.68 ! [M: nat,N: nat,X: rat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 13.25/7.68 = ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6905_sum__gp__multiplied,axiom,
% 13.25/7.68 ! [M: nat,N: nat,X: int] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 13.25/7.68 = ( minus_minus_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6906_sum__gp__multiplied,axiom,
% 13.25/7.68 ! [M: nat,N: nat,X: real] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.68 => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 13.25/7.68 = ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6907_sum_Oin__pairs,axiom,
% 13.25/7.68 ! [G: nat > rat,M: nat,N: nat] :
% 13.25/7.68 ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 13.25/7.68 = ( groups2906978787729119204at_rat
% 13.25/7.68 @ ^ [I4: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6908_sum_Oin__pairs,axiom,
% 13.25/7.68 ! [G: nat > int,M: nat,N: nat] :
% 13.25/7.68 ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 13.25/7.68 = ( groups3539618377306564664at_int
% 13.25/7.68 @ ^ [I4: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6909_sum_Oin__pairs,axiom,
% 13.25/7.68 ! [G: nat > nat,M: nat,N: nat] :
% 13.25/7.68 ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 13.25/7.68 = ( groups3542108847815614940at_nat
% 13.25/7.68 @ ^ [I4: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6910_sum_Oin__pairs,axiom,
% 13.25/7.68 ! [G: nat > real,M: nat,N: nat] :
% 13.25/7.68 ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 13.25/7.68 = ( groups6591440286371151544t_real
% 13.25/7.68 @ ^ [I4: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6911_mask__eq__sum__exp__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 13.25/7.68 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.68 @ ( collect_nat
% 13.25/7.68 @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6912_gauss__sum__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( groups3542108847815614940at_nat
% 13.25/7.68 @ ^ [X2: nat] : X2
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.68 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6913_arith__series__nat,axiom,
% 13.25/7.68 ! [A: nat,D: nat,N: nat] :
% 13.25/7.68 ( ( groups3542108847815614940at_nat
% 13.25/7.68 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I4 @ D ) )
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.68 = ( 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 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6914_Sum__Icc__nat,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( groups3542108847815614940at_nat
% 13.25/7.68 @ ^ [X2: nat] : X2
% 13.25/7.68 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( 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 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6915_divmod__divmod__step,axiom,
% 13.25/7.68 ( unique5055182867167087721od_nat
% 13.25/7.68 = ( ^ [M3: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M3 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6916_divmod__divmod__step,axiom,
% 13.25/7.68 ( unique5052692396658037445od_int
% 13.25/7.68 = ( ^ [M3: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M3 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M3 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6917_divmod__divmod__step,axiom,
% 13.25/7.68 ( unique3479559517661332726nteger
% 13.25/7.68 = ( ^ [M3: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M3 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M3 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6918_take__bit__numeral__bit1,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6919_take__bit__numeral__bit1,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 13.25/7.68 = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L2 ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6920_take__bit__numeral__minus__bit1,axiom,
% 13.25/7.68 ! [L2: num,K: num] :
% 13.25/7.68 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 13.25/7.68 = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6921_divmod__nat__if,axiom,
% 13.25/7.68 ( divmod_nat
% 13.25/7.68 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.68 ( if_Pro6206227464963214023at_nat
% 13.25/7.68 @ ( ( N2 = zero_zero_nat )
% 13.25/7.68 | ( ord_less_nat @ M3 @ N2 ) )
% 13.25/7.68 @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
% 13.25/7.68 @ ( produc2626176000494625587at_nat
% 13.25/7.68 @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
% 13.25/7.68 @ ( divmod_nat @ ( minus_minus_nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6922_take__bit__Suc__minus__bit1,axiom,
% 13.25/7.68 ! [N: nat,K: num] :
% 13.25/7.68 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 13.25/7.68 = ( 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 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6923_infinite__int__iff__unbounded,axiom,
% 13.25/7.68 ! [S2: set_int] :
% 13.25/7.68 ( ( ~ ( finite_finite_int @ S2 ) )
% 13.25/7.68 = ( ! [M3: int] :
% 13.25/7.68 ? [N2: int] :
% 13.25/7.68 ( ( ord_less_int @ M3 @ ( abs_abs_int @ N2 ) )
% 13.25/7.68 & ( member_int @ N2 @ S2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6924_infinite__int__iff__unbounded__le,axiom,
% 13.25/7.68 ! [S2: set_int] :
% 13.25/7.68 ( ( ~ ( finite_finite_int @ S2 ) )
% 13.25/7.68 = ( ! [M3: int] :
% 13.25/7.68 ? [N2: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ M3 @ ( abs_abs_int @ N2 ) )
% 13.25/7.68 & ( member_int @ N2 @ S2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6925_dbl__inc__simps_I3_J,axiom,
% 13.25/7.68 ( ( neg_nu8557863876264182079omplex @ one_one_complex )
% 13.25/7.68 = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6926_dbl__inc__simps_I3_J,axiom,
% 13.25/7.68 ( ( neg_nu8295874005876285629c_real @ one_one_real )
% 13.25/7.68 = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6927_dbl__inc__simps_I3_J,axiom,
% 13.25/7.68 ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
% 13.25/7.68 = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6928_dbl__inc__simps_I3_J,axiom,
% 13.25/7.68 ( ( neg_nu5851722552734809277nc_int @ one_one_int )
% 13.25/7.68 = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6929_pred__numeral__inc,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( pred_numeral @ ( inc @ K ) )
% 13.25/7.68 = ( numeral_numeral_nat @ K ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6930_dbl__inc__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 13.25/7.68 = one_one_complex ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6931_dbl__inc__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 13.25/7.68 = one_one_real ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6932_dbl__inc__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 13.25/7.68 = one_one_rat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6933_dbl__inc__simps_I2_J,axiom,
% 13.25/7.68 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 13.25/7.68 = one_one_int ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6934_dbl__inc__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.68 = ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6935_dbl__inc__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.68 = ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6936_dbl__inc__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.68 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6937_dbl__inc__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.68 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6938_dbl__inc__simps_I4_J,axiom,
% 13.25/7.68 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.68 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6939_dbl__inc__simps_I5_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
% 13.25/7.68 = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6940_dbl__inc__simps_I5_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
% 13.25/7.68 = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6941_dbl__inc__simps_I5_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) )
% 13.25/7.68 = ( numeral_numeral_rat @ ( bit1 @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6942_dbl__inc__simps_I5_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
% 13.25/7.68 = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6943_dbl__dec__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 13.25/7.68 = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6944_dbl__dec__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6945_dbl__dec__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 13.25/7.68 = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6946_dbl__dec__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 13.25/7.68 = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6947_dbl__dec__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 13.25/7.68 = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6948_dbl__inc__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
% 13.25/7.68 = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6949_dbl__inc__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6950_dbl__inc__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
% 13.25/7.68 = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6951_dbl__inc__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
% 13.25/7.68 = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6952_dbl__inc__simps_I1_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 13.25/7.68 = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6953_add__neg__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.68 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6954_add__neg__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6955_add__neg__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.68 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6956_add__neg__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.68 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6957_add__neg__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.68 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6958_add__neg__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.68 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6959_add__neg__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6960_add__neg__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
% 13.25/7.68 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6961_add__neg__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 13.25/7.68 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6962_add__neg__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.68 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6963_diff__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.68 = ( numeral_numeral_real @ ( inc @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6964_diff__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.68 = ( numeral_numeral_int @ ( inc @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6965_diff__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.68 = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6966_diff__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 13.25/7.68 = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6967_diff__numeral__special_I6_J,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 13.25/7.68 = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6968_diff__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
% 13.25/7.68 = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6969_diff__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.68 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6970_diff__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
% 13.25/7.68 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6971_diff__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
% 13.25/7.68 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6972_diff__numeral__special_I5_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
% 13.25/7.68 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6973_num__induct,axiom,
% 13.25/7.68 ! [P: num > $o,X: num] :
% 13.25/7.68 ( ( P @ one )
% 13.25/7.68 => ( ! [X3: num] :
% 13.25/7.68 ( ( P @ X3 )
% 13.25/7.68 => ( P @ ( inc @ X3 ) ) )
% 13.25/7.68 => ( P @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6974_add__inc,axiom,
% 13.25/7.68 ! [X: num,Y3: num] :
% 13.25/7.68 ( ( plus_plus_num @ X @ ( inc @ Y3 ) )
% 13.25/7.68 = ( inc @ ( plus_plus_num @ X @ Y3 ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6975_inc_Osimps_I1_J,axiom,
% 13.25/7.68 ( ( inc @ one )
% 13.25/7.68 = ( bit0 @ one ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6976_inc_Osimps_I3_J,axiom,
% 13.25/7.68 ! [X: num] :
% 13.25/7.68 ( ( inc @ ( bit1 @ X ) )
% 13.25/7.68 = ( bit0 @ ( inc @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6977_inc_Osimps_I2_J,axiom,
% 13.25/7.68 ! [X: num] :
% 13.25/7.68 ( ( inc @ ( bit0 @ X ) )
% 13.25/7.68 = ( bit1 @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6978_add__One,axiom,
% 13.25/7.68 ! [X: num] :
% 13.25/7.68 ( ( plus_plus_num @ X @ one )
% 13.25/7.68 = ( inc @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6979_mult__inc,axiom,
% 13.25/7.68 ! [X: num,Y3: num] :
% 13.25/7.68 ( ( times_times_num @ X @ ( inc @ Y3 ) )
% 13.25/7.68 = ( plus_plus_num @ ( times_times_num @ X @ Y3 ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6980_numeral__inc,axiom,
% 13.25/7.68 ! [X: num] :
% 13.25/7.68 ( ( numera6690914467698888265omplex @ ( inc @ X ) )
% 13.25/7.68 = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6981_numeral__inc,axiom,
% 13.25/7.68 ! [X: num] :
% 13.25/7.68 ( ( numeral_numeral_real @ ( inc @ X ) )
% 13.25/7.68 = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6982_numeral__inc,axiom,
% 13.25/7.68 ! [X: num] :
% 13.25/7.68 ( ( numeral_numeral_rat @ ( inc @ X ) )
% 13.25/7.68 = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6983_numeral__inc,axiom,
% 13.25/7.68 ! [X: num] :
% 13.25/7.68 ( ( numeral_numeral_nat @ ( inc @ X ) )
% 13.25/7.68 = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6984_numeral__inc,axiom,
% 13.25/7.68 ! [X: num] :
% 13.25/7.68 ( ( numeral_numeral_int @ ( inc @ X ) )
% 13.25/7.68 = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6985_dbl__inc__def,axiom,
% 13.25/7.68 ( neg_nu8557863876264182079omplex
% 13.25/7.68 = ( ^ [X2: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X2 @ X2 ) @ one_one_complex ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6986_dbl__inc__def,axiom,
% 13.25/7.68 ( neg_nu8295874005876285629c_real
% 13.25/7.68 = ( ^ [X2: real] : ( plus_plus_real @ ( plus_plus_real @ X2 @ X2 ) @ one_one_real ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6987_dbl__inc__def,axiom,
% 13.25/7.68 ( neg_nu5219082963157363817nc_rat
% 13.25/7.68 = ( ^ [X2: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X2 @ X2 ) @ one_one_rat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6988_dbl__inc__def,axiom,
% 13.25/7.68 ( neg_nu5851722552734809277nc_int
% 13.25/7.68 = ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6989_divmod__nat__def,axiom,
% 13.25/7.68 ( divmod_nat
% 13.25/7.68 = ( ^ [M3: nat,N2: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M3 @ N2 ) @ ( modulo_modulo_nat @ M3 @ N2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6990_infinite__nat__iff__unbounded,axiom,
% 13.25/7.68 ! [S2: set_nat] :
% 13.25/7.68 ( ( ~ ( finite_finite_nat @ S2 ) )
% 13.25/7.68 = ( ! [M3: nat] :
% 13.25/7.68 ? [N2: nat] :
% 13.25/7.68 ( ( ord_less_nat @ M3 @ N2 )
% 13.25/7.68 & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6991_unbounded__k__infinite,axiom,
% 13.25/7.68 ! [K: nat,S2: set_nat] :
% 13.25/7.68 ( ! [M4: nat] :
% 13.25/7.68 ( ( ord_less_nat @ K @ M4 )
% 13.25/7.68 => ? [N7: nat] :
% 13.25/7.68 ( ( ord_less_nat @ M4 @ N7 )
% 13.25/7.68 & ( member_nat @ N7 @ S2 ) ) )
% 13.25/7.68 => ~ ( finite_finite_nat @ S2 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6992_infinite__nat__iff__unbounded__le,axiom,
% 13.25/7.68 ! [S2: set_nat] :
% 13.25/7.68 ( ( ~ ( finite_finite_nat @ S2 ) )
% 13.25/7.68 = ( ! [M3: nat] :
% 13.25/7.68 ? [N2: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.68 & ( member_nat @ N2 @ S2 ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6993_sum__gp,axiom,
% 13.25/7.68 ! [N: nat,M: nat,X: complex] :
% 13.25/7.68 ( ( ( ord_less_nat @ N @ M )
% 13.25/7.68 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = zero_zero_complex ) )
% 13.25/7.68 & ( ~ ( ord_less_nat @ N @ M )
% 13.25/7.68 => ( ( ( X = one_one_complex )
% 13.25/7.68 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 13.25/7.68 & ( ( X != one_one_complex )
% 13.25/7.68 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6994_sum__gp,axiom,
% 13.25/7.68 ! [N: nat,M: nat,X: rat] :
% 13.25/7.68 ( ( ( ord_less_nat @ N @ M )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = zero_zero_rat ) )
% 13.25/7.68 & ( ~ ( ord_less_nat @ N @ M )
% 13.25/7.68 => ( ( ( X = one_one_rat )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 13.25/7.68 & ( ( X != one_one_rat )
% 13.25/7.68 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6995_sum__gp,axiom,
% 13.25/7.68 ! [N: nat,M: nat,X: real] :
% 13.25/7.68 ( ( ( ord_less_nat @ N @ M )
% 13.25/7.68 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = zero_zero_real ) )
% 13.25/7.68 & ( ~ ( ord_less_nat @ N @ M )
% 13.25/7.68 => ( ( ( X = one_one_real )
% 13.25/7.68 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 13.25/7.68 & ( ( X != one_one_real )
% 13.25/7.68 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 13.25/7.68 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6996_divmod__BitM__2__eq,axiom,
% 13.25/7.68 ! [M: num] :
% 13.25/7.68 ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 13.25/7.68 = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6997_gauss__sum__from__Suc__0,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 13.25/7.68 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6998_gauss__sum__from__Suc__0,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 13.25/7.68 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_6999_of__int__code__if,axiom,
% 13.25/7.68 ( ring_1_of_int_real
% 13.25/7.68 = ( ^ [K2: int] :
% 13.25/7.68 ( if_real @ ( K2 = zero_zero_int ) @ zero_zero_real
% 13.25/7.68 @ ( if_real @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K2 ) ) )
% 13.25/7.68 @ ( if_real
% 13.25/7.68 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.68 = zero_zero_int )
% 13.25/7.68 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 13.25/7.68 @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7000_of__int__code__if,axiom,
% 13.25/7.68 ( ring_1_of_int_int
% 13.25/7.68 = ( ^ [K2: int] :
% 13.25/7.68 ( if_int @ ( K2 = zero_zero_int ) @ zero_zero_int
% 13.25/7.68 @ ( if_int @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K2 ) ) )
% 13.25/7.68 @ ( if_int
% 13.25/7.68 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.68 = zero_zero_int )
% 13.25/7.68 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 13.25/7.68 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7001_of__int__code__if,axiom,
% 13.25/7.68 ( ring_17405671764205052669omplex
% 13.25/7.68 = ( ^ [K2: int] :
% 13.25/7.68 ( if_complex @ ( K2 = zero_zero_int ) @ zero_zero_complex
% 13.25/7.68 @ ( if_complex @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K2 ) ) )
% 13.25/7.68 @ ( if_complex
% 13.25/7.68 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.68 = zero_zero_int )
% 13.25/7.68 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 13.25/7.68 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7002_of__int__code__if,axiom,
% 13.25/7.68 ( ring_18347121197199848620nteger
% 13.25/7.68 = ( ^ [K2: int] :
% 13.25/7.68 ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 13.25/7.68 @ ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K2 ) ) )
% 13.25/7.68 @ ( if_Code_integer
% 13.25/7.68 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.68 = zero_zero_int )
% 13.25/7.68 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 13.25/7.68 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7003_of__int__code__if,axiom,
% 13.25/7.68 ( ring_1_of_int_rat
% 13.25/7.68 = ( ^ [K2: int] :
% 13.25/7.68 ( if_rat @ ( K2 = zero_zero_int ) @ zero_zero_rat
% 13.25/7.68 @ ( if_rat @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K2 ) ) )
% 13.25/7.68 @ ( if_rat
% 13.25/7.68 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.68 = zero_zero_int )
% 13.25/7.68 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 13.25/7.68 @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7004_signed__take__bit__eq__take__bit__minus,axiom,
% 13.25/7.68 ( bit_ri631733984087533419it_int
% 13.25/7.68 = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K2 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7005_of__nat__eq__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ( semiri1314217659103216013at_int @ M )
% 13.25/7.68 = ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.68 = ( M = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7006_of__nat__eq__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ( semiri5074537144036343181t_real @ M )
% 13.25/7.68 = ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.68 = ( M = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7007_of__nat__eq__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ( semiri1316708129612266289at_nat @ M )
% 13.25/7.68 = ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.68 = ( M = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7008_of__nat__eq__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ( semiri8010041392384452111omplex @ M )
% 13.25/7.68 = ( semiri8010041392384452111omplex @ N ) )
% 13.25/7.68 = ( M = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7009_of__nat__eq__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ( semiri681578069525770553at_rat @ M )
% 13.25/7.68 = ( semiri681578069525770553at_rat @ N ) )
% 13.25/7.68 = ( M = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7010_int__eq__iff__numeral,axiom,
% 13.25/7.68 ! [M: nat,V: num] :
% 13.25/7.68 ( ( ( semiri1314217659103216013at_int @ M )
% 13.25/7.68 = ( numeral_numeral_int @ V ) )
% 13.25/7.68 = ( M
% 13.25/7.68 = ( numeral_numeral_nat @ V ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7011_bit__0__eq,axiom,
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ zero_zero_int )
% 13.25/7.68 = bot_bot_nat_o ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7012_bit__0__eq,axiom,
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
% 13.25/7.68 = bot_bot_nat_o ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7013_abs__of__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 13.25/7.68 = ( semiri4939895301339042750nteger @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7014_abs__of__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.68 = ( semiri1314217659103216013at_int @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7015_abs__of__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.68 = ( semiri5074537144036343181t_real @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7016_abs__of__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 13.25/7.68 = ( semiri681578069525770553at_rat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7017_negative__zle,axiom,
% 13.25/7.68 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7018_of__nat__eq__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ( semiri1314217659103216013at_int @ M )
% 13.25/7.68 = zero_zero_int )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7019_of__nat__eq__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ( semiri5074537144036343181t_real @ M )
% 13.25/7.68 = zero_zero_real )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7020_of__nat__eq__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ( semiri1316708129612266289at_nat @ M )
% 13.25/7.68 = zero_zero_nat )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7021_of__nat__eq__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ( semiri8010041392384452111omplex @ M )
% 13.25/7.68 = zero_zero_complex )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7022_of__nat__eq__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ( semiri681578069525770553at_rat @ M )
% 13.25/7.68 = zero_zero_rat )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7023_of__nat__0__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( zero_zero_int
% 13.25/7.68 = ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.68 = ( zero_zero_nat = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7024_of__nat__0__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( zero_zero_real
% 13.25/7.68 = ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.68 = ( zero_zero_nat = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7025_of__nat__0__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( zero_zero_nat
% 13.25/7.68 = ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.68 = ( zero_zero_nat = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7026_of__nat__0__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( zero_zero_complex
% 13.25/7.68 = ( semiri8010041392384452111omplex @ N ) )
% 13.25/7.68 = ( zero_zero_nat = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7027_of__nat__0__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( zero_zero_rat
% 13.25/7.68 = ( semiri681578069525770553at_rat @ N ) )
% 13.25/7.68 = ( zero_zero_nat = N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7028_of__nat__0,axiom,
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 13.25/7.68 = zero_zero_int ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7029_of__nat__0,axiom,
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 13.25/7.68 = zero_zero_real ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7030_of__nat__0,axiom,
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 13.25/7.68 = zero_zero_nat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7031_of__nat__0,axiom,
% 13.25/7.68 ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 13.25/7.68 = zero_zero_complex ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7032_of__nat__0,axiom,
% 13.25/7.68 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 13.25/7.68 = zero_zero_rat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7033_of__nat__less__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.68 = ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7034_of__nat__less__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.68 = ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7035_of__nat__less__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.68 = ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7036_of__nat__less__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 13.25/7.68 = ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7037_of__nat__numeral,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( numeral_numeral_int @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7038_of__nat__numeral,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( numeral_numeral_real @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7039_of__nat__numeral,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( numeral_numeral_nat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7040_of__nat__numeral,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( numera6690914467698888265omplex @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7041_of__nat__numeral,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( numeral_numeral_rat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7042_of__nat__le__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7043_of__nat__le__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7044_of__nat__le__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7045_of__nat__le__iff,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7046_of__nat__add,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7047_of__nat__add,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7048_of__nat__add,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7049_of__nat__add,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7050_of__nat__add,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.68 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7051_of__nat__mult,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 13.25/7.68 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7052_of__nat__mult,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 13.25/7.68 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7053_of__nat__mult,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 13.25/7.68 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7054_of__nat__mult,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 13.25/7.68 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7055_of__nat__mult,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 13.25/7.68 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7056_of__nat__1,axiom,
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 13.25/7.68 = one_one_int ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7057_of__nat__1,axiom,
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 13.25/7.68 = one_one_real ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7058_of__nat__1,axiom,
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 13.25/7.68 = one_one_nat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7059_of__nat__1,axiom,
% 13.25/7.68 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 13.25/7.68 = one_one_complex ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7060_of__nat__1,axiom,
% 13.25/7.68 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 13.25/7.68 = one_one_rat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7061_of__nat__1__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( one_one_int
% 13.25/7.68 = ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7062_of__nat__1__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( one_one_real
% 13.25/7.68 = ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7063_of__nat__1__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( one_one_nat
% 13.25/7.68 = ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7064_of__nat__1__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( one_one_complex
% 13.25/7.68 = ( semiri8010041392384452111omplex @ N ) )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7065_of__nat__1__eq__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( one_one_rat
% 13.25/7.68 = ( semiri681578069525770553at_rat @ N ) )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7066_of__nat__eq__1__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ( semiri1314217659103216013at_int @ N )
% 13.25/7.68 = one_one_int )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7067_of__nat__eq__1__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ( semiri5074537144036343181t_real @ N )
% 13.25/7.68 = one_one_real )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7068_of__nat__eq__1__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ( semiri1316708129612266289at_nat @ N )
% 13.25/7.68 = one_one_nat )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7069_of__nat__eq__1__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ( semiri8010041392384452111omplex @ N )
% 13.25/7.68 = one_one_complex )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7070_of__nat__eq__1__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ( semiri681578069525770553at_rat @ N )
% 13.25/7.68 = one_one_rat )
% 13.25/7.68 = ( N = one_one_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7071_of__int__le__iff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ W @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7072_of__int__le__iff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ W @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7073_of__int__le__iff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ W @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7074_of__int__eq__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ( ring_17405671764205052669omplex @ Z )
% 13.25/7.68 = ( numera6690914467698888265omplex @ N ) )
% 13.25/7.68 = ( Z
% 13.25/7.68 = ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7075_of__int__eq__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ( ring_1_of_int_real @ Z )
% 13.25/7.68 = ( numeral_numeral_real @ N ) )
% 13.25/7.68 = ( Z
% 13.25/7.68 = ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7076_of__int__eq__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ( ring_1_of_int_rat @ Z )
% 13.25/7.68 = ( numeral_numeral_rat @ N ) )
% 13.25/7.68 = ( Z
% 13.25/7.68 = ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7077_of__int__eq__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ( ring_1_of_int_int @ Z )
% 13.25/7.68 = ( numeral_numeral_int @ N ) )
% 13.25/7.68 = ( Z
% 13.25/7.68 = ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7078_of__int__numeral,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
% 13.25/7.68 = ( numera6690914467698888265omplex @ K ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7079_of__int__numeral,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 13.25/7.68 = ( numeral_numeral_real @ K ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7080_of__int__numeral,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 13.25/7.68 = ( numeral_numeral_rat @ K ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7081_of__int__numeral,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 13.25/7.68 = ( numeral_numeral_int @ K ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7082_of__int__less__iff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
% 13.25/7.68 = ( ord_less_int @ W @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7083_of__int__less__iff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
% 13.25/7.68 = ( ord_less_int @ W @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7084_of__int__less__iff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
% 13.25/7.68 = ( ord_less_int @ W @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7085_of__int__1,axiom,
% 13.25/7.68 ( ( ring_17405671764205052669omplex @ one_one_int )
% 13.25/7.68 = one_one_complex ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7086_of__int__1,axiom,
% 13.25/7.68 ( ( ring_1_of_int_int @ one_one_int )
% 13.25/7.68 = one_one_int ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7087_of__int__1,axiom,
% 13.25/7.68 ( ( ring_1_of_int_real @ one_one_int )
% 13.25/7.68 = one_one_real ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7088_of__int__1,axiom,
% 13.25/7.68 ( ( ring_1_of_int_rat @ one_one_int )
% 13.25/7.68 = one_one_rat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7089_of__int__eq__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ( ring_17405671764205052669omplex @ Z )
% 13.25/7.68 = one_one_complex )
% 13.25/7.68 = ( Z = one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7090_of__int__eq__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ( ring_1_of_int_int @ Z )
% 13.25/7.68 = one_one_int )
% 13.25/7.68 = ( Z = one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7091_of__int__eq__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ( ring_1_of_int_real @ Z )
% 13.25/7.68 = one_one_real )
% 13.25/7.68 = ( Z = one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7092_of__int__eq__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ( ring_1_of_int_rat @ Z )
% 13.25/7.68 = one_one_rat )
% 13.25/7.68 = ( Z = one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7093_of__nat__power,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N ) )
% 13.25/7.68 = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7094_of__nat__power,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N ) )
% 13.25/7.68 = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7095_of__nat__power,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N ) )
% 13.25/7.68 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7096_of__nat__power,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N ) )
% 13.25/7.68 = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7097_of__nat__power,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( semiri681578069525770553at_rat @ ( power_power_nat @ M @ N ) )
% 13.25/7.68 = ( power_power_rat @ ( semiri681578069525770553at_rat @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7098_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
% 13.25/7.68 = ( semiri1314217659103216013at_int @ X ) )
% 13.25/7.68 = ( ( power_power_nat @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7099_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
% 13.25/7.68 = ( semiri5074537144036343181t_real @ X ) )
% 13.25/7.68 = ( ( power_power_nat @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7100_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
% 13.25/7.68 = ( semiri1316708129612266289at_nat @ X ) )
% 13.25/7.68 = ( ( power_power_nat @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7101_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
% 13.25/7.68 = ( semiri8010041392384452111omplex @ X ) )
% 13.25/7.68 = ( ( power_power_nat @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7102_of__nat__eq__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W )
% 13.25/7.68 = ( semiri681578069525770553at_rat @ X ) )
% 13.25/7.68 = ( ( power_power_nat @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7103_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ( semiri1314217659103216013at_int @ X )
% 13.25/7.68 = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7104_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ( semiri5074537144036343181t_real @ X )
% 13.25/7.68 = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7105_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ( semiri1316708129612266289at_nat @ X )
% 13.25/7.68 = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7106_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ( semiri8010041392384452111omplex @ X )
% 13.25/7.68 = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7107_of__nat__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ( semiri681578069525770553at_rat @ X )
% 13.25/7.68 = ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7108_of__int__mult,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_real @ ( times_times_int @ W @ Z ) )
% 13.25/7.68 = ( times_times_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7109_of__int__mult,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_rat @ ( times_times_int @ W @ Z ) )
% 13.25/7.68 = ( times_times_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7110_of__int__mult,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_int @ ( times_times_int @ W @ Z ) )
% 13.25/7.68 = ( times_times_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7111_of__int__add,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
% 13.25/7.68 = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7112_of__int__add,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
% 13.25/7.68 = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7113_of__int__add,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
% 13.25/7.68 = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7114_negative__zless,axiom,
% 13.25/7.68 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7115_of__int__diff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_real @ ( minus_minus_int @ W @ Z ) )
% 13.25/7.68 = ( minus_minus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7116_of__int__diff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_rat @ ( minus_minus_int @ W @ Z ) )
% 13.25/7.68 = ( minus_minus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7117_of__int__diff,axiom,
% 13.25/7.68 ! [W: int,Z: int] :
% 13.25/7.68 ( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
% 13.25/7.68 = ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7118_of__int__power,axiom,
% 13.25/7.68 ! [Z: int,N: nat] :
% 13.25/7.68 ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N ) )
% 13.25/7.68 = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7119_of__int__power,axiom,
% 13.25/7.68 ! [Z: int,N: nat] :
% 13.25/7.68 ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N ) )
% 13.25/7.68 = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7120_of__int__power,axiom,
% 13.25/7.68 ! [Z: int,N: nat] :
% 13.25/7.68 ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
% 13.25/7.68 = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7121_of__int__power,axiom,
% 13.25/7.68 ! [Z: int,N: nat] :
% 13.25/7.68 ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N ) )
% 13.25/7.68 = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7122_of__int__eq__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
% 13.25/7.68 = ( ring_1_of_int_rat @ X ) )
% 13.25/7.68 = ( ( power_power_int @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7123_of__int__eq__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
% 13.25/7.68 = ( ring_1_of_int_real @ X ) )
% 13.25/7.68 = ( ( power_power_int @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7124_of__int__eq__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
% 13.25/7.68 = ( ring_1_of_int_int @ X ) )
% 13.25/7.68 = ( ( power_power_int @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7125_of__int__eq__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
% 13.25/7.68 = ( ring_17405671764205052669omplex @ X ) )
% 13.25/7.68 = ( ( power_power_int @ B @ W )
% 13.25/7.68 = X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7126_of__int__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_rat @ X )
% 13.25/7.68 = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7127_of__int__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_real @ X )
% 13.25/7.68 = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7128_of__int__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_int @ X )
% 13.25/7.68 = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7129_of__int__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ( ring_17405671764205052669omplex @ X )
% 13.25/7.68 = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
% 13.25/7.68 = ( X
% 13.25/7.68 = ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7130_of__nat__of__bool,axiom,
% 13.25/7.68 ! [P: $o] :
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 13.25/7.68 = ( zero_n3304061248610475627l_real @ P ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7131_of__nat__of__bool,axiom,
% 13.25/7.68 ! [P: $o] :
% 13.25/7.68 ( ( semiri8010041392384452111omplex @ ( zero_n2687167440665602831ol_nat @ P ) )
% 13.25/7.68 = ( zero_n1201886186963655149omplex @ P ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7132_of__nat__of__bool,axiom,
% 13.25/7.68 ! [P: $o] :
% 13.25/7.68 ( ( semiri681578069525770553at_rat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 13.25/7.68 = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7133_of__nat__of__bool,axiom,
% 13.25/7.68 ! [P: $o] :
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 13.25/7.68 = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7134_of__nat__of__bool,axiom,
% 13.25/7.68 ! [P: $o] :
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 13.25/7.68 = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7135_of__nat__of__bool,axiom,
% 13.25/7.68 ! [P: $o] :
% 13.25/7.68 ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
% 13.25/7.68 = ( zero_n356916108424825756nteger @ P ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7136_dbl__dec__simps_I5_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
% 13.25/7.68 = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7137_dbl__dec__simps_I5_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
% 13.25/7.68 = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7138_dbl__dec__simps_I5_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) )
% 13.25/7.68 = ( numeral_numeral_rat @ ( bitM @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7139_dbl__dec__simps_I5_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
% 13.25/7.68 = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7140_of__nat__le__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7141_of__nat__le__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7142_of__nat__le__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7143_of__nat__le__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 13.25/7.68 = ( M = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7144_of__nat__Suc,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 13.25/7.68 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7145_of__nat__Suc,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 13.25/7.68 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7146_of__nat__Suc,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 13.25/7.68 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7147_of__nat__Suc,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 13.25/7.68 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7148_of__nat__Suc,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 13.25/7.68 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7149_bit__numeral__Bit0__Suc__iff,axiom,
% 13.25/7.68 ! [M: num,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 13.25/7.68 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7150_bit__numeral__Bit0__Suc__iff,axiom,
% 13.25/7.68 ! [M: num,N: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N ) )
% 13.25/7.68 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7151_real__of__nat__less__numeral__iff,axiom,
% 13.25/7.68 ! [N: nat,W: num] :
% 13.25/7.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W ) )
% 13.25/7.68 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7152_numeral__less__real__of__nat__iff,axiom,
% 13.25/7.68 ! [W: num,N: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.68 = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7153_bit__numeral__Bit1__Suc__iff,axiom,
% 13.25/7.68 ! [M: num,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 13.25/7.68 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7154_bit__numeral__Bit1__Suc__iff,axiom,
% 13.25/7.68 ! [M: num,N: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N ) )
% 13.25/7.68 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7155_numeral__le__real__of__nat__iff,axiom,
% 13.25/7.68 ! [N: num,M: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7156_pred__numeral__simps_I2_J,axiom,
% 13.25/7.68 ! [K: num] :
% 13.25/7.68 ( ( pred_numeral @ ( bit0 @ K ) )
% 13.25/7.68 = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7157_signed__take__bit__nonnegative__iff,axiom,
% 13.25/7.68 ! [N: nat,K: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 13.25/7.68 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7158_signed__take__bit__negative__iff,axiom,
% 13.25/7.68 ! [N: nat,K: int] :
% 13.25/7.68 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 13.25/7.68 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7159_of__nat__0__less__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.68 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7160_of__nat__0__less__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.68 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7161_of__nat__0__less__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.68 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7162_of__nat__0__less__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 13.25/7.68 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7163_of__int__0__le__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7164_of__int__0__le__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7165_of__int__0__le__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7166_of__int__le__0__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7167_of__int__le__0__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7168_of__int__le__0__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7169_of__int__less__0__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 13.25/7.68 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7170_of__int__less__0__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 13.25/7.68 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7171_of__int__less__0__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 13.25/7.68 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7172_of__int__0__less__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 13.25/7.68 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7173_of__int__0__less__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 13.25/7.68 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7174_of__int__0__less__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 13.25/7.68 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7175_of__nat__less__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 13.25/7.68 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7176_of__nat__less__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 13.25/7.68 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7177_of__nat__less__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 13.25/7.68 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7178_of__nat__less__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 13.25/7.68 = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7179_of__nat__power__less__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7180_of__nat__power__less__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7181_of__nat__power__less__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7182_of__nat__power__less__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7183_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: nat,X: num,N: nat] :
% 13.25/7.68 ( ( ( semiri1314217659103216013at_int @ Y3 )
% 13.25/7.68 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7184_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: nat,X: num,N: nat] :
% 13.25/7.68 ( ( ( semiri5074537144036343181t_real @ Y3 )
% 13.25/7.68 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7185_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: nat,X: num,N: nat] :
% 13.25/7.68 ( ( ( semiri1316708129612266289at_nat @ Y3 )
% 13.25/7.68 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7186_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: nat,X: num,N: nat] :
% 13.25/7.68 ( ( ( semiri8010041392384452111omplex @ Y3 )
% 13.25/7.68 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7187_real__of__nat__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: nat,X: num,N: nat] :
% 13.25/7.68 ( ( ( semiri681578069525770553at_rat @ Y3 )
% 13.25/7.68 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7188_numeral__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: nat] :
% 13.25/7.68 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 13.25/7.68 = ( semiri1314217659103216013at_int @ Y3 ) )
% 13.25/7.68 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7189_numeral__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: nat] :
% 13.25/7.68 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 13.25/7.68 = ( semiri5074537144036343181t_real @ Y3 ) )
% 13.25/7.68 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7190_numeral__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: nat] :
% 13.25/7.68 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 13.25/7.68 = ( semiri1316708129612266289at_nat @ Y3 ) )
% 13.25/7.68 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7191_numeral__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: nat] :
% 13.25/7.68 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 13.25/7.68 = ( semiri8010041392384452111omplex @ Y3 ) )
% 13.25/7.68 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7192_numeral__power__eq__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: nat] :
% 13.25/7.68 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 13.25/7.68 = ( semiri681578069525770553at_rat @ Y3 ) )
% 13.25/7.68 = ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7193_of__nat__le__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7194_of__nat__le__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7195_of__nat__le__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7196_of__nat__le__of__nat__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: nat,W: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7197_of__nat__power__le__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7198_of__nat__power__le__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7199_of__nat__power__le__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7200_of__nat__power__le__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,B: nat,W: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7201_of__int__le__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7202_of__int__le__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7203_of__int__le__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7204_of__int__numeral__le__iff,axiom,
% 13.25/7.68 ! [N: num,Z: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7205_of__int__numeral__le__iff,axiom,
% 13.25/7.68 ! [N: num,Z: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7206_of__int__numeral__le__iff,axiom,
% 13.25/7.68 ! [N: num,Z: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7207_of__int__numeral__less__iff,axiom,
% 13.25/7.68 ! [N: num,Z: int] :
% 13.25/7.68 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 13.25/7.68 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7208_of__int__numeral__less__iff,axiom,
% 13.25/7.68 ! [N: num,Z: int] :
% 13.25/7.68 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 13.25/7.68 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7209_of__int__numeral__less__iff,axiom,
% 13.25/7.68 ! [N: num,Z: int] :
% 13.25/7.68 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 13.25/7.68 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7210_of__int__less__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 13.25/7.68 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7211_of__int__less__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 13.25/7.68 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7212_of__int__less__numeral__iff,axiom,
% 13.25/7.68 ! [Z: int,N: num] :
% 13.25/7.68 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.68 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7213_of__int__le__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7214_of__int__le__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7215_of__int__le__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 13.25/7.68 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7216_of__int__1__le__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7217_of__int__1__le__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7218_of__int__1__le__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 13.25/7.68 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7219_of__int__1__less__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 13.25/7.68 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7220_of__int__1__less__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 13.25/7.68 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7221_of__int__1__less__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 13.25/7.68 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7222_of__int__less__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 13.25/7.68 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7223_of__int__less__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 13.25/7.68 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7224_of__int__less__1__iff,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 13.25/7.68 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7225_of__int__le__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7226_of__int__le__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7227_of__int__le__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7228_of__int__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7229_of__int__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7230_of__int__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7231_numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N )
% 13.25/7.68 = ( ring_17405671764205052669omplex @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7232_numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_power_real @ ( numeral_numeral_real @ X ) @ N )
% 13.25/7.68 = ( ring_1_of_int_real @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7233_numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N )
% 13.25/7.68 = ( ring_1_of_int_rat @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7234_numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 13.25/7.68 = ( ring_1_of_int_int @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7235_of__int__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 13.25/7.68 = ( power_power_complex @ ( numera6690914467698888265omplex @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7236_of__int__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_real @ Y3 )
% 13.25/7.68 = ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7237_of__int__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_rat @ Y3 )
% 13.25/7.68 = ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7238_of__int__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_int @ Y3 )
% 13.25/7.68 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7239_of__int__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( ring_1_of_int_real @ X ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7240_of__int__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7241_of__int__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: int,B: int,W: nat] :
% 13.25/7.68 ( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
% 13.25/7.68 = ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7242_of__int__less__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7243_of__int__less__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7244_of__int__less__of__int__power__cancel__iff,axiom,
% 13.25/7.68 ! [B: int,W: nat,X: int] :
% 13.25/7.68 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7245_bit__numeral__simps_I2_J,axiom,
% 13.25/7.68 ! [W: num,N: num] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7246_bit__numeral__simps_I2_J,axiom,
% 13.25/7.68 ! [W: num,N: num] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7247_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 13.25/7.68 ! [W: num,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
% 13.25/7.68 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7248_bit__numeral__simps_I3_J,axiom,
% 13.25/7.68 ! [W: num,N: num] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7249_bit__numeral__simps_I3_J,axiom,
% 13.25/7.68 ! [W: num,N: num] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7250_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 13.25/7.68 ! [W: num,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
% 13.25/7.68 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7251_of__nat__zero__less__power__iff,axiom,
% 13.25/7.68 ! [X: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
% 13.25/7.68 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 13.25/7.68 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7252_of__nat__zero__less__power__iff,axiom,
% 13.25/7.68 ! [X: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
% 13.25/7.68 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 13.25/7.68 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7253_of__nat__zero__less__power__iff,axiom,
% 13.25/7.68 ! [X: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
% 13.25/7.68 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 13.25/7.68 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7254_of__nat__zero__less__power__iff,axiom,
% 13.25/7.68 ! [X: nat,N: nat] :
% 13.25/7.68 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
% 13.25/7.68 = ( ( ord_less_nat @ zero_zero_nat @ X )
% 13.25/7.68 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7255_bit__0,axiom,
% 13.25/7.68 ! [A: code_integer] :
% 13.25/7.68 ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 13.25/7.68 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7256_bit__0,axiom,
% 13.25/7.68 ! [A: int] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 13.25/7.68 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7257_bit__0,axiom,
% 13.25/7.68 ! [A: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 13.25/7.68 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7258_bit__minus__numeral__int_I1_J,axiom,
% 13.25/7.68 ! [W: num,N: num] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7259_bit__minus__numeral__int_I2_J,axiom,
% 13.25/7.68 ! [W: num,N: num] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.68 = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7260_even__of__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N ) )
% 13.25/7.68 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7261_even__of__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.68 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7262_even__of__nat,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.68 = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7263_numeral__power__less__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [I: num,N: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 13.25/7.68 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7264_numeral__power__less__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [I: num,N: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 13.25/7.68 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7265_numeral__power__less__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [I: num,N: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 13.25/7.68 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7266_numeral__power__less__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [I: num,N: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 13.25/7.68 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7267_of__nat__less__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,I: num,N: nat] :
% 13.25/7.68 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 13.25/7.68 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7268_of__nat__less__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,I: num,N: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 13.25/7.68 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7269_of__nat__less__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,I: num,N: nat] :
% 13.25/7.68 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 13.25/7.68 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7270_of__nat__less__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,I: num,N: nat] :
% 13.25/7.68 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 13.25/7.68 = ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7271_numeral__power__le__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [I: num,N: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7272_numeral__power__le__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [I: num,N: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7273_numeral__power__le__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [I: num,N: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7274_numeral__power__le__of__nat__cancel__iff,axiom,
% 13.25/7.68 ! [I: num,N: nat,X: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
% 13.25/7.68 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7275_of__nat__le__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,I: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7276_of__nat__le__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,I: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7277_of__nat__le__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,I: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7278_of__nat__le__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [X: nat,I: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7279_bit__mod__2__iff,axiom,
% 13.25/7.68 ! [A: code_integer,N: nat] :
% 13.25/7.68 ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
% 13.25/7.68 = ( ( N = zero_zero_nat )
% 13.25/7.68 & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7280_bit__mod__2__iff,axiom,
% 13.25/7.68 ! [A: int,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
% 13.25/7.68 = ( ( N = zero_zero_nat )
% 13.25/7.68 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7281_bit__mod__2__iff,axiom,
% 13.25/7.68 ! [A: nat,N: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 13.25/7.68 = ( ( N = zero_zero_nat )
% 13.25/7.68 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7282_numeral__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7283_numeral__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7284_numeral__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7285_of__int__le__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7286_of__int__le__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7287_of__int__le__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7288_numeral__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7289_numeral__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7290_numeral__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7291_of__int__less__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X ) @ N ) )
% 13.25/7.68 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7292_of__int__less__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X ) @ N ) )
% 13.25/7.68 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7293_of__int__less__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) )
% 13.25/7.68 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7294_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N )
% 13.25/7.68 = ( ring_1_of_int_real @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7295_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 13.25/7.68 = ( ring_1_of_int_int @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7296_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N )
% 13.25/7.68 = ( ring_17405671764205052669omplex @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7297_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N )
% 13.25/7.68 = ( ring_18347121197199848620nteger @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7298_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,Y3: int] :
% 13.25/7.68 ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N )
% 13.25/7.68 = ( ring_1_of_int_rat @ Y3 ) )
% 13.25/7.68 = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N )
% 13.25/7.68 = Y3 ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7299_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_real @ Y3 )
% 13.25/7.68 = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7300_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_int @ Y3 )
% 13.25/7.68 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7301_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_17405671764205052669omplex @ Y3 )
% 13.25/7.68 = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X ) ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7302_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_18347121197199848620nteger @ Y3 )
% 13.25/7.68 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7303_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [Y3: int,X: num,N: nat] :
% 13.25/7.68 ( ( ( ring_1_of_int_rat @ Y3 )
% 13.25/7.68 = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 13.25/7.68 = ( Y3
% 13.25/7.68 = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7304_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7305_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7306_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7307_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 13.25/7.68 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7308_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7309_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7310_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7311_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 13.25/7.68 = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7312_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) )
% 13.25/7.68 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7313_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) )
% 13.25/7.68 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7314_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) )
% 13.25/7.68 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7315_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 13.25/7.68 ! [A: int,X: num,N: nat] :
% 13.25/7.68 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) )
% 13.25/7.68 = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7316_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7317_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7318_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7319_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 13.25/7.68 ! [X: num,N: nat,A: int] :
% 13.25/7.68 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
% 13.25/7.68 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X ) ) @ N ) @ A ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7320_bit__numeral__iff,axiom,
% 13.25/7.68 ! [M: num,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N )
% 13.25/7.68 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7321_bit__numeral__iff,axiom,
% 13.25/7.68 ! [M: num,N: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N )
% 13.25/7.68 = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7322_of__nat__less__of__int__iff,axiom,
% 13.25/7.68 ! [N: nat,X: int] :
% 13.25/7.68 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
% 13.25/7.68 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7323_of__nat__less__of__int__iff,axiom,
% 13.25/7.68 ! [N: nat,X: int] :
% 13.25/7.68 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X ) )
% 13.25/7.68 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7324_of__nat__less__of__int__iff,axiom,
% 13.25/7.68 ! [N: nat,X: int] :
% 13.25/7.68 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X ) )
% 13.25/7.68 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7325_bit__of__nat__iff__bit,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N )
% 13.25/7.68 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7326_bit__of__nat__iff__bit,axiom,
% 13.25/7.68 ! [M: nat,N: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N )
% 13.25/7.68 = ( bit_se1148574629649215175it_nat @ M @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7327_bit__disjunctive__add__iff,axiom,
% 13.25/7.68 ! [A: int,B: int,N: nat] :
% 13.25/7.68 ( ! [N4: nat] :
% 13.25/7.68 ( ~ ( bit_se1146084159140164899it_int @ A @ N4 )
% 13.25/7.68 | ~ ( bit_se1146084159140164899it_int @ B @ N4 ) )
% 13.25/7.68 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
% 13.25/7.68 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 13.25/7.68 | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7328_bit__disjunctive__add__iff,axiom,
% 13.25/7.68 ! [A: nat,B: nat,N: nat] :
% 13.25/7.68 ( ! [N4: nat] :
% 13.25/7.68 ( ~ ( bit_se1148574629649215175it_nat @ A @ N4 )
% 13.25/7.68 | ~ ( bit_se1148574629649215175it_nat @ B @ N4 ) )
% 13.25/7.68 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 13.25/7.68 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 13.25/7.68 | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7329_mult__of__int__commute,axiom,
% 13.25/7.68 ! [X: int,Y3: real] :
% 13.25/7.68 ( ( times_times_real @ ( ring_1_of_int_real @ X ) @ Y3 )
% 13.25/7.68 = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7330_mult__of__int__commute,axiom,
% 13.25/7.68 ! [X: int,Y3: rat] :
% 13.25/7.68 ( ( times_times_rat @ ( ring_1_of_int_rat @ X ) @ Y3 )
% 13.25/7.68 = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7331_mult__of__int__commute,axiom,
% 13.25/7.68 ! [X: int,Y3: int] :
% 13.25/7.68 ( ( times_times_int @ ( ring_1_of_int_int @ X ) @ Y3 )
% 13.25/7.68 = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7332_mult__of__nat__commute,axiom,
% 13.25/7.68 ! [X: nat,Y3: int] :
% 13.25/7.68 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y3 )
% 13.25/7.68 = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7333_mult__of__nat__commute,axiom,
% 13.25/7.68 ! [X: nat,Y3: real] :
% 13.25/7.68 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y3 )
% 13.25/7.68 = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7334_mult__of__nat__commute,axiom,
% 13.25/7.68 ! [X: nat,Y3: nat] :
% 13.25/7.68 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y3 )
% 13.25/7.68 = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7335_mult__of__nat__commute,axiom,
% 13.25/7.68 ! [X: nat,Y3: complex] :
% 13.25/7.68 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X ) @ Y3 )
% 13.25/7.68 = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7336_mult__of__nat__commute,axiom,
% 13.25/7.68 ! [X: nat,Y3: rat] :
% 13.25/7.68 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y3 )
% 13.25/7.68 = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7337_int__diff__cases,axiom,
% 13.25/7.68 ! [Z: int] :
% 13.25/7.68 ~ ! [M4: nat,N4: nat] :
% 13.25/7.68 ( Z
% 13.25/7.68 != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7338_bit__unset__bit__iff,axiom,
% 13.25/7.68 ! [M: nat,A: int,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N )
% 13.25/7.68 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 13.25/7.68 & ( M != N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7339_bit__unset__bit__iff,axiom,
% 13.25/7.68 ! [M: nat,A: nat,N: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N )
% 13.25/7.68 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 13.25/7.68 & ( M != N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7340_semiring__norm_I26_J,axiom,
% 13.25/7.68 ( ( bitM @ one )
% 13.25/7.68 = one ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7341_not__bit__1__Suc,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7342_not__bit__1__Suc,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7343_bit__1__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ one_one_int @ N )
% 13.25/7.68 = ( N = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7344_bit__1__iff,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N )
% 13.25/7.68 = ( N = zero_zero_nat ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7345_bit__numeral__simps_I1_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7346_bit__numeral__simps_I1_J,axiom,
% 13.25/7.68 ! [N: num] :
% 13.25/7.68 ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7347_of__nat__0__le__iff,axiom,
% 13.25/7.68 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7348_of__nat__0__le__iff,axiom,
% 13.25/7.68 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7349_of__nat__0__le__iff,axiom,
% 13.25/7.68 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7350_of__nat__0__le__iff,axiom,
% 13.25/7.68 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7351_of__nat__less__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7352_of__nat__less__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7353_of__nat__less__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7354_of__nat__less__0__iff,axiom,
% 13.25/7.68 ! [M: nat] :
% 13.25/7.68 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7355_bit__take__bit__iff,axiom,
% 13.25/7.68 ! [M: nat,A: int,N: nat] :
% 13.25/7.68 ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N )
% 13.25/7.68 = ( ( ord_less_nat @ N @ M )
% 13.25/7.68 & ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7356_bit__take__bit__iff,axiom,
% 13.25/7.68 ! [M: nat,A: nat,N: nat] :
% 13.25/7.68 ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N )
% 13.25/7.68 = ( ( ord_less_nat @ N @ M )
% 13.25/7.68 & ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7357_of__nat__neq__0,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 13.25/7.68 != zero_zero_int ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7358_of__nat__neq__0,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 13.25/7.68 != zero_zero_real ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7359_of__nat__neq__0,axiom,
% 13.25/7.68 ! [N: nat] :
% 13.25/7.68 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 13.25/7.68 != zero_zero_nat ) ).
% 13.25/7.68
% 13.25/7.68 thf(fact_7360_of__nat__neq__0,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
% 13.25/7.69 != zero_zero_complex ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7361_of__nat__neq__0,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 13.25/7.69 != zero_zero_rat ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7362_bit__of__bool__iff,axiom,
% 13.25/7.69 ! [B: $o,N: nat] :
% 13.25/7.69 ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N )
% 13.25/7.69 = ( B
% 13.25/7.69 & ( N = zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7363_bit__of__bool__iff,axiom,
% 13.25/7.69 ! [B: $o,N: nat] :
% 13.25/7.69 ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N )
% 13.25/7.69 = ( B
% 13.25/7.69 & ( N = zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7364_bit__of__bool__iff,axiom,
% 13.25/7.69 ! [B: $o,N: nat] :
% 13.25/7.69 ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N )
% 13.25/7.69 = ( B
% 13.25/7.69 & ( N = zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7365_div__mult2__eq_H,axiom,
% 13.25/7.69 ! [A: int,M: nat,N: nat] :
% 13.25/7.69 ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 13.25/7.69 = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7366_div__mult2__eq_H,axiom,
% 13.25/7.69 ! [A: nat,M: nat,N: nat] :
% 13.25/7.69 ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 13.25/7.69 = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7367_less__imp__of__nat__less,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_nat @ M @ N )
% 13.25/7.69 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7368_less__imp__of__nat__less,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_nat @ M @ N )
% 13.25/7.69 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7369_less__imp__of__nat__less,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_nat @ M @ N )
% 13.25/7.69 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7370_less__imp__of__nat__less,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_nat @ M @ N )
% 13.25/7.69 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7371_of__nat__less__imp__less,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.69 => ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7372_of__nat__less__imp__less,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.69 => ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7373_of__nat__less__imp__less,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.69 => ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7374_of__nat__less__imp__less,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 13.25/7.69 => ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7375_of__nat__mono,axiom,
% 13.25/7.69 ! [I: nat,J: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.69 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7376_of__nat__mono,axiom,
% 13.25/7.69 ! [I: nat,J: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.69 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7377_of__nat__mono,axiom,
% 13.25/7.69 ! [I: nat,J: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.69 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7378_of__nat__mono,axiom,
% 13.25/7.69 ! [I: nat,J: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.69 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7379_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) )
% 13.25/7.69 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7380_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N ) )
% 13.25/7.69 = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7381_of__nat__dvd__iff,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) )
% 13.25/7.69 = ( dvd_dvd_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7382_of__nat__dvd__iff,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.69 = ( dvd_dvd_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7383_of__nat__dvd__iff,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 13.25/7.69 = ( dvd_dvd_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7384_int__ops_I1_J,axiom,
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 13.25/7.69 = zero_zero_int ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7385_signed__take__bit__eq__if__positive,axiom,
% 13.25/7.69 ! [A: int,N: nat] :
% 13.25/7.69 ( ~ ( bit_se1146084159140164899it_int @ A @ N )
% 13.25/7.69 => ( ( bit_ri631733984087533419it_int @ N @ A )
% 13.25/7.69 = ( bit_se2923211474154528505it_int @ N @ A ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7386_int__ops_I3_J,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
% 13.25/7.69 = ( numeral_numeral_int @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7387_nat__int__comparison_I2_J,axiom,
% 13.25/7.69 ( ord_less_nat
% 13.25/7.69 = ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7388_zle__int,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.69 = ( ord_less_eq_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7389_nat__int__comparison_I3_J,axiom,
% 13.25/7.69 ( ord_less_eq_nat
% 13.25/7.69 = ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7390_zero__le__imp__eq__int,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.69 => ? [N4: nat] :
% 13.25/7.69 ( K
% 13.25/7.69 = ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7391_nonneg__int__cases,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.69 => ~ ! [N4: nat] :
% 13.25/7.69 ( K
% 13.25/7.69 != ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7392_of__nat__mod,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N ) )
% 13.25/7.69 = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7393_of__nat__mod,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) )
% 13.25/7.69 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7394_of__nat__mod,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N ) )
% 13.25/7.69 = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7395_zadd__int__left,axiom,
% 13.25/7.69 ! [M: nat,N: nat,Z: int] :
% 13.25/7.69 ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
% 13.25/7.69 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7396_int__plus,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
% 13.25/7.69 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7397_int__ops_I5_J,axiom,
% 13.25/7.69 ! [A: nat,B: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 13.25/7.69 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7398_int__ops_I2_J,axiom,
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 13.25/7.69 = one_one_int ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7399_int__ops_I7_J,axiom,
% 13.25/7.69 ! [A: nat,B: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
% 13.25/7.69 = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7400_zle__iff__zadd,axiom,
% 13.25/7.69 ( ord_less_eq_int
% 13.25/7.69 = ( ^ [W3: int,Z2: int] :
% 13.25/7.69 ? [N2: nat] :
% 13.25/7.69 ( Z2
% 13.25/7.69 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7401_not__int__zless__negative,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7402_take__bit__of__int,axiom,
% 13.25/7.69 ! [N: nat,K: int] :
% 13.25/7.69 ( ( bit_se2923211474154528505it_int @ N @ ( ring_1_of_int_int @ K ) )
% 13.25/7.69 = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7403_zdiv__int,axiom,
% 13.25/7.69 ! [A: nat,B: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
% 13.25/7.69 = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7404_of__nat__max,axiom,
% 13.25/7.69 ! [X: nat,Y3: nat] :
% 13.25/7.69 ( ( semiri4216267220026989637d_enat @ ( ord_max_nat @ X @ Y3 ) )
% 13.25/7.69 = ( ord_ma741700101516333627d_enat @ ( semiri4216267220026989637d_enat @ X ) @ ( semiri4216267220026989637d_enat @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7405_of__nat__max,axiom,
% 13.25/7.69 ! [X: nat,Y3: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y3 ) )
% 13.25/7.69 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7406_of__nat__max,axiom,
% 13.25/7.69 ! [X: nat,Y3: nat] :
% 13.25/7.69 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y3 ) )
% 13.25/7.69 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7407_of__nat__max,axiom,
% 13.25/7.69 ! [X: nat,Y3: nat] :
% 13.25/7.69 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y3 ) )
% 13.25/7.69 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7408_of__nat__max,axiom,
% 13.25/7.69 ! [X: nat,Y3: nat] :
% 13.25/7.69 ( ( semiri681578069525770553at_rat @ ( ord_max_nat @ X @ Y3 ) )
% 13.25/7.69 = ( ord_max_rat @ ( semiri681578069525770553at_rat @ X ) @ ( semiri681578069525770553at_rat @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7409_zmod__int,axiom,
% 13.25/7.69 ! [A: nat,B: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
% 13.25/7.69 = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7410_take__bit__of__nat,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( bit_se2923211474154528505it_int @ N @ ( semiri1314217659103216013at_int @ M ) )
% 13.25/7.69 = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7411_take__bit__of__nat,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( bit_se2925701944663578781it_nat @ N @ ( semiri1316708129612266289at_nat @ M ) )
% 13.25/7.69 = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7412_semiring__norm_I27_J,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( bitM @ ( bit0 @ N ) )
% 13.25/7.69 = ( bit1 @ ( bitM @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7413_semiring__norm_I28_J,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( bitM @ ( bit1 @ N ) )
% 13.25/7.69 = ( bit1 @ ( bit0 @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7414_nat__less__as__int,axiom,
% 13.25/7.69 ( ord_less_nat
% 13.25/7.69 = ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7415_nat__leq__as__int,axiom,
% 13.25/7.69 ( ord_less_eq_nat
% 13.25/7.69 = ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7416_inc__BitM__eq,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( inc @ ( bitM @ N ) )
% 13.25/7.69 = ( bit0 @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7417_BitM__inc__eq,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( bitM @ ( inc @ N ) )
% 13.25/7.69 = ( bit1 @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7418_of__nat__diff,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.69 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.69 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7419_of__nat__diff,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.69 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.69 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7420_of__nat__diff,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.69 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.69 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7421_of__nat__diff,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.69 => ( ( semiri8010041392384452111omplex @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.69 = ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7422_of__nat__diff,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.69 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 13.25/7.69 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7423_bit__not__int__iff_H,axiom,
% 13.25/7.69 ! [K: int,N: nat] :
% 13.25/7.69 ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N )
% 13.25/7.69 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7424_reals__Archimedean3,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.69 => ! [Y5: real] :
% 13.25/7.69 ? [N4: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7425_real__of__int__div4,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7426_int__cases4,axiom,
% 13.25/7.69 ! [M: int] :
% 13.25/7.69 ( ! [N4: nat] :
% 13.25/7.69 ( M
% 13.25/7.69 != ( semiri1314217659103216013at_int @ N4 ) )
% 13.25/7.69 => ~ ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 13.25/7.69 => ( M
% 13.25/7.69 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7427_real__of__nat__div4,axiom,
% 13.25/7.69 ! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7428_int__zle__neg,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 13.25/7.69 = ( ( N = zero_zero_nat )
% 13.25/7.69 & ( M = zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7429_int__ops_I4_J,axiom,
% 13.25/7.69 ! [A: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 13.25/7.69 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7430_int__Suc,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 13.25/7.69 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7431_zless__iff__Suc__zadd,axiom,
% 13.25/7.69 ( ord_less_int
% 13.25/7.69 = ( ^ [W3: int,Z2: int] :
% 13.25/7.69 ? [N2: nat] :
% 13.25/7.69 ( Z2
% 13.25/7.69 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7432_nonpos__int__cases,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 13.25/7.69 => ~ ! [N4: nat] :
% 13.25/7.69 ( K
% 13.25/7.69 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7433_negative__zle__0,axiom,
% 13.25/7.69 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7434_real__of__nat__div,axiom,
% 13.25/7.69 ! [D: nat,N: nat] :
% 13.25/7.69 ( ( dvd_dvd_nat @ D @ N )
% 13.25/7.69 => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ D ) )
% 13.25/7.69 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7435_real__of__int__div,axiom,
% 13.25/7.69 ! [D: int,N: int] :
% 13.25/7.69 ( ( dvd_dvd_int @ D @ N )
% 13.25/7.69 => ( ( ring_1_of_int_real @ ( divide_divide_int @ N @ D ) )
% 13.25/7.69 = ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ D ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7436_flip__bit__eq__if,axiom,
% 13.25/7.69 ( bit_se2159334234014336723it_int
% 13.25/7.69 = ( ^ [N2: nat,A3: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A3 @ N2 ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N2 @ A3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7437_flip__bit__eq__if,axiom,
% 13.25/7.69 ( bit_se2161824704523386999it_nat
% 13.25/7.69 = ( ^ [N2: nat,A3: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A3 @ N2 ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N2 @ A3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7438_eval__nat__numeral_I2_J,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( numeral_numeral_nat @ ( bit0 @ N ) )
% 13.25/7.69 = ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7439_one__plus__BitM,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( plus_plus_num @ one @ ( bitM @ N ) )
% 13.25/7.69 = ( bit0 @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7440_BitM__plus__one,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( plus_plus_num @ ( bitM @ N ) @ one )
% 13.25/7.69 = ( bit0 @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7441_of__int__nonneg,axiom,
% 13.25/7.69 ! [Z: int] :
% 13.25/7.69 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.69 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7442_of__int__nonneg,axiom,
% 13.25/7.69 ! [Z: int] :
% 13.25/7.69 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7443_of__int__nonneg,axiom,
% 13.25/7.69 ! [Z: int] :
% 13.25/7.69 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.69 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7444_of__int__leD,axiom,
% 13.25/7.69 ! [N: int,X: code_integer] :
% 13.25/7.69 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 13.25/7.69 => ( ( N = zero_zero_int )
% 13.25/7.69 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7445_of__int__leD,axiom,
% 13.25/7.69 ! [N: int,X: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 13.25/7.69 => ( ( N = zero_zero_int )
% 13.25/7.69 | ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7446_of__int__leD,axiom,
% 13.25/7.69 ! [N: int,X: rat] :
% 13.25/7.69 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 13.25/7.69 => ( ( N = zero_zero_int )
% 13.25/7.69 | ( ord_less_eq_rat @ one_one_rat @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7447_of__int__leD,axiom,
% 13.25/7.69 ! [N: int,X: int] :
% 13.25/7.69 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 13.25/7.69 => ( ( N = zero_zero_int )
% 13.25/7.69 | ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7448_of__int__pos,axiom,
% 13.25/7.69 ! [Z: int] :
% 13.25/7.69 ( ( ord_less_int @ zero_zero_int @ Z )
% 13.25/7.69 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7449_of__int__pos,axiom,
% 13.25/7.69 ! [Z: int] :
% 13.25/7.69 ( ( ord_less_int @ zero_zero_int @ Z )
% 13.25/7.69 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7450_of__int__pos,axiom,
% 13.25/7.69 ! [Z: int] :
% 13.25/7.69 ( ( ord_less_int @ zero_zero_int @ Z )
% 13.25/7.69 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7451_of__int__lessD,axiom,
% 13.25/7.69 ! [N: int,X: code_integer] :
% 13.25/7.69 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X )
% 13.25/7.69 => ( ( N = zero_zero_int )
% 13.25/7.69 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7452_of__int__lessD,axiom,
% 13.25/7.69 ! [N: int,X: real] :
% 13.25/7.69 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X )
% 13.25/7.69 => ( ( N = zero_zero_int )
% 13.25/7.69 | ( ord_less_real @ one_one_real @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7453_of__int__lessD,axiom,
% 13.25/7.69 ! [N: int,X: rat] :
% 13.25/7.69 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X )
% 13.25/7.69 => ( ( N = zero_zero_int )
% 13.25/7.69 | ( ord_less_rat @ one_one_rat @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7454_of__int__lessD,axiom,
% 13.25/7.69 ! [N: int,X: int] :
% 13.25/7.69 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
% 13.25/7.69 => ( ( N = zero_zero_int )
% 13.25/7.69 | ( ord_less_int @ one_one_int @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7455_mod__mult2__eq_H,axiom,
% 13.25/7.69 ! [A: code_integer,M: nat,N: nat] :
% 13.25/7.69 ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N ) ) )
% 13.25/7.69 = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7456_mod__mult2__eq_H,axiom,
% 13.25/7.69 ! [A: int,M: nat,N: nat] :
% 13.25/7.69 ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 13.25/7.69 = ( 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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7457_mod__mult2__eq_H,axiom,
% 13.25/7.69 ! [A: nat,M: nat,N: nat] :
% 13.25/7.69 ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) )
% 13.25/7.69 = ( 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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7458_of__int__neg__numeral,axiom,
% 13.25/7.69 ! [K: num] :
% 13.25/7.69 ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.69 = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7459_of__int__neg__numeral,axiom,
% 13.25/7.69 ! [K: num] :
% 13.25/7.69 ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.69 = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7460_of__int__neg__numeral,axiom,
% 13.25/7.69 ! [K: num] :
% 13.25/7.69 ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.69 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7461_of__int__neg__numeral,axiom,
% 13.25/7.69 ! [K: num] :
% 13.25/7.69 ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.69 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7462_of__int__neg__numeral,axiom,
% 13.25/7.69 ! [K: num] :
% 13.25/7.69 ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.69 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7463_bit__imp__take__bit__positive,axiom,
% 13.25/7.69 ! [N: nat,M: nat,K: int] :
% 13.25/7.69 ( ( ord_less_nat @ N @ M )
% 13.25/7.69 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 13.25/7.69 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7464_field__char__0__class_Oof__nat__div,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N ) )
% 13.25/7.69 = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7465_field__char__0__class_Oof__nat__div,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N ) )
% 13.25/7.69 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7466_field__char__0__class_Oof__nat__div,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N ) )
% 13.25/7.69 = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N ) ) ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7467_int__le__real__less,axiom,
% 13.25/7.69 ( ord_less_eq_int
% 13.25/7.69 = ( ^ [N2: int,M3: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M3 ) @ one_one_real ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7468_int__less__real__le,axiom,
% 13.25/7.69 ( ord_less_int
% 13.25/7.69 = ( ^ [N2: int,M3: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7469_bit__concat__bit__iff,axiom,
% 13.25/7.69 ! [M: nat,K: int,L2: int,N: nat] :
% 13.25/7.69 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L2 ) @ N )
% 13.25/7.69 = ( ( ( ord_less_nat @ N @ M )
% 13.25/7.69 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 13.25/7.69 | ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.69 & ( bit_se1146084159140164899it_int @ L2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7470_zero__less__imp__eq__int,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( ord_less_int @ zero_zero_int @ K )
% 13.25/7.69 => ? [N4: nat] :
% 13.25/7.69 ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 13.25/7.69 & ( K
% 13.25/7.69 = ( semiri1314217659103216013at_int @ N4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7471_pos__int__cases,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( ord_less_int @ zero_zero_int @ K )
% 13.25/7.69 => ~ ! [N4: nat] :
% 13.25/7.69 ( ( K
% 13.25/7.69 = ( semiri1314217659103216013at_int @ N4 ) )
% 13.25/7.69 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7472_int__cases3,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( K != zero_zero_int )
% 13.25/7.69 => ( ! [N4: nat] :
% 13.25/7.69 ( ( K
% 13.25/7.69 = ( semiri1314217659103216013at_int @ N4 ) )
% 13.25/7.69 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) )
% 13.25/7.69 => ~ ! [N4: nat] :
% 13.25/7.69 ( ( K
% 13.25/7.69 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 13.25/7.69 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7473_nat__less__real__le,axiom,
% 13.25/7.69 ( ord_less_nat
% 13.25/7.69 = ( ^ [N2: nat,M3: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7474_nat__le__real__less,axiom,
% 13.25/7.69 ( ord_less_eq_nat
% 13.25/7.69 = ( ^ [N2: nat,M3: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M3 ) @ one_one_real ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7475_zmult__zless__mono2__lemma,axiom,
% 13.25/7.69 ! [I: int,J: int,K: nat] :
% 13.25/7.69 ( ( ord_less_int @ I @ J )
% 13.25/7.69 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.25/7.69 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7476_not__zle__0__negative,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7477_negative__zless__0,axiom,
% 13.25/7.69 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7478_negD,axiom,
% 13.25/7.69 ! [X: int] :
% 13.25/7.69 ( ( ord_less_int @ X @ zero_zero_int )
% 13.25/7.69 => ? [N4: nat] :
% 13.25/7.69 ( X
% 13.25/7.69 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7479_int__ops_I6_J,axiom,
% 13.25/7.69 ! [A: nat,B: nat] :
% 13.25/7.69 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 13.25/7.69 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 13.25/7.69 = zero_zero_int ) )
% 13.25/7.69 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 13.25/7.69 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 13.25/7.69 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7480_real__of__int__div__aux,axiom,
% 13.25/7.69 ! [X: int,D: int] :
% 13.25/7.69 ( ( divide_divide_real @ ( ring_1_of_int_real @ X ) @ ( ring_1_of_int_real @ D ) )
% 13.25/7.69 = ( 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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7481_real__of__nat__div__aux,axiom,
% 13.25/7.69 ! [X: nat,D: nat] :
% 13.25/7.69 ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ D ) )
% 13.25/7.69 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7482_signed__take__bit__eq__concat__bit,axiom,
% 13.25/7.69 ( bit_ri631733984087533419it_int
% 13.25/7.69 = ( ^ [N2: nat,K2: int] : ( bit_concat_bit @ N2 @ K2 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7483_exp__eq__0__imp__not__bit,axiom,
% 13.25/7.69 ! [N: nat,A: int] :
% 13.25/7.69 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 13.25/7.69 = zero_zero_int )
% 13.25/7.69 => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7484_exp__eq__0__imp__not__bit,axiom,
% 13.25/7.69 ! [N: nat,A: nat] :
% 13.25/7.69 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.69 = zero_zero_nat )
% 13.25/7.69 => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7485_bit__Suc,axiom,
% 13.25/7.69 ! [A: int,N: nat] :
% 13.25/7.69 ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
% 13.25/7.69 = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7486_bit__Suc,axiom,
% 13.25/7.69 ! [A: nat,N: nat] :
% 13.25/7.69 ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
% 13.25/7.69 = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7487_stable__imp__bit__iff__odd,axiom,
% 13.25/7.69 ! [A: code_integer,N: nat] :
% 13.25/7.69 ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.69 = A )
% 13.25/7.69 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 13.25/7.69 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7488_stable__imp__bit__iff__odd,axiom,
% 13.25/7.69 ! [A: int,N: nat] :
% 13.25/7.69 ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.69 = A )
% 13.25/7.69 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 13.25/7.69 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7489_stable__imp__bit__iff__odd,axiom,
% 13.25/7.69 ! [A: nat,N: nat] :
% 13.25/7.69 ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.69 = A )
% 13.25/7.69 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 13.25/7.69 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7490_bit__iff__idd__imp__stable,axiom,
% 13.25/7.69 ! [A: code_integer] :
% 13.25/7.69 ( ! [N4: nat] :
% 13.25/7.69 ( ( bit_se9216721137139052372nteger @ A @ N4 )
% 13.25/7.69 = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 13.25/7.69 => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.69 = A ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7491_bit__iff__idd__imp__stable,axiom,
% 13.25/7.69 ! [A: int] :
% 13.25/7.69 ( ! [N4: nat] :
% 13.25/7.69 ( ( bit_se1146084159140164899it_int @ A @ N4 )
% 13.25/7.69 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 13.25/7.69 => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.69 = A ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7492_bit__iff__idd__imp__stable,axiom,
% 13.25/7.69 ! [A: nat] :
% 13.25/7.69 ( ! [N4: nat] :
% 13.25/7.69 ( ( bit_se1148574629649215175it_nat @ A @ N4 )
% 13.25/7.69 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 13.25/7.69 => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.69 = A ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7493_numeral__BitM,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
% 13.25/7.69 = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7494_numeral__BitM,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( numeral_numeral_real @ ( bitM @ N ) )
% 13.25/7.69 = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7495_numeral__BitM,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( numeral_numeral_rat @ ( bitM @ N ) )
% 13.25/7.69 = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7496_numeral__BitM,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( numeral_numeral_int @ ( bitM @ N ) )
% 13.25/7.69 = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7497_odd__numeral__BitM,axiom,
% 13.25/7.69 ! [W: num] :
% 13.25/7.69 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7498_odd__numeral__BitM,axiom,
% 13.25/7.69 ! [W: num] :
% 13.25/7.69 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7499_odd__numeral__BitM,axiom,
% 13.25/7.69 ! [W: num] :
% 13.25/7.69 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7500_of__nat__less__two__power,axiom,
% 13.25/7.69 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7501_of__nat__less__two__power,axiom,
% 13.25/7.69 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7502_of__nat__less__two__power,axiom,
% 13.25/7.69 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7503_int__bit__bound,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ~ ! [N4: nat] :
% 13.25/7.69 ( ! [M2: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N4 @ M2 )
% 13.25/7.69 => ( ( bit_se1146084159140164899it_int @ K @ M2 )
% 13.25/7.69 = ( bit_se1146084159140164899it_int @ K @ N4 ) ) )
% 13.25/7.69 => ~ ( ( ord_less_nat @ zero_zero_nat @ N4 )
% 13.25/7.69 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N4 @ one_one_nat ) )
% 13.25/7.69 = ( ~ ( bit_se1146084159140164899it_int @ K @ N4 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7504_inverse__of__nat__le,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.69 => ( ( N != zero_zero_nat )
% 13.25/7.69 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7505_inverse__of__nat__le,axiom,
% 13.25/7.69 ! [N: nat,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.69 => ( ( N != zero_zero_nat )
% 13.25/7.69 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7506_real__archimedian__rdiv__eq__0,axiom,
% 13.25/7.69 ! [X: real,C: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.69 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 13.25/7.69 => ( ! [M4: nat] :
% 13.25/7.69 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
% 13.25/7.69 => ( X = zero_zero_real ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7507_real__of__int__div2,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7508_real__of__int__div3,axiom,
% 13.25/7.69 ! [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 ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7509_neg__int__cases,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( ord_less_int @ K @ zero_zero_int )
% 13.25/7.69 => ~ ! [N4: nat] :
% 13.25/7.69 ( ( K
% 13.25/7.69 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N4 ) ) )
% 13.25/7.69 => ~ ( ord_less_nat @ zero_zero_nat @ N4 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7510_zdiff__int__split,axiom,
% 13.25/7.69 ! [P: int > $o,X: nat,Y3: nat] :
% 13.25/7.69 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y3 ) ) )
% 13.25/7.69 = ( ( ( ord_less_eq_nat @ Y3 @ X )
% 13.25/7.69 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 13.25/7.69 & ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.69 => ( P @ zero_zero_int ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7511_real__of__nat__div2,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7512_real__of__nat__div3,axiom,
% 13.25/7.69 ! [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 ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7513_ln__realpow,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.69 => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
% 13.25/7.69 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7514_bit__iff__odd,axiom,
% 13.25/7.69 ( bit_se9216721137139052372nteger
% 13.25/7.69 = ( ^ [A3: code_integer,N2: nat] :
% 13.25/7.69 ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7515_bit__iff__odd,axiom,
% 13.25/7.69 ( bit_se1146084159140164899it_int
% 13.25/7.69 = ( ^ [A3: int,N2: nat] :
% 13.25/7.69 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7516_bit__iff__odd,axiom,
% 13.25/7.69 ( bit_se1148574629649215175it_nat
% 13.25/7.69 = ( ^ [A3: nat,N2: nat] :
% 13.25/7.69 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7517_even__of__int__iff,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
% 13.25/7.69 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7518_even__of__int__iff,axiom,
% 13.25/7.69 ! [K: int] :
% 13.25/7.69 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
% 13.25/7.69 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7519_bit__int__def,axiom,
% 13.25/7.69 ( bit_se1146084159140164899it_int
% 13.25/7.69 = ( ^ [K2: int,N2: nat] :
% 13.25/7.69 ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7520_linear__plus__1__le__power,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.69 => ( 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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7521_Bernoulli__inequality,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.69 => ( 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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7522_even__bit__succ__iff,axiom,
% 13.25/7.69 ! [A: code_integer,N: nat] :
% 13.25/7.69 ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.69 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
% 13.25/7.69 = ( ( bit_se9216721137139052372nteger @ A @ N )
% 13.25/7.69 | ( N = zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7523_even__bit__succ__iff,axiom,
% 13.25/7.69 ! [A: int,N: nat] :
% 13.25/7.69 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.69 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
% 13.25/7.69 = ( ( bit_se1146084159140164899it_int @ A @ N )
% 13.25/7.69 | ( N = zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7524_even__bit__succ__iff,axiom,
% 13.25/7.69 ! [A: nat,N: nat] :
% 13.25/7.69 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.69 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
% 13.25/7.69 = ( ( bit_se1148574629649215175it_nat @ A @ N )
% 13.25/7.69 | ( N = zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7525_odd__bit__iff__bit__pred,axiom,
% 13.25/7.69 ! [A: code_integer,N: nat] :
% 13.25/7.69 ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 13.25/7.69 => ( ( bit_se9216721137139052372nteger @ A @ N )
% 13.25/7.69 = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
% 13.25/7.69 | ( N = zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7526_odd__bit__iff__bit__pred,axiom,
% 13.25/7.69 ! [A: int,N: nat] :
% 13.25/7.69 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 13.25/7.69 => ( ( bit_se1146084159140164899it_int @ A @ N )
% 13.25/7.69 = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
% 13.25/7.69 | ( N = zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7527_odd__bit__iff__bit__pred,axiom,
% 13.25/7.69 ! [A: nat,N: nat] :
% 13.25/7.69 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 13.25/7.69 => ( ( bit_se1148574629649215175it_nat @ A @ N )
% 13.25/7.69 = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
% 13.25/7.69 | ( N = zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7528_bit__sum__mult__2__cases,axiom,
% 13.25/7.69 ! [A: code_integer,B: code_integer,N: nat] :
% 13.25/7.69 ( ! [J2: nat] :
% 13.25/7.69 ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 13.25/7.69 => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
% 13.25/7.69 = ( ( ( N = zero_zero_nat )
% 13.25/7.69 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.69 & ( ( N != zero_zero_nat )
% 13.25/7.69 => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7529_bit__sum__mult__2__cases,axiom,
% 13.25/7.69 ! [A: int,B: int,N: nat] :
% 13.25/7.69 ( ! [J2: nat] :
% 13.25/7.69 ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 13.25/7.69 => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
% 13.25/7.69 = ( ( ( N = zero_zero_nat )
% 13.25/7.69 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.69 & ( ( N != zero_zero_nat )
% 13.25/7.69 => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7530_bit__sum__mult__2__cases,axiom,
% 13.25/7.69 ! [A: nat,B: nat,N: nat] :
% 13.25/7.69 ( ! [J2: nat] :
% 13.25/7.69 ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 13.25/7.69 => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
% 13.25/7.69 = ( ( ( N = zero_zero_nat )
% 13.25/7.69 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 13.25/7.69 & ( ( N != zero_zero_nat )
% 13.25/7.69 => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7531_bit__rec,axiom,
% 13.25/7.69 ( bit_se9216721137139052372nteger
% 13.25/7.69 = ( ^ [A3: code_integer,N2: nat] :
% 13.25/7.69 ( ( ( N2 = zero_zero_nat )
% 13.25/7.69 => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
% 13.25/7.69 & ( ( N2 != zero_zero_nat )
% 13.25/7.69 => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7532_bit__rec,axiom,
% 13.25/7.69 ( bit_se1146084159140164899it_int
% 13.25/7.69 = ( ^ [A3: int,N2: nat] :
% 13.25/7.69 ( ( ( N2 = zero_zero_nat )
% 13.25/7.69 => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
% 13.25/7.69 & ( ( N2 != zero_zero_nat )
% 13.25/7.69 => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7533_bit__rec,axiom,
% 13.25/7.69 ( bit_se1148574629649215175it_nat
% 13.25/7.69 = ( ^ [A3: nat,N2: nat] :
% 13.25/7.69 ( ( ( N2 = zero_zero_nat )
% 13.25/7.69 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
% 13.25/7.69 & ( ( N2 != zero_zero_nat )
% 13.25/7.69 => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7534_set__bit__eq,axiom,
% 13.25/7.69 ( bit_se7879613467334960850it_int
% 13.25/7.69 = ( ^ [N2: nat,K2: int] :
% 13.25/7.69 ( plus_plus_int @ K2
% 13.25/7.69 @ ( times_times_int
% 13.25/7.69 @ ( zero_n2684676970156552555ol_int
% 13.25/7.69 @ ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) )
% 13.25/7.69 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7535_unset__bit__eq,axiom,
% 13.25/7.69 ( bit_se4203085406695923979it_int
% 13.25/7.69 = ( ^ [N2: nat,K2: int] : ( minus_minus_int @ K2 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7536_take__bit__Suc__from__most,axiom,
% 13.25/7.69 ! [N: nat,K: int] :
% 13.25/7.69 ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K )
% 13.25/7.69 = ( 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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7537_Bernoulli__inequality__even,axiom,
% 13.25/7.69 ! [N: nat,X: real] :
% 13.25/7.69 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.69 => ( 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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7538_double__arith__series,axiom,
% 13.25/7.69 ! [A: int,D: int,N: nat] :
% 13.25/7.69 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 13.25/7.69 @ ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 13.25/7.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7539_double__arith__series,axiom,
% 13.25/7.69 ! [A: complex,D: complex,N: nat] :
% 13.25/7.69 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 13.25/7.69 @ ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [I4: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 13.25/7.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7540_double__arith__series,axiom,
% 13.25/7.69 ! [A: rat,D: rat,N: nat] :
% 13.25/7.69 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 13.25/7.69 @ ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [I4: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 13.25/7.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7541_double__arith__series,axiom,
% 13.25/7.69 ! [A: nat,D: nat,N: nat] :
% 13.25/7.69 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 13.25/7.69 @ ( groups3542108847815614940at_nat
% 13.25/7.69 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 13.25/7.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7542_double__arith__series,axiom,
% 13.25/7.69 ! [A: real,D: real,N: nat] :
% 13.25/7.69 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 13.25/7.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7543_double__gauss__sum,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7544_double__gauss__sum,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7545_double__gauss__sum,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7546_double__gauss__sum,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7547_double__gauss__sum,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.69 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7548_arith__series,axiom,
% 13.25/7.69 ! [A: int,D: int,N: nat] :
% 13.25/7.69 ( ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [I4: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 13.25/7.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.69 = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7549_arith__series,axiom,
% 13.25/7.69 ! [A: nat,D: nat,N: nat] :
% 13.25/7.69 ( ( groups3542108847815614940at_nat
% 13.25/7.69 @ ^ [I4: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 13.25/7.69 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.69 = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7550_gauss__sum,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.69 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7551_gauss__sum,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 13.25/7.69 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7552_double__gauss__sum__from__Suc__0,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 13.25/7.69 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7553_double__gauss__sum__from__Suc__0,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 13.25/7.69 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7554_double__gauss__sum__from__Suc__0,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 13.25/7.69 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7555_double__gauss__sum__from__Suc__0,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 13.25/7.69 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7556_double__gauss__sum__from__Suc__0,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 13.25/7.69 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7557_sum__gp__offset,axiom,
% 13.25/7.69 ! [X: complex,M: nat,N: nat] :
% 13.25/7.69 ( ( ( X = one_one_complex )
% 13.25/7.69 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 13.25/7.69 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
% 13.25/7.69 & ( ( X != one_one_complex )
% 13.25/7.69 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 13.25/7.69 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7558_sum__gp__offset,axiom,
% 13.25/7.69 ! [X: rat,M: nat,N: nat] :
% 13.25/7.69 ( ( ( X = one_one_rat )
% 13.25/7.69 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 13.25/7.69 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
% 13.25/7.69 & ( ( X != one_one_rat )
% 13.25/7.69 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 13.25/7.69 = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7559_sum__gp__offset,axiom,
% 13.25/7.69 ! [X: real,M: nat,N: nat] :
% 13.25/7.69 ( ( ( X = one_one_real )
% 13.25/7.69 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 13.25/7.69 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
% 13.25/7.69 & ( ( X != one_one_real )
% 13.25/7.69 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 13.25/7.69 = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7560_of__nat__code__if,axiom,
% 13.25/7.69 ( semiri1314217659103216013at_int
% 13.25/7.69 = ( ^ [N2: nat] :
% 13.25/7.69 ( if_int @ ( N2 = zero_zero_nat ) @ zero_zero_int
% 13.25/7.69 @ ( produc6840382203811409530at_int
% 13.25/7.69 @ ^ [M3: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M3 ) ) @ one_one_int ) )
% 13.25/7.69 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7561_of__nat__code__if,axiom,
% 13.25/7.69 ( semiri5074537144036343181t_real
% 13.25/7.69 = ( ^ [N2: nat] :
% 13.25/7.69 ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 13.25/7.69 @ ( produc1703576794950452218t_real
% 13.25/7.69 @ ^ [M3: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M3 ) ) @ one_one_real ) )
% 13.25/7.69 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7562_of__nat__code__if,axiom,
% 13.25/7.69 ( semiri8010041392384452111omplex
% 13.25/7.69 = ( ^ [N2: nat] :
% 13.25/7.69 ( if_complex @ ( N2 = zero_zero_nat ) @ zero_zero_complex
% 13.25/7.69 @ ( produc1917071388513777916omplex
% 13.25/7.69 @ ^ [M3: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M3 ) ) @ one_one_complex ) )
% 13.25/7.69 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7563_of__nat__code__if,axiom,
% 13.25/7.69 ( semiri681578069525770553at_rat
% 13.25/7.69 = ( ^ [N2: nat] :
% 13.25/7.69 ( if_rat @ ( N2 = zero_zero_nat ) @ zero_zero_rat
% 13.25/7.69 @ ( produc6207742614233964070at_rat
% 13.25/7.69 @ ^ [M3: nat,Q5: nat] : ( if_rat @ ( Q5 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M3 ) ) @ one_one_rat ) )
% 13.25/7.69 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7564_of__nat__code__if,axiom,
% 13.25/7.69 ( semiri1316708129612266289at_nat
% 13.25/7.69 = ( ^ [N2: nat] :
% 13.25/7.69 ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 13.25/7.69 @ ( produc6842872674320459806at_nat
% 13.25/7.69 @ ^ [M3: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M3 ) ) @ one_one_nat ) )
% 13.25/7.69 @ ( divmod_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7565_nat__approx__posE,axiom,
% 13.25/7.69 ! [E: real] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ E )
% 13.25/7.69 => ~ ! [N4: nat] :
% 13.25/7.69 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ E ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7566_nat__approx__posE,axiom,
% 13.25/7.69 ! [E: rat] :
% 13.25/7.69 ( ( ord_less_rat @ zero_zero_rat @ E )
% 13.25/7.69 => ~ ! [N4: nat] :
% 13.25/7.69 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N4 ) ) ) @ E ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7567_floor__exists1,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ? [X3: int] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ X )
% 13.25/7.69 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 13.25/7.69 & ! [Y5: int] :
% 13.25/7.69 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X )
% 13.25/7.69 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 13.25/7.69 => ( Y5 = X3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7568_floor__exists1,axiom,
% 13.25/7.69 ! [X: rat] :
% 13.25/7.69 ? [X3: int] :
% 13.25/7.69 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ X )
% 13.25/7.69 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ X3 @ one_one_int ) ) )
% 13.25/7.69 & ! [Y5: int] :
% 13.25/7.69 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X )
% 13.25/7.69 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 13.25/7.69 => ( Y5 = X3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7569_floor__exists,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ? [Z5: int] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
% 13.25/7.69 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7570_floor__exists,axiom,
% 13.25/7.69 ! [X: rat] :
% 13.25/7.69 ? [Z5: int] :
% 13.25/7.69 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
% 13.25/7.69 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7571_monoseq__arctan__series,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.69 => ( topolo6980174941875973593q_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7572_lemma__termdiff3,axiom,
% 13.25/7.69 ! [H2: real,Z: real,K5: real,N: nat] :
% 13.25/7.69 ( ( H2 != zero_zero_real )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K5 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7573_lemma__termdiff3,axiom,
% 13.25/7.69 ! [H2: complex,Z: complex,K5: real,N: nat] :
% 13.25/7.69 ( ( H2 != zero_zero_complex )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K5 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7574_int__if,axiom,
% 13.25/7.69 ! [P: $o,A: nat,B: nat] :
% 13.25/7.69 ( ( P
% 13.25/7.69 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 13.25/7.69 = ( semiri1314217659103216013at_int @ A ) ) )
% 13.25/7.69 & ( ~ P
% 13.25/7.69 => ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
% 13.25/7.69 = ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7575_nat__int__comparison_I1_J,axiom,
% 13.25/7.69 ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.69 = ( ^ [A3: nat,B4: nat] :
% 13.25/7.69 ( ( semiri1314217659103216013at_int @ A3 )
% 13.25/7.69 = ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7576_bit__Suc__0__iff,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.69 = ( N = zero_zero_nat ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7577_not__bit__Suc__0__Suc,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7578_complex__mod__minus__le__complex__mod,axiom,
% 13.25/7.69 ! [X: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7579_complex__mod__triangle__ineq2,axiom,
% 13.25/7.69 ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7580_not__bit__Suc__0__numeral,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7581_bit__nat__def,axiom,
% 13.25/7.69 ( bit_se1148574629649215175it_nat
% 13.25/7.69 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.69 ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7582_monoseq__realpow,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.69 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.69 => ( topolo6980174941875973593q_real @ ( power_power_real @ X ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7583_real__arch__simple,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ? [N4: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7584_real__arch__simple,axiom,
% 13.25/7.69 ! [X: rat] :
% 13.25/7.69 ? [N4: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7585_reals__Archimedean2,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ? [N4: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N4 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7586_reals__Archimedean2,axiom,
% 13.25/7.69 ! [X: rat] :
% 13.25/7.69 ? [N4: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N4 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7587_ex__le__of__int,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ? [Z5: int] : ( ord_less_eq_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7588_ex__le__of__int,axiom,
% 13.25/7.69 ! [X: rat] :
% 13.25/7.69 ? [Z5: int] : ( ord_less_eq_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7589_ex__of__int__less,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ? [Z5: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z5 ) @ X ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7590_ex__of__int__less,axiom,
% 13.25/7.69 ! [X: rat] :
% 13.25/7.69 ? [Z5: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z5 ) @ X ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7591_ex__less__of__int,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ? [Z5: int] : ( ord_less_real @ X @ ( ring_1_of_int_real @ Z5 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7592_ex__less__of__int,axiom,
% 13.25/7.69 ! [X: rat] :
% 13.25/7.69 ? [Z5: int] : ( ord_less_rat @ X @ ( ring_1_of_int_rat @ Z5 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7593_ex__less__of__nat__mult,axiom,
% 13.25/7.69 ! [X: real,Y3: real] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.69 => ? [N4: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7594_ex__less__of__nat__mult,axiom,
% 13.25/7.69 ! [X: rat,Y3: rat] :
% 13.25/7.69 ( ( ord_less_rat @ zero_zero_rat @ X )
% 13.25/7.69 => ? [N4: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N4 ) @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7595_norm__divide__numeral,axiom,
% 13.25/7.69 ! [A: real,W: num] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.69 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7596_norm__divide__numeral,axiom,
% 13.25/7.69 ! [A: complex,W: num] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.69 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7597_norm__mult__numeral2,axiom,
% 13.25/7.69 ! [A: real,W: num] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.69 = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7598_norm__mult__numeral2,axiom,
% 13.25/7.69 ! [A: complex,W: num] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.69 = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7599_norm__mult__numeral1,axiom,
% 13.25/7.69 ! [W: num,A: real] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
% 13.25/7.69 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7600_norm__mult__numeral1,axiom,
% 13.25/7.69 ! [W: num,A: complex] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
% 13.25/7.69 = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7601_norm__neg__numeral,axiom,
% 13.25/7.69 ! [W: num] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.69 = ( numeral_numeral_real @ W ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7602_norm__neg__numeral,axiom,
% 13.25/7.69 ! [W: num] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.69 = ( numeral_numeral_real @ W ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7603_norm__le__zero__iff,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
% 13.25/7.69 = ( X = zero_zero_real ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7604_norm__le__zero__iff,axiom,
% 13.25/7.69 ! [X: complex] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
% 13.25/7.69 = ( X = zero_zero_complex ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7605_norm__one,axiom,
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ one_one_real )
% 13.25/7.69 = one_one_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7606_norm__one,axiom,
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ one_one_complex )
% 13.25/7.69 = one_one_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7607_norm__numeral,axiom,
% 13.25/7.69 ! [W: num] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
% 13.25/7.69 = ( numeral_numeral_real @ W ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7608_norm__numeral,axiom,
% 13.25/7.69 ! [W: num] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.69 = ( numeral_numeral_real @ W ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7609_zero__less__norm__iff,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
% 13.25/7.69 = ( X != zero_zero_real ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7610_zero__less__norm__iff,axiom,
% 13.25/7.69 ! [X: complex] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
% 13.25/7.69 = ( X != zero_zero_complex ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7611_norm__minus__commute,axiom,
% 13.25/7.69 ! [A: real,B: real] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) )
% 13.25/7.69 = ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7612_norm__minus__commute,axiom,
% 13.25/7.69 ! [A: complex,B: complex] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) )
% 13.25/7.69 = ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ A ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7613_norm__not__less__zero,axiom,
% 13.25/7.69 ! [X: complex] :
% 13.25/7.69 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7614_norm__ge__zero,axiom,
% 13.25/7.69 ! [X: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7615_norm__mult,axiom,
% 13.25/7.69 ! [X: real,Y3: real] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y3 ) )
% 13.25/7.69 = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7616_norm__mult,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y3 ) )
% 13.25/7.69 = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7617_sum__norm__le,axiom,
% 13.25/7.69 ! [S2: set_real,F: real > complex,G: real > real] :
% 13.25/7.69 ( ! [X3: real] :
% 13.25/7.69 ( ( member_real @ X3 @ S2 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7618_sum__norm__le,axiom,
% 13.25/7.69 ! [S2: set_int,F: int > complex,G: int > real] :
% 13.25/7.69 ( ! [X3: int] :
% 13.25/7.69 ( ( member_int @ X3 @ S2 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7619_sum__norm__le,axiom,
% 13.25/7.69 ! [S2: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > complex,G: produc859450856879609959at_nat > real] :
% 13.25/7.69 ( ! [X3: produc859450856879609959at_nat] :
% 13.25/7.69 ( ( member8206827879206165904at_nat @ X3 @ S2 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7963564606188603611omplex @ F @ S2 ) ) @ ( groups7817252763086508633t_real @ G @ S2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7620_sum__norm__le,axiom,
% 13.25/7.69 ! [S2: set_nat,F: nat > complex,G: nat > real] :
% 13.25/7.69 ( ! [X3: nat] :
% 13.25/7.69 ( ( member_nat @ X3 @ S2 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7621_sum__norm__le,axiom,
% 13.25/7.69 ! [S2: set_complex,F: complex > complex,G: complex > real] :
% 13.25/7.69 ( ! [X3: complex] :
% 13.25/7.69 ( ( member_complex @ X3 @ S2 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7622_sum__norm__le,axiom,
% 13.25/7.69 ! [S2: set_nat,F: nat > real,G: nat > real] :
% 13.25/7.69 ( ! [X3: nat] :
% 13.25/7.69 ( ( member_nat @ X3 @ S2 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X3 ) ) @ ( G @ X3 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7623_norm__divide,axiom,
% 13.25/7.69 ! [A: real,B: real] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 13.25/7.69 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7624_norm__divide,axiom,
% 13.25/7.69 ! [A: complex,B: complex] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.69 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7625_norm__power,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) )
% 13.25/7.69 = ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7626_norm__power,axiom,
% 13.25/7.69 ! [X: complex,N: nat] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) )
% 13.25/7.69 = ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7627_norm__sum,axiom,
% 13.25/7.69 ! [F: nat > complex,A2: set_nat] :
% 13.25/7.69 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 13.25/7.69 @ A2 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7628_norm__sum,axiom,
% 13.25/7.69 ! [F: complex > complex,A2: set_complex] :
% 13.25/7.69 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
% 13.25/7.69 @ ( groups5808333547571424918x_real
% 13.25/7.69 @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 13.25/7.69 @ A2 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7629_norm__sum,axiom,
% 13.25/7.69 ! [F: nat > real,A2: set_nat] :
% 13.25/7.69 ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
% 13.25/7.69 @ A2 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7630_norm__uminus__minus,axiom,
% 13.25/7.69 ! [X: real,Y3: real] :
% 13.25/7.69 ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ Y3 ) )
% 13.25/7.69 = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7631_norm__uminus__minus,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex] :
% 13.25/7.69 ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ Y3 ) )
% 13.25/7.69 = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7632_nonzero__norm__divide,axiom,
% 13.25/7.69 ! [B: real,A: real] :
% 13.25/7.69 ( ( B != zero_zero_real )
% 13.25/7.69 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 13.25/7.69 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7633_nonzero__norm__divide,axiom,
% 13.25/7.69 ! [B: complex,A: complex] :
% 13.25/7.69 ( ( B != zero_zero_complex )
% 13.25/7.69 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.69 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7634_power__eq__imp__eq__norm,axiom,
% 13.25/7.69 ! [W: real,N: nat,Z: real] :
% 13.25/7.69 ( ( ( power_power_real @ W @ N )
% 13.25/7.69 = ( power_power_real @ Z @ N ) )
% 13.25/7.69 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.69 => ( ( real_V7735802525324610683m_real @ W )
% 13.25/7.69 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7635_power__eq__imp__eq__norm,axiom,
% 13.25/7.69 ! [W: complex,N: nat,Z: complex] :
% 13.25/7.69 ( ( ( power_power_complex @ W @ N )
% 13.25/7.69 = ( power_power_complex @ Z @ N ) )
% 13.25/7.69 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.69 => ( ( real_V1022390504157884413omplex @ W )
% 13.25/7.69 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7636_norm__mult__less,axiom,
% 13.25/7.69 ! [X: real,R: real,Y3: real,S: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 13.25/7.69 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y3 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7637_norm__mult__less,axiom,
% 13.25/7.69 ! [X: complex,R: real,Y3: complex,S: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 13.25/7.69 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y3 ) ) @ ( times_times_real @ R @ S ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7638_norm__mult__ineq,axiom,
% 13.25/7.69 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7639_norm__mult__ineq,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7640_norm__triangle__lt,axiom,
% 13.25/7.69 ! [X: real,Y3: real,E: real] :
% 13.25/7.69 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 13.25/7.69 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) @ E ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7641_norm__triangle__lt,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex,E: real] :
% 13.25/7.69 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 13.25/7.69 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) @ E ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7642_norm__add__less,axiom,
% 13.25/7.69 ! [X: real,R: real,Y3: real,S: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 13.25/7.69 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7643_norm__add__less,axiom,
% 13.25/7.69 ! [X: complex,R: real,Y3: complex,S: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 13.25/7.69 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7644_norm__power__ineq,axiom,
% 13.25/7.69 ! [X: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7645_norm__power__ineq,axiom,
% 13.25/7.69 ! [X: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X ) @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7646_norm__triangle__mono,axiom,
% 13.25/7.69 ! [A: real,R: real,B: real,S: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7647_norm__triangle__mono,axiom,
% 13.25/7.69 ! [A: complex,R: real,B: complex,S: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R @ S ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7648_norm__triangle__ineq,axiom,
% 13.25/7.69 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7649_norm__triangle__ineq,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7650_norm__triangle__le,axiom,
% 13.25/7.69 ! [X: real,Y3: real,E: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X @ Y3 ) ) @ E ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7651_norm__triangle__le,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex,E: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X @ Y3 ) ) @ E ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7652_norm__add__leD,axiom,
% 13.25/7.69 ! [A: real,B: real,C: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7653_norm__add__leD,axiom,
% 13.25/7.69 ! [A: complex,B: complex,C: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7654_norm__diff__triangle__less,axiom,
% 13.25/7.69 ! [X: real,Y3: real,E1: real,Z: real,E22: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y3 ) ) @ E1 )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 13.25/7.69 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7655_norm__diff__triangle__less,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y3 ) ) @ E1 )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 13.25/7.69 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7656_norm__triangle__sub,axiom,
% 13.25/7.69 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7657_norm__triangle__sub,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7658_norm__triangle__ineq4,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7659_norm__triangle__ineq4,axiom,
% 13.25/7.69 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7660_norm__diff__triangle__le,axiom,
% 13.25/7.69 ! [X: real,Y3: real,E1: real,Z: real,E22: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y3 ) ) @ E1 )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7661_norm__diff__triangle__le,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y3 ) ) @ E1 )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7662_norm__triangle__le__diff,axiom,
% 13.25/7.69 ! [X: real,Y3: real,E: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X @ Y3 ) ) @ E ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7663_norm__triangle__le__diff,axiom,
% 13.25/7.69 ! [X: complex,Y3: complex,E: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X @ Y3 ) ) @ E ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7664_norm__diff__ineq,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7665_norm__diff__ineq,axiom,
% 13.25/7.69 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7666_norm__triangle__ineq2,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7667_norm__triangle__ineq2,axiom,
% 13.25/7.69 ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7668_power__eq__1__iff,axiom,
% 13.25/7.69 ! [W: real,N: nat] :
% 13.25/7.69 ( ( ( power_power_real @ W @ N )
% 13.25/7.69 = one_one_real )
% 13.25/7.69 => ( ( ( real_V7735802525324610683m_real @ W )
% 13.25/7.69 = one_one_real )
% 13.25/7.69 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7669_power__eq__1__iff,axiom,
% 13.25/7.69 ! [W: complex,N: nat] :
% 13.25/7.69 ( ( ( power_power_complex @ W @ N )
% 13.25/7.69 = one_one_complex )
% 13.25/7.69 => ( ( ( real_V1022390504157884413omplex @ W )
% 13.25/7.69 = one_one_real )
% 13.25/7.69 | ( N = zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7670_norm__diff__triangle__ineq,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7671_norm__diff__triangle__ineq,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7672_norm__triangle__ineq3,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7673_norm__triangle__ineq3,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7674_square__norm__one,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.69 = one_one_real )
% 13.25/7.69 => ( ( real_V7735802525324610683m_real @ X )
% 13.25/7.69 = one_one_real ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7675_square__norm__one,axiom,
% 13.25/7.69 ! [X: complex] :
% 13.25/7.69 ( ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.69 = one_one_complex )
% 13.25/7.69 => ( ( real_V1022390504157884413omplex @ X )
% 13.25/7.69 = one_one_real ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7676_norm__power__diff,axiom,
% 13.25/7.69 ! [Z: real,W: real,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
% 13.25/7.69 => ( 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 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7677_norm__power__diff,axiom,
% 13.25/7.69 ! [Z: complex,W: complex,M: nat] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
% 13.25/7.69 => ( 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 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7678_ln__series,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.69 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.69 => ( ( ln_ln_real @ X )
% 13.25/7.69 = ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7679_arctan__series,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.69 => ( ( arctan @ X )
% 13.25/7.69 = ( suminf_real
% 13.25/7.69 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7680_round__unique,axiom,
% 13.25/7.69 ! [X: real,Y3: int] :
% 13.25/7.69 ( ( ord_less_real @ ( minus_minus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 13.25/7.69 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 13.25/7.69 => ( ( archim8280529875227126926d_real @ X )
% 13.25/7.69 = Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7681_round__unique,axiom,
% 13.25/7.69 ! [X: rat,Y3: int] :
% 13.25/7.69 ( ( ord_less_rat @ ( minus_minus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 13.25/7.69 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.69 => ( ( archim7778729529865785530nd_rat @ X )
% 13.25/7.69 = Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7682_lemma__termdiff2,axiom,
% 13.25/7.69 ! [H2: complex,Z: complex,N: nat] :
% 13.25/7.69 ( ( H2 != zero_zero_complex )
% 13.25/7.69 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 13.25/7.69 = ( times_times_complex @ H2
% 13.25/7.69 @ ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [P4: nat] :
% 13.25/7.69 ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q5 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7683_lemma__termdiff2,axiom,
% 13.25/7.69 ! [H2: rat,Z: rat,N: nat] :
% 13.25/7.69 ( ( H2 != zero_zero_rat )
% 13.25/7.69 => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 13.25/7.69 = ( times_times_rat @ H2
% 13.25/7.69 @ ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [P4: nat] :
% 13.25/7.69 ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q5 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7684_lemma__termdiff2,axiom,
% 13.25/7.69 ! [H2: real,Z: real,N: nat] :
% 13.25/7.69 ( ( H2 != zero_zero_real )
% 13.25/7.69 => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 13.25/7.69 = ( times_times_real @ H2
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [P4: nat] :
% 13.25/7.69 ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q5 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7685_summable__arctan__series,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7686_lessThan__iff,axiom,
% 13.25/7.69 ! [I: rat,K: rat] :
% 13.25/7.69 ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 13.25/7.69 = ( ord_less_rat @ I @ K ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7687_lessThan__iff,axiom,
% 13.25/7.69 ! [I: num,K: num] :
% 13.25/7.69 ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 13.25/7.69 = ( ord_less_num @ I @ K ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7688_lessThan__iff,axiom,
% 13.25/7.69 ! [I: nat,K: nat] :
% 13.25/7.69 ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 13.25/7.69 = ( ord_less_nat @ I @ K ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7689_lessThan__iff,axiom,
% 13.25/7.69 ! [I: int,K: int] :
% 13.25/7.69 ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 13.25/7.69 = ( ord_less_int @ I @ K ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7690_lessThan__iff,axiom,
% 13.25/7.69 ! [I: real,K: real] :
% 13.25/7.69 ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 13.25/7.69 = ( ord_less_real @ I @ K ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7691_lessThan__subset__iff,axiom,
% 13.25/7.69 ! [X: rat,Y3: rat] :
% 13.25/7.69 ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 13.25/7.69 = ( ord_less_eq_rat @ X @ Y3 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7692_lessThan__subset__iff,axiom,
% 13.25/7.69 ! [X: num,Y3: num] :
% 13.25/7.69 ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y3 ) )
% 13.25/7.69 = ( ord_less_eq_num @ X @ Y3 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7693_lessThan__subset__iff,axiom,
% 13.25/7.69 ! [X: nat,Y3: nat] :
% 13.25/7.69 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 13.25/7.69 = ( ord_less_eq_nat @ X @ Y3 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7694_lessThan__subset__iff,axiom,
% 13.25/7.69 ! [X: int,Y3: int] :
% 13.25/7.69 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y3 ) )
% 13.25/7.69 = ( ord_less_eq_int @ X @ Y3 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7695_lessThan__subset__iff,axiom,
% 13.25/7.69 ! [X: real,Y3: real] :
% 13.25/7.69 ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 13.25/7.69 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7696_round__numeral,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
% 13.25/7.69 = ( numeral_numeral_int @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7697_round__numeral,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
% 13.25/7.69 = ( numeral_numeral_int @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7698_round__1,axiom,
% 13.25/7.69 ( ( archim8280529875227126926d_real @ one_one_real )
% 13.25/7.69 = one_one_int ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7699_round__1,axiom,
% 13.25/7.69 ( ( archim7778729529865785530nd_rat @ one_one_rat )
% 13.25/7.69 = one_one_int ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7700_sum_OlessThan__Suc,axiom,
% 13.25/7.69 ! [G: nat > rat,N: nat] :
% 13.25/7.69 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7701_sum_OlessThan__Suc,axiom,
% 13.25/7.69 ! [G: nat > int,N: nat] :
% 13.25/7.69 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7702_sum_OlessThan__Suc,axiom,
% 13.25/7.69 ! [G: nat > nat,N: nat] :
% 13.25/7.69 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7703_sum_OlessThan__Suc,axiom,
% 13.25/7.69 ! [G: nat > real,N: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7704_round__neg__numeral,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.69 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7705_round__neg__numeral,axiom,
% 13.25/7.69 ! [N: num] :
% 13.25/7.69 ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 13.25/7.69 = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7706_powser__zero,axiom,
% 13.25/7.69 ! [F: nat > complex] :
% 13.25/7.69 ( ( suminf_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 13.25/7.69 = ( F @ zero_zero_nat ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7707_powser__zero,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 13.25/7.69 = ( F @ zero_zero_nat ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7708_lessThan__def,axiom,
% 13.25/7.69 ( set_ord_lessThan_rat
% 13.25/7.69 = ( ^ [U3: rat] :
% 13.25/7.69 ( collect_rat
% 13.25/7.69 @ ^ [X2: rat] : ( ord_less_rat @ X2 @ U3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7709_lessThan__def,axiom,
% 13.25/7.69 ( set_ord_lessThan_num
% 13.25/7.69 = ( ^ [U3: num] :
% 13.25/7.69 ( collect_num
% 13.25/7.69 @ ^ [X2: num] : ( ord_less_num @ X2 @ U3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7710_lessThan__def,axiom,
% 13.25/7.69 ( set_ord_lessThan_nat
% 13.25/7.69 = ( ^ [U3: nat] :
% 13.25/7.69 ( collect_nat
% 13.25/7.69 @ ^ [X2: nat] : ( ord_less_nat @ X2 @ U3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7711_lessThan__def,axiom,
% 13.25/7.69 ( set_ord_lessThan_int
% 13.25/7.69 = ( ^ [U3: int] :
% 13.25/7.69 ( collect_int
% 13.25/7.69 @ ^ [X2: int] : ( ord_less_int @ X2 @ U3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7712_lessThan__def,axiom,
% 13.25/7.69 ( set_or5984915006950818249n_real
% 13.25/7.69 = ( ^ [U3: real] :
% 13.25/7.69 ( collect_real
% 13.25/7.69 @ ^ [X2: real] : ( ord_less_real @ X2 @ U3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7713_powser__insidea,axiom,
% 13.25/7.69 ! [F: nat > real,X: real,Z: real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7714_powser__insidea,axiom,
% 13.25/7.69 ! [F: nat > complex,X: complex,Z: complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7715_lessThan__strict__subset__iff,axiom,
% 13.25/7.69 ! [M: rat,N: rat] :
% 13.25/7.69 ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 13.25/7.69 = ( ord_less_rat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7716_lessThan__strict__subset__iff,axiom,
% 13.25/7.69 ! [M: num,N: num] :
% 13.25/7.69 ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 13.25/7.69 = ( ord_less_num @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7717_lessThan__strict__subset__iff,axiom,
% 13.25/7.69 ! [M: nat,N: nat] :
% 13.25/7.69 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( ord_less_nat @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7718_lessThan__strict__subset__iff,axiom,
% 13.25/7.69 ! [M: int,N: int] :
% 13.25/7.69 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 13.25/7.69 = ( ord_less_int @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7719_lessThan__strict__subset__iff,axiom,
% 13.25/7.69 ! [M: real,N: real] :
% 13.25/7.69 ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 13.25/7.69 = ( ord_less_real @ M @ N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7720_sum__pos__lt__pair,axiom,
% 13.25/7.69 ! [F: nat > real,K: nat] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
% 13.25/7.69 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7721_finite__nat__iff__bounded,axiom,
% 13.25/7.69 ( finite_finite_nat
% 13.25/7.69 = ( ^ [S5: set_nat] :
% 13.25/7.69 ? [K2: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7722_finite__nat__bounded,axiom,
% 13.25/7.69 ! [S2: set_nat] :
% 13.25/7.69 ( ( finite_finite_nat @ S2 )
% 13.25/7.69 => ? [K3: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7723_round__mono,axiom,
% 13.25/7.69 ! [X: rat,Y3: rat] :
% 13.25/7.69 ( ( ord_less_eq_rat @ X @ Y3 )
% 13.25/7.69 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7724_sum_Onat__diff__reindex,axiom,
% 13.25/7.69 ! [G: nat > nat,N: nat] :
% 13.25/7.69 ( ( groups3542108847815614940at_nat
% 13.25/7.69 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7725_sum_Onat__diff__reindex,axiom,
% 13.25/7.69 ! [G: nat > real,N: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7726_sum__diff__distrib,axiom,
% 13.25/7.69 ! [Q: int > nat,P: int > nat,N: int] :
% 13.25/7.69 ( ! [X3: int] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
% 13.25/7.69 => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 13.25/7.69 = ( groups4541462559716669496nt_nat
% 13.25/7.69 @ ^ [X2: int] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 13.25/7.69 @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7727_sum__diff__distrib,axiom,
% 13.25/7.69 ! [Q: real > nat,P: real > nat,N: real] :
% 13.25/7.69 ( ! [X3: real] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
% 13.25/7.69 => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 13.25/7.69 = ( groups1935376822645274424al_nat
% 13.25/7.69 @ ^ [X2: real] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 13.25/7.69 @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7728_sum__diff__distrib,axiom,
% 13.25/7.69 ! [Q: nat > nat,P: nat > nat,N: nat] :
% 13.25/7.69 ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P @ X3 ) )
% 13.25/7.69 => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 13.25/7.69 = ( groups3542108847815614940at_nat
% 13.25/7.69 @ ^ [X2: nat] : ( minus_minus_nat @ ( P @ X2 ) @ ( Q @ X2 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7729_powser__inside,axiom,
% 13.25/7.69 ! [F: nat > real,X: real,Z: real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X ) )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7730_powser__inside,axiom,
% 13.25/7.69 ! [F: nat > complex,X: complex,Z: complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 13.25/7.69 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X ) )
% 13.25/7.69 => ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7731_sum_OlessThan__Suc__shift,axiom,
% 13.25/7.69 ! [G: nat > rat,N: nat] :
% 13.25/7.69 ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 13.25/7.69 @ ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7732_sum_OlessThan__Suc__shift,axiom,
% 13.25/7.69 ! [G: nat > int,N: nat] :
% 13.25/7.69 ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = ( plus_plus_int @ ( G @ zero_zero_nat )
% 13.25/7.69 @ ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7733_sum_OlessThan__Suc__shift,axiom,
% 13.25/7.69 ! [G: nat > nat,N: nat] :
% 13.25/7.69 ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 13.25/7.69 @ ( groups3542108847815614940at_nat
% 13.25/7.69 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7734_sum_OlessThan__Suc__shift,axiom,
% 13.25/7.69 ! [G: nat > real,N: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = ( plus_plus_real @ ( G @ zero_zero_nat )
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7735_sumr__diff__mult__const2,axiom,
% 13.25/7.69 ! [F: nat > int,N: nat,R: int] :
% 13.25/7.69 ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R ) )
% 13.25/7.69 = ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ I4 ) @ R )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7736_sumr__diff__mult__const2,axiom,
% 13.25/7.69 ! [F: nat > complex,N: nat,R: complex] :
% 13.25/7.69 ( ( minus_minus_complex @ ( groups2073611262835488442omplex @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ R ) )
% 13.25/7.69 = ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [I4: nat] : ( minus_minus_complex @ ( F @ I4 ) @ R )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7737_sumr__diff__mult__const2,axiom,
% 13.25/7.69 ! [F: nat > rat,N: nat,R: rat] :
% 13.25/7.69 ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R ) )
% 13.25/7.69 = ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ I4 ) @ R )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7738_sumr__diff__mult__const2,axiom,
% 13.25/7.69 ! [F: nat > real,N: nat,R: real] :
% 13.25/7.69 ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R ) )
% 13.25/7.69 = ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ I4 ) @ R )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7739_sum__lessThan__telescope_H,axiom,
% 13.25/7.69 ! [F: nat > rat,M: nat] :
% 13.25/7.69 ( ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7740_sum__lessThan__telescope_H,axiom,
% 13.25/7.69 ! [F: nat > int,M: nat] :
% 13.25/7.69 ( ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7741_sum__lessThan__telescope_H,axiom,
% 13.25/7.69 ! [F: nat > real,M: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7742_sum__lessThan__telescope,axiom,
% 13.25/7.69 ! [F: nat > rat,M: nat] :
% 13.25/7.69 ( ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7743_sum__lessThan__telescope,axiom,
% 13.25/7.69 ! [F: nat > int,M: nat] :
% 13.25/7.69 ( ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7744_sum__lessThan__telescope,axiom,
% 13.25/7.69 ! [F: nat > real,M: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7745_one__diff__power__eq,axiom,
% 13.25/7.69 ! [X: complex,N: nat] :
% 13.25/7.69 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
% 13.25/7.69 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7746_one__diff__power__eq,axiom,
% 13.25/7.69 ! [X: rat,N: nat] :
% 13.25/7.69 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
% 13.25/7.69 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7747_one__diff__power__eq,axiom,
% 13.25/7.69 ! [X: int,N: nat] :
% 13.25/7.69 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
% 13.25/7.69 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7748_one__diff__power__eq,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
% 13.25/7.69 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7749_power__diff__1__eq,axiom,
% 13.25/7.69 ! [X: complex,N: nat] :
% 13.25/7.69 ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex )
% 13.25/7.69 = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7750_power__diff__1__eq,axiom,
% 13.25/7.69 ! [X: rat,N: nat] :
% 13.25/7.69 ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
% 13.25/7.69 = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7751_power__diff__1__eq,axiom,
% 13.25/7.69 ! [X: int,N: nat] :
% 13.25/7.69 ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
% 13.25/7.69 = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7752_power__diff__1__eq,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real )
% 13.25/7.69 = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7753_geometric__sum,axiom,
% 13.25/7.69 ! [X: complex,N: nat] :
% 13.25/7.69 ( ( X != one_one_complex )
% 13.25/7.69 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7754_geometric__sum,axiom,
% 13.25/7.69 ! [X: rat,N: nat] :
% 13.25/7.69 ( ( X != one_one_rat )
% 13.25/7.69 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7755_geometric__sum,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( X != one_one_real )
% 13.25/7.69 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7756_round__diff__minimal,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7757_round__diff__minimal,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7758_sum__gp__strict,axiom,
% 13.25/7.69 ! [X: complex,N: nat] :
% 13.25/7.69 ( ( ( X = one_one_complex )
% 13.25/7.69 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( semiri8010041392384452111omplex @ N ) ) )
% 13.25/7.69 & ( ( X != one_one_complex )
% 13.25/7.69 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7759_sum__gp__strict,axiom,
% 13.25/7.69 ! [X: rat,N: nat] :
% 13.25/7.69 ( ( ( X = one_one_rat )
% 13.25/7.69 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( semiri681578069525770553at_rat @ N ) ) )
% 13.25/7.69 & ( ( X != one_one_rat )
% 13.25/7.69 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7760_sum__gp__strict,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( ( X = one_one_real )
% 13.25/7.69 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( semiri5074537144036343181t_real @ N ) ) )
% 13.25/7.69 & ( ( X != one_one_real )
% 13.25/7.69 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7761_lemma__termdiff1,axiom,
% 13.25/7.69 ! [Z: complex,H2: complex,M: nat] :
% 13.25/7.69 ( ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [P4: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ P4 ) ) @ ( power_power_complex @ Z @ M ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P4 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7762_lemma__termdiff1,axiom,
% 13.25/7.69 ! [Z: rat,H2: rat,M: nat] :
% 13.25/7.69 ( ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [P4: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ P4 ) ) @ ( power_power_rat @ Z @ M ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P4 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7763_lemma__termdiff1,axiom,
% 13.25/7.69 ! [Z: int,H2: int,M: nat] :
% 13.25/7.69 ( ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [P4: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ P4 ) ) @ ( power_power_int @ Z @ M ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ Z @ P4 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7764_lemma__termdiff1,axiom,
% 13.25/7.69 ! [Z: real,H2: real,M: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [P4: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ P4 ) ) @ ( power_power_real @ Z @ M ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) )
% 13.25/7.69 = ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ Z @ P4 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ ( minus_minus_nat @ M @ P4 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P4 ) ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7765_diff__power__eq__sum,axiom,
% 13.25/7.69 ! [X: complex,N: nat,Y3: complex] :
% 13.25/7.69 ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
% 13.25/7.69 = ( times_times_complex @ ( minus_minus_complex @ X @ Y3 )
% 13.25/7.69 @ ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [P4: nat] : ( times_times_complex @ ( power_power_complex @ X @ P4 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7766_diff__power__eq__sum,axiom,
% 13.25/7.69 ! [X: rat,N: nat,Y3: rat] :
% 13.25/7.69 ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
% 13.25/7.69 = ( times_times_rat @ ( minus_minus_rat @ X @ Y3 )
% 13.25/7.69 @ ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [P4: nat] : ( times_times_rat @ ( power_power_rat @ X @ P4 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7767_diff__power__eq__sum,axiom,
% 13.25/7.69 ! [X: int,N: nat,Y3: int] :
% 13.25/7.69 ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
% 13.25/7.69 = ( times_times_int @ ( minus_minus_int @ X @ Y3 )
% 13.25/7.69 @ ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [P4: nat] : ( times_times_int @ ( power_power_int @ X @ P4 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7768_diff__power__eq__sum,axiom,
% 13.25/7.69 ! [X: real,N: nat,Y3: real] :
% 13.25/7.69 ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
% 13.25/7.69 = ( times_times_real @ ( minus_minus_real @ X @ Y3 )
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [P4: nat] : ( times_times_real @ ( power_power_real @ X @ P4 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7769_power__diff__sumr2,axiom,
% 13.25/7.69 ! [X: complex,N: nat,Y3: complex] :
% 13.25/7.69 ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 13.25/7.69 = ( times_times_complex @ ( minus_minus_complex @ X @ Y3 )
% 13.25/7.69 @ ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7770_power__diff__sumr2,axiom,
% 13.25/7.69 ! [X: rat,N: nat,Y3: rat] :
% 13.25/7.69 ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 13.25/7.69 = ( times_times_rat @ ( minus_minus_rat @ X @ Y3 )
% 13.25/7.69 @ ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7771_power__diff__sumr2,axiom,
% 13.25/7.69 ! [X: int,N: nat,Y3: int] :
% 13.25/7.69 ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y3 @ N ) )
% 13.25/7.69 = ( times_times_int @ ( minus_minus_int @ X @ Y3 )
% 13.25/7.69 @ ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7772_power__diff__sumr2,axiom,
% 13.25/7.69 ! [X: real,N: nat,Y3: real] :
% 13.25/7.69 ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y3 @ N ) )
% 13.25/7.69 = ( times_times_real @ ( minus_minus_real @ X @ Y3 )
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7773_real__sum__nat__ivl__bounded2,axiom,
% 13.25/7.69 ! [N: nat,F: nat > rat,K5: rat,K: nat] :
% 13.25/7.69 ( ! [P7: nat] :
% 13.25/7.69 ( ( ord_less_nat @ P7 @ N )
% 13.25/7.69 => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 13.25/7.69 => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 13.25/7.69 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7774_real__sum__nat__ivl__bounded2,axiom,
% 13.25/7.69 ! [N: nat,F: nat > int,K5: int,K: nat] :
% 13.25/7.69 ( ! [P7: nat] :
% 13.25/7.69 ( ( ord_less_nat @ P7 @ N )
% 13.25/7.69 => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 13.25/7.69 => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 13.25/7.69 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7775_real__sum__nat__ivl__bounded2,axiom,
% 13.25/7.69 ! [N: nat,F: nat > nat,K5: nat,K: nat] :
% 13.25/7.69 ( ! [P7: nat] :
% 13.25/7.69 ( ( ord_less_nat @ P7 @ N )
% 13.25/7.69 => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 13.25/7.69 => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 13.25/7.69 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7776_real__sum__nat__ivl__bounded2,axiom,
% 13.25/7.69 ! [N: nat,F: nat > real,K5: real,K: nat] :
% 13.25/7.69 ( ! [P7: nat] :
% 13.25/7.69 ( ( ord_less_nat @ P7 @ N )
% 13.25/7.69 => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 13.25/7.69 => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 13.25/7.69 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7777_one__diff__power__eq_H,axiom,
% 13.25/7.69 ! [X: complex,N: nat] :
% 13.25/7.69 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
% 13.25/7.69 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
% 13.25/7.69 @ ( groups2073611262835488442omplex
% 13.25/7.69 @ ^ [I4: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7778_one__diff__power__eq_H,axiom,
% 13.25/7.69 ! [X: rat,N: nat] :
% 13.25/7.69 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
% 13.25/7.69 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
% 13.25/7.69 @ ( groups2906978787729119204at_rat
% 13.25/7.69 @ ^ [I4: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7779_one__diff__power__eq_H,axiom,
% 13.25/7.69 ! [X: int,N: nat] :
% 13.25/7.69 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
% 13.25/7.69 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
% 13.25/7.69 @ ( groups3539618377306564664at_int
% 13.25/7.69 @ ^ [I4: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7780_one__diff__power__eq_H,axiom,
% 13.25/7.69 ! [X: real,N: nat] :
% 13.25/7.69 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
% 13.25/7.69 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7781_sum__split__even__odd,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real,N: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ ( F @ I4 ) @ ( G @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.69 = ( plus_plus_real
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) @ one_one_nat ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7782_of__int__round__le,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7783_of__int__round__le,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7784_of__int__round__ge,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7785_of__int__round__ge,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7786_of__int__round__gt,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7787_of__int__round__gt,axiom,
% 13.25/7.69 ! [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 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7788_of__int__round__abs__le,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7789_of__int__round__abs__le,axiom,
% 13.25/7.69 ! [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 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7790_round__unique_H,axiom,
% 13.25/7.69 ! [X: real,N: int] :
% 13.25/7.69 ( ( 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 ) ) ) )
% 13.25/7.69 => ( ( archim8280529875227126926d_real @ X )
% 13.25/7.69 = N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7791_round__unique_H,axiom,
% 13.25/7.69 ! [X: rat,N: int] :
% 13.25/7.69 ( ( 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 ) ) ) )
% 13.25/7.69 => ( ( archim7778729529865785530nd_rat @ X )
% 13.25/7.69 = N ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7792_summable__geometric__iff,axiom,
% 13.25/7.69 ! [C: real] :
% 13.25/7.69 ( ( summable_real @ ( power_power_real @ C ) )
% 13.25/7.69 = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7793_summable__geometric__iff,axiom,
% 13.25/7.69 ! [C: complex] :
% 13.25/7.69 ( ( summable_complex @ ( power_power_complex @ C ) )
% 13.25/7.69 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7794_summable__divide__iff,axiom,
% 13.25/7.69 ! [F: nat > complex,C: complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 13.25/7.69 = ( ( C = zero_zero_complex )
% 13.25/7.69 | ( summable_complex @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7795_summable__divide__iff,axiom,
% 13.25/7.69 ! [F: nat > real,C: real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 13.25/7.69 = ( ( C = zero_zero_real )
% 13.25/7.69 | ( summable_real @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7796_suminf__geometric,axiom,
% 13.25/7.69 ! [C: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 13.25/7.69 => ( ( suminf_real @ ( power_power_real @ C ) )
% 13.25/7.69 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7797_suminf__geometric,axiom,
% 13.25/7.69 ! [C: complex] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 13.25/7.69 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 13.25/7.69 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7798_summable__cmult__iff,axiom,
% 13.25/7.69 ! [C: complex,F: nat > complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 13.25/7.69 = ( ( C = zero_zero_complex )
% 13.25/7.69 | ( summable_complex @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7799_summable__cmult__iff,axiom,
% 13.25/7.69 ! [C: real,F: nat > real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 13.25/7.69 = ( ( C = zero_zero_real )
% 13.25/7.69 | ( summable_real @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7800_sum__less__suminf2,axiom,
% 13.25/7.69 ! [F: nat > int,N: nat,I: nat] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ! [M4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M4 )
% 13.25/7.69 => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
% 13.25/7.69 => ( ( ord_less_eq_nat @ N @ I )
% 13.25/7.69 => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 13.25/7.69 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7801_sum__less__suminf2,axiom,
% 13.25/7.69 ! [F: nat > nat,N: nat,I: nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ! [M4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M4 )
% 13.25/7.69 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 13.25/7.69 => ( ( ord_less_eq_nat @ N @ I )
% 13.25/7.69 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 13.25/7.69 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7802_sum__less__suminf2,axiom,
% 13.25/7.69 ! [F: nat > real,N: nat,I: nat] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [M4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M4 )
% 13.25/7.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
% 13.25/7.69 => ( ( ord_less_eq_nat @ N @ I )
% 13.25/7.69 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 13.25/7.69 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7803_summable__iff__shift,axiom,
% 13.25/7.69 ! [F: nat > real,K: nat] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 13.25/7.69 = ( summable_real @ F ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7804_summable__mult2,axiom,
% 13.25/7.69 ! [F: nat > real,C: real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7805_summable__mult,axiom,
% 13.25/7.69 ! [F: nat > real,C: real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7806_summable__add,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7807_summable__add,axiom,
% 13.25/7.69 ! [F: nat > nat,G: nat > nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ( summable_nat @ G )
% 13.25/7.69 => ( summable_nat
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7808_summable__add,axiom,
% 13.25/7.69 ! [F: nat > int,G: nat > int] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ( summable_int @ G )
% 13.25/7.69 => ( summable_int
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7809_summable__diff,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7810_summable__divide,axiom,
% 13.25/7.69 ! [F: nat > complex,C: complex] :
% 13.25/7.69 ( ( summable_complex @ F )
% 13.25/7.69 => ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7811_summable__divide,axiom,
% 13.25/7.69 ! [F: nat > real,C: real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7812_summable__ignore__initial__segment,axiom,
% 13.25/7.69 ! [F: nat > real,K: nat] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7813_suminf__le,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 13.25/7.69 => ( ( summable_real @ F )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7814_suminf__le,axiom,
% 13.25/7.69 ! [F: nat > nat,G: nat > nat] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
% 13.25/7.69 => ( ( summable_nat @ F )
% 13.25/7.69 => ( ( summable_nat @ G )
% 13.25/7.69 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7815_suminf__le,axiom,
% 13.25/7.69 ! [F: nat > int,G: nat > int] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
% 13.25/7.69 => ( ( summable_int @ F )
% 13.25/7.69 => ( ( summable_int @ G )
% 13.25/7.69 => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7816_summable__mult__D,axiom,
% 13.25/7.69 ! [C: complex,F: nat > complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 13.25/7.69 => ( ( C != zero_zero_complex )
% 13.25/7.69 => ( summable_complex @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7817_summable__mult__D,axiom,
% 13.25/7.69 ! [C: real,F: nat > real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 13.25/7.69 => ( ( C != zero_zero_real )
% 13.25/7.69 => ( summable_real @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7818_summable__zero__power,axiom,
% 13.25/7.69 summable_real @ ( power_power_real @ zero_zero_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7819_summable__zero__power,axiom,
% 13.25/7.69 summable_int @ ( power_power_int @ zero_zero_int ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7820_summable__zero__power,axiom,
% 13.25/7.69 summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7821_suminf__mult2,axiom,
% 13.25/7.69 ! [F: nat > real,C: real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( times_times_real @ ( suminf_real @ F ) @ C )
% 13.25/7.69 = ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7822_suminf__mult,axiom,
% 13.25/7.69 ! [F: nat > real,C: real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 13.25/7.69 = ( times_times_real @ C @ ( suminf_real @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7823_suminf__add,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 13.25/7.69 = ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7824_suminf__add,axiom,
% 13.25/7.69 ! [F: nat > nat,G: nat > nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ( summable_nat @ G )
% 13.25/7.69 => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 13.25/7.69 = ( suminf_nat
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7825_suminf__add,axiom,
% 13.25/7.69 ! [F: nat > int,G: nat > int] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ( summable_int @ G )
% 13.25/7.69 => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 13.25/7.69 = ( suminf_int
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7826_suminf__diff,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( ( minus_minus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 13.25/7.69 = ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7827_suminf__divide,axiom,
% 13.25/7.69 ! [F: nat > complex,C: complex] :
% 13.25/7.69 ( ( summable_complex @ F )
% 13.25/7.69 => ( ( suminf_complex
% 13.25/7.69 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 13.25/7.69 = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7828_suminf__divide,axiom,
% 13.25/7.69 ! [F: nat > real,C: real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 13.25/7.69 = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7829_suminf__eq__zero__iff,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ( suminf_real @ F )
% 13.25/7.69 = zero_zero_real )
% 13.25/7.69 = ( ! [N2: nat] :
% 13.25/7.69 ( ( F @ N2 )
% 13.25/7.69 = zero_zero_real ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7830_suminf__eq__zero__iff,axiom,
% 13.25/7.69 ! [F: nat > nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ( suminf_nat @ F )
% 13.25/7.69 = zero_zero_nat )
% 13.25/7.69 = ( ! [N2: nat] :
% 13.25/7.69 ( ( F @ N2 )
% 13.25/7.69 = zero_zero_nat ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7831_suminf__eq__zero__iff,axiom,
% 13.25/7.69 ! [F: nat > int] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ( suminf_int @ F )
% 13.25/7.69 = zero_zero_int )
% 13.25/7.69 = ( ! [N2: nat] :
% 13.25/7.69 ( ( F @ N2 )
% 13.25/7.69 = zero_zero_int ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7832_suminf__nonneg,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 13.25/7.69 => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7833_suminf__nonneg,axiom,
% 13.25/7.69 ! [F: nat > nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 13.25/7.69 => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7834_suminf__nonneg,axiom,
% 13.25/7.69 ! [F: nat > int] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 13.25/7.69 => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7835_suminf__pos,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N4 ) )
% 13.25/7.69 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7836_suminf__pos,axiom,
% 13.25/7.69 ! [F: nat > nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N4 ) )
% 13.25/7.69 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7837_suminf__pos,axiom,
% 13.25/7.69 ! [F: nat > int] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N4 ) )
% 13.25/7.69 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7838_summable__zero__power_H,axiom,
% 13.25/7.69 ! [F: nat > complex] :
% 13.25/7.69 ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7839_summable__zero__power_H,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7840_summable__zero__power_H,axiom,
% 13.25/7.69 ! [F: nat > int] :
% 13.25/7.69 ( summable_int
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7841_summable__0__powser,axiom,
% 13.25/7.69 ! [F: nat > complex] :
% 13.25/7.69 ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7842_summable__0__powser,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7843_powser__split__head_I3_J,axiom,
% 13.25/7.69 ! [F: nat > complex,Z: complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 => ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7844_powser__split__head_I3_J,axiom,
% 13.25/7.69 ! [F: nat > real,Z: real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7845_summable__powser__split__head,axiom,
% 13.25/7.69 ! [F: nat > complex,Z: complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 = ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7846_summable__powser__split__head,axiom,
% 13.25/7.69 ! [F: nat > real,Z: real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 = ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7847_summable__powser__ignore__initial__segment,axiom,
% 13.25/7.69 ! [F: nat > complex,M: nat,Z: complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 = ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7848_summable__powser__ignore__initial__segment,axiom,
% 13.25/7.69 ! [F: nat > real,M: nat,Z: real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N2 @ M ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 = ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7849_summable__norm__comparison__test,axiom,
% 13.25/7.69 ! [F: nat > complex,G: nat > real] :
% 13.25/7.69 ( ? [N8: nat] :
% 13.25/7.69 ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N8 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7850_summable__rabs__comparison__test,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real] :
% 13.25/7.69 ( ? [N8: nat] :
% 13.25/7.69 ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N8 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7851_summable__rabs,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 13.25/7.69 @ ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7852_suminf__pos__iff,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 13.25/7.69 = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7853_suminf__pos__iff,axiom,
% 13.25/7.69 ! [F: nat > nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 13.25/7.69 = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7854_suminf__pos__iff,axiom,
% 13.25/7.69 ! [F: nat > int] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 13.25/7.69 = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7855_suminf__pos2,axiom,
% 13.25/7.69 ! [F: nat > real,I: nat] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 13.25/7.69 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7856_suminf__pos2,axiom,
% 13.25/7.69 ! [F: nat > nat,I: nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 13.25/7.69 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7857_suminf__pos2,axiom,
% 13.25/7.69 ! [F: nat > int,I: nat] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 13.25/7.69 => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 13.25/7.69 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7858_suminf__le__const,axiom,
% 13.25/7.69 ! [F: nat > int,X: int] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 13.25/7.69 => ( ord_less_eq_int @ ( suminf_int @ F ) @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7859_suminf__le__const,axiom,
% 13.25/7.69 ! [F: nat > nat,X: nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 13.25/7.69 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7860_suminf__le__const,axiom,
% 13.25/7.69 ! [F: nat > real,X: real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 13.25/7.69 => ( ord_less_eq_real @ ( suminf_real @ F ) @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7861_summable__comparison__test,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real] :
% 13.25/7.69 ( ? [N8: nat] :
% 13.25/7.69 ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N8 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( summable_real @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7862_summable__comparison__test,axiom,
% 13.25/7.69 ! [F: nat > complex,G: nat > real] :
% 13.25/7.69 ( ? [N8: nat] :
% 13.25/7.69 ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N8 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 13.25/7.69 => ( ( summable_real @ G )
% 13.25/7.69 => ( summable_complex @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7863_summable__comparison__test_H,axiom,
% 13.25/7.69 ! [G: nat > real,N3: nat,F: nat > real] :
% 13.25/7.69 ( ( summable_real @ G )
% 13.25/7.69 => ( ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N3 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 13.25/7.69 => ( summable_real @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7864_summable__comparison__test_H,axiom,
% 13.25/7.69 ! [G: nat > real,N3: nat,F: nat > complex] :
% 13.25/7.69 ( ( summable_real @ G )
% 13.25/7.69 => ( ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N3 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) @ ( G @ N4 ) ) )
% 13.25/7.69 => ( summable_complex @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7865_summableI__nonneg__bounded,axiom,
% 13.25/7.69 ! [F: nat > int,X: int] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 13.25/7.69 => ( summable_int @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7866_summableI__nonneg__bounded,axiom,
% 13.25/7.69 ! [F: nat > nat,X: nat] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 13.25/7.69 => ( summable_nat @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7867_summableI__nonneg__bounded,axiom,
% 13.25/7.69 ! [F: nat > real,X: real] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N4 ) ) @ X )
% 13.25/7.69 => ( summable_real @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7868_complete__algebra__summable__geometric,axiom,
% 13.25/7.69 ! [X: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ one_one_real )
% 13.25/7.69 => ( summable_real @ ( power_power_real @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7869_complete__algebra__summable__geometric,axiom,
% 13.25/7.69 ! [X: complex] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ one_one_real )
% 13.25/7.69 => ( summable_complex @ ( power_power_complex @ X ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7870_summable__geometric,axiom,
% 13.25/7.69 ! [C: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 13.25/7.69 => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7871_summable__geometric,axiom,
% 13.25/7.69 ! [C: complex] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 13.25/7.69 => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7872_suminf__split__head,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 13.25/7.69 = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7873_summable__norm,axiom,
% 13.25/7.69 ! [F: nat > real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 13.25/7.69 @ ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7874_summable__norm,axiom,
% 13.25/7.69 ! [F: nat > complex] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 13.25/7.69 @ ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7875_sum__le__suminf,axiom,
% 13.25/7.69 ! [F: nat > int,I5: set_nat] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ( finite_finite_nat @ I5 )
% 13.25/7.69 => ( ! [N4: nat] :
% 13.25/7.69 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 13.25/7.69 => ( ord_less_eq_int @ zero_zero_int @ ( F @ N4 ) ) )
% 13.25/7.69 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7876_sum__le__suminf,axiom,
% 13.25/7.69 ! [F: nat > nat,I5: set_nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ( finite_finite_nat @ I5 )
% 13.25/7.69 => ( ! [N4: nat] :
% 13.25/7.69 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 13.25/7.69 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N4 ) ) )
% 13.25/7.69 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7877_sum__le__suminf,axiom,
% 13.25/7.69 ! [F: nat > real,I5: set_nat] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( finite_finite_nat @ I5 )
% 13.25/7.69 => ( ! [N4: nat] :
% 13.25/7.69 ( ( member_nat @ N4 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 13.25/7.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ N4 ) ) )
% 13.25/7.69 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7878_suminf__split__initial__segment,axiom,
% 13.25/7.69 ! [F: nat > real,K: nat] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( suminf_real @ F )
% 13.25/7.69 = ( plus_plus_real
% 13.25/7.69 @ ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 13.25/7.69 @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7879_suminf__minus__initial__segment,axiom,
% 13.25/7.69 ! [F: nat > real,K: nat] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ K ) ) )
% 13.25/7.69 = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7880_sum__less__suminf,axiom,
% 13.25/7.69 ! [F: nat > int,N: nat] :
% 13.25/7.69 ( ( summable_int @ F )
% 13.25/7.69 => ( ! [M4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M4 )
% 13.25/7.69 => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
% 13.25/7.69 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7881_sum__less__suminf,axiom,
% 13.25/7.69 ! [F: nat > nat,N: nat] :
% 13.25/7.69 ( ( summable_nat @ F )
% 13.25/7.69 => ( ! [M4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M4 )
% 13.25/7.69 => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 13.25/7.69 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7882_sum__less__suminf,axiom,
% 13.25/7.69 ! [F: nat > real,N: nat] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ! [M4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N @ M4 )
% 13.25/7.69 => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
% 13.25/7.69 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7883_powser__split__head_I1_J,axiom,
% 13.25/7.69 ! [F: nat > complex,Z: complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 => ( ( suminf_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 13.25/7.69 @ ( times_times_complex
% 13.25/7.69 @ ( suminf_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 @ Z ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7884_powser__split__head_I1_J,axiom,
% 13.25/7.69 ! [F: nat > real,Z: real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 => ( ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 = ( plus_plus_real @ ( F @ zero_zero_nat )
% 13.25/7.69 @ ( times_times_real
% 13.25/7.69 @ ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 @ Z ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7885_powser__split__head_I2_J,axiom,
% 13.25/7.69 ! [F: nat > complex,Z: complex] :
% 13.25/7.69 ( ( summable_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 => ( ( times_times_complex
% 13.25/7.69 @ ( suminf_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 @ Z )
% 13.25/7.69 = ( minus_minus_complex
% 13.25/7.69 @ ( suminf_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 13.25/7.69 @ ( F @ zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7886_powser__split__head_I2_J,axiom,
% 13.25/7.69 ! [F: nat > real,Z: real] :
% 13.25/7.69 ( ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 => ( ( times_times_real
% 13.25/7.69 @ ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 @ Z )
% 13.25/7.69 = ( minus_minus_real
% 13.25/7.69 @ ( suminf_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 13.25/7.69 @ ( F @ zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7887_summable__partial__sum__bound,axiom,
% 13.25/7.69 ! [F: nat > complex,E: real] :
% 13.25/7.69 ( ( summable_complex @ F )
% 13.25/7.69 => ( ( ord_less_real @ zero_zero_real @ E )
% 13.25/7.69 => ~ ! [N9: nat] :
% 13.25/7.69 ~ ! [M2: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N9 @ M2 )
% 13.25/7.69 => ! [N7: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7888_summable__partial__sum__bound,axiom,
% 13.25/7.69 ! [F: nat > real,E: real] :
% 13.25/7.69 ( ( summable_real @ F )
% 13.25/7.69 => ( ( ord_less_real @ zero_zero_real @ E )
% 13.25/7.69 => ~ ! [N9: nat] :
% 13.25/7.69 ~ ! [M2: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N9 @ M2 )
% 13.25/7.69 => ! [N7: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N7 ) ) ) @ E ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7889_suminf__exist__split,axiom,
% 13.25/7.69 ! [R: real,F: nat > real] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ R )
% 13.25/7.69 => ( ( summable_real @ F )
% 13.25/7.69 => ? [N9: nat] :
% 13.25/7.69 ! [N7: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N9 @ N7 )
% 13.25/7.69 => ( ord_less_real
% 13.25/7.69 @ ( real_V7735802525324610683m_real
% 13.25/7.69 @ ( suminf_real
% 13.25/7.69 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N7 ) ) ) )
% 13.25/7.69 @ R ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7890_suminf__exist__split,axiom,
% 13.25/7.69 ! [R: real,F: nat > complex] :
% 13.25/7.69 ( ( ord_less_real @ zero_zero_real @ R )
% 13.25/7.69 => ( ( summable_complex @ F )
% 13.25/7.69 => ? [N9: nat] :
% 13.25/7.69 ! [N7: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N9 @ N7 )
% 13.25/7.69 => ( ord_less_real
% 13.25/7.69 @ ( real_V1022390504157884413omplex
% 13.25/7.69 @ ( suminf_complex
% 13.25/7.69 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N7 ) ) ) )
% 13.25/7.69 @ R ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7891_summable__power__series,axiom,
% 13.25/7.69 ! [F: nat > real,Z: real] :
% 13.25/7.69 ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 13.25/7.69 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 13.25/7.69 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 13.25/7.69 => ( ( ord_less_real @ Z @ one_one_real )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7892_Abel__lemma,axiom,
% 13.25/7.69 ! [R: real,R0: real,A: nat > complex,M5: real] :
% 13.25/7.69 ( ( ord_less_eq_real @ zero_zero_real @ R )
% 13.25/7.69 => ( ( ord_less_real @ R @ R0 )
% 13.25/7.69 => ( ! [N4: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N4 ) ) @ ( power_power_real @ R0 @ N4 ) ) @ M5 )
% 13.25/7.69 => ( summable_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R @ N2 ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7893_summable__ratio__test,axiom,
% 13.25/7.69 ! [C: real,N3: nat,F: nat > real] :
% 13.25/7.69 ( ( ord_less_real @ C @ one_one_real )
% 13.25/7.69 => ( ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N3 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N4 ) ) ) ) )
% 13.25/7.69 => ( summable_real @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7894_summable__ratio__test,axiom,
% 13.25/7.69 ! [C: real,N3: nat,F: nat > complex] :
% 13.25/7.69 ( ( ord_less_real @ C @ one_one_real )
% 13.25/7.69 => ( ! [N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ N3 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N4 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N4 ) ) ) ) )
% 13.25/7.69 => ( summable_complex @ F ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7895_sum__bounds__lt__plus1,axiom,
% 13.25/7.69 ! [F: nat > nat,Mm: nat] :
% 13.25/7.69 ( ( groups3542108847815614940at_nat
% 13.25/7.69 @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ Mm ) )
% 13.25/7.69 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7896_sum__bounds__lt__plus1,axiom,
% 13.25/7.69 ! [F: nat > real,Mm: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ Mm ) )
% 13.25/7.69 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7897_sumr__cos__zero__one,axiom,
% 13.25/7.69 ! [N: nat] :
% 13.25/7.69 ( ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ zero_zero_real @ M3 ) )
% 13.25/7.69 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.69 = one_one_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7898_pi__series,axiom,
% 13.25/7.69 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.69 = ( suminf_real
% 13.25/7.69 @ ^ [K2: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7899_geometric__deriv__sums,axiom,
% 13.25/7.69 ! [Z: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
% 13.25/7.69 @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7900_geometric__deriv__sums,axiom,
% 13.25/7.69 ! [Z: complex] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 13.25/7.69 => ( sums_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
% 13.25/7.69 @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7901_monoI1,axiom,
% 13.25/7.69 ! [X7: nat > real] :
% 13.25/7.69 ( ! [M4: nat,N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.69 => ( ord_less_eq_real @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 13.25/7.69 => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7902_monoI1,axiom,
% 13.25/7.69 ! [X7: nat > set_int] :
% 13.25/7.69 ( ! [M4: nat,N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.69 => ( ord_less_eq_set_int @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 13.25/7.69 => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7903_monoI1,axiom,
% 13.25/7.69 ! [X7: nat > rat] :
% 13.25/7.69 ( ! [M4: nat,N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.69 => ( ord_less_eq_rat @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 13.25/7.69 => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7904_monoI1,axiom,
% 13.25/7.69 ! [X7: nat > num] :
% 13.25/7.69 ( ! [M4: nat,N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.69 => ( ord_less_eq_num @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 13.25/7.69 => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7905_monoI1,axiom,
% 13.25/7.69 ! [X7: nat > nat] :
% 13.25/7.69 ( ! [M4: nat,N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.69 => ( ord_less_eq_nat @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 13.25/7.69 => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7906_monoI1,axiom,
% 13.25/7.69 ! [X7: nat > int] :
% 13.25/7.69 ( ! [M4: nat,N4: nat] :
% 13.25/7.69 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.69 => ( ord_less_eq_int @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 13.25/7.69 => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7907_cos__coeff__0,axiom,
% 13.25/7.69 ( ( cos_coeff @ zero_zero_nat )
% 13.25/7.69 = one_one_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7908_powser__sums__zero__iff,axiom,
% 13.25/7.69 ! [A: nat > complex,X: complex] :
% 13.25/7.69 ( ( sums_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 13.25/7.69 @ X )
% 13.25/7.69 = ( ( A @ zero_zero_nat )
% 13.25/7.69 = X ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7909_powser__sums__zero__iff,axiom,
% 13.25/7.69 ! [A: nat > real,X: real] :
% 13.25/7.69 ( ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 13.25/7.69 @ X )
% 13.25/7.69 = ( ( A @ zero_zero_nat )
% 13.25/7.69 = X ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7910_sums__le,axiom,
% 13.25/7.69 ! [F: nat > real,G: nat > real,S: real,T: real] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 13.25/7.69 => ( ( sums_real @ F @ S )
% 13.25/7.69 => ( ( sums_real @ G @ T )
% 13.25/7.69 => ( ord_less_eq_real @ S @ T ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7911_sums__le,axiom,
% 13.25/7.69 ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( G @ N4 ) )
% 13.25/7.69 => ( ( sums_nat @ F @ S )
% 13.25/7.69 => ( ( sums_nat @ G @ T )
% 13.25/7.69 => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7912_sums__le,axiom,
% 13.25/7.69 ! [F: nat > int,G: nat > int,S: int,T: int] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_int @ ( F @ N4 ) @ ( G @ N4 ) )
% 13.25/7.69 => ( ( sums_int @ F @ S )
% 13.25/7.69 => ( ( sums_int @ G @ T )
% 13.25/7.69 => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7913_sums__mult,axiom,
% 13.25/7.69 ! [F: nat > real,A: real,C: real] :
% 13.25/7.69 ( ( sums_real @ F @ A )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 13.25/7.69 @ ( times_times_real @ C @ A ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7914_sums__mult2,axiom,
% 13.25/7.69 ! [F: nat > real,A: real,C: real] :
% 13.25/7.69 ( ( sums_real @ F @ A )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 13.25/7.69 @ ( times_times_real @ A @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7915_sums__add,axiom,
% 13.25/7.69 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 13.25/7.69 ( ( sums_real @ F @ A )
% 13.25/7.69 => ( ( sums_real @ G @ B )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 13.25/7.69 @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7916_sums__add,axiom,
% 13.25/7.69 ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
% 13.25/7.69 ( ( sums_nat @ F @ A )
% 13.25/7.69 => ( ( sums_nat @ G @ B )
% 13.25/7.69 => ( sums_nat
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_nat @ ( F @ N2 ) @ ( G @ N2 ) )
% 13.25/7.69 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7917_sums__add,axiom,
% 13.25/7.69 ! [F: nat > int,A: int,G: nat > int,B: int] :
% 13.25/7.69 ( ( sums_int @ F @ A )
% 13.25/7.69 => ( ( sums_int @ G @ B )
% 13.25/7.69 => ( sums_int
% 13.25/7.69 @ ^ [N2: nat] : ( plus_plus_int @ ( F @ N2 ) @ ( G @ N2 ) )
% 13.25/7.69 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7918_sums__diff,axiom,
% 13.25/7.69 ! [F: nat > real,A: real,G: nat > real,B: real] :
% 13.25/7.69 ( ( sums_real @ F @ A )
% 13.25/7.69 => ( ( sums_real @ G @ B )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 13.25/7.69 @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7919_sums__divide,axiom,
% 13.25/7.69 ! [F: nat > complex,A: complex,C: complex] :
% 13.25/7.69 ( ( sums_complex @ F @ A )
% 13.25/7.69 => ( sums_complex
% 13.25/7.69 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C )
% 13.25/7.69 @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7920_sums__divide,axiom,
% 13.25/7.69 ! [F: nat > real,A: real,C: real] :
% 13.25/7.69 ( ( sums_real @ F @ A )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C )
% 13.25/7.69 @ ( divide_divide_real @ A @ C ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7921_sums__mult__iff,axiom,
% 13.25/7.69 ! [C: complex,F: nat > complex,D: complex] :
% 13.25/7.69 ( ( C != zero_zero_complex )
% 13.25/7.69 => ( ( sums_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 13.25/7.69 @ ( times_times_complex @ C @ D ) )
% 13.25/7.69 = ( sums_complex @ F @ D ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7922_sums__mult__iff,axiom,
% 13.25/7.69 ! [C: real,F: nat > real,D: real] :
% 13.25/7.69 ( ( C != zero_zero_real )
% 13.25/7.69 => ( ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 13.25/7.69 @ ( times_times_real @ C @ D ) )
% 13.25/7.69 = ( sums_real @ F @ D ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7923_sums__mult2__iff,axiom,
% 13.25/7.69 ! [C: complex,F: nat > complex,D: complex] :
% 13.25/7.69 ( ( C != zero_zero_complex )
% 13.25/7.69 => ( ( sums_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
% 13.25/7.69 @ ( times_times_complex @ D @ C ) )
% 13.25/7.69 = ( sums_complex @ F @ D ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7924_sums__mult2__iff,axiom,
% 13.25/7.69 ! [C: real,F: nat > real,D: real] :
% 13.25/7.69 ( ( C != zero_zero_real )
% 13.25/7.69 => ( ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 13.25/7.69 @ ( times_times_real @ D @ C ) )
% 13.25/7.69 = ( sums_real @ F @ D ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7925_pi__gt__zero,axiom,
% 13.25/7.69 ord_less_real @ zero_zero_real @ pi ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7926_pi__not__less__zero,axiom,
% 13.25/7.69 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7927_pi__ge__zero,axiom,
% 13.25/7.69 ord_less_eq_real @ zero_zero_real @ pi ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7928_sums__mult__D,axiom,
% 13.25/7.69 ! [C: complex,F: nat > complex,A: complex] :
% 13.25/7.69 ( ( sums_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 13.25/7.69 @ A )
% 13.25/7.69 => ( ( C != zero_zero_complex )
% 13.25/7.69 => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7929_sums__mult__D,axiom,
% 13.25/7.69 ! [C: real,F: nat > real,A: real] :
% 13.25/7.69 ( ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 13.25/7.69 @ A )
% 13.25/7.69 => ( ( C != zero_zero_real )
% 13.25/7.69 => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7930_sums__Suc,axiom,
% 13.25/7.69 ! [F: nat > real,L2: real] :
% 13.25/7.69 ( ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 13.25/7.69 @ L2 )
% 13.25/7.69 => ( sums_real @ F @ ( plus_plus_real @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7931_sums__Suc,axiom,
% 13.25/7.69 ! [F: nat > nat,L2: nat] :
% 13.25/7.69 ( ( sums_nat
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 13.25/7.69 @ L2 )
% 13.25/7.69 => ( sums_nat @ F @ ( plus_plus_nat @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7932_sums__Suc,axiom,
% 13.25/7.69 ! [F: nat > int,L2: int] :
% 13.25/7.69 ( ( sums_int
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 13.25/7.69 @ L2 )
% 13.25/7.69 => ( sums_int @ F @ ( plus_plus_int @ L2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7933_sums__Suc__iff,axiom,
% 13.25/7.69 ! [F: nat > real,S: real] :
% 13.25/7.69 ( ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 13.25/7.69 @ S )
% 13.25/7.69 = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7934_sums__zero__iff__shift,axiom,
% 13.25/7.69 ! [N: nat,F: nat > complex,S: complex] :
% 13.25/7.69 ( ! [I2: nat] :
% 13.25/7.69 ( ( ord_less_nat @ I2 @ N )
% 13.25/7.69 => ( ( F @ I2 )
% 13.25/7.69 = zero_zero_complex ) )
% 13.25/7.69 => ( ( sums_complex
% 13.25/7.69 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 13.25/7.69 @ S )
% 13.25/7.69 = ( sums_complex @ F @ S ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7935_sums__zero__iff__shift,axiom,
% 13.25/7.69 ! [N: nat,F: nat > real,S: real] :
% 13.25/7.69 ( ! [I2: nat] :
% 13.25/7.69 ( ( ord_less_nat @ I2 @ N )
% 13.25/7.69 => ( ( F @ I2 )
% 13.25/7.69 = zero_zero_real ) )
% 13.25/7.69 => ( ( sums_real
% 13.25/7.69 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 13.25/7.69 @ S )
% 13.25/7.69 = ( sums_real @ F @ S ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7936_powser__sums__if,axiom,
% 13.25/7.69 ! [M: nat,Z: complex] :
% 13.25/7.69 ( sums_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
% 13.25/7.69 @ ( power_power_complex @ Z @ M ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7937_powser__sums__if,axiom,
% 13.25/7.69 ! [M: nat,Z: real] :
% 13.25/7.69 ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
% 13.25/7.69 @ ( power_power_real @ Z @ M ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7938_powser__sums__if,axiom,
% 13.25/7.69 ! [M: nat,Z: int] :
% 13.25/7.69 ( sums_int
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
% 13.25/7.69 @ ( power_power_int @ Z @ M ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7939_powser__sums__zero,axiom,
% 13.25/7.69 ! [A: nat > complex] :
% 13.25/7.69 ( sums_complex
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_complex @ ( A @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 13.25/7.69 @ ( A @ zero_zero_nat ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7940_powser__sums__zero,axiom,
% 13.25/7.69 ! [A: nat > real] :
% 13.25/7.69 ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( times_times_real @ ( A @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 13.25/7.69 @ ( A @ zero_zero_nat ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7941_sums__iff__shift,axiom,
% 13.25/7.69 ! [F: nat > real,N: nat,S: real] :
% 13.25/7.69 ( ( sums_real
% 13.25/7.69 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 13.25/7.69 @ S )
% 13.25/7.69 = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7942_sums__iff__shift_H,axiom,
% 13.25/7.69 ! [F: nat > real,N: nat,S: real] :
% 13.25/7.69 ( ( sums_real
% 13.25/7.69 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 13.25/7.69 @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) )
% 13.25/7.69 = ( sums_real @ F @ S ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7943_sums__split__initial__segment,axiom,
% 13.25/7.69 ! [F: nat > real,S: real,N: nat] :
% 13.25/7.69 ( ( sums_real @ F @ S )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 13.25/7.69 @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7944_sums__If__finite__set_H,axiom,
% 13.25/7.69 ! [G: nat > real,S2: real,A2: set_nat,S4: real,F: nat > real] :
% 13.25/7.69 ( ( sums_real @ G @ S2 )
% 13.25/7.69 => ( ( finite_finite_nat @ A2 )
% 13.25/7.69 => ( ( S4
% 13.25/7.69 = ( plus_plus_real @ S2
% 13.25/7.69 @ ( groups6591440286371151544t_real
% 13.25/7.69 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 13.25/7.69 @ A2 ) ) )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A2 ) @ ( F @ N2 ) @ ( G @ N2 ) )
% 13.25/7.69 @ S4 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7945_pi__less__4,axiom,
% 13.25/7.69 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7946_pi__ge__two,axiom,
% 13.25/7.69 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7947_pi__half__neq__two,axiom,
% 13.25/7.69 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.69 != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7948_pi__half__neq__zero,axiom,
% 13.25/7.69 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.69 != zero_zero_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7949_pi__half__less__two,axiom,
% 13.25/7.69 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7950_pi__half__le__two,axiom,
% 13.25/7.69 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7951_pi__half__gt__zero,axiom,
% 13.25/7.69 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7952_pi__half__ge__zero,axiom,
% 13.25/7.69 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7953_m2pi__less__pi,axiom,
% 13.25/7.69 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7954_arctan__ubound,axiom,
% 13.25/7.69 ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7955_arctan__one,axiom,
% 13.25/7.69 ( ( arctan @ one_one_real )
% 13.25/7.69 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7956_geometric__sums,axiom,
% 13.25/7.69 ! [C: real] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 13.25/7.69 => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7957_geometric__sums,axiom,
% 13.25/7.69 ! [C: complex] :
% 13.25/7.69 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 13.25/7.69 => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7958_power__half__series,axiom,
% 13.25/7.69 ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N2 ) )
% 13.25/7.69 @ one_one_real ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7959_minus__pi__half__less__zero,axiom,
% 13.25/7.69 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7960_arctan__bounded,axiom,
% 13.25/7.69 ! [Y3: real] :
% 13.25/7.69 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 13.25/7.69 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7961_arctan__lbound,axiom,
% 13.25/7.69 ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7962_sums__if_H,axiom,
% 13.25/7.69 ! [G: nat > real,X: real] :
% 13.25/7.69 ( ( sums_real @ G @ X )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.69 @ X ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7963_sums__if,axiom,
% 13.25/7.69 ! [G: nat > real,X: real,F: nat > real,Y3: real] :
% 13.25/7.69 ( ( sums_real @ G @ X )
% 13.25/7.69 => ( ( sums_real @ F @ Y3 )
% 13.25/7.69 => ( sums_real
% 13.25/7.69 @ ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( F @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.69 @ ( plus_plus_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7964_machin__Euler,axiom,
% 13.25/7.69 ( ( 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 ) ) ) ) ) ) ) ) ) ) )
% 13.25/7.69 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7965_machin,axiom,
% 13.25/7.69 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.69 = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7966_monoseq__Suc,axiom,
% 13.25/7.69 ( topolo6980174941875973593q_real
% 13.25/7.69 = ( ^ [X6: nat > real] :
% 13.25/7.69 ( ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 13.25/7.69 | ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7967_monoseq__Suc,axiom,
% 13.25/7.69 ( topolo3100542954746470799et_int
% 13.25/7.69 = ( ^ [X6: nat > set_int] :
% 13.25/7.69 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 13.25/7.69 | ! [N2: nat] : ( ord_less_eq_set_int @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7968_monoseq__Suc,axiom,
% 13.25/7.69 ( topolo4267028734544971653eq_rat
% 13.25/7.69 = ( ^ [X6: nat > rat] :
% 13.25/7.69 ( ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 13.25/7.69 | ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7969_monoseq__Suc,axiom,
% 13.25/7.69 ( topolo1459490580787246023eq_num
% 13.25/7.69 = ( ^ [X6: nat > num] :
% 13.25/7.69 ( ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 13.25/7.69 | ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7970_monoseq__Suc,axiom,
% 13.25/7.69 ( topolo4902158794631467389eq_nat
% 13.25/7.69 = ( ^ [X6: nat > nat] :
% 13.25/7.69 ( ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 13.25/7.69 | ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7971_monoseq__Suc,axiom,
% 13.25/7.69 ( topolo4899668324122417113eq_int
% 13.25/7.69 = ( ^ [X6: nat > int] :
% 13.25/7.69 ( ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
% 13.25/7.69 | ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7972_mono__SucI2,axiom,
% 13.25/7.69 ! [X7: nat > real] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 13.25/7.69 => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7973_mono__SucI2,axiom,
% 13.25/7.69 ! [X7: nat > set_int] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 13.25/7.69 => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7974_mono__SucI2,axiom,
% 13.25/7.69 ! [X7: nat > rat] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 13.25/7.69 => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 13.25/7.69
% 13.25/7.69 thf(fact_7975_mono__SucI2,axiom,
% 13.25/7.69 ! [X7: nat > num] :
% 13.25/7.69 ( ! [N4: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 13.25/7.70 => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7976_mono__SucI2,axiom,
% 13.25/7.70 ! [X7: nat > nat] :
% 13.25/7.70 ( ! [N4: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 13.25/7.70 => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7977_mono__SucI2,axiom,
% 13.25/7.70 ! [X7: nat > int] :
% 13.25/7.70 ( ! [N4: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N4 ) ) @ ( X7 @ N4 ) )
% 13.25/7.70 => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7978_mono__SucI1,axiom,
% 13.25/7.70 ! [X7: nat > real] :
% 13.25/7.70 ( ! [N4: nat] : ( ord_less_eq_real @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 13.25/7.70 => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7979_mono__SucI1,axiom,
% 13.25/7.70 ! [X7: nat > set_int] :
% 13.25/7.70 ( ! [N4: nat] : ( ord_less_eq_set_int @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 13.25/7.70 => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7980_mono__SucI1,axiom,
% 13.25/7.70 ! [X7: nat > rat] :
% 13.25/7.70 ( ! [N4: nat] : ( ord_less_eq_rat @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 13.25/7.70 => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7981_mono__SucI1,axiom,
% 13.25/7.70 ! [X7: nat > num] :
% 13.25/7.70 ( ! [N4: nat] : ( ord_less_eq_num @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 13.25/7.70 => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7982_mono__SucI1,axiom,
% 13.25/7.70 ! [X7: nat > nat] :
% 13.25/7.70 ( ! [N4: nat] : ( ord_less_eq_nat @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 13.25/7.70 => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7983_mono__SucI1,axiom,
% 13.25/7.70 ! [X7: nat > int] :
% 13.25/7.70 ( ! [N4: nat] : ( ord_less_eq_int @ ( X7 @ N4 ) @ ( X7 @ ( suc @ N4 ) ) )
% 13.25/7.70 => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7984_monoseq__def,axiom,
% 13.25/7.70 ( topolo6980174941875973593q_real
% 13.25/7.70 = ( ^ [X6: nat > real] :
% 13.25/7.70 ( ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_real @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 13.25/7.70 | ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7985_monoseq__def,axiom,
% 13.25/7.70 ( topolo3100542954746470799et_int
% 13.25/7.70 = ( ^ [X6: nat > set_int] :
% 13.25/7.70 ( ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_set_int @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 13.25/7.70 | ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_set_int @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7986_monoseq__def,axiom,
% 13.25/7.70 ( topolo4267028734544971653eq_rat
% 13.25/7.70 = ( ^ [X6: nat > rat] :
% 13.25/7.70 ( ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_rat @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 13.25/7.70 | ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7987_monoseq__def,axiom,
% 13.25/7.70 ( topolo1459490580787246023eq_num
% 13.25/7.70 = ( ^ [X6: nat > num] :
% 13.25/7.70 ( ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_num @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 13.25/7.70 | ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7988_monoseq__def,axiom,
% 13.25/7.70 ( topolo4902158794631467389eq_nat
% 13.25/7.70 = ( ^ [X6: nat > nat] :
% 13.25/7.70 ( ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_nat @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 13.25/7.70 | ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7989_monoseq__def,axiom,
% 13.25/7.70 ( topolo4899668324122417113eq_int
% 13.25/7.70 = ( ^ [X6: nat > int] :
% 13.25/7.70 ( ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_int @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) )
% 13.25/7.70 | ! [M3: nat,N2: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M3 @ N2 )
% 13.25/7.70 => ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ M3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7990_monoI2,axiom,
% 13.25/7.70 ! [X7: nat > real] :
% 13.25/7.70 ( ! [M4: nat,N4: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.70 => ( ord_less_eq_real @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 13.25/7.70 => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7991_monoI2,axiom,
% 13.25/7.70 ! [X7: nat > set_int] :
% 13.25/7.70 ( ! [M4: nat,N4: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.70 => ( ord_less_eq_set_int @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 13.25/7.70 => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7992_monoI2,axiom,
% 13.25/7.70 ! [X7: nat > rat] :
% 13.25/7.70 ( ! [M4: nat,N4: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.70 => ( ord_less_eq_rat @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 13.25/7.70 => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7993_monoI2,axiom,
% 13.25/7.70 ! [X7: nat > num] :
% 13.25/7.70 ( ! [M4: nat,N4: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.70 => ( ord_less_eq_num @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 13.25/7.70 => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7994_monoI2,axiom,
% 13.25/7.70 ! [X7: nat > nat] :
% 13.25/7.70 ( ! [M4: nat,N4: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.70 => ( ord_less_eq_nat @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 13.25/7.70 => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7995_monoI2,axiom,
% 13.25/7.70 ! [X7: nat > int] :
% 13.25/7.70 ( ! [M4: nat,N4: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.70 => ( ord_less_eq_int @ ( X7 @ N4 ) @ ( X7 @ M4 ) ) )
% 13.25/7.70 => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7996_sin__cos__npi,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( 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 ) ) ) )
% 13.25/7.70 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7997_diffs__equiv,axiom,
% 13.25/7.70 ! [C: nat > real,X: real] :
% 13.25/7.70 ( ( summable_real
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) )
% 13.25/7.70 => ( sums_real
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 13.25/7.70 @ ( suminf_real
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7998_diffs__equiv,axiom,
% 13.25/7.70 ! [C: nat > complex,X: complex] :
% 13.25/7.70 ( ( summable_complex
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) )
% 13.25/7.70 => ( sums_complex
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 13.25/7.70 @ ( suminf_complex
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_7999_cos__pi__eq__zero,axiom,
% 13.25/7.70 ! [M: nat] :
% 13.25/7.70 ( ( 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 ) ) ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8000_pochhammer__double,axiom,
% 13.25/7.70 ! [Z: real,N: nat] :
% 13.25/7.70 ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.70 = ( 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 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8001_pochhammer__double,axiom,
% 13.25/7.70 ! [Z: complex,N: nat] :
% 13.25/7.70 ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.70 = ( 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 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8002_pochhammer__double,axiom,
% 13.25/7.70 ! [Z: rat,N: nat] :
% 13.25/7.70 ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.70 = ( 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 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8003_of__nat__code,axiom,
% 13.25/7.70 ( semiri1314217659103216013at_int
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri8420488043553186161ux_int
% 13.25/7.70 @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 13.25/7.70 @ N2
% 13.25/7.70 @ zero_zero_int ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8004_of__nat__code,axiom,
% 13.25/7.70 ( semiri5074537144036343181t_real
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri7260567687927622513x_real
% 13.25/7.70 @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 13.25/7.70 @ N2
% 13.25/7.70 @ zero_zero_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8005_of__nat__code,axiom,
% 13.25/7.70 ( semiri1316708129612266289at_nat
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri8422978514062236437ux_nat
% 13.25/7.70 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 13.25/7.70 @ N2
% 13.25/7.70 @ zero_zero_nat ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8006_of__nat__code,axiom,
% 13.25/7.70 ( semiri8010041392384452111omplex
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri2816024913162550771omplex
% 13.25/7.70 @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
% 13.25/7.70 @ N2
% 13.25/7.70 @ zero_zero_complex ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8007_of__nat__code,axiom,
% 13.25/7.70 ( semiri681578069525770553at_rat
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri7787848453975740701ux_rat
% 13.25/7.70 @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 13.25/7.70 @ N2
% 13.25/7.70 @ zero_zero_rat ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8008_cos__zero,axiom,
% 13.25/7.70 ( ( cos_complex @ zero_zero_complex )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8009_cos__zero,axiom,
% 13.25/7.70 ( ( cos_real @ zero_zero_real )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8010_pochhammer__0,axiom,
% 13.25/7.70 ! [A: complex] :
% 13.25/7.70 ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8011_pochhammer__0,axiom,
% 13.25/7.70 ! [A: real] :
% 13.25/7.70 ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8012_pochhammer__0,axiom,
% 13.25/7.70 ! [A: rat] :
% 13.25/7.70 ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 13.25/7.70 = one_one_rat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8013_pochhammer__0,axiom,
% 13.25/7.70 ! [A: nat] :
% 13.25/7.70 ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 13.25/7.70 = one_one_nat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8014_pochhammer__0,axiom,
% 13.25/7.70 ! [A: int] :
% 13.25/7.70 ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 13.25/7.70 = one_one_int ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8015_sin__pi__minus,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( sin_real @ ( minus_minus_real @ pi @ X ) )
% 13.25/7.70 = ( sin_real @ X ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8016_cos__pi,axiom,
% 13.25/7.70 ( ( cos_real @ pi )
% 13.25/7.70 = ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8017_cos__periodic__pi2,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( cos_real @ ( plus_plus_real @ pi @ X ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8018_cos__periodic__pi,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( cos_real @ ( plus_plus_real @ X @ pi ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8019_sin__periodic__pi2,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( sin_real @ ( plus_plus_real @ pi @ X ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8020_sin__periodic__pi,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( sin_real @ ( plus_plus_real @ X @ pi ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8021_cos__minus__pi,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( cos_real @ ( minus_minus_real @ X @ pi ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8022_cos__pi__minus,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( cos_real @ ( minus_minus_real @ pi @ X ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( cos_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8023_sin__minus__pi,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( sin_real @ ( minus_minus_real @ X @ pi ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8024_sin__cos__squared__add3,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8025_sin__cos__squared__add3,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8026_sin__npi2,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8027_sin__npi,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8028_sin__npi__int,axiom,
% 13.25/7.70 ! [N: int] :
% 13.25/7.70 ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8029_cos__pi__half,axiom,
% 13.25/7.70 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8030_sin__two__pi,axiom,
% 13.25/7.70 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8031_sin__pi__half,axiom,
% 13.25/7.70 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8032_cos__two__pi,axiom,
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8033_cos__periodic,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 13.25/7.70 = ( cos_real @ X ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8034_sin__periodic,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 13.25/7.70 = ( sin_real @ X ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8035_cos__2pi__minus,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 13.25/7.70 = ( cos_real @ X ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8036_cos__npi,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 13.25/7.70 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8037_cos__npi2,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
% 13.25/7.70 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8038_sin__cos__squared__add,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( 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 ) ) ) )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8039_sin__cos__squared__add,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( 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 ) ) ) )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8040_sin__cos__squared__add2,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( 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 ) ) ) )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8041_sin__cos__squared__add2,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( 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 ) ) ) )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8042_sin__2npi,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8043_cos__2npi,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8044_sin__2pi__minus,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8045_sin__int__2pin,axiom,
% 13.25/7.70 ! [N: int] :
% 13.25/7.70 ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8046_cos__int__2pin,axiom,
% 13.25/7.70 ! [N: int] :
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N ) ) )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8047_cos__3over2__pi,axiom,
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 13.25/7.70 = zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8048_sin__3over2__pi,axiom,
% 13.25/7.70 ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 13.25/7.70 = ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8049_cos__npi__int,axiom,
% 13.25/7.70 ! [N: int] :
% 13.25/7.70 ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 13.25/7.70 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 13.25/7.70 = one_one_real ) )
% 13.25/7.70 & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 13.25/7.70 => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
% 13.25/7.70 = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8050_cos__one__sin__zero,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( ( cos_complex @ X )
% 13.25/7.70 = one_one_complex )
% 13.25/7.70 => ( ( sin_complex @ X )
% 13.25/7.70 = zero_zero_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8051_cos__one__sin__zero,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( cos_real @ X )
% 13.25/7.70 = one_one_real )
% 13.25/7.70 => ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8052_sin__diff,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( sin_real @ ( minus_minus_real @ X @ Y3 ) )
% 13.25/7.70 = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y3 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8053_polar__Ex,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ? [R2: real,A5: real] :
% 13.25/7.70 ( ( X
% 13.25/7.70 = ( times_times_real @ R2 @ ( cos_real @ A5 ) ) )
% 13.25/7.70 & ( Y3
% 13.25/7.70 = ( times_times_real @ R2 @ ( sin_real @ A5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8054_sin__add,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( sin_real @ ( plus_plus_real @ X @ Y3 ) )
% 13.25/7.70 = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y3 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8055_cos__diff,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( cos_real @ ( minus_minus_real @ X @ Y3 ) )
% 13.25/7.70 = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8056_cos__add,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( cos_real @ ( plus_plus_real @ X @ Y3 ) )
% 13.25/7.70 = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8057_sin__zero__norm__cos__one,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
% 13.25/7.70 = one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8058_sin__zero__norm__cos__one,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( ( sin_complex @ X )
% 13.25/7.70 = zero_zero_complex )
% 13.25/7.70 => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
% 13.25/7.70 = one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8059_sin__zero__abs__cos__one,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 => ( ( abs_abs_real @ ( cos_real @ X ) )
% 13.25/7.70 = one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8060_sin__double,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 13.25/7.70 = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8061_sin__double,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 13.25/7.70 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8062_sincos__principal__value,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ? [Y4: real] :
% 13.25/7.70 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 13.25/7.70 & ( ord_less_eq_real @ Y4 @ pi )
% 13.25/7.70 & ( ( sin_real @ Y4 )
% 13.25/7.70 = ( sin_real @ X ) )
% 13.25/7.70 & ( ( cos_real @ Y4 )
% 13.25/7.70 = ( cos_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8063_sin__x__le__x,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ord_less_eq_real @ ( sin_real @ X ) @ X ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8064_sin__le__one,axiom,
% 13.25/7.70 ! [X: real] : ( ord_less_eq_real @ ( sin_real @ X ) @ one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8065_cos__le__one,axiom,
% 13.25/7.70 ! [X: real] : ( ord_less_eq_real @ ( cos_real @ X ) @ one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8066_abs__sin__x__le__abs__x,axiom,
% 13.25/7.70 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ ( abs_abs_real @ X ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8067_pochhammer__pos,axiom,
% 13.25/7.70 ! [X: real,N: nat] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8068_pochhammer__pos,axiom,
% 13.25/7.70 ! [X: rat,N: nat] :
% 13.25/7.70 ( ( ord_less_rat @ zero_zero_rat @ X )
% 13.25/7.70 => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8069_pochhammer__pos,axiom,
% 13.25/7.70 ! [X: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_nat @ zero_zero_nat @ X )
% 13.25/7.70 => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8070_pochhammer__pos,axiom,
% 13.25/7.70 ! [X: int,N: nat] :
% 13.25/7.70 ( ( ord_less_int @ zero_zero_int @ X )
% 13.25/7.70 => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8071_pochhammer__eq__0__mono,axiom,
% 13.25/7.70 ! [A: complex,N: nat,M: nat] :
% 13.25/7.70 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 13.25/7.70 = zero_zero_complex )
% 13.25/7.70 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( ( comm_s2602460028002588243omplex @ A @ M )
% 13.25/7.70 = zero_zero_complex ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8072_pochhammer__eq__0__mono,axiom,
% 13.25/7.70 ! [A: real,N: nat,M: nat] :
% 13.25/7.70 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( ( comm_s7457072308508201937r_real @ A @ M )
% 13.25/7.70 = zero_zero_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8073_pochhammer__eq__0__mono,axiom,
% 13.25/7.70 ! [A: rat,N: nat,M: nat] :
% 13.25/7.70 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 13.25/7.70 = zero_zero_rat )
% 13.25/7.70 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 13.25/7.70 = zero_zero_rat ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8074_pochhammer__neq__0__mono,axiom,
% 13.25/7.70 ! [A: complex,M: nat,N: nat] :
% 13.25/7.70 ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 13.25/7.70 != zero_zero_complex )
% 13.25/7.70 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( ( comm_s2602460028002588243omplex @ A @ N )
% 13.25/7.70 != zero_zero_complex ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8075_pochhammer__neq__0__mono,axiom,
% 13.25/7.70 ! [A: real,M: nat,N: nat] :
% 13.25/7.70 ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 13.25/7.70 != zero_zero_real )
% 13.25/7.70 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( ( comm_s7457072308508201937r_real @ A @ N )
% 13.25/7.70 != zero_zero_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8076_pochhammer__neq__0__mono,axiom,
% 13.25/7.70 ! [A: rat,M: nat,N: nat] :
% 13.25/7.70 ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 13.25/7.70 != zero_zero_rat )
% 13.25/7.70 => ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( ( comm_s4028243227959126397er_rat @ A @ N )
% 13.25/7.70 != zero_zero_rat ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8077_sin__cos__le1,axiom,
% 13.25/7.70 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8078_cos__squared__eq,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8079_cos__squared__eq,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8080_sin__squared__eq,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( power_power_complex @ ( sin_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8081_sin__squared__eq,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( power_power_real @ ( sin_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8082_sin__gt__zero,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ pi )
% 13.25/7.70 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8083_sin__x__ge__neg__x,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ord_less_eq_real @ ( uminus_uminus_real @ X ) @ ( sin_real @ X ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8084_sin__ge__zero,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ pi )
% 13.25/7.70 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8085_sin__ge__minus__one,axiom,
% 13.25/7.70 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8086_cos__monotone__0__pi__le,axiom,
% 13.25/7.70 ! [Y3: real,X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ pi )
% 13.25/7.70 => ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8087_cos__mono__le__eq,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ pi )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ pi )
% 13.25/7.70 => ( ( ord_less_eq_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) )
% 13.25/7.70 = ( ord_less_eq_real @ Y3 @ X ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8088_cos__inj__pi,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ pi )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ pi )
% 13.25/7.70 => ( ( ( cos_real @ X )
% 13.25/7.70 = ( cos_real @ Y3 ) )
% 13.25/7.70 => ( X = Y3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8089_cos__ge__minus__one,axiom,
% 13.25/7.70 ! [X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8090_abs__sin__le__one,axiom,
% 13.25/7.70 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X ) ) @ one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8091_abs__cos__le__one,axiom,
% 13.25/7.70 ! [X: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X ) ) @ one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8092_sin__times__sin,axiom,
% 13.25/7.70 ! [W: complex,Z: complex] :
% 13.25/7.70 ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 13.25/7.70 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8093_sin__times__sin,axiom,
% 13.25/7.70 ! [W: real,Z: real] :
% 13.25/7.70 ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8094_sin__times__cos,axiom,
% 13.25/7.70 ! [W: complex,Z: complex] :
% 13.25/7.70 ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
% 13.25/7.70 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8095_sin__times__cos,axiom,
% 13.25/7.70 ! [W: real,Z: real] :
% 13.25/7.70 ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8096_cos__times__sin,axiom,
% 13.25/7.70 ! [W: complex,Z: complex] :
% 13.25/7.70 ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
% 13.25/7.70 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8097_cos__times__sin,axiom,
% 13.25/7.70 ! [W: real,Z: real] :
% 13.25/7.70 ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8098_sin__plus__sin,axiom,
% 13.25/7.70 ! [W: complex,Z: complex] :
% 13.25/7.70 ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8099_sin__plus__sin,axiom,
% 13.25/7.70 ! [W: real,Z: real] :
% 13.25/7.70 ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8100_sin__diff__sin,axiom,
% 13.25/7.70 ! [W: complex,Z: complex] :
% 13.25/7.70 ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8101_sin__diff__sin,axiom,
% 13.25/7.70 ! [W: real,Z: real] :
% 13.25/7.70 ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8102_cos__diff__cos,axiom,
% 13.25/7.70 ! [W: complex,Z: complex] :
% 13.25/7.70 ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8103_cos__diff__cos,axiom,
% 13.25/7.70 ! [W: real,Z: real] :
% 13.25/7.70 ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8104_pochhammer__nonneg,axiom,
% 13.25/7.70 ! [X: real,N: nat] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8105_pochhammer__nonneg,axiom,
% 13.25/7.70 ! [X: rat,N: nat] :
% 13.25/7.70 ( ( ord_less_rat @ zero_zero_rat @ X )
% 13.25/7.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8106_pochhammer__nonneg,axiom,
% 13.25/7.70 ! [X: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_nat @ zero_zero_nat @ X )
% 13.25/7.70 => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8107_pochhammer__nonneg,axiom,
% 13.25/7.70 ! [X: int,N: nat] :
% 13.25/7.70 ( ( ord_less_int @ zero_zero_int @ X )
% 13.25/7.70 => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8108_cos__double,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8109_cos__double,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8110_pochhammer__0__left,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( ( N = zero_zero_nat )
% 13.25/7.70 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 & ( ( N != zero_zero_nat )
% 13.25/7.70 => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
% 13.25/7.70 = zero_zero_complex ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8111_pochhammer__0__left,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( ( N = zero_zero_nat )
% 13.25/7.70 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 13.25/7.70 = one_one_real ) )
% 13.25/7.70 & ( ( N != zero_zero_nat )
% 13.25/7.70 => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
% 13.25/7.70 = zero_zero_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8112_pochhammer__0__left,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( ( N = zero_zero_nat )
% 13.25/7.70 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 13.25/7.70 = one_one_rat ) )
% 13.25/7.70 & ( ( N != zero_zero_nat )
% 13.25/7.70 => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
% 13.25/7.70 = zero_zero_rat ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8113_pochhammer__0__left,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( ( N = zero_zero_nat )
% 13.25/7.70 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 13.25/7.70 = one_one_nat ) )
% 13.25/7.70 & ( ( N != zero_zero_nat )
% 13.25/7.70 => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
% 13.25/7.70 = zero_zero_nat ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8114_pochhammer__0__left,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( ( N = zero_zero_nat )
% 13.25/7.70 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 13.25/7.70 = one_one_int ) )
% 13.25/7.70 & ( ( N != zero_zero_nat )
% 13.25/7.70 => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
% 13.25/7.70 = zero_zero_int ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8115_cos__double__sin,axiom,
% 13.25/7.70 ! [W: complex] :
% 13.25/7.70 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 13.25/7.70 = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8116_cos__double__sin,axiom,
% 13.25/7.70 ! [W: real] :
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8117_cos__two__neq__zero,axiom,
% 13.25/7.70 ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.70 != zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8118_cos__monotone__0__pi,axiom,
% 13.25/7.70 ! [Y3: real,X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.70 => ( ( ord_less_real @ Y3 @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ pi )
% 13.25/7.70 => ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8119_cos__mono__less__eq,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ pi )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ pi )
% 13.25/7.70 => ( ( ord_less_real @ ( cos_real @ X ) @ ( cos_real @ Y3 ) )
% 13.25/7.70 = ( ord_less_real @ Y3 @ X ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8120_sin__eq__0__pi,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ pi )
% 13.25/7.70 => ( ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 => ( X = zero_zero_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8121_sin__zero__pi__iff,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ ( abs_abs_real @ X ) @ pi )
% 13.25/7.70 => ( ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 = ( X = zero_zero_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8122_cos__monotone__minus__pi__0_H,axiom,
% 13.25/7.70 ! [Y3: real,X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.25/7.70 => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8123_sin__zero__iff__int2,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 = ( ? [I4: int] :
% 13.25/7.70 ( X
% 13.25/7.70 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ pi ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8124_pochhammer__rec,axiom,
% 13.25/7.70 ! [A: complex,N: nat] :
% 13.25/7.70 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8125_pochhammer__rec,axiom,
% 13.25/7.70 ! [A: real,N: nat] :
% 13.25/7.70 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8126_pochhammer__rec,axiom,
% 13.25/7.70 ! [A: rat,N: nat] :
% 13.25/7.70 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8127_pochhammer__rec,axiom,
% 13.25/7.70 ! [A: nat,N: nat] :
% 13.25/7.70 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8128_pochhammer__rec,axiom,
% 13.25/7.70 ! [A: int,N: nat] :
% 13.25/7.70 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8129_diffs__def,axiom,
% 13.25/7.70 ( diffs_int
% 13.25/7.70 = ( ^ [C3: nat > int,N2: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8130_diffs__def,axiom,
% 13.25/7.70 ( diffs_real
% 13.25/7.70 = ( ^ [C3: nat > real,N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8131_diffs__def,axiom,
% 13.25/7.70 ( diffs_complex
% 13.25/7.70 = ( ^ [C3: nat > complex,N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8132_diffs__def,axiom,
% 13.25/7.70 ( diffs_rat
% 13.25/7.70 = ( ^ [C3: nat > rat,N2: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8133_pochhammer__Suc,axiom,
% 13.25/7.70 ! [A: int,N: nat] :
% 13.25/7.70 ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8134_pochhammer__Suc,axiom,
% 13.25/7.70 ! [A: real,N: nat] :
% 13.25/7.70 ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8135_pochhammer__Suc,axiom,
% 13.25/7.70 ! [A: nat,N: nat] :
% 13.25/7.70 ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8136_pochhammer__Suc,axiom,
% 13.25/7.70 ! [A: complex,N: nat] :
% 13.25/7.70 ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8137_pochhammer__Suc,axiom,
% 13.25/7.70 ! [A: rat,N: nat] :
% 13.25/7.70 ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8138_pochhammer__rec_H,axiom,
% 13.25/7.70 ! [Z: int,N: nat] :
% 13.25/7.70 ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8139_pochhammer__rec_H,axiom,
% 13.25/7.70 ! [Z: real,N: nat] :
% 13.25/7.70 ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8140_pochhammer__rec_H,axiom,
% 13.25/7.70 ! [Z: nat,N: nat] :
% 13.25/7.70 ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8141_pochhammer__rec_H,axiom,
% 13.25/7.70 ! [Z: complex,N: nat] :
% 13.25/7.70 ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ ( comm_s2602460028002588243omplex @ Z @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8142_pochhammer__rec_H,axiom,
% 13.25/7.70 ! [Z: rat,N: nat] :
% 13.25/7.70 ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8143_pochhammer__of__nat__eq__0__lemma,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ord_less_nat @ N @ K )
% 13.25/7.70 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 13.25/7.70 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8144_pochhammer__of__nat__eq__0__lemma,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ord_less_nat @ N @ K )
% 13.25/7.70 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 13.25/7.70 = zero_zero_int ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8145_pochhammer__of__nat__eq__0__lemma,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ord_less_nat @ N @ K )
% 13.25/7.70 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 13.25/7.70 = zero_zero_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8146_pochhammer__of__nat__eq__0__lemma,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ord_less_nat @ N @ K )
% 13.25/7.70 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 13.25/7.70 = zero_zero_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8147_pochhammer__of__nat__eq__0__lemma,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ord_less_nat @ N @ K )
% 13.25/7.70 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 13.25/7.70 = zero_zero_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8148_pochhammer__of__nat__eq__0__iff,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 13.25/7.70 = zero_z3403309356797280102nteger )
% 13.25/7.70 = ( ord_less_nat @ N @ K ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8149_pochhammer__of__nat__eq__0__iff,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 13.25/7.70 = zero_zero_int )
% 13.25/7.70 = ( ord_less_nat @ N @ K ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8150_pochhammer__of__nat__eq__0__iff,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 = ( ord_less_nat @ N @ K ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8151_pochhammer__of__nat__eq__0__iff,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 13.25/7.70 = zero_zero_complex )
% 13.25/7.70 = ( ord_less_nat @ N @ K ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8152_pochhammer__of__nat__eq__0__iff,axiom,
% 13.25/7.70 ! [N: nat,K: nat] :
% 13.25/7.70 ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 13.25/7.70 = zero_zero_rat )
% 13.25/7.70 = ( ord_less_nat @ N @ K ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8153_pochhammer__eq__0__iff,axiom,
% 13.25/7.70 ! [A: real,N: nat] :
% 13.25/7.70 ( ( ( comm_s7457072308508201937r_real @ A @ N )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 = ( ? [K2: nat] :
% 13.25/7.70 ( ( ord_less_nat @ K2 @ N )
% 13.25/7.70 & ( A
% 13.25/7.70 = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8154_pochhammer__eq__0__iff,axiom,
% 13.25/7.70 ! [A: complex,N: nat] :
% 13.25/7.70 ( ( ( comm_s2602460028002588243omplex @ A @ N )
% 13.25/7.70 = zero_zero_complex )
% 13.25/7.70 = ( ? [K2: nat] :
% 13.25/7.70 ( ( ord_less_nat @ K2 @ N )
% 13.25/7.70 & ( A
% 13.25/7.70 = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8155_pochhammer__eq__0__iff,axiom,
% 13.25/7.70 ! [A: rat,N: nat] :
% 13.25/7.70 ( ( ( comm_s4028243227959126397er_rat @ A @ N )
% 13.25/7.70 = zero_zero_rat )
% 13.25/7.70 = ( ? [K2: nat] :
% 13.25/7.70 ( ( ord_less_nat @ K2 @ N )
% 13.25/7.70 & ( A
% 13.25/7.70 = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8156_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 13.25/7.70 ! [K: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.70 => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K )
% 13.25/7.70 != zero_z3403309356797280102nteger ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8157_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 13.25/7.70 ! [K: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.70 => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K )
% 13.25/7.70 != zero_zero_int ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8158_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 13.25/7.70 ! [K: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.70 => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K )
% 13.25/7.70 != zero_zero_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8159_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 13.25/7.70 ! [K: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.70 => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K )
% 13.25/7.70 != zero_zero_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8160_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 13.25/7.70 ! [K: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.70 => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K )
% 13.25/7.70 != zero_zero_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8161_sincos__total__pi,axiom,
% 13.25/7.70 ! [Y3: real,X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.70 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.70 = one_one_real )
% 13.25/7.70 => ? [T5: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 13.25/7.70 & ( ord_less_eq_real @ T5 @ pi )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( cos_real @ T5 ) )
% 13.25/7.70 & ( Y3
% 13.25/7.70 = ( sin_real @ T5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8162_pochhammer__product_H,axiom,
% 13.25/7.70 ! [Z: int,N: nat,M: nat] :
% 13.25/7.70 ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N @ M ) )
% 13.25/7.70 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N ) ) @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8163_pochhammer__product_H,axiom,
% 13.25/7.70 ! [Z: real,N: nat,M: nat] :
% 13.25/7.70 ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N @ M ) )
% 13.25/7.70 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N ) ) @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8164_pochhammer__product_H,axiom,
% 13.25/7.70 ! [Z: nat,N: nat,M: nat] :
% 13.25/7.70 ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N @ M ) )
% 13.25/7.70 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N ) ) @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8165_pochhammer__product_H,axiom,
% 13.25/7.70 ! [Z: complex,N: nat,M: nat] :
% 13.25/7.70 ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N @ M ) )
% 13.25/7.70 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N ) ) @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8166_pochhammer__product_H,axiom,
% 13.25/7.70 ! [Z: rat,N: nat,M: nat] :
% 13.25/7.70 ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N @ M ) )
% 13.25/7.70 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N ) ) @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8167_sin__expansion__lemma,axiom,
% 13.25/7.70 ! [X: real,M: nat] :
% 13.25/7.70 ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 13.25/7.70 = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8168_cos__expansion__lemma,axiom,
% 13.25/7.70 ! [X: real,M: nat] :
% 13.25/7.70 ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 13.25/7.70 = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8169_termdiff__converges__all,axiom,
% 13.25/7.70 ! [C: nat > complex,X: complex] :
% 13.25/7.70 ( ! [X3: complex] :
% 13.25/7.70 ( summable_complex
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) )
% 13.25/7.70 => ( summable_complex
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8170_termdiff__converges__all,axiom,
% 13.25/7.70 ! [C: nat > real,X: real] :
% 13.25/7.70 ( ! [X3: real] :
% 13.25/7.70 ( summable_real
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) )
% 13.25/7.70 => ( summable_real
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8171_sin__gt__zero__02,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.70 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8172_cos__two__less__zero,axiom,
% 13.25/7.70 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8173_cos__is__zero,axiom,
% 13.25/7.70 ? [X3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 13.25/7.70 & ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.70 & ( ( cos_real @ X3 )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 & ! [Y5: real] :
% 13.25/7.70 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 13.25/7.70 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.70 & ( ( cos_real @ Y5 )
% 13.25/7.70 = zero_zero_real ) )
% 13.25/7.70 => ( Y5 = X3 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8174_cos__two__le__zero,axiom,
% 13.25/7.70 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8175_cos__monotone__minus__pi__0,axiom,
% 13.25/7.70 ! [Y3: real,X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_real @ Y3 @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.25/7.70 => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8176_cos__total,axiom,
% 13.25/7.70 ! [Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.70 => ? [X3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 13.25/7.70 & ( ord_less_eq_real @ X3 @ pi )
% 13.25/7.70 & ( ( cos_real @ X3 )
% 13.25/7.70 = Y3 )
% 13.25/7.70 & ! [Y5: real] :
% 13.25/7.70 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 13.25/7.70 & ( ord_less_eq_real @ Y5 @ pi )
% 13.25/7.70 & ( ( cos_real @ Y5 )
% 13.25/7.70 = Y3 ) )
% 13.25/7.70 => ( Y5 = X3 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8177_sincos__total__pi__half,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.70 => ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.70 = one_one_real )
% 13.25/7.70 => ? [T5: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 13.25/7.70 & ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( cos_real @ T5 ) )
% 13.25/7.70 & ( Y3
% 13.25/7.70 = ( sin_real @ T5 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8178_sincos__total__2pi__le,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.70 = one_one_real )
% 13.25/7.70 => ? [T5: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 13.25/7.70 & ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( cos_real @ T5 ) )
% 13.25/7.70 & ( Y3
% 13.25/7.70 = ( sin_real @ T5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8179_pochhammer__product,axiom,
% 13.25/7.70 ! [M: nat,N: nat,Z: int] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ( comm_s4660882817536571857er_int @ Z @ N )
% 13.25/7.70 = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8180_pochhammer__product,axiom,
% 13.25/7.70 ! [M: nat,N: nat,Z: real] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ( comm_s7457072308508201937r_real @ Z @ N )
% 13.25/7.70 = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8181_pochhammer__product,axiom,
% 13.25/7.70 ! [M: nat,N: nat,Z: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ( comm_s4663373288045622133er_nat @ Z @ N )
% 13.25/7.70 = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8182_pochhammer__product,axiom,
% 13.25/7.70 ! [M: nat,N: nat,Z: complex] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ( comm_s2602460028002588243omplex @ Z @ N )
% 13.25/7.70 = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8183_pochhammer__product,axiom,
% 13.25/7.70 ! [M: nat,N: nat,Z: rat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ( comm_s4028243227959126397er_rat @ Z @ N )
% 13.25/7.70 = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8184_sincos__total__2pi,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.70 = one_one_real )
% 13.25/7.70 => ~ ! [T5: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 13.25/7.70 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 13.25/7.70 => ( ( X
% 13.25/7.70 = ( cos_real @ T5 ) )
% 13.25/7.70 => ( Y3
% 13.25/7.70 != ( sin_real @ T5 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8185_sin__pi__divide__n__ge__0,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( N != zero_zero_nat )
% 13.25/7.70 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8186_pochhammer__absorb__comp,axiom,
% 13.25/7.70 ! [R: code_integer,K: nat] :
% 13.25/7.70 ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R ) @ K ) )
% 13.25/7.70 = ( times_3573771949741848930nteger @ R @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8187_pochhammer__absorb__comp,axiom,
% 13.25/7.70 ! [R: int,K: nat] :
% 13.25/7.70 ( ( times_times_int @ ( minus_minus_int @ R @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R ) @ K ) )
% 13.25/7.70 = ( times_times_int @ R @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R ) @ one_one_int ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8188_pochhammer__absorb__comp,axiom,
% 13.25/7.70 ! [R: real,K: nat] :
% 13.25/7.70 ( ( times_times_real @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R ) @ K ) )
% 13.25/7.70 = ( times_times_real @ R @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R ) @ one_one_real ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8189_pochhammer__absorb__comp,axiom,
% 13.25/7.70 ! [R: complex,K: nat] :
% 13.25/7.70 ( ( times_times_complex @ ( minus_minus_complex @ R @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R ) @ K ) )
% 13.25/7.70 = ( times_times_complex @ R @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R ) @ one_one_complex ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8190_pochhammer__absorb__comp,axiom,
% 13.25/7.70 ! [R: rat,K: nat] :
% 13.25/7.70 ( ( times_times_rat @ ( minus_minus_rat @ R @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R ) @ K ) )
% 13.25/7.70 = ( times_times_rat @ R @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R ) @ one_one_rat ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8191_cos__times__cos,axiom,
% 13.25/7.70 ! [W: complex,Z: complex] :
% 13.25/7.70 ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 13.25/7.70 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8192_cos__times__cos,axiom,
% 13.25/7.70 ! [W: real,Z: real] :
% 13.25/7.70 ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8193_cos__plus__cos,axiom,
% 13.25/7.70 ! [W: complex,Z: complex] :
% 13.25/7.70 ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8194_cos__plus__cos,axiom,
% 13.25/7.70 ! [W: real,Z: real] :
% 13.25/7.70 ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
% 13.25/7.70 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8195_sin__gt__zero2,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8196_sin__lt__zero,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ pi @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 13.25/7.70 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8197_cos__double__less__one,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.70 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8198_sin__30,axiom,
% 13.25/7.70 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 13.25/7.70 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8199_cos__gt__zero,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8200_sin__inj__pi,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ( ( sin_real @ X )
% 13.25/7.70 = ( sin_real @ Y3 ) )
% 13.25/7.70 => ( X = Y3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8201_sin__mono__le__eq,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ( ord_less_eq_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) )
% 13.25/7.70 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8202_sin__monotone__2pi__le,axiom,
% 13.25/7.70 ! [Y3: real,X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8203_cos__60,axiom,
% 13.25/7.70 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 13.25/7.70 = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8204_cos__one__2pi__int,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( cos_real @ X )
% 13.25/7.70 = one_one_real )
% 13.25/7.70 = ( ? [X2: int] :
% 13.25/7.70 ( X
% 13.25/7.70 = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8205_cos__double__cos,axiom,
% 13.25/7.70 ! [W: complex] :
% 13.25/7.70 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
% 13.25/7.70 = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8206_cos__double__cos,axiom,
% 13.25/7.70 ! [W: real] :
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
% 13.25/7.70 = ( 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 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8207_cos__treble__cos,axiom,
% 13.25/7.70 ! [X: complex] :
% 13.25/7.70 ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
% 13.25/7.70 = ( 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 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8208_cos__treble__cos,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
% 13.25/7.70 = ( 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 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8209_pochhammer__minus_H,axiom,
% 13.25/7.70 ! [B: code_integer,K: nat] :
% 13.25/7.70 ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 13.25/7.70 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8210_pochhammer__minus_H,axiom,
% 13.25/7.70 ! [B: int,K: nat] :
% 13.25/7.70 ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 13.25/7.70 = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8211_pochhammer__minus_H,axiom,
% 13.25/7.70 ! [B: real,K: nat] :
% 13.25/7.70 ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 13.25/7.70 = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8212_pochhammer__minus_H,axiom,
% 13.25/7.70 ! [B: complex,K: nat] :
% 13.25/7.70 ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 13.25/7.70 = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8213_pochhammer__minus_H,axiom,
% 13.25/7.70 ! [B: rat,K: nat] :
% 13.25/7.70 ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 13.25/7.70 = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8214_pochhammer__minus,axiom,
% 13.25/7.70 ! [B: code_integer,K: nat] :
% 13.25/7.70 ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 13.25/7.70 = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8215_pochhammer__minus,axiom,
% 13.25/7.70 ! [B: int,K: nat] :
% 13.25/7.70 ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 13.25/7.70 = ( 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 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8216_pochhammer__minus,axiom,
% 13.25/7.70 ! [B: real,K: nat] :
% 13.25/7.70 ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 13.25/7.70 = ( 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 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8217_pochhammer__minus,axiom,
% 13.25/7.70 ! [B: complex,K: nat] :
% 13.25/7.70 ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 13.25/7.70 = ( 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 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8218_pochhammer__minus,axiom,
% 13.25/7.70 ! [B: rat,K: nat] :
% 13.25/7.70 ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 13.25/7.70 = ( 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 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8219_termdiff__converges,axiom,
% 13.25/7.70 ! [X: real,K5: real,C: nat > real] :
% 13.25/7.70 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ K5 )
% 13.25/7.70 => ( ! [X3: real] :
% 13.25/7.70 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
% 13.25/7.70 => ( summable_real
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X3 @ N2 ) ) ) )
% 13.25/7.70 => ( summable_real
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8220_termdiff__converges,axiom,
% 13.25/7.70 ! [X: complex,K5: real,C: nat > complex] :
% 13.25/7.70 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ K5 )
% 13.25/7.70 => ( ! [X3: complex] :
% 13.25/7.70 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
% 13.25/7.70 => ( summable_complex
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X3 @ N2 ) ) ) )
% 13.25/7.70 => ( summable_complex
% 13.25/7.70 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8221_sin__le__zero,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ pi @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8222_sin__less__zero,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ zero_zero_real )
% 13.25/7.70 => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8223_sin__monotone__2pi,axiom,
% 13.25/7.70 ! [Y3: real,X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_real @ Y3 @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8224_sin__mono__less__eq,axiom,
% 13.25/7.70 ! [X: real,Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ( ord_less_real @ ( sin_real @ X ) @ ( sin_real @ Y3 ) )
% 13.25/7.70 = ( ord_less_real @ X @ Y3 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8225_sin__total,axiom,
% 13.25/7.70 ! [Y3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.70 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.70 => ? [X3: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 13.25/7.70 & ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 & ( ( sin_real @ X3 )
% 13.25/7.70 = Y3 )
% 13.25/7.70 & ! [Y5: real] :
% 13.25/7.70 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 13.25/7.70 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 & ( ( sin_real @ Y5 )
% 13.25/7.70 = Y3 ) )
% 13.25/7.70 => ( Y5 = X3 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8226_cos__gt__zero__pi,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.70 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8227_cos__ge__zero,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.70 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8228_cos__one__2pi,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( cos_real @ X )
% 13.25/7.70 = one_one_real )
% 13.25/7.70 = ( ? [X2: nat] :
% 13.25/7.70 ( X
% 13.25/7.70 = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
% 13.25/7.70 | ? [X2: nat] :
% 13.25/7.70 ( X
% 13.25/7.70 = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8229_sin__pi__divide__n__gt__0,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.70 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8230_sin__zero__iff__int,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 = ( ? [I4: int] :
% 13.25/7.70 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8231_cos__zero__iff__int,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( cos_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 = ( ? [I4: int] :
% 13.25/7.70 ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I4 )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( times_times_real @ ( ring_1_of_int_real @ I4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8232_sin__zero__lemma,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 => ? [N4: nat] :
% 13.25/7.70 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8233_sin__zero__iff,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( sin_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 = ( ? [N2: nat] :
% 13.25/7.70 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.70 | ? [N2: nat] :
% 13.25/7.70 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8234_cos__zero__lemma,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ( cos_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 => ? [N4: nat] :
% 13.25/7.70 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8235_cos__zero__iff,axiom,
% 13.25/7.70 ! [X: real] :
% 13.25/7.70 ( ( ( cos_real @ X )
% 13.25/7.70 = zero_zero_real )
% 13.25/7.70 = ( ? [N2: nat] :
% 13.25/7.70 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.70 | ? [N2: nat] :
% 13.25/7.70 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 13.25/7.70 & ( X
% 13.25/7.70 = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8236_Maclaurin__cos__expansion2,axiom,
% 13.25/7.70 ! [X: real,N: nat] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.70 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.70 => ? [T5: real] :
% 13.25/7.70 ( ( ord_less_real @ zero_zero_real @ T5 )
% 13.25/7.70 & ( ord_less_real @ T5 @ X )
% 13.25/7.70 & ( ( cos_real @ X )
% 13.25/7.70 = ( plus_plus_real
% 13.25/7.70 @ ( groups6591440286371151544t_real
% 13.25/7.70 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.70 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.70 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8237_Maclaurin__minus__cos__expansion,axiom,
% 13.25/7.70 ! [N: nat,X: real] :
% 13.25/7.70 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.70 => ( ( ord_less_real @ X @ zero_zero_real )
% 13.25/7.70 => ? [T5: real] :
% 13.25/7.70 ( ( ord_less_real @ X @ T5 )
% 13.25/7.70 & ( ord_less_real @ T5 @ zero_zero_real )
% 13.25/7.70 & ( ( cos_real @ X )
% 13.25/7.70 = ( plus_plus_real
% 13.25/7.70 @ ( groups6591440286371151544t_real
% 13.25/7.70 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.70 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.70 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8238_Maclaurin__cos__expansion,axiom,
% 13.25/7.70 ! [X: real,N: nat] :
% 13.25/7.70 ? [T5: real] :
% 13.25/7.70 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 13.25/7.70 & ( ( cos_real @ X )
% 13.25/7.70 = ( plus_plus_real
% 13.25/7.70 @ ( groups6591440286371151544t_real
% 13.25/7.70 @ ^ [M3: nat] : ( times_times_real @ ( cos_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.70 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.70 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8239_pochhammer__times__pochhammer__half,axiom,
% 13.25/7.70 ! [Z: real,N: nat] :
% 13.25/7.70 ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 13.25/7.70 = ( groups129246275422532515t_real
% 13.25/7.70 @ ^ [K2: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8240_pochhammer__times__pochhammer__half,axiom,
% 13.25/7.70 ! [Z: complex,N: nat] :
% 13.25/7.70 ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 13.25/7.70 = ( groups6464643781859351333omplex
% 13.25/7.70 @ ^ [K2: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8241_pochhammer__times__pochhammer__half,axiom,
% 13.25/7.70 ! [Z: rat,N: nat] :
% 13.25/7.70 ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
% 13.25/7.70 = ( groups73079841787564623at_rat
% 13.25/7.70 @ ^ [K2: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8242_pochhammer__code,axiom,
% 13.25/7.70 ( comm_s4660882817536571857er_int
% 13.25/7.70 = ( ^ [A3: int,N2: nat] :
% 13.25/7.70 ( if_int @ ( N2 = zero_zero_nat ) @ one_one_int
% 13.25/7.70 @ ( set_fo2581907887559384638at_int
% 13.25/7.70 @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
% 13.25/7.70 @ zero_zero_nat
% 13.25/7.70 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 13.25/7.70 @ one_one_int ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8243_pochhammer__code,axiom,
% 13.25/7.70 ( comm_s7457072308508201937r_real
% 13.25/7.70 = ( ^ [A3: real,N2: nat] :
% 13.25/7.70 ( if_real @ ( N2 = zero_zero_nat ) @ one_one_real
% 13.25/7.70 @ ( set_fo3111899725591712190t_real
% 13.25/7.70 @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ O ) ) )
% 13.25/7.70 @ zero_zero_nat
% 13.25/7.70 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 13.25/7.70 @ one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8244_pochhammer__code,axiom,
% 13.25/7.70 ( comm_s2602460028002588243omplex
% 13.25/7.70 = ( ^ [A3: complex,N2: nat] :
% 13.25/7.70 ( if_complex @ ( N2 = zero_zero_nat ) @ one_one_complex
% 13.25/7.70 @ ( set_fo1517530859248394432omplex
% 13.25/7.70 @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ O ) ) )
% 13.25/7.70 @ zero_zero_nat
% 13.25/7.70 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 13.25/7.70 @ one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8245_pochhammer__code,axiom,
% 13.25/7.70 ( comm_s4028243227959126397er_rat
% 13.25/7.70 = ( ^ [A3: rat,N2: nat] :
% 13.25/7.70 ( if_rat @ ( N2 = zero_zero_nat ) @ one_one_rat
% 13.25/7.70 @ ( set_fo1949268297981939178at_rat
% 13.25/7.70 @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ O ) ) )
% 13.25/7.70 @ zero_zero_nat
% 13.25/7.70 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 13.25/7.70 @ one_one_rat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8246_pochhammer__code,axiom,
% 13.25/7.70 ( comm_s4663373288045622133er_nat
% 13.25/7.70 = ( ^ [A3: nat,N2: nat] :
% 13.25/7.70 ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat
% 13.25/7.70 @ ( set_fo2584398358068434914at_nat
% 13.25/7.70 @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 13.25/7.70 @ zero_zero_nat
% 13.25/7.70 @ ( minus_minus_nat @ N2 @ one_one_nat )
% 13.25/7.70 @ one_one_nat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8247_prod_Oneutral__const,axiom,
% 13.25/7.70 ! [A2: set_nat] :
% 13.25/7.70 ( ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [Uu3: nat] : one_one_nat
% 13.25/7.70 @ A2 )
% 13.25/7.70 = one_one_nat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8248_prod_Oneutral__const,axiom,
% 13.25/7.70 ! [A2: set_nat] :
% 13.25/7.70 ( ( groups705719431365010083at_int
% 13.25/7.70 @ ^ [Uu3: nat] : one_one_int
% 13.25/7.70 @ A2 )
% 13.25/7.70 = one_one_int ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8249_prod_Oneutral__const,axiom,
% 13.25/7.70 ! [A2: set_int] :
% 13.25/7.70 ( ( groups1705073143266064639nt_int
% 13.25/7.70 @ ^ [Uu3: int] : one_one_int
% 13.25/7.70 @ A2 )
% 13.25/7.70 = one_one_int ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8250_prod_Oempty,axiom,
% 13.25/7.70 ! [G: nat > complex] :
% 13.25/7.70 ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8251_prod_Oempty,axiom,
% 13.25/7.70 ! [G: nat > real] :
% 13.25/7.70 ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8252_prod_Oempty,axiom,
% 13.25/7.70 ! [G: nat > rat] :
% 13.25/7.70 ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
% 13.25/7.70 = one_one_rat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8253_prod_Oempty,axiom,
% 13.25/7.70 ! [G: int > complex] :
% 13.25/7.70 ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8254_prod_Oempty,axiom,
% 13.25/7.70 ! [G: int > real] :
% 13.25/7.70 ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8255_prod_Oempty,axiom,
% 13.25/7.70 ! [G: int > rat] :
% 13.25/7.70 ( ( groups1072433553688619179nt_rat @ G @ bot_bot_set_int )
% 13.25/7.70 = one_one_rat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8256_prod_Oempty,axiom,
% 13.25/7.70 ! [G: int > nat] :
% 13.25/7.70 ( ( groups1707563613775114915nt_nat @ G @ bot_bot_set_int )
% 13.25/7.70 = one_one_nat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8257_prod_Oempty,axiom,
% 13.25/7.70 ! [G: real > complex] :
% 13.25/7.70 ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8258_prod_Oempty,axiom,
% 13.25/7.70 ! [G: real > real] :
% 13.25/7.70 ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8259_prod_Oempty,axiom,
% 13.25/7.70 ! [G: real > rat] :
% 13.25/7.70 ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 13.25/7.70 = one_one_rat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8260_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_nat,G: nat > complex] :
% 13.25/7.70 ( ~ ( finite_finite_nat @ A2 )
% 13.25/7.70 => ( ( groups6464643781859351333omplex @ G @ A2 )
% 13.25/7.70 = one_one_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8261_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_int,G: int > complex] :
% 13.25/7.70 ( ~ ( finite_finite_int @ A2 )
% 13.25/7.70 => ( ( groups7440179247065528705omplex @ G @ A2 )
% 13.25/7.70 = one_one_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8262_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_complex,G: complex > complex] :
% 13.25/7.70 ( ~ ( finite3207457112153483333omplex @ A2 )
% 13.25/7.70 => ( ( groups3708469109370488835omplex @ G @ A2 )
% 13.25/7.70 = one_one_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8263_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_nat,G: nat > real] :
% 13.25/7.70 ( ~ ( finite_finite_nat @ A2 )
% 13.25/7.70 => ( ( groups129246275422532515t_real @ G @ A2 )
% 13.25/7.70 = one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8264_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_int,G: int > real] :
% 13.25/7.70 ( ~ ( finite_finite_int @ A2 )
% 13.25/7.70 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 13.25/7.70 = one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8265_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_complex,G: complex > real] :
% 13.25/7.70 ( ~ ( finite3207457112153483333omplex @ A2 )
% 13.25/7.70 => ( ( groups766887009212190081x_real @ G @ A2 )
% 13.25/7.70 = one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8266_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_nat,G: nat > rat] :
% 13.25/7.70 ( ~ ( finite_finite_nat @ A2 )
% 13.25/7.70 => ( ( groups73079841787564623at_rat @ G @ A2 )
% 13.25/7.70 = one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8267_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_int,G: int > rat] :
% 13.25/7.70 ( ~ ( finite_finite_int @ A2 )
% 13.25/7.70 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 13.25/7.70 = one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8268_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_complex,G: complex > rat] :
% 13.25/7.70 ( ~ ( finite3207457112153483333omplex @ A2 )
% 13.25/7.70 => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 13.25/7.70 = one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8269_prod_Oinfinite,axiom,
% 13.25/7.70 ! [A2: set_int,G: int > nat] :
% 13.25/7.70 ( ~ ( finite_finite_int @ A2 )
% 13.25/7.70 => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 13.25/7.70 = one_one_nat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8270_fact__0,axiom,
% 13.25/7.70 ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8271_fact__0,axiom,
% 13.25/7.70 ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 13.25/7.70 = one_one_rat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8272_fact__0,axiom,
% 13.25/7.70 ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 13.25/7.70 = one_one_int ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8273_fact__0,axiom,
% 13.25/7.70 ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8274_fact__0,axiom,
% 13.25/7.70 ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 13.25/7.70 = one_one_nat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8275_fact__1,axiom,
% 13.25/7.70 ( ( semiri5044797733671781792omplex @ one_one_nat )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8276_fact__1,axiom,
% 13.25/7.70 ( ( semiri773545260158071498ct_rat @ one_one_nat )
% 13.25/7.70 = one_one_rat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8277_fact__1,axiom,
% 13.25/7.70 ( ( semiri1406184849735516958ct_int @ one_one_nat )
% 13.25/7.70 = one_one_int ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8278_fact__1,axiom,
% 13.25/7.70 ( ( semiri2265585572941072030t_real @ one_one_nat )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8279_fact__1,axiom,
% 13.25/7.70 ( ( semiri1408675320244567234ct_nat @ one_one_nat )
% 13.25/7.70 = one_one_nat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8280_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_real,A: real,B: real > complex] :
% 13.25/7.70 ( ( finite_finite_real @ S2 )
% 13.25/7.70 => ( ( ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups713298508707869441omplex
% 13.25/7.70 @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups713298508707869441omplex
% 13.25/7.70 @ ^ [K2: real] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8281_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_nat,A: nat,B: nat > complex] :
% 13.25/7.70 ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ( ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups6464643781859351333omplex
% 13.25/7.70 @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups6464643781859351333omplex
% 13.25/7.70 @ ^ [K2: nat] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8282_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_int,A: int,B: int > complex] :
% 13.25/7.70 ( ( finite_finite_int @ S2 )
% 13.25/7.70 => ( ( ( member_int @ A @ S2 )
% 13.25/7.70 => ( ( groups7440179247065528705omplex
% 13.25/7.70 @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_int @ A @ S2 )
% 13.25/7.70 => ( ( groups7440179247065528705omplex
% 13.25/7.70 @ ^ [K2: int] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8283_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_complex,A: complex,B: complex > complex] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.70 => ( ( ( member_complex @ A @ S2 )
% 13.25/7.70 => ( ( groups3708469109370488835omplex
% 13.25/7.70 @ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_complex @ A @ S2 )
% 13.25/7.70 => ( ( groups3708469109370488835omplex
% 13.25/7.70 @ ^ [K2: complex] : ( if_complex @ ( A = K2 ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8284_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_real,A: real,B: real > real] :
% 13.25/7.70 ( ( finite_finite_real @ S2 )
% 13.25/7.70 => ( ( ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups1681761925125756287l_real
% 13.25/7.70 @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups1681761925125756287l_real
% 13.25/7.70 @ ^ [K2: real] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8285_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_nat,A: nat,B: nat > real] :
% 13.25/7.70 ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ( ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups129246275422532515t_real
% 13.25/7.70 @ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups129246275422532515t_real
% 13.25/7.70 @ ^ [K2: nat] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8286_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_int,A: int,B: int > real] :
% 13.25/7.70 ( ( finite_finite_int @ S2 )
% 13.25/7.70 => ( ( ( member_int @ A @ S2 )
% 13.25/7.70 => ( ( groups2316167850115554303t_real
% 13.25/7.70 @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_int @ A @ S2 )
% 13.25/7.70 => ( ( groups2316167850115554303t_real
% 13.25/7.70 @ ^ [K2: int] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8287_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_complex,A: complex,B: complex > real] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.70 => ( ( ( member_complex @ A @ S2 )
% 13.25/7.70 => ( ( groups766887009212190081x_real
% 13.25/7.70 @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_complex @ A @ S2 )
% 13.25/7.70 => ( ( groups766887009212190081x_real
% 13.25/7.70 @ ^ [K2: complex] : ( if_real @ ( A = K2 ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8288_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_real,A: real,B: real > rat] :
% 13.25/7.70 ( ( finite_finite_real @ S2 )
% 13.25/7.70 => ( ( ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups4061424788464935467al_rat
% 13.25/7.70 @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups4061424788464935467al_rat
% 13.25/7.70 @ ^ [K2: real] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_rat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8289_prod_Odelta_H,axiom,
% 13.25/7.70 ! [S2: set_nat,A: nat,B: nat > rat] :
% 13.25/7.70 ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ( ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups73079841787564623at_rat
% 13.25/7.70 @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups73079841787564623at_rat
% 13.25/7.70 @ ^ [K2: nat] : ( if_rat @ ( A = K2 ) @ ( B @ K2 ) @ one_one_rat )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_rat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8290_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_real,A: real,B: real > complex] :
% 13.25/7.70 ( ( finite_finite_real @ S2 )
% 13.25/7.70 => ( ( ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups713298508707869441omplex
% 13.25/7.70 @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups713298508707869441omplex
% 13.25/7.70 @ ^ [K2: real] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8291_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_nat,A: nat,B: nat > complex] :
% 13.25/7.70 ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ( ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups6464643781859351333omplex
% 13.25/7.70 @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups6464643781859351333omplex
% 13.25/7.70 @ ^ [K2: nat] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8292_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_int,A: int,B: int > complex] :
% 13.25/7.70 ( ( finite_finite_int @ S2 )
% 13.25/7.70 => ( ( ( member_int @ A @ S2 )
% 13.25/7.70 => ( ( groups7440179247065528705omplex
% 13.25/7.70 @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_int @ A @ S2 )
% 13.25/7.70 => ( ( groups7440179247065528705omplex
% 13.25/7.70 @ ^ [K2: int] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8293_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_complex,A: complex,B: complex > complex] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.70 => ( ( ( member_complex @ A @ S2 )
% 13.25/7.70 => ( ( groups3708469109370488835omplex
% 13.25/7.70 @ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_complex @ A @ S2 )
% 13.25/7.70 => ( ( groups3708469109370488835omplex
% 13.25/7.70 @ ^ [K2: complex] : ( if_complex @ ( K2 = A ) @ ( B @ K2 ) @ one_one_complex )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_complex ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8294_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_real,A: real,B: real > real] :
% 13.25/7.70 ( ( finite_finite_real @ S2 )
% 13.25/7.70 => ( ( ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups1681761925125756287l_real
% 13.25/7.70 @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups1681761925125756287l_real
% 13.25/7.70 @ ^ [K2: real] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8295_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_nat,A: nat,B: nat > real] :
% 13.25/7.70 ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ( ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups129246275422532515t_real
% 13.25/7.70 @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups129246275422532515t_real
% 13.25/7.70 @ ^ [K2: nat] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8296_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_int,A: int,B: int > real] :
% 13.25/7.70 ( ( finite_finite_int @ S2 )
% 13.25/7.70 => ( ( ( member_int @ A @ S2 )
% 13.25/7.70 => ( ( groups2316167850115554303t_real
% 13.25/7.70 @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_int @ A @ S2 )
% 13.25/7.70 => ( ( groups2316167850115554303t_real
% 13.25/7.70 @ ^ [K2: int] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8297_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_complex,A: complex,B: complex > real] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.70 => ( ( ( member_complex @ A @ S2 )
% 13.25/7.70 => ( ( groups766887009212190081x_real
% 13.25/7.70 @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_complex @ A @ S2 )
% 13.25/7.70 => ( ( groups766887009212190081x_real
% 13.25/7.70 @ ^ [K2: complex] : ( if_real @ ( K2 = A ) @ ( B @ K2 ) @ one_one_real )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_real ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8298_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_real,A: real,B: real > rat] :
% 13.25/7.70 ( ( finite_finite_real @ S2 )
% 13.25/7.70 => ( ( ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups4061424788464935467al_rat
% 13.25/7.70 @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_real @ A @ S2 )
% 13.25/7.70 => ( ( groups4061424788464935467al_rat
% 13.25/7.70 @ ^ [K2: real] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_rat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8299_prod_Odelta,axiom,
% 13.25/7.70 ! [S2: set_nat,A: nat,B: nat > rat] :
% 13.25/7.70 ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ( ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups73079841787564623at_rat
% 13.25/7.70 @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = ( B @ A ) ) )
% 13.25/7.70 & ( ~ ( member_nat @ A @ S2 )
% 13.25/7.70 => ( ( groups73079841787564623at_rat
% 13.25/7.70 @ ^ [K2: nat] : ( if_rat @ ( K2 = A ) @ ( B @ K2 ) @ one_one_rat )
% 13.25/7.70 @ S2 )
% 13.25/7.70 = one_one_rat ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8300_fact__Suc__0,axiom,
% 13.25/7.70 ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 13.25/7.70 = one_one_complex ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8301_fact__Suc__0,axiom,
% 13.25/7.70 ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 13.25/7.70 = one_one_rat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8302_fact__Suc__0,axiom,
% 13.25/7.70 ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 13.25/7.70 = one_one_int ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8303_fact__Suc__0,axiom,
% 13.25/7.70 ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 13.25/7.70 = one_one_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8304_fact__Suc__0,axiom,
% 13.25/7.70 ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 13.25/7.70 = one_one_nat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8305_fact__Suc,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8306_fact__Suc,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( semiri5044797733671781792omplex @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( semiri5044797733671781792omplex @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8307_fact__Suc,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8308_fact__Suc,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8309_fact__Suc,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
% 13.25/7.70 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8310_prod_OlessThan__Suc,axiom,
% 13.25/7.70 ! [G: nat > real,N: nat] :
% 13.25/7.70 ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8311_prod_OlessThan__Suc,axiom,
% 13.25/7.70 ! [G: nat > rat,N: nat] :
% 13.25/7.70 ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8312_prod_OlessThan__Suc,axiom,
% 13.25/7.70 ! [G: nat > nat,N: nat] :
% 13.25/7.70 ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8313_prod_OlessThan__Suc,axiom,
% 13.25/7.70 ! [G: nat > int,N: nat] :
% 13.25/7.70 ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8314_fact__2,axiom,
% 13.25/7.70 ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8315_fact__2,axiom,
% 13.25/7.70 ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8316_fact__2,axiom,
% 13.25/7.70 ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8317_fact__2,axiom,
% 13.25/7.70 ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8318_fact__2,axiom,
% 13.25/7.70 ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.70 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8319_prod_Ocl__ivl__Suc,axiom,
% 13.25/7.70 ! [N: nat,M: nat,G: nat > complex] :
% 13.25/7.70 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8320_prod_Ocl__ivl__Suc,axiom,
% 13.25/7.70 ! [N: nat,M: nat,G: nat > real] :
% 13.25/7.70 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = one_one_real ) )
% 13.25/7.70 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8321_prod_Ocl__ivl__Suc,axiom,
% 13.25/7.70 ! [N: nat,M: nat,G: nat > rat] :
% 13.25/7.70 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = one_one_rat ) )
% 13.25/7.70 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8322_prod_Ocl__ivl__Suc,axiom,
% 13.25/7.70 ! [N: nat,M: nat,G: nat > nat] :
% 13.25/7.70 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = one_one_nat ) )
% 13.25/7.70 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8323_prod_Ocl__ivl__Suc,axiom,
% 13.25/7.70 ! [N: nat,M: nat,G: nat > int] :
% 13.25/7.70 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = one_one_int ) )
% 13.25/7.70 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 13.25/7.70 => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 13.25/7.70 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8324_prod_Oneutral,axiom,
% 13.25/7.70 ! [A2: set_nat,G: nat > nat] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ( G @ X3 )
% 13.25/7.70 = one_one_nat ) )
% 13.25/7.70 => ( ( groups708209901874060359at_nat @ G @ A2 )
% 13.25/7.70 = one_one_nat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8325_prod_Oneutral,axiom,
% 13.25/7.70 ! [A2: set_nat,G: nat > int] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ( G @ X3 )
% 13.25/7.70 = one_one_int ) )
% 13.25/7.70 => ( ( groups705719431365010083at_int @ G @ A2 )
% 13.25/7.70 = one_one_int ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8326_prod_Oneutral,axiom,
% 13.25/7.70 ! [A2: set_int,G: int > int] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ( G @ X3 )
% 13.25/7.70 = one_one_int ) )
% 13.25/7.70 => ( ( groups1705073143266064639nt_int @ G @ A2 )
% 13.25/7.70 = one_one_int ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8327_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: nat > complex,A2: set_nat] :
% 13.25/7.70 ( ( ( groups6464643781859351333omplex @ G @ A2 )
% 13.25/7.70 != one_one_complex )
% 13.25/7.70 => ~ ! [A5: nat] :
% 13.25/7.70 ( ( member_nat @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8328_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: real > complex,A2: set_real] :
% 13.25/7.70 ( ( ( groups713298508707869441omplex @ G @ A2 )
% 13.25/7.70 != one_one_complex )
% 13.25/7.70 => ~ ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8329_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: int > complex,A2: set_int] :
% 13.25/7.70 ( ( ( groups7440179247065528705omplex @ G @ A2 )
% 13.25/7.70 != one_one_complex )
% 13.25/7.70 => ~ ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8330_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: complex > complex,A2: set_complex] :
% 13.25/7.70 ( ( ( groups3708469109370488835omplex @ G @ A2 )
% 13.25/7.70 != one_one_complex )
% 13.25/7.70 => ~ ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8331_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: nat > real,A2: set_nat] :
% 13.25/7.70 ( ( ( groups129246275422532515t_real @ G @ A2 )
% 13.25/7.70 != one_one_real )
% 13.25/7.70 => ~ ! [A5: nat] :
% 13.25/7.70 ( ( member_nat @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8332_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: real > real,A2: set_real] :
% 13.25/7.70 ( ( ( groups1681761925125756287l_real @ G @ A2 )
% 13.25/7.70 != one_one_real )
% 13.25/7.70 => ~ ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8333_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: int > real,A2: set_int] :
% 13.25/7.70 ( ( ( groups2316167850115554303t_real @ G @ A2 )
% 13.25/7.70 != one_one_real )
% 13.25/7.70 => ~ ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8334_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: complex > real,A2: set_complex] :
% 13.25/7.70 ( ( ( groups766887009212190081x_real @ G @ A2 )
% 13.25/7.70 != one_one_real )
% 13.25/7.70 => ~ ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_real ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8335_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: nat > rat,A2: set_nat] :
% 13.25/7.70 ( ( ( groups73079841787564623at_rat @ G @ A2 )
% 13.25/7.70 != one_one_rat )
% 13.25/7.70 => ~ ! [A5: nat] :
% 13.25/7.70 ( ( member_nat @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_rat ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8336_prod_Onot__neutral__contains__not__neutral,axiom,
% 13.25/7.70 ! [G: real > rat,A2: set_real] :
% 13.25/7.70 ( ( ( groups4061424788464935467al_rat @ G @ A2 )
% 13.25/7.70 != one_one_rat )
% 13.25/7.70 => ~ ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ A2 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_rat ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8337_prod_Odistrib,axiom,
% 13.25/7.70 ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 13.25/7.70 ( ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [X2: nat] : ( times_times_nat @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 13.25/7.70 @ A2 )
% 13.25/7.70 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ A2 ) @ ( groups708209901874060359at_nat @ H2 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8338_prod_Odistrib,axiom,
% 13.25/7.70 ! [G: nat > int,H2: nat > int,A2: set_nat] :
% 13.25/7.70 ( ( groups705719431365010083at_int
% 13.25/7.70 @ ^ [X2: nat] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 13.25/7.70 @ A2 )
% 13.25/7.70 = ( times_times_int @ ( groups705719431365010083at_int @ G @ A2 ) @ ( groups705719431365010083at_int @ H2 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8339_prod_Odistrib,axiom,
% 13.25/7.70 ! [G: int > int,H2: int > int,A2: set_int] :
% 13.25/7.70 ( ( groups1705073143266064639nt_int
% 13.25/7.70 @ ^ [X2: int] : ( times_times_int @ ( G @ X2 ) @ ( H2 @ X2 ) )
% 13.25/7.70 @ A2 )
% 13.25/7.70 = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ A2 ) @ ( groups1705073143266064639nt_int @ H2 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8340_prod__power__distrib,axiom,
% 13.25/7.70 ! [F: nat > nat,A2: set_nat,N: nat] :
% 13.25/7.70 ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N )
% 13.25/7.70 = ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [X2: nat] : ( power_power_nat @ ( F @ X2 ) @ N )
% 13.25/7.70 @ A2 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8341_prod__power__distrib,axiom,
% 13.25/7.70 ! [F: nat > int,A2: set_nat,N: nat] :
% 13.25/7.70 ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N )
% 13.25/7.70 = ( groups705719431365010083at_int
% 13.25/7.70 @ ^ [X2: nat] : ( power_power_int @ ( F @ X2 ) @ N )
% 13.25/7.70 @ A2 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8342_prod__power__distrib,axiom,
% 13.25/7.70 ! [F: int > int,A2: set_int,N: nat] :
% 13.25/7.70 ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N )
% 13.25/7.70 = ( groups1705073143266064639nt_int
% 13.25/7.70 @ ^ [X2: int] : ( power_power_int @ ( F @ X2 ) @ N )
% 13.25/7.70 @ A2 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8343_mod__prod__eq,axiom,
% 13.25/7.70 ! [F: nat > nat,A: nat,A2: set_nat] :
% 13.25/7.70 ( ( modulo_modulo_nat
% 13.25/7.70 @ ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [I4: nat] : ( modulo_modulo_nat @ ( F @ I4 ) @ A )
% 13.25/7.70 @ A2 )
% 13.25/7.70 @ A )
% 13.25/7.70 = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8344_mod__prod__eq,axiom,
% 13.25/7.70 ! [F: nat > int,A: int,A2: set_nat] :
% 13.25/7.70 ( ( modulo_modulo_int
% 13.25/7.70 @ ( groups705719431365010083at_int
% 13.25/7.70 @ ^ [I4: nat] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 13.25/7.70 @ A2 )
% 13.25/7.70 @ A )
% 13.25/7.70 = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8345_mod__prod__eq,axiom,
% 13.25/7.70 ! [F: int > int,A: int,A2: set_int] :
% 13.25/7.70 ( ( modulo_modulo_int
% 13.25/7.70 @ ( groups1705073143266064639nt_int
% 13.25/7.70 @ ^ [I4: int] : ( modulo_modulo_int @ ( F @ I4 ) @ A )
% 13.25/7.70 @ A2 )
% 13.25/7.70 @ A )
% 13.25/7.70 = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8346_fact__prod,axiom,
% 13.25/7.70 ( semiri1406184849735516958ct_int
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri1314217659103216013at_int
% 13.25/7.70 @ ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [X2: nat] : X2
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8347_fact__prod,axiom,
% 13.25/7.70 ( semiri5044797733671781792omplex
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri8010041392384452111omplex
% 13.25/7.70 @ ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [X2: nat] : X2
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8348_fact__prod,axiom,
% 13.25/7.70 ( semiri773545260158071498ct_rat
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri681578069525770553at_rat
% 13.25/7.70 @ ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [X2: nat] : X2
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8349_fact__prod,axiom,
% 13.25/7.70 ( semiri2265585572941072030t_real
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri5074537144036343181t_real
% 13.25/7.70 @ ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [X2: nat] : X2
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8350_fact__prod,axiom,
% 13.25/7.70 ( semiri1408675320244567234ct_nat
% 13.25/7.70 = ( ^ [N2: nat] :
% 13.25/7.70 ( semiri1316708129612266289at_nat
% 13.25/7.70 @ ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [X2: nat] : X2
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8351_prod__nonneg,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > nat] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8352_prod__nonneg,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > int] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8353_prod__nonneg,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > int] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8354_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > real,G: nat > real] :
% 13.25/7.70 ( ! [I2: nat] :
% 13.25/7.70 ( ( member_nat @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8355_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > real,G: real > real] :
% 13.25/7.70 ( ! [I2: real] :
% 13.25/7.70 ( ( member_real @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8356_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > real,G: int > real] :
% 13.25/7.70 ( ! [I2: int] :
% 13.25/7.70 ( ( member_int @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8357_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_complex,F: complex > real,G: complex > real] :
% 13.25/7.70 ( ! [I2: complex] :
% 13.25/7.70 ( ( member_complex @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_real @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8358_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 13.25/7.70 ( ! [I2: nat] :
% 13.25/7.70 ( ( member_nat @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8359_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > rat,G: real > rat] :
% 13.25/7.70 ( ! [I2: real] :
% 13.25/7.70 ( ( member_real @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8360_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > rat,G: int > rat] :
% 13.25/7.70 ( ! [I2: int] :
% 13.25/7.70 ( ( member_int @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8361_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 13.25/7.70 ( ! [I2: complex] :
% 13.25/7.70 ( ( member_complex @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_rat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8362_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > nat,G: real > nat] :
% 13.25/7.70 ( ! [I2: real] :
% 13.25/7.70 ( ( member_real @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8363_prod__mono,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > nat,G: int > nat] :
% 13.25/7.70 ( ! [I2: int] :
% 13.25/7.70 ( ( member_int @ I2 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) )
% 13.25/7.70 & ( ord_less_eq_nat @ ( F @ I2 ) @ ( G @ I2 ) ) ) )
% 13.25/7.70 => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8364_prod__pos,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > nat] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8365_prod__pos,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > int] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8366_prod__pos,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > int] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_int @ zero_zero_int @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8367_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > real] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8368_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > real] :
% 13.25/7.70 ( ! [X3: real] :
% 13.25/7.70 ( ( member_real @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8369_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > real] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8370_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_complex,F: complex > real] :
% 13.25/7.70 ( ! [X3: complex] :
% 13.25/7.70 ( ( member_complex @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8371_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > rat] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8372_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > rat] :
% 13.25/7.70 ( ! [X3: real] :
% 13.25/7.70 ( ( member_real @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8373_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > rat] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8374_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_complex,F: complex > rat] :
% 13.25/7.70 ( ! [X3: complex] :
% 13.25/7.70 ( ( member_complex @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_rat @ one_one_rat @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8375_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > nat] :
% 13.25/7.70 ( ! [X3: real] :
% 13.25/7.70 ( ( member_real @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_nat @ one_one_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8376_prod__ge__1,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > nat] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ord_less_eq_nat @ one_one_nat @ ( F @ X3 ) ) )
% 13.25/7.70 => ( ord_less_eq_nat @ one_one_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8377_prod__atLeastAtMost__code,axiom,
% 13.25/7.70 ! [F: nat > complex,A: nat,B: nat] :
% 13.25/7.70 ( ( groups6464643781859351333omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 13.25/7.70 = ( set_fo1517530859248394432omplex
% 13.25/7.70 @ ^ [A3: nat] : ( times_times_complex @ ( F @ A3 ) )
% 13.25/7.70 @ A
% 13.25/7.70 @ B
% 13.25/7.70 @ one_one_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8378_prod__atLeastAtMost__code,axiom,
% 13.25/7.70 ! [F: nat > real,A: nat,B: nat] :
% 13.25/7.70 ( ( groups129246275422532515t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 13.25/7.70 = ( set_fo3111899725591712190t_real
% 13.25/7.70 @ ^ [A3: nat] : ( times_times_real @ ( F @ A3 ) )
% 13.25/7.70 @ A
% 13.25/7.70 @ B
% 13.25/7.70 @ one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8379_prod__atLeastAtMost__code,axiom,
% 13.25/7.70 ! [F: nat > rat,A: nat,B: nat] :
% 13.25/7.70 ( ( groups73079841787564623at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 13.25/7.70 = ( set_fo1949268297981939178at_rat
% 13.25/7.70 @ ^ [A3: nat] : ( times_times_rat @ ( F @ A3 ) )
% 13.25/7.70 @ A
% 13.25/7.70 @ B
% 13.25/7.70 @ one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8380_prod__atLeastAtMost__code,axiom,
% 13.25/7.70 ! [F: nat > nat,A: nat,B: nat] :
% 13.25/7.70 ( ( groups708209901874060359at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 13.25/7.70 = ( set_fo2584398358068434914at_nat
% 13.25/7.70 @ ^ [A3: nat] : ( times_times_nat @ ( F @ A3 ) )
% 13.25/7.70 @ A
% 13.25/7.70 @ B
% 13.25/7.70 @ one_one_nat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8381_prod__atLeastAtMost__code,axiom,
% 13.25/7.70 ! [F: nat > int,A: nat,B: nat] :
% 13.25/7.70 ( ( groups705719431365010083at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
% 13.25/7.70 = ( set_fo2581907887559384638at_int
% 13.25/7.70 @ ^ [A3: nat] : ( times_times_int @ ( F @ A3 ) )
% 13.25/7.70 @ A
% 13.25/7.70 @ B
% 13.25/7.70 @ one_one_int ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8382_fact__ge__zero,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8383_fact__ge__zero,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8384_fact__ge__zero,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8385_fact__ge__zero,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8386_fact__gt__zero,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8387_fact__gt__zero,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8388_fact__gt__zero,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8389_fact__gt__zero,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8390_fact__not__neg,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8391_fact__not__neg,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8392_fact__not__neg,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8393_fact__not__neg,axiom,
% 13.25/7.70 ! [N: nat] :
% 13.25/7.70 ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8394_fact__ge__1,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8395_fact__ge__1,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8396_fact__ge__1,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8397_fact__ge__1,axiom,
% 13.25/7.70 ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8398_fact__mono,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8399_fact__mono,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8400_fact__mono,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8401_fact__mono,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8402_fact__dvd,axiom,
% 13.25/7.70 ! [N: nat,M: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8403_fact__dvd,axiom,
% 13.25/7.70 ! [N: nat,M: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8404_fact__dvd,axiom,
% 13.25/7.70 ! [N: nat,M: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8405_fact__dvd,axiom,
% 13.25/7.70 ! [N: nat,M: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.70 => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8406_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_real,G: real > complex,P: real > $o] :
% 13.25/7.70 ( ( finite_finite_real @ A2 )
% 13.25/7.70 => ( ( groups713298508707869441omplex @ G
% 13.25/7.70 @ ( collect_real
% 13.25/7.70 @ ^ [X2: real] :
% 13.25/7.70 ( ( member_real @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups713298508707869441omplex
% 13.25/7.70 @ ^ [X2: real] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8407_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 13.25/7.70 ( ( finite_finite_nat @ A2 )
% 13.25/7.70 => ( ( groups6464643781859351333omplex @ G
% 13.25/7.70 @ ( collect_nat
% 13.25/7.70 @ ^ [X2: nat] :
% 13.25/7.70 ( ( member_nat @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups6464643781859351333omplex
% 13.25/7.70 @ ^ [X2: nat] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8408_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_int,G: int > complex,P: int > $o] :
% 13.25/7.70 ( ( finite_finite_int @ A2 )
% 13.25/7.70 => ( ( groups7440179247065528705omplex @ G
% 13.25/7.70 @ ( collect_int
% 13.25/7.70 @ ^ [X2: int] :
% 13.25/7.70 ( ( member_int @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups7440179247065528705omplex
% 13.25/7.70 @ ^ [X2: int] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8409_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_complex,G: complex > complex,P: complex > $o] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.70 => ( ( groups3708469109370488835omplex @ G
% 13.25/7.70 @ ( collect_complex
% 13.25/7.70 @ ^ [X2: complex] :
% 13.25/7.70 ( ( member_complex @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups3708469109370488835omplex
% 13.25/7.70 @ ^ [X2: complex] : ( if_complex @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_complex )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8410_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_real,G: real > real,P: real > $o] :
% 13.25/7.70 ( ( finite_finite_real @ A2 )
% 13.25/7.70 => ( ( groups1681761925125756287l_real @ G
% 13.25/7.70 @ ( collect_real
% 13.25/7.70 @ ^ [X2: real] :
% 13.25/7.70 ( ( member_real @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups1681761925125756287l_real
% 13.25/7.70 @ ^ [X2: real] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8411_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_nat,G: nat > real,P: nat > $o] :
% 13.25/7.70 ( ( finite_finite_nat @ A2 )
% 13.25/7.70 => ( ( groups129246275422532515t_real @ G
% 13.25/7.70 @ ( collect_nat
% 13.25/7.70 @ ^ [X2: nat] :
% 13.25/7.70 ( ( member_nat @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups129246275422532515t_real
% 13.25/7.70 @ ^ [X2: nat] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8412_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_int,G: int > real,P: int > $o] :
% 13.25/7.70 ( ( finite_finite_int @ A2 )
% 13.25/7.70 => ( ( groups2316167850115554303t_real @ G
% 13.25/7.70 @ ( collect_int
% 13.25/7.70 @ ^ [X2: int] :
% 13.25/7.70 ( ( member_int @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups2316167850115554303t_real
% 13.25/7.70 @ ^ [X2: int] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8413_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.70 => ( ( groups766887009212190081x_real @ G
% 13.25/7.70 @ ( collect_complex
% 13.25/7.70 @ ^ [X2: complex] :
% 13.25/7.70 ( ( member_complex @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups766887009212190081x_real
% 13.25/7.70 @ ^ [X2: complex] : ( if_real @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_real )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8414_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_real,G: real > rat,P: real > $o] :
% 13.25/7.70 ( ( finite_finite_real @ A2 )
% 13.25/7.70 => ( ( groups4061424788464935467al_rat @ G
% 13.25/7.70 @ ( collect_real
% 13.25/7.70 @ ^ [X2: real] :
% 13.25/7.70 ( ( member_real @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups4061424788464935467al_rat
% 13.25/7.70 @ ^ [X2: real] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8415_prod_Ointer__filter,axiom,
% 13.25/7.70 ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 13.25/7.70 ( ( finite_finite_nat @ A2 )
% 13.25/7.70 => ( ( groups73079841787564623at_rat @ G
% 13.25/7.70 @ ( collect_nat
% 13.25/7.70 @ ^ [X2: nat] :
% 13.25/7.70 ( ( member_nat @ X2 @ A2 )
% 13.25/7.70 & ( P @ X2 ) ) ) )
% 13.25/7.70 = ( groups73079841787564623at_rat
% 13.25/7.70 @ ^ [X2: nat] : ( if_rat @ ( P @ X2 ) @ ( G @ X2 ) @ one_one_rat )
% 13.25/7.70 @ A2 ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8416_pochhammer__fact,axiom,
% 13.25/7.70 ( semiri5044797733671781792omplex
% 13.25/7.70 = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8417_pochhammer__fact,axiom,
% 13.25/7.70 ( semiri773545260158071498ct_rat
% 13.25/7.70 = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8418_pochhammer__fact,axiom,
% 13.25/7.70 ( semiri1406184849735516958ct_int
% 13.25/7.70 = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8419_pochhammer__fact,axiom,
% 13.25/7.70 ( semiri2265585572941072030t_real
% 13.25/7.70 = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8420_pochhammer__fact,axiom,
% 13.25/7.70 ( semiri1408675320244567234ct_nat
% 13.25/7.70 = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8421_power__sum,axiom,
% 13.25/7.70 ! [C: real,F: nat > nat,A2: set_nat] :
% 13.25/7.70 ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 13.25/7.70 = ( groups129246275422532515t_real
% 13.25/7.70 @ ^ [A3: nat] : ( power_power_real @ C @ ( F @ A3 ) )
% 13.25/7.70 @ A2 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8422_power__sum,axiom,
% 13.25/7.70 ! [C: complex,F: nat > nat,A2: set_nat] :
% 13.25/7.70 ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 13.25/7.70 = ( groups6464643781859351333omplex
% 13.25/7.70 @ ^ [A3: nat] : ( power_power_complex @ C @ ( F @ A3 ) )
% 13.25/7.70 @ A2 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8423_power__sum,axiom,
% 13.25/7.70 ! [C: nat,F: nat > nat,A2: set_nat] :
% 13.25/7.70 ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 13.25/7.70 = ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [A3: nat] : ( power_power_nat @ C @ ( F @ A3 ) )
% 13.25/7.70 @ A2 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8424_power__sum,axiom,
% 13.25/7.70 ! [C: int,F: nat > nat,A2: set_nat] :
% 13.25/7.70 ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 13.25/7.70 = ( groups705719431365010083at_int
% 13.25/7.70 @ ^ [A3: nat] : ( power_power_int @ C @ ( F @ A3 ) )
% 13.25/7.70 @ A2 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8425_power__sum,axiom,
% 13.25/7.70 ! [C: int,F: int > nat,A2: set_int] :
% 13.25/7.70 ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A2 ) )
% 13.25/7.70 = ( groups1705073143266064639nt_int
% 13.25/7.70 @ ^ [A3: int] : ( power_power_int @ C @ ( F @ A3 ) )
% 13.25/7.70 @ A2 ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8426_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 13.25/7.70 ! [G: nat > nat,M: nat,K: nat,N: nat] :
% 13.25/7.70 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 13.25/7.70 = ( groups708209901874060359at_nat
% 13.25/7.70 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8427_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 13.25/7.70 ! [G: nat > int,M: nat,K: nat,N: nat] :
% 13.25/7.70 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 13.25/7.70 = ( groups705719431365010083at_int
% 13.25/7.70 @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K ) )
% 13.25/7.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8428_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > real] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F @ A2 ) @ one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8429_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > real] :
% 13.25/7.70 ( ! [X3: real] :
% 13.25/7.70 ( ( member_real @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F @ A2 ) @ one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8430_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > real] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F @ A2 ) @ one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8431_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_complex,F: complex > real] :
% 13.25/7.70 ( ! [X3: complex] :
% 13.25/7.70 ( ( member_complex @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_real @ ( F @ X3 ) @ one_one_real ) ) )
% 13.25/7.70 => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F @ A2 ) @ one_one_real ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8432_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_nat,F: nat > rat] :
% 13.25/7.70 ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8433_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > rat] :
% 13.25/7.70 ( ! [X3: real] :
% 13.25/7.70 ( ( member_real @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8434_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > rat] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8435_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_complex,F: complex > rat] :
% 13.25/7.70 ( ! [X3: complex] :
% 13.25/7.70 ( ( member_complex @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_rat @ ( F @ X3 ) @ one_one_rat ) ) )
% 13.25/7.70 => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F @ A2 ) @ one_one_rat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8436_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_real,F: real > nat] :
% 13.25/7.70 ( ! [X3: real] :
% 13.25/7.70 ( ( member_real @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
% 13.25/7.70 => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8437_prod__le__1,axiom,
% 13.25/7.70 ! [A2: set_int,F: int > nat] :
% 13.25/7.70 ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ A2 )
% 13.25/7.70 => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) )
% 13.25/7.70 & ( ord_less_eq_nat @ ( F @ X3 ) @ one_one_nat ) ) )
% 13.25/7.70 => ( ord_less_eq_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ one_one_nat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8438_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: complex > complex > $o,S2: set_nat,H2: nat > complex,G: nat > complex] :
% 13.25/7.70 ( ( R4 @ one_one_complex @ one_one_complex )
% 13.25/7.70 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups6464643781859351333omplex @ H2 @ S2 ) @ ( groups6464643781859351333omplex @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8439_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: complex > complex > $o,S2: set_int,H2: int > complex,G: int > complex] :
% 13.25/7.70 ( ( R4 @ one_one_complex @ one_one_complex )
% 13.25/7.70 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite_finite_int @ S2 )
% 13.25/7.70 => ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups7440179247065528705omplex @ H2 @ S2 ) @ ( groups7440179247065528705omplex @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8440_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: complex > complex > $o,S2: set_complex,H2: complex > complex,G: complex > complex] :
% 13.25/7.70 ( ( R4 @ one_one_complex @ one_one_complex )
% 13.25/7.70 => ( ! [X1: complex,Y1: complex,X23: complex,Y22: complex] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_complex @ X1 @ Y1 ) @ ( times_times_complex @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.70 => ( ! [X3: complex] :
% 13.25/7.70 ( ( member_complex @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups3708469109370488835omplex @ H2 @ S2 ) @ ( groups3708469109370488835omplex @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8441_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: real > real > $o,S2: set_nat,H2: nat > real,G: nat > real] :
% 13.25/7.70 ( ( R4 @ one_one_real @ one_one_real )
% 13.25/7.70 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups129246275422532515t_real @ H2 @ S2 ) @ ( groups129246275422532515t_real @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8442_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: real > real > $o,S2: set_int,H2: int > real,G: int > real] :
% 13.25/7.70 ( ( R4 @ one_one_real @ one_one_real )
% 13.25/7.70 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite_finite_int @ S2 )
% 13.25/7.70 => ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups2316167850115554303t_real @ H2 @ S2 ) @ ( groups2316167850115554303t_real @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8443_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: real > real > $o,S2: set_complex,H2: complex > real,G: complex > real] :
% 13.25/7.70 ( ( R4 @ one_one_real @ one_one_real )
% 13.25/7.70 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_real @ X1 @ Y1 ) @ ( times_times_real @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.70 => ( ! [X3: complex] :
% 13.25/7.70 ( ( member_complex @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups766887009212190081x_real @ H2 @ S2 ) @ ( groups766887009212190081x_real @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8444_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: rat > rat > $o,S2: set_nat,H2: nat > rat,G: nat > rat] :
% 13.25/7.70 ( ( R4 @ one_one_rat @ one_one_rat )
% 13.25/7.70 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite_finite_nat @ S2 )
% 13.25/7.70 => ( ! [X3: nat] :
% 13.25/7.70 ( ( member_nat @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups73079841787564623at_rat @ H2 @ S2 ) @ ( groups73079841787564623at_rat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8445_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: rat > rat > $o,S2: set_int,H2: int > rat,G: int > rat] :
% 13.25/7.70 ( ( R4 @ one_one_rat @ one_one_rat )
% 13.25/7.70 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite_finite_int @ S2 )
% 13.25/7.70 => ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups1072433553688619179nt_rat @ H2 @ S2 ) @ ( groups1072433553688619179nt_rat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8446_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: rat > rat > $o,S2: set_complex,H2: complex > rat,G: complex > rat] :
% 13.25/7.70 ( ( R4 @ one_one_rat @ one_one_rat )
% 13.25/7.70 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_rat @ X1 @ Y1 ) @ ( times_times_rat @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite3207457112153483333omplex @ S2 )
% 13.25/7.70 => ( ! [X3: complex] :
% 13.25/7.70 ( ( member_complex @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups225925009352817453ex_rat @ H2 @ S2 ) @ ( groups225925009352817453ex_rat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8447_prod_Orelated,axiom,
% 13.25/7.70 ! [R4: nat > nat > $o,S2: set_int,H2: int > nat,G: int > nat] :
% 13.25/7.70 ( ( R4 @ one_one_nat @ one_one_nat )
% 13.25/7.70 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 13.25/7.70 ( ( ( R4 @ X1 @ X23 )
% 13.25/7.70 & ( R4 @ Y1 @ Y22 ) )
% 13.25/7.70 => ( R4 @ ( times_times_nat @ X1 @ Y1 ) @ ( times_times_nat @ X23 @ Y22 ) ) )
% 13.25/7.70 => ( ( finite_finite_int @ S2 )
% 13.25/7.70 => ( ! [X3: int] :
% 13.25/7.70 ( ( member_int @ X3 @ S2 )
% 13.25/7.70 => ( R4 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
% 13.25/7.70 => ( R4 @ ( groups1707563613775114915nt_nat @ H2 @ S2 ) @ ( groups1707563613775114915nt_nat @ G @ S2 ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8448_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_complex,A2: set_complex,F: complex > nat] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.70 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8449_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_complex,A2: set_complex,F: complex > int] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.70 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8450_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_nat,A2: set_nat,F: nat > code_integer] :
% 13.25/7.70 ( ( finite_finite_nat @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8451_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_complex,A2: set_complex,F: complex > code_integer] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.70 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8452_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_int,A2: set_int,F: int > nat] :
% 13.25/7.70 ( ( finite_finite_int @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8453_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_int,A2: set_int,F: int > code_integer] :
% 13.25/7.70 ( ( finite_finite_int @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8454_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_nat,A2: set_nat,F: nat > nat] :
% 13.25/7.70 ( ( finite_finite_nat @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8455_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_nat,A2: set_nat,F: nat > int] :
% 13.25/7.70 ( ( finite_finite_nat @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A2 ) @ ( groups705719431365010083at_int @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8456_prod__dvd__prod__subset,axiom,
% 13.25/7.70 ! [B5: set_int,A2: set_int,F: int > int] :
% 13.25/7.70 ( ( finite_finite_int @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.70 => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ F @ B5 ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8457_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_real,A2: set_real,F: real > nat,G: real > nat] :
% 13.25/7.70 ( ( finite_finite_real @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A2 ) @ ( groups4696554848551431203al_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8458_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_complex,A2: set_complex,F: complex > nat,G: complex > nat] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.70 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A2 ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8459_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_real,A2: set_real,F: real > int,G: real > int] :
% 13.25/7.70 ( ( finite_finite_real @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A2 ) @ ( groups4694064378042380927al_int @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8460_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_complex,A2: set_complex,F: complex > int,G: complex > int] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.70 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_int @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A2 ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8461_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_real,A2: set_real,F: real > code_integer,G: real > code_integer] :
% 13.25/7.70 ( ( finite_finite_real @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_real @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A2 ) @ ( groups6225526099057966256nteger @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8462_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_nat,A2: set_nat,F: nat > code_integer,G: nat > code_integer] :
% 13.25/7.70 ( ( finite_finite_nat @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: nat] :
% 13.25/7.70 ( ( member_nat @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A2 ) @ ( groups3455450783089532116nteger @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8463_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_complex,A2: set_complex,F: complex > code_integer,G: complex > code_integer] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ B5 )
% 13.25/7.70 => ( ( ord_le211207098394363844omplex @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A2 ) @ ( groups8682486955453173170nteger @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8464_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_int,A2: set_int,F: int > nat,G: int > nat] :
% 13.25/7.70 ( ( finite_finite_int @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A2 ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8465_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_int,A2: set_int,F: int > code_integer,G: int > code_integer] :
% 13.25/7.70 ( ( finite_finite_int @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_int @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A2 ) @ ( groups3827104343326376752nteger @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8466_prod__dvd__prod__subset2,axiom,
% 13.25/7.70 ! [B5: set_nat,A2: set_nat,F: nat > nat,G: nat > nat] :
% 13.25/7.70 ( ( finite_finite_nat @ B5 )
% 13.25/7.70 => ( ( ord_less_eq_set_nat @ A2 @ B5 )
% 13.25/7.70 => ( ! [A5: nat] :
% 13.25/7.70 ( ( member_nat @ A5 @ A2 )
% 13.25/7.70 => ( dvd_dvd_nat @ ( F @ A5 ) @ ( G @ A5 ) ) )
% 13.25/7.70 => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8467_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 13.25/7.70 ( ( finite_finite_real @ S4 )
% 13.25/7.70 => ( ( finite_finite_real @ T4 )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.70 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.70 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups713298508707869441omplex @ G @ S2 )
% 13.25/7.70 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8468_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_real,T4: set_int,S2: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 13.25/7.70 ( ( finite_finite_real @ S4 )
% 13.25/7.70 => ( ( finite_finite_int @ T4 )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.70 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.70 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups713298508707869441omplex @ G @ S2 )
% 13.25/7.70 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8469_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_real,T4: set_complex,S2: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > complex,H2: complex > complex] :
% 13.25/7.70 ( ( finite_finite_real @ S4 )
% 13.25/7.70 => ( ( finite3207457112153483333omplex @ T4 )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.70 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.70 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups713298508707869441omplex @ G @ S2 )
% 13.25/7.70 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8470_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_int,T4: set_real,S2: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 13.25/7.70 ( ( finite_finite_int @ S4 )
% 13.25/7.70 => ( ( finite_finite_real @ T4 )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.70 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.70 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups7440179247065528705omplex @ G @ S2 )
% 13.25/7.70 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8471_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_int,T4: set_int,S2: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 13.25/7.70 ( ( finite_finite_int @ S4 )
% 13.25/7.70 => ( ( finite_finite_int @ T4 )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.70 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.70 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups7440179247065528705omplex @ G @ S2 )
% 13.25/7.70 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8472_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_int,T4: set_complex,S2: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > complex,H2: complex > complex] :
% 13.25/7.70 ( ( finite_finite_int @ S4 )
% 13.25/7.70 => ( ( finite3207457112153483333omplex @ T4 )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ ( minus_minus_set_int @ S2 @ S4 ) )
% 13.25/7.70 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.70 => ( member_int @ ( I @ B3 ) @ ( minus_minus_set_int @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: int] :
% 13.25/7.70 ( ( member_int @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups7440179247065528705omplex @ G @ S2 )
% 13.25/7.70 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8473_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_complex,T4: set_real,S2: set_complex,I: real > complex,J: complex > real,T3: set_real,G: complex > complex,H2: real > complex] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ S4 )
% 13.25/7.70 => ( ( finite_finite_real @ T4 )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 13.25/7.70 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.70 => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups3708469109370488835omplex @ G @ S2 )
% 13.25/7.70 = ( groups713298508707869441omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8474_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_complex,T4: set_int,S2: set_complex,I: int > complex,J: complex > int,T3: set_int,G: complex > complex,H2: int > complex] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ S4 )
% 13.25/7.70 => ( ( finite_finite_int @ T4 )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 13.25/7.70 => ( member_int @ ( J @ A5 ) @ ( minus_minus_set_int @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ ( minus_minus_set_int @ T3 @ T4 ) )
% 13.25/7.70 => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: int] :
% 13.25/7.70 ( ( member_int @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups3708469109370488835omplex @ G @ S2 )
% 13.25/7.70 = ( groups7440179247065528705omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8475_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_complex,T4: set_complex,S2: set_complex,I: complex > complex,J: complex > complex,T3: set_complex,G: complex > complex,H2: complex > complex] :
% 13.25/7.70 ( ( finite3207457112153483333omplex @ S4 )
% 13.25/7.70 => ( ( finite3207457112153483333omplex @ T4 )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ ( minus_811609699411566653omplex @ S2 @ S4 ) )
% 13.25/7.70 => ( member_complex @ ( J @ A5 ) @ ( minus_811609699411566653omplex @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T3 @ T4 ) )
% 13.25/7.70 => ( member_complex @ ( I @ B3 ) @ ( minus_811609699411566653omplex @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [B3: complex] :
% 13.25/7.70 ( ( member_complex @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_complex ) )
% 13.25/7.70 => ( ! [A5: complex] :
% 13.25/7.70 ( ( member_complex @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups3708469109370488835omplex @ G @ S2 )
% 13.25/7.70 = ( groups3708469109370488835omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8476_prod_Oreindex__bij__witness__not__neutral,axiom,
% 13.25/7.70 ! [S4: set_real,T4: set_real,S2: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 13.25/7.70 ( ( finite_finite_real @ S4 )
% 13.25/7.70 => ( ( finite_finite_real @ T4 )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.70 => ( ( I @ ( J @ A5 ) )
% 13.25/7.70 = A5 ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ ( minus_minus_set_real @ S2 @ S4 ) )
% 13.25/7.70 => ( member_real @ ( J @ A5 ) @ ( minus_minus_set_real @ T3 @ T4 ) ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.70 => ( ( J @ ( I @ B3 ) )
% 13.25/7.70 = B3 ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ ( minus_minus_set_real @ T3 @ T4 ) )
% 13.25/7.70 => ( member_real @ ( I @ B3 ) @ ( minus_minus_set_real @ S2 @ S4 ) ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ S4 )
% 13.25/7.70 => ( ( G @ A5 )
% 13.25/7.70 = one_one_real ) )
% 13.25/7.70 => ( ! [B3: real] :
% 13.25/7.70 ( ( member_real @ B3 @ T4 )
% 13.25/7.70 => ( ( H2 @ B3 )
% 13.25/7.70 = one_one_real ) )
% 13.25/7.70 => ( ! [A5: real] :
% 13.25/7.70 ( ( member_real @ A5 @ S2 )
% 13.25/7.70 => ( ( H2 @ ( J @ A5 ) )
% 13.25/7.70 = ( G @ A5 ) ) )
% 13.25/7.70 => ( ( groups1681761925125756287l_real @ G @ S2 )
% 13.25/7.70 = ( groups1681761925125756287l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8477_fact__less__mono,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.70 => ( ( ord_less_nat @ M @ N )
% 13.25/7.70 => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8478_fact__less__mono,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.70 => ( ( ord_less_nat @ M @ N )
% 13.25/7.70 => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8479_fact__less__mono,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.70 => ( ( ord_less_nat @ M @ N )
% 13.25/7.70 => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8480_fact__less__mono,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.70 => ( ( ord_less_nat @ M @ N )
% 13.25/7.70 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8481_fact__mod,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M ) )
% 13.25/7.70 = zero_zero_int ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8482_fact__mod,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M ) )
% 13.25/7.70 = zero_z3403309356797280102nteger ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8483_fact__mod,axiom,
% 13.25/7.70 ! [M: nat,N: nat] :
% 13.25/7.70 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.70 => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 13.25/7.70 = zero_zero_nat ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8484_fact__fact__dvd__fact,axiom,
% 13.25/7.70 ! [K: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8485_fact__fact__dvd__fact,axiom,
% 13.25/7.70 ! [K: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8486_fact__fact__dvd__fact,axiom,
% 13.25/7.70 ! [K: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N ) ) ) ).
% 13.25/7.70
% 13.25/7.70 thf(fact_8487_fact__fact__dvd__fact,axiom,
% 13.25/7.71 ! [K: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8488_fact__fact__dvd__fact,axiom,
% 13.25/7.71 ! [K: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8489_fact__le__power,axiom,
% 13.25/7.71 ! [N: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8490_fact__le__power,axiom,
% 13.25/7.71 ! [N: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8491_fact__le__power,axiom,
% 13.25/7.71 ! [N: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8492_fact__le__power,axiom,
% 13.25/7.71 ! [N: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8493_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_real,G: real > complex] :
% 13.25/7.71 ( ( finite_finite_real @ A2 )
% 13.25/7.71 => ( ( groups713298508707869441omplex @ G
% 13.25/7.71 @ ( minus_minus_set_real @ A2
% 13.25/7.71 @ ( collect_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_complex ) ) ) )
% 13.25/7.71 = ( groups713298508707869441omplex @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8494_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_int,G: int > complex] :
% 13.25/7.71 ( ( finite_finite_int @ A2 )
% 13.25/7.71 => ( ( groups7440179247065528705omplex @ G
% 13.25/7.71 @ ( minus_minus_set_int @ A2
% 13.25/7.71 @ ( collect_int
% 13.25/7.71 @ ^ [X2: int] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_complex ) ) ) )
% 13.25/7.71 = ( groups7440179247065528705omplex @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8495_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_complex,G: complex > complex] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.71 => ( ( groups3708469109370488835omplex @ G
% 13.25/7.71 @ ( minus_811609699411566653omplex @ A2
% 13.25/7.71 @ ( collect_complex
% 13.25/7.71 @ ^ [X2: complex] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_complex ) ) ) )
% 13.25/7.71 = ( groups3708469109370488835omplex @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8496_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_real,G: real > real] :
% 13.25/7.71 ( ( finite_finite_real @ A2 )
% 13.25/7.71 => ( ( groups1681761925125756287l_real @ G
% 13.25/7.71 @ ( minus_minus_set_real @ A2
% 13.25/7.71 @ ( collect_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_real ) ) ) )
% 13.25/7.71 = ( groups1681761925125756287l_real @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8497_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_int,G: int > real] :
% 13.25/7.71 ( ( finite_finite_int @ A2 )
% 13.25/7.71 => ( ( groups2316167850115554303t_real @ G
% 13.25/7.71 @ ( minus_minus_set_int @ A2
% 13.25/7.71 @ ( collect_int
% 13.25/7.71 @ ^ [X2: int] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_real ) ) ) )
% 13.25/7.71 = ( groups2316167850115554303t_real @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8498_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_complex,G: complex > real] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.71 => ( ( groups766887009212190081x_real @ G
% 13.25/7.71 @ ( minus_811609699411566653omplex @ A2
% 13.25/7.71 @ ( collect_complex
% 13.25/7.71 @ ^ [X2: complex] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_real ) ) ) )
% 13.25/7.71 = ( groups766887009212190081x_real @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8499_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_real,G: real > rat] :
% 13.25/7.71 ( ( finite_finite_real @ A2 )
% 13.25/7.71 => ( ( groups4061424788464935467al_rat @ G
% 13.25/7.71 @ ( minus_minus_set_real @ A2
% 13.25/7.71 @ ( collect_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_rat ) ) ) )
% 13.25/7.71 = ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8500_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_int,G: int > rat] :
% 13.25/7.71 ( ( finite_finite_int @ A2 )
% 13.25/7.71 => ( ( groups1072433553688619179nt_rat @ G
% 13.25/7.71 @ ( minus_minus_set_int @ A2
% 13.25/7.71 @ ( collect_int
% 13.25/7.71 @ ^ [X2: int] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_rat ) ) ) )
% 13.25/7.71 = ( groups1072433553688619179nt_rat @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8501_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_complex,G: complex > rat] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.71 => ( ( groups225925009352817453ex_rat @ G
% 13.25/7.71 @ ( minus_811609699411566653omplex @ A2
% 13.25/7.71 @ ( collect_complex
% 13.25/7.71 @ ^ [X2: complex] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_rat ) ) ) )
% 13.25/7.71 = ( groups225925009352817453ex_rat @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8502_prod_Osetdiff__irrelevant,axiom,
% 13.25/7.71 ! [A2: set_real,G: real > nat] :
% 13.25/7.71 ( ( finite_finite_real @ A2 )
% 13.25/7.71 => ( ( groups4696554848551431203al_nat @ G
% 13.25/7.71 @ ( minus_minus_set_real @ A2
% 13.25/7.71 @ ( collect_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( G @ X2 )
% 13.25/7.71 = one_one_nat ) ) ) )
% 13.25/7.71 = ( groups4696554848551431203al_nat @ G @ A2 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8503_prod_Onat__diff__reindex,axiom,
% 13.25/7.71 ! [G: nat > nat,N: nat] :
% 13.25/7.71 ( ( groups708209901874060359at_nat
% 13.25/7.71 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 = ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8504_prod_Onat__diff__reindex,axiom,
% 13.25/7.71 ! [G: nat > int,N: nat] :
% 13.25/7.71 ( ( groups705719431365010083at_int
% 13.25/7.71 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 = ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8505_prod_OatLeastAtMost__rev,axiom,
% 13.25/7.71 ! [G: nat > nat,N: nat,M: nat] :
% 13.25/7.71 ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 13.25/7.71 = ( groups708209901874060359at_nat
% 13.25/7.71 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 13.25/7.71 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8506_prod_OatLeastAtMost__rev,axiom,
% 13.25/7.71 ! [G: nat > int,N: nat,M: nat] :
% 13.25/7.71 ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 13.25/7.71 = ( groups705719431365010083at_int
% 13.25/7.71 @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 13.25/7.71 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8507_fact__code,axiom,
% 13.25/7.71 ( semiri1406184849735516958ct_int
% 13.25/7.71 = ( ^ [N2: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8508_fact__code,axiom,
% 13.25/7.71 ( semiri5044797733671781792omplex
% 13.25/7.71 = ( ^ [N2: nat] : ( semiri8010041392384452111omplex @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8509_fact__code,axiom,
% 13.25/7.71 ( semiri773545260158071498ct_rat
% 13.25/7.71 = ( ^ [N2: nat] : ( semiri681578069525770553at_rat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8510_fact__code,axiom,
% 13.25/7.71 ( semiri2265585572941072030t_real
% 13.25/7.71 = ( ^ [N2: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8511_fact__code,axiom,
% 13.25/7.71 ( semiri1408675320244567234ct_nat
% 13.25/7.71 = ( ^ [N2: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 @ one_one_nat ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8512_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_real,I: real,F: real > real] :
% 13.25/7.71 ( ( finite_finite_real @ I5 )
% 13.25/7.71 => ( ( member_real @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: real] :
% 13.25/7.71 ( ( member_real @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8513_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_nat,I: nat,F: nat > real] :
% 13.25/7.71 ( ( finite_finite_nat @ I5 )
% 13.25/7.71 => ( ( member_nat @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: nat] :
% 13.25/7.71 ( ( member_nat @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8514_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_int,I: int,F: int > real] :
% 13.25/7.71 ( ( finite_finite_int @ I5 )
% 13.25/7.71 => ( ( member_int @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: int] :
% 13.25/7.71 ( ( member_int @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8515_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_complex,I: complex,F: complex > real] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.71 => ( ( member_complex @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: complex] :
% 13.25/7.71 ( ( member_complex @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_real @ one_one_real @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8516_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_real,I: real,F: real > rat] :
% 13.25/7.71 ( ( finite_finite_real @ I5 )
% 13.25/7.71 => ( ( member_real @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: real] :
% 13.25/7.71 ( ( member_real @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8517_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_nat,I: nat,F: nat > rat] :
% 13.25/7.71 ( ( finite_finite_nat @ I5 )
% 13.25/7.71 => ( ( member_nat @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: nat] :
% 13.25/7.71 ( ( member_nat @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8518_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_int,I: int,F: int > rat] :
% 13.25/7.71 ( ( finite_finite_int @ I5 )
% 13.25/7.71 => ( ( member_int @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: int] :
% 13.25/7.71 ( ( member_int @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8519_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_complex,I: complex,F: complex > rat] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.71 => ( ( member_complex @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: complex] :
% 13.25/7.71 ( ( member_complex @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_rat @ one_one_rat @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8520_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_real,I: real,F: real > int] :
% 13.25/7.71 ( ( finite_finite_real @ I5 )
% 13.25/7.71 => ( ( member_real @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: real] :
% 13.25/7.71 ( ( member_real @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8521_less__1__prod2,axiom,
% 13.25/7.71 ! [I5: set_complex,I: complex,F: complex > int] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.71 => ( ( member_complex @ I @ I5 )
% 13.25/7.71 => ( ( ord_less_int @ one_one_int @ ( F @ I ) )
% 13.25/7.71 => ( ! [I2: complex] :
% 13.25/7.71 ( ( member_complex @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_eq_int @ one_one_int @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8522_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_complex,F: complex > real] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_complex )
% 13.25/7.71 => ( ! [I2: complex] :
% 13.25/7.71 ( ( member_complex @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8523_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_nat,F: nat > real] :
% 13.25/7.71 ( ( finite_finite_nat @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_nat )
% 13.25/7.71 => ( ! [I2: nat] :
% 13.25/7.71 ( ( member_nat @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8524_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_int,F: int > real] :
% 13.25/7.71 ( ( finite_finite_int @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_int )
% 13.25/7.71 => ( ! [I2: int] :
% 13.25/7.71 ( ( member_int @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8525_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_real,F: real > real] :
% 13.25/7.71 ( ( finite_finite_real @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_real )
% 13.25/7.71 => ( ! [I2: real] :
% 13.25/7.71 ( ( member_real @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8526_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_complex,F: complex > rat] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_complex )
% 13.25/7.71 => ( ! [I2: complex] :
% 13.25/7.71 ( ( member_complex @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8527_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_nat,F: nat > rat] :
% 13.25/7.71 ( ( finite_finite_nat @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_nat )
% 13.25/7.71 => ( ! [I2: nat] :
% 13.25/7.71 ( ( member_nat @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8528_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_int,F: int > rat] :
% 13.25/7.71 ( ( finite_finite_int @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_int )
% 13.25/7.71 => ( ! [I2: int] :
% 13.25/7.71 ( ( member_int @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( groups1072433553688619179nt_rat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8529_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_real,F: real > rat] :
% 13.25/7.71 ( ( finite_finite_real @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_real )
% 13.25/7.71 => ( ! [I2: real] :
% 13.25/7.71 ( ( member_real @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8530_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_complex,F: complex > int] :
% 13.25/7.71 ( ( finite3207457112153483333omplex @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_complex )
% 13.25/7.71 => ( ! [I2: complex] :
% 13.25/7.71 ( ( member_complex @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_int @ one_one_int @ ( groups858564598930262913ex_int @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8531_less__1__prod,axiom,
% 13.25/7.71 ! [I5: set_real,F: real > int] :
% 13.25/7.71 ( ( finite_finite_real @ I5 )
% 13.25/7.71 => ( ( I5 != bot_bot_set_real )
% 13.25/7.71 => ( ! [I2: real] :
% 13.25/7.71 ( ( member_real @ I2 @ I5 )
% 13.25/7.71 => ( ord_less_int @ one_one_int @ ( F @ I2 ) ) )
% 13.25/7.71 => ( ord_less_int @ one_one_int @ ( groups4694064378042380927al_int @ F @ I5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8532_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_nat,A2: set_nat,G: nat > rat] :
% 13.25/7.71 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.71 => ( ( finite_finite_nat @ A2 )
% 13.25/7.71 => ( ( groups73079841787564623at_rat @ G @ A2 )
% 13.25/7.71 = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups73079841787564623at_rat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8533_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_complex,A2: set_complex,G: complex > nat] :
% 13.25/7.71 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.71 => ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.71 => ( ( groups861055069439313189ex_nat @ G @ A2 )
% 13.25/7.71 = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8534_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_complex,A2: set_complex,G: complex > int] :
% 13.25/7.71 ( ( ord_le211207098394363844omplex @ B5 @ A2 )
% 13.25/7.71 => ( ( finite3207457112153483333omplex @ A2 )
% 13.25/7.71 => ( ( groups858564598930262913ex_int @ G @ A2 )
% 13.25/7.71 = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B5 ) ) @ ( groups858564598930262913ex_int @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8535_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_int,A2: set_int,G: int > real] :
% 13.25/7.71 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.71 => ( ( finite_finite_int @ A2 )
% 13.25/7.71 => ( ( groups2316167850115554303t_real @ G @ A2 )
% 13.25/7.71 = ( times_times_real @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups2316167850115554303t_real @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8536_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_int,A2: set_int,G: int > rat] :
% 13.25/7.71 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.71 => ( ( finite_finite_int @ A2 )
% 13.25/7.71 => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 13.25/7.71 = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8537_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_int,A2: set_int,G: int > nat] :
% 13.25/7.71 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.71 => ( ( finite_finite_int @ A2 )
% 13.25/7.71 => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 13.25/7.71 = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8538_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_nat,A2: set_nat,G: nat > nat] :
% 13.25/7.71 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.71 => ( ( finite_finite_nat @ A2 )
% 13.25/7.71 => ( ( groups708209901874060359at_nat @ G @ A2 )
% 13.25/7.71 = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups708209901874060359at_nat @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8539_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_nat,A2: set_nat,G: nat > int] :
% 13.25/7.71 ( ( ord_less_eq_set_nat @ B5 @ A2 )
% 13.25/7.71 => ( ( finite_finite_nat @ A2 )
% 13.25/7.71 => ( ( groups705719431365010083at_int @ G @ A2 )
% 13.25/7.71 = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A2 @ B5 ) ) @ ( groups705719431365010083at_int @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8540_prod_Osubset__diff,axiom,
% 13.25/7.71 ! [B5: set_int,A2: set_int,G: int > int] :
% 13.25/7.71 ( ( ord_less_eq_set_int @ B5 @ A2 )
% 13.25/7.71 => ( ( finite_finite_int @ A2 )
% 13.25/7.71 => ( ( groups1705073143266064639nt_int @ G @ A2 )
% 13.25/7.71 = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ ( minus_minus_set_int @ A2 @ B5 ) ) @ ( groups1705073143266064639nt_int @ G @ B5 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8541_square__fact__le__2__fact,axiom,
% 13.25/7.71 ! [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 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8542_Maclaurin__lemma,axiom,
% 13.25/7.71 ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ H2 )
% 13.25/7.71 => ? [B9: real] :
% 13.25/7.71 ( ( F @ H2 )
% 13.25/7.71 = ( plus_plus_real
% 13.25/7.71 @ ( groups6591440286371151544t_real
% 13.25/7.71 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8543_cos__coeff__def,axiom,
% 13.25/7.71 ( cos_coeff
% 13.25/7.71 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8544_cos__paired,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( sums_real
% 13.25/7.71 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 13.25/7.71 @ ( cos_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8545_sin__paired,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( sums_real
% 13.25/7.71 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 13.25/7.71 @ ( sin_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8546_Maclaurin__sin__expansion3,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ? [T5: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ T5 )
% 13.25/7.71 & ( ord_less_real @ T5 @ X )
% 13.25/7.71 & ( ( sin_real @ X )
% 13.25/7.71 = ( plus_plus_real
% 13.25/7.71 @ ( groups6591440286371151544t_real
% 13.25/7.71 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8547_Maclaurin__sin__expansion4,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ? [T5: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ T5 )
% 13.25/7.71 & ( ord_less_eq_real @ T5 @ X )
% 13.25/7.71 & ( ( sin_real @ X )
% 13.25/7.71 = ( plus_plus_real
% 13.25/7.71 @ ( groups6591440286371151544t_real
% 13.25/7.71 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8548_Maclaurin__sin__expansion2,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ? [T5: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 13.25/7.71 & ( ( sin_real @ X )
% 13.25/7.71 = ( plus_plus_real
% 13.25/7.71 @ ( groups6591440286371151544t_real
% 13.25/7.71 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8549_Maclaurin__sin__expansion,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ? [T5: real] :
% 13.25/7.71 ( ( sin_real @ X )
% 13.25/7.71 = ( plus_plus_real
% 13.25/7.71 @ ( groups6591440286371151544t_real
% 13.25/7.71 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8550_sin__coeff__def,axiom,
% 13.25/7.71 ( sin_coeff
% 13.25/7.71 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8551_fact__mono__nat,axiom,
% 13.25/7.71 ! [M: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.71 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8552_fact__ge__self,axiom,
% 13.25/7.71 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8553_fact__less__mono__nat,axiom,
% 13.25/7.71 ! [M: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.71 => ( ( ord_less_nat @ M @ N )
% 13.25/7.71 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8554_fact__ge__Suc__0__nat,axiom,
% 13.25/7.71 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8555_dvd__fact,axiom,
% 13.25/7.71 ! [M: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 13.25/7.71 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.71 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8556_fact__diff__Suc,axiom,
% 13.25/7.71 ! [N: nat,M: nat] :
% 13.25/7.71 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 13.25/7.71 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 13.25/7.71 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8557_fact__div__fact__le__pow,axiom,
% 13.25/7.71 ! [R: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ R @ N )
% 13.25/7.71 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R ) ) ) @ ( power_power_nat @ N @ R ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8558_fact__eq__fact__times,axiom,
% 13.25/7.71 ! [N: nat,M: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.71 => ( ( semiri1408675320244567234ct_nat @ M )
% 13.25/7.71 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 13.25/7.71 @ ( groups708209901874060359at_nat
% 13.25/7.71 @ ^ [X2: nat] : X2
% 13.25/7.71 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8559_prod__int__plus__eq,axiom,
% 13.25/7.71 ! [I: nat,J: nat] :
% 13.25/7.71 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 13.25/7.71 = ( groups1705073143266064639nt_int
% 13.25/7.71 @ ^ [X2: int] : X2
% 13.25/7.71 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8560_fact__div__fact,axiom,
% 13.25/7.71 ! [N: nat,M: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.71 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 13.25/7.71 = ( groups708209901874060359at_nat
% 13.25/7.71 @ ^ [X2: nat] : X2
% 13.25/7.71 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8561_sin__coeff__Suc,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( sin_coeff @ ( suc @ N ) )
% 13.25/7.71 = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8562_cos__coeff__Suc,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( cos_coeff @ ( suc @ N ) )
% 13.25/7.71 = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8563_complex__unimodular__polar,axiom,
% 13.25/7.71 ! [Z: complex] :
% 13.25/7.71 ( ( ( real_V1022390504157884413omplex @ Z )
% 13.25/7.71 = one_one_real )
% 13.25/7.71 => ~ ! [T5: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ T5 )
% 13.25/7.71 => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 13.25/7.71 => ( Z
% 13.25/7.71 != ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8564_Maclaurin__exp__lt,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ( ( X != zero_zero_real )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ? [T5: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 13.25/7.71 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 13.25/7.71 & ( ( exp_real @ X )
% 13.25/7.71 = ( plus_plus_real
% 13.25/7.71 @ ( groups6591440286371151544t_real
% 13.25/7.71 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8565_binomial__code,axiom,
% 13.25/7.71 ( binomial
% 13.25/7.71 = ( ^ [N2: nat,K2: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K2 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K2 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K2 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K2 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8566_tan__periodic__pi,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( tan_real @ ( plus_plus_real @ X @ pi ) )
% 13.25/7.71 = ( tan_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8567_exp__less__cancel__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ X @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8568_exp__less__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8569_exp__le__cancel__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( exp_real @ X ) @ ( exp_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8570_binomial__n__n,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( binomial @ N @ N )
% 13.25/7.71 = one_one_nat ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8571_binomial__eq__0__iff,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( ( binomial @ N @ K )
% 13.25/7.71 = zero_zero_nat )
% 13.25/7.71 = ( ord_less_nat @ N @ K ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8572_binomial__Suc__Suc,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 13.25/7.71 = ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8573_binomial__n__0,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( binomial @ N @ zero_zero_nat )
% 13.25/7.71 = one_one_nat ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8574_exp__eq__one__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ( exp_real @ X )
% 13.25/7.71 = one_one_real )
% 13.25/7.71 = ( X = zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8575_zero__less__binomial__iff,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 13.25/7.71 = ( ord_less_eq_nat @ K @ N ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8576_tan__npi,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 13.25/7.71 = zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8577_exp__less__one__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( exp_real @ X ) @ one_one_real )
% 13.25/7.71 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8578_one__less__exp__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ ( exp_real @ X ) )
% 13.25/7.71 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8579_exp__le__one__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( exp_real @ X ) @ one_one_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8580_one__le__exp__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X ) )
% 13.25/7.71 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8581_tan__periodic__n,axiom,
% 13.25/7.71 ! [X: real,N: num] :
% 13.25/7.71 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
% 13.25/7.71 = ( tan_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8582_tan__periodic__nat,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
% 13.25/7.71 = ( tan_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8583_exp__ln__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ( exp_real @ ( ln_ln_real @ X ) )
% 13.25/7.71 = X )
% 13.25/7.71 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8584_exp__ln,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( exp_real @ ( ln_ln_real @ X ) )
% 13.25/7.71 = X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8585_tan__periodic__int,axiom,
% 13.25/7.71 ! [X: real,I: int] :
% 13.25/7.71 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) )
% 13.25/7.71 = ( tan_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8586_norm__cos__sin,axiom,
% 13.25/7.71 ! [T: real] :
% 13.25/7.71 ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T ) @ ( sin_real @ T ) ) )
% 13.25/7.71 = one_one_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8587_tan__periodic,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 13.25/7.71 = ( tan_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8588_complex__diff,axiom,
% 13.25/7.71 ! [A: real,B: real,C: real,D: real] :
% 13.25/7.71 ( ( minus_minus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 13.25/7.71 = ( complex2 @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8589_exp__less__cancel,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( exp_real @ X ) @ ( exp_real @ Y3 ) )
% 13.25/7.71 => ( ord_less_real @ X @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8590_choose__one,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( binomial @ N @ one_one_nat )
% 13.25/7.71 = N ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8591_binomial__eq__0,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( ord_less_nat @ N @ K )
% 13.25/7.71 => ( ( binomial @ N @ K )
% 13.25/7.71 = zero_zero_nat ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8592_Suc__times__binomial__eq,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
% 13.25/7.71 = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8593_Suc__times__binomial,axiom,
% 13.25/7.71 ! [K: nat,N: nat] :
% 13.25/7.71 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
% 13.25/7.71 = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8594_binomial__symmetric,axiom,
% 13.25/7.71 ! [K: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.71 => ( ( binomial @ N @ K )
% 13.25/7.71 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8595_exp__total,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ? [X3: real] :
% 13.25/7.71 ( ( exp_real @ X3 )
% 13.25/7.71 = Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8596_exp__gt__zero,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8597_not__exp__less__zero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ~ ( ord_less_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8598_not__exp__le__zero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ~ ( ord_less_eq_real @ ( exp_real @ X ) @ zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8599_exp__ge__zero,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8600_choose__mult__lemma,axiom,
% 13.25/7.71 ! [M: nat,R: nat,K: nat] :
% 13.25/7.71 ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 13.25/7.71 = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R ) @ M ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8601_binomial__le__pow,axiom,
% 13.25/7.71 ! [R: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ R @ N )
% 13.25/7.71 => ( ord_less_eq_nat @ ( binomial @ N @ R ) @ ( power_power_nat @ N @ R ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8602_Complex__eq__numeral,axiom,
% 13.25/7.71 ! [A: real,B: real,W: num] :
% 13.25/7.71 ( ( ( complex2 @ A @ B )
% 13.25/7.71 = ( numera6690914467698888265omplex @ W ) )
% 13.25/7.71 = ( ( A
% 13.25/7.71 = ( numeral_numeral_real @ W ) )
% 13.25/7.71 & ( B = zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8603_complex__add,axiom,
% 13.25/7.71 ! [A: real,B: real,C: real,D: real] :
% 13.25/7.71 ( ( plus_plus_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 13.25/7.71 = ( complex2 @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8604_zero__less__binomial,axiom,
% 13.25/7.71 ! [K: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.71 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8605_Suc__times__binomial__add,axiom,
% 13.25/7.71 ! [A: nat,B: nat] :
% 13.25/7.71 ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 13.25/7.71 = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8606_binomial__Suc__Suc__eq__times,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 13.25/7.71 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8607_choose__mult,axiom,
% 13.25/7.71 ! [K: nat,M: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ K @ M )
% 13.25/7.71 => ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.71 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 13.25/7.71 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8608_binomial__fact__lemma,axiom,
% 13.25/7.71 ! [K: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.71 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 13.25/7.71 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8609_exp__gt__one,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( exp_real @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8610_binomial__absorb__comp,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
% 13.25/7.71 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8611_exp__ge__add__one__self,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8612_Complex__eq__neg__numeral,axiom,
% 13.25/7.71 ! [A: real,B: real,W: num] :
% 13.25/7.71 ( ( ( complex2 @ A @ B )
% 13.25/7.71 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.71 = ( ( A
% 13.25/7.71 = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
% 13.25/7.71 & ( B = zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8613_complex__mult,axiom,
% 13.25/7.71 ! [A: real,B: real,C: real,D: real] :
% 13.25/7.71 ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C @ D ) )
% 13.25/7.71 = ( 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 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8614_one__complex_Ocode,axiom,
% 13.25/7.71 ( one_one_complex
% 13.25/7.71 = ( complex2 @ one_one_real @ zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8615_Complex__eq__1,axiom,
% 13.25/7.71 ! [A: real,B: real] :
% 13.25/7.71 ( ( ( complex2 @ A @ B )
% 13.25/7.71 = one_one_complex )
% 13.25/7.71 = ( ( A = one_one_real )
% 13.25/7.71 & ( B = zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8616_binomial__absorption,axiom,
% 13.25/7.71 ! [K: nat,N: nat] :
% 13.25/7.71 ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 13.25/7.71 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8617_binomial__altdef__nat,axiom,
% 13.25/7.71 ! [K: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ K @ N )
% 13.25/7.71 => ( ( binomial @ N @ K )
% 13.25/7.71 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8618_exp__ge__add__one__self__aux,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X ) @ ( exp_real @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8619_lemma__exp__total,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 13.25/7.71 => ? [X3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 13.25/7.71 & ( ord_less_eq_real @ X3 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 13.25/7.71 & ( ( exp_real @ X3 )
% 13.25/7.71 = Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8620_ln__ge__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X ) )
% 13.25/7.71 = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8621_ln__x__over__x__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X ) @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8622_Complex__eq__neg__1,axiom,
% 13.25/7.71 ! [A: real,B: real] :
% 13.25/7.71 ( ( ( complex2 @ A @ B )
% 13.25/7.71 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 13.25/7.71 = ( ( A
% 13.25/7.71 = ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.71 & ( B = zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8623_binomial__mono,axiom,
% 13.25/7.71 ! [K: nat,K6: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ K @ K6 )
% 13.25/7.71 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 13.25/7.71 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8624_binomial__maximum_H,axiom,
% 13.25/7.71 ! [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 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8625_binomial__maximum,axiom,
% 13.25/7.71 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8626_binomial__antimono,axiom,
% 13.25/7.71 ! [K: nat,K6: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ K @ K6 )
% 13.25/7.71 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 13.25/7.71 => ( ( ord_less_eq_nat @ K6 @ N )
% 13.25/7.71 => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8627_binomial__le__pow2,axiom,
% 13.25/7.71 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8628_choose__reduce__nat,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.25/7.71 => ( ( binomial @ N @ K )
% 13.25/7.71 = ( 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 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8629_times__binomial__minus1__eq,axiom,
% 13.25/7.71 ! [K: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.25/7.71 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 13.25/7.71 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8630_exp__le,axiom,
% 13.25/7.71 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8631_binomial__less__binomial__Suc,axiom,
% 13.25/7.71 ! [K: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8632_binomial__strict__mono,axiom,
% 13.25/7.71 ! [K: nat,K6: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ K @ K6 )
% 13.25/7.71 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
% 13.25/7.71 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K6 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8633_binomial__strict__antimono,axiom,
% 13.25/7.71 ! [K: nat,K6: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ K @ K6 )
% 13.25/7.71 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 13.25/7.71 => ( ( ord_less_eq_nat @ K6 @ N )
% 13.25/7.71 => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8634_central__binomial__odd,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.71 => ( ( binomial @ N @ ( suc @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 = ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8635_binomial__addition__formula,axiom,
% 13.25/7.71 ! [N: nat,K: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( binomial @ N @ ( suc @ K ) )
% 13.25/7.71 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8636_exp__half__le2,axiom,
% 13.25/7.71 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8637_tan__45,axiom,
% 13.25/7.71 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 = one_one_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8638_choose__two,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( binomial @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 = ( divide_divide_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8639_lemma__tan__total,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ? [X3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X3 )
% 13.25/7.71 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ord_less_real @ Y3 @ ( tan_real @ X3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8640_tan__gt__zero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8641_tan__total,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ? [X3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 13.25/7.71 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ( tan_real @ X3 )
% 13.25/7.71 = Y3 )
% 13.25/7.71 & ! [Y5: real] :
% 13.25/7.71 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 13.25/7.71 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ( tan_real @ Y5 )
% 13.25/7.71 = Y3 ) )
% 13.25/7.71 => ( Y5 = X3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8642_tan__monotone,axiom,
% 13.25/7.71 ! [Y3: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8643_tan__monotone_H,axiom,
% 13.25/7.71 ! [Y3: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ X )
% 13.25/7.71 = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8644_tan__mono__lt__eq,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ ( tan_real @ X ) @ ( tan_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ X @ Y3 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8645_lemma__tan__total1,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ? [X3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 13.25/7.71 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ( tan_real @ X3 )
% 13.25/7.71 = Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8646_tan__minus__45,axiom,
% 13.25/7.71 ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.71 = ( uminus_uminus_real @ one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8647_tan__inverse,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y3 ) )
% 13.25/7.71 = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8648_exp__bound,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.71 => ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8649_tan__pos__pi2__le,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8650_tan__total__pos,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ? [X3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 13.25/7.71 & ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ( tan_real @ X3 )
% 13.25/7.71 = Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8651_tan__less__zero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ zero_zero_real )
% 13.25/7.71 => ( ord_less_real @ ( tan_real @ X ) @ zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8652_tan__mono__le,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8653_tan__mono__le__eq,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( tan_real @ X ) @ ( tan_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8654_tan__bound__pi2,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X ) ) @ one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8655_arctan,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 13.25/7.71 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ( tan_real @ ( arctan @ Y3 ) )
% 13.25/7.71 = Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8656_arctan__tan,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( arctan @ ( tan_real @ X ) )
% 13.25/7.71 = X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8657_arctan__unique,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ( tan_real @ X )
% 13.25/7.71 = Y3 )
% 13.25/7.71 => ( ( arctan @ Y3 )
% 13.25/7.71 = X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8658_real__exp__bound__lemma,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8659_exp__ge__one__plus__x__over__n__power__n,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8660_exp__ge__one__minus__x__over__n__power__n,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( 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 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8661_tan__total__pi4,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.71 => ? [Z5: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z5 )
% 13.25/7.71 & ( ord_less_real @ Z5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 & ( ( tan_real @ Z5 )
% 13.25/7.71 = X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8662_Maclaurin__exp__le,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ? [T5: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 13.25/7.71 & ( ( exp_real @ X )
% 13.25/7.71 = ( plus_plus_real
% 13.25/7.71 @ ( groups6591440286371151544t_real
% 13.25/7.71 @ ^ [M3: nat] : ( divide_divide_real @ ( power_power_real @ X @ M3 ) @ ( semiri2265585572941072030t_real @ M3 ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.71 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8663_exp__lower__Taylor__quadratic,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( 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 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8664_tanh__real__altdef,axiom,
% 13.25/7.71 ( tanh_real
% 13.25/7.71 = ( ^ [X2: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8665_central__binomial__lower__bound,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( 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 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8666_sin__tan,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( sin_real @ X )
% 13.25/7.71 = ( divide_divide_real @ ( tan_real @ X ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8667_real__sqrt__eq__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ( sqrt @ X )
% 13.25/7.71 = ( sqrt @ Y3 ) )
% 13.25/7.71 = ( X = Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8668_real__sqrt__zero,axiom,
% 13.25/7.71 ( ( sqrt @ zero_zero_real )
% 13.25/7.71 = zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8669_real__sqrt__eq__zero__cancel__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ( sqrt @ X )
% 13.25/7.71 = zero_zero_real )
% 13.25/7.71 = ( X = zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8670_real__sqrt__less__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ X @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8671_real__sqrt__le__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8672_real__sqrt__one,axiom,
% 13.25/7.71 ( ( sqrt @ one_one_real )
% 13.25/7.71 = one_one_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8673_real__sqrt__eq__1__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ( sqrt @ X )
% 13.25/7.71 = one_one_real )
% 13.25/7.71 = ( X = one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8674_real__sqrt__lt__0__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( sqrt @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8675_real__sqrt__gt__0__iff,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8676_real__sqrt__ge__0__iff,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8677_real__sqrt__le__0__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( sqrt @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8678_real__sqrt__gt__1__iff,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8679_real__sqrt__lt__1__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( sqrt @ X ) @ one_one_real )
% 13.25/7.71 = ( ord_less_real @ X @ one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8680_real__sqrt__le__1__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( sqrt @ X ) @ one_one_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8681_real__sqrt__ge__1__iff,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8682_real__sqrt__mult__self,axiom,
% 13.25/7.71 ! [A: real] :
% 13.25/7.71 ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
% 13.25/7.71 = ( abs_abs_real @ A ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8683_real__sqrt__abs2,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( sqrt @ ( times_times_real @ X @ X ) )
% 13.25/7.71 = ( abs_abs_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8684_real__sqrt__four,axiom,
% 13.25/7.71 ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 13.25/7.71 = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8685_real__sqrt__abs,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( sqrt @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.71 = ( abs_abs_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8686_real__sqrt__pow2__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 = X )
% 13.25/7.71 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8687_real__sqrt__pow2,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( power_power_real @ ( sqrt @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 = X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8688_real__sqrt__sum__squares__mult__squared__eq,axiom,
% 13.25/7.71 ! [X: real,Y3: real,Xa: real,Ya: real] :
% 13.25/7.71 ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8689_real__sqrt__minus,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( sqrt @ ( uminus_uminus_real @ X ) )
% 13.25/7.71 = ( uminus_uminus_real @ ( sqrt @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8690_real__sqrt__divide,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( sqrt @ ( divide_divide_real @ X @ Y3 ) )
% 13.25/7.71 = ( divide_divide_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8691_real__sqrt__mult,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( sqrt @ ( times_times_real @ X @ Y3 ) )
% 13.25/7.71 = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8692_real__sqrt__less__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8693_real__sqrt__le__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8694_real__sqrt__power,axiom,
% 13.25/7.71 ! [X: real,K: nat] :
% 13.25/7.71 ( ( sqrt @ ( power_power_real @ X @ K ) )
% 13.25/7.71 = ( power_power_real @ ( sqrt @ X ) @ K ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8695_real__sqrt__gt__zero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8696_real__sqrt__ge__zero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8697_real__sqrt__eq__zero__cancel,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ( sqrt @ X )
% 13.25/7.71 = zero_zero_real )
% 13.25/7.71 => ( X = zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8698_real__sqrt__ge__one,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ one_one_real @ X )
% 13.25/7.71 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8699_finite__nat__iff__bounded__le,axiom,
% 13.25/7.71 ( finite_finite_nat
% 13.25/7.71 = ( ^ [S5: set_nat] :
% 13.25/7.71 ? [K2: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8700_real__div__sqrt,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( divide_divide_real @ X @ ( sqrt @ X ) )
% 13.25/7.71 = ( sqrt @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8701_sqrt__add__le__add__sqrt,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8702_le__real__sqrt__sumsq,axiom,
% 13.25/7.71 ! [X: real,Y3: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8703_sqrt2__less__2,axiom,
% 13.25/7.71 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8704_sum__choose__lower,axiom,
% 13.25/7.71 ! [R: nat,N: nat] :
% 13.25/7.71 ( ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [K2: nat] : ( binomial @ ( plus_plus_nat @ R @ K2 ) @ K2 )
% 13.25/7.71 @ ( set_ord_atMost_nat @ N ) )
% 13.25/7.71 = ( binomial @ ( suc @ ( plus_plus_nat @ R @ N ) ) @ N ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8705_choose__rising__sum_I2_J,axiom,
% 13.25/7.71 ! [N: nat,M: nat] :
% 13.25/7.71 ( ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 13.25/7.71 @ ( set_ord_atMost_nat @ M ) )
% 13.25/7.71 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ M ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8706_choose__rising__sum_I1_J,axiom,
% 13.25/7.71 ! [N: nat,M: nat] :
% 13.25/7.71 ( ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
% 13.25/7.71 @ ( set_ord_atMost_nat @ M ) )
% 13.25/7.71 = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8707_real__less__rsqrt,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 13.25/7.71 => ( ord_less_real @ X @ ( sqrt @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8708_real__le__rsqrt,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ X @ ( sqrt @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8709_sqrt__le__D,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( sqrt @ X ) @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ X @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8710_tan__60,axiom,
% 13.25/7.71 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 13.25/7.71 = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8711_sum__choose__diagonal,axiom,
% 13.25/7.71 ! [M: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.71 => ( ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [K2: nat] : ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M @ K2 ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ M ) )
% 13.25/7.71 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8712_vandermonde,axiom,
% 13.25/7.71 ! [M: nat,N: nat,R: nat] :
% 13.25/7.71 ( ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [K2: nat] : ( times_times_nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus_nat @ R @ K2 ) ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ R ) )
% 13.25/7.71 = ( binomial @ ( plus_plus_nat @ M @ N ) @ R ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8713_real__sqrt__unique,axiom,
% 13.25/7.71 ! [Y3: real,X: real] :
% 13.25/7.71 ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 = X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( sqrt @ X )
% 13.25/7.71 = Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8714_real__le__lsqrt,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_eq_real @ ( sqrt @ X ) @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8715_lemma__real__divide__sqrt__less,axiom,
% 13.25/7.71 ! [U: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ U )
% 13.25/7.71 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8716_real__sqrt__sum__squares__eq__cancel2,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 = Y3 )
% 13.25/7.71 => ( X = zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8717_real__sqrt__sum__squares__eq__cancel,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 = X )
% 13.25/7.71 => ( Y3 = zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8718_real__sqrt__sum__squares__triangle__ineq,axiom,
% 13.25/7.71 ! [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 ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8719_real__sqrt__sum__squares__ge2,axiom,
% 13.25/7.71 ! [Y3: real,X: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8720_real__sqrt__sum__squares__ge1,axiom,
% 13.25/7.71 ! [X: real,Y3: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8721_sqrt__ge__absD,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( sqrt @ Y3 ) )
% 13.25/7.71 => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8722_cos__45,axiom,
% 13.25/7.71 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8723_sin__45,axiom,
% 13.25/7.71 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8724_choose__row__sum,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( groups3542108847815614940at_nat @ ( binomial @ N ) @ ( set_ord_atMost_nat @ N ) )
% 13.25/7.71 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8725_real__less__lsqrt,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_real @ ( sqrt @ X ) @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8726_binomial,axiom,
% 13.25/7.71 ! [A: nat,B: nat,N: nat] :
% 13.25/7.71 ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
% 13.25/7.71 = ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [K2: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K2 ) ) @ ( power_power_nat @ A @ K2 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K2 ) ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8727_sqrt__sum__squares__le__sum,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8728_tan__30,axiom,
% 13.25/7.71 ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 13.25/7.71 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8729_sqrt__even__pow2,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.71 => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.71 = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8730_sqrt__sum__squares__le__sum__abs,axiom,
% 13.25/7.71 ! [X: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8731_real__sqrt__ge__abs2,axiom,
% 13.25/7.71 ! [Y3: real,X: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8732_real__sqrt__ge__abs1,axiom,
% 13.25/7.71 ! [X: real,Y3: 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 @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8733_ln__sqrt,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ln_ln_real @ ( sqrt @ X ) )
% 13.25/7.71 = ( divide_divide_real @ ( ln_ln_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8734_cos__30,axiom,
% 13.25/7.71 ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
% 13.25/7.71 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8735_sin__60,axiom,
% 13.25/7.71 ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
% 13.25/7.71 = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8736_complex__norm,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( real_V1022390504157884413omplex @ ( complex2 @ X @ Y3 ) )
% 13.25/7.71 = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8737_polynomial__product__nat,axiom,
% 13.25/7.71 ! [M: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
% 13.25/7.71 ( ! [I2: nat] :
% 13.25/7.71 ( ( ord_less_nat @ M @ I2 )
% 13.25/7.71 => ( ( A @ I2 )
% 13.25/7.71 = zero_zero_nat ) )
% 13.25/7.71 => ( ! [J2: nat] :
% 13.25/7.71 ( ( ord_less_nat @ N @ J2 )
% 13.25/7.71 => ( ( B @ J2 )
% 13.25/7.71 = zero_zero_nat ) )
% 13.25/7.71 => ( ( times_times_nat
% 13.25/7.71 @ ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( power_power_nat @ X @ I4 ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ M ) )
% 13.25/7.71 @ ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X @ J3 ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ N ) ) )
% 13.25/7.71 = ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [R5: nat] :
% 13.25/7.71 ( times_times_nat
% 13.25/7.71 @ ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [K2: nat] : ( times_times_nat @ ( A @ K2 ) @ ( B @ ( minus_minus_nat @ R5 @ K2 ) ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ R5 ) )
% 13.25/7.71 @ ( power_power_nat @ X @ R5 ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8738_choose__square__sum,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [K2: nat] : ( power_power_nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ N ) )
% 13.25/7.71 = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8739_arsinh__real__aux,axiom,
% 13.25/7.71 ! [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 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8740_real__sqrt__power__even,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( power_power_real @ ( sqrt @ X ) @ N )
% 13.25/7.71 = ( power_power_real @ X @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8741_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 13.25/7.71 ! [X: real,Y3: 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 @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8742_arith__geo__mean__sqrt,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8743_cos__x__y__le__one,axiom,
% 13.25/7.71 ! [X: real,Y3: 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 @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8744_real__sqrt__sum__squares__less,axiom,
% 13.25/7.71 ! [X: real,U: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8745_arcosh__real__def,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ one_one_real @ X )
% 13.25/7.71 => ( ( arcosh_real @ X )
% 13.25/7.71 = ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8746_cos__arctan,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( cos_real @ ( arctan @ X ) )
% 13.25/7.71 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8747_sin__arctan,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( sin_real @ ( arctan @ X ) )
% 13.25/7.71 = ( divide_divide_real @ X @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8748_binomial__r__part__sum,axiom,
% 13.25/7.71 ! [M: nat] :
% 13.25/7.71 ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 13.25/7.71 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8749_choose__linear__sum,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( groups3542108847815614940at_nat
% 13.25/7.71 @ ^ [I4: nat] : ( times_times_nat @ I4 @ ( binomial @ N @ I4 ) )
% 13.25/7.71 @ ( set_ord_atMost_nat @ N ) )
% 13.25/7.71 = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8750_sqrt__sum__squares__half__less,axiom,
% 13.25/7.71 ! [X: real,U: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ X @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8751_sin__cos__sqrt,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X ) )
% 13.25/7.71 => ( ( sin_real @ X )
% 13.25/7.71 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8752_arctan__half,axiom,
% 13.25/7.71 ( arctan
% 13.25/7.71 = ( ^ [X2: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X2 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8753_cos__tan,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( cos_real @ X )
% 13.25/7.71 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8754_arsinh__real__def,axiom,
% 13.25/7.71 ( arsinh_real
% 13.25/7.71 = ( ^ [X2: real] : ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8755_cos__arcsin,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.71 => ( ( cos_real @ ( arcsin @ X ) )
% 13.25/7.71 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8756_of__nat__id,axiom,
% 13.25/7.71 ( semiri1316708129612266289at_nat
% 13.25/7.71 = ( ^ [N2: nat] : N2 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8757_sin__arcsin,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( sin_real @ ( arcsin @ Y3 ) )
% 13.25/7.71 = Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8758_arcsin__1,axiom,
% 13.25/7.71 ( ( arcsin @ one_one_real )
% 13.25/7.71 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8759_arcsin__minus__1,axiom,
% 13.25/7.71 ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.71 = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8760_real__scaleR__def,axiom,
% 13.25/7.71 real_V1485227260804924795R_real = times_times_real ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8761_complex__scaleR,axiom,
% 13.25/7.71 ! [R: real,A: real,B: real] :
% 13.25/7.71 ( ( real_V2046097035970521341omplex @ R @ ( complex2 @ A @ B ) )
% 13.25/7.71 = ( complex2 @ ( times_times_real @ R @ A ) @ ( times_times_real @ R @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8762_arcsin__minus,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.71 => ( ( arcsin @ ( uminus_uminus_real @ X ) )
% 13.25/7.71 = ( uminus_uminus_real @ ( arcsin @ X ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8763_arcsin__le__arcsin,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8764_arcsin__eq__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 13.25/7.71 => ( ( ( arcsin @ X )
% 13.25/7.71 = ( arcsin @ Y3 ) )
% 13.25/7.71 = ( X = Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8765_arcsin__le__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ ( arcsin @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8766_arcsin__less__arcsin,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8767_arcsin__less__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ ( arcsin @ X ) @ ( arcsin @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8768_cos__arcsin__nonzero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.71 => ( ( cos_real @ ( arcsin @ X ) )
% 13.25/7.71 != zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8769_arcsin__lt__bounded,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 13.25/7.71 & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8770_arcsin__bounded,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 13.25/7.71 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8771_arcsin__ubound,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8772_arcsin__lbound,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8773_arcsin__sin,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( arcsin @ ( sin_real @ X ) )
% 13.25/7.71 = X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8774_arcsin,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 13.25/7.71 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 13.25/7.71 = Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8775_arcsin__pi,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 13.25/7.71 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 13.25/7.71 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 13.25/7.71 = Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8776_arcsin__le__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( arcsin @ X ) @ Y3 )
% 13.25/7.71 = ( ord_less_eq_real @ X @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8777_le__arcsin__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X ) )
% 13.25/7.71 = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8778_sin__arccos__abs,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 13.25/7.71 => ( ( sin_real @ ( arccos @ Y3 ) )
% 13.25/7.71 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8779_sin__arccos,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.71 => ( ( sin_real @ ( arccos @ X ) )
% 13.25/7.71 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8780_Maclaurin__sin__bound,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ( ord_less_eq_real
% 13.25/7.71 @ ( abs_abs_real
% 13.25/7.71 @ ( minus_minus_real @ ( sin_real @ X )
% 13.25/7.71 @ ( groups6591440286371151544t_real
% 13.25/7.71 @ ^ [M3: nat] : ( times_times_real @ ( sin_coeff @ M3 ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 13.25/7.71 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8781_arccos__1,axiom,
% 13.25/7.71 ( ( arccos @ one_one_real )
% 13.25/7.71 = zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8782_arccos__minus__1,axiom,
% 13.25/7.71 ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.71 = pi ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8783_cos__arccos,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( cos_real @ ( arccos @ Y3 ) )
% 13.25/7.71 = Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8784_arccos__0,axiom,
% 13.25/7.71 ( ( arccos @ zero_zero_real )
% 13.25/7.71 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8785_real__sqrt__inverse,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( sqrt @ ( inverse_inverse_real @ X ) )
% 13.25/7.71 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8786_divide__real__def,axiom,
% 13.25/7.71 ( divide_divide_real
% 13.25/7.71 = ( ^ [X2: real,Y: real] : ( times_times_real @ X2 @ ( inverse_inverse_real @ Y ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8787_arccos__le__arccos,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8788_arccos__eq__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.71 & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 13.25/7.71 => ( ( ( arccos @ X )
% 13.25/7.71 = ( arccos @ Y3 ) )
% 13.25/7.71 = ( X = Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8789_arccos__le__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( arccos @ X ) @ ( arccos @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8790_forall__pos__mono__1,axiom,
% 13.25/7.71 ! [P: real > $o,E: real] :
% 13.25/7.71 ( ! [D3: real,E2: real] :
% 13.25/7.71 ( ( ord_less_real @ D3 @ E2 )
% 13.25/7.71 => ( ( P @ D3 )
% 13.25/7.71 => ( P @ E2 ) ) )
% 13.25/7.71 => ( ! [N4: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ E )
% 13.25/7.71 => ( P @ E ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8791_forall__pos__mono,axiom,
% 13.25/7.71 ! [P: real > $o,E: real] :
% 13.25/7.71 ( ! [D3: real,E2: real] :
% 13.25/7.71 ( ( ord_less_real @ D3 @ E2 )
% 13.25/7.71 => ( ( P @ D3 )
% 13.25/7.71 => ( P @ E2 ) ) )
% 13.25/7.71 => ( ! [N4: nat] :
% 13.25/7.71 ( ( N4 != zero_zero_nat )
% 13.25/7.71 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) ) )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ E )
% 13.25/7.71 => ( P @ E ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8792_real__arch__inverse,axiom,
% 13.25/7.71 ! [E: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ E )
% 13.25/7.71 = ( ? [N2: nat] :
% 13.25/7.71 ( ( N2 != zero_zero_nat )
% 13.25/7.71 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 13.25/7.71 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8793_sqrt__divide__self__eq,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( divide_divide_real @ ( sqrt @ X ) @ X )
% 13.25/7.71 = ( inverse_inverse_real @ ( sqrt @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8794_ln__inverse,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ln_ln_real @ ( inverse_inverse_real @ X ) )
% 13.25/7.71 = ( uminus_uminus_real @ ( ln_ln_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8795_arccos__lbound,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8796_arccos__less__arccos,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8797_arccos__less__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ ( arccos @ X ) @ ( arccos @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ Y3 @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8798_arccos__ubound,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8799_arccos__cos,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ pi )
% 13.25/7.71 => ( ( arccos @ ( cos_real @ X ) )
% 13.25/7.71 = X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8800_cos__arccos__abs,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 13.25/7.71 => ( ( cos_real @ ( arccos @ Y3 ) )
% 13.25/7.71 = Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8801_arccos__cos__eq__abs,axiom,
% 13.25/7.71 ! [Theta: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 13.25/7.71 => ( ( arccos @ ( cos_real @ Theta ) )
% 13.25/7.71 = ( abs_abs_real @ Theta ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8802_arccos__lt__bounded,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 13.25/7.71 & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8803_arccos__bounded,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 13.25/7.71 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8804_sin__arccos__nonzero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.71 => ( ( sin_real @ ( arccos @ X ) )
% 13.25/7.71 != zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8805_arccos__cos2,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X )
% 13.25/7.71 => ( ( arccos @ ( cos_real @ X ) )
% 13.25/7.71 = ( uminus_uminus_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8806_arccos__minus,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.71 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 13.25/7.71 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8807_exp__plus__inverse__exp,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( inverse_inverse_real @ ( exp_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8808_plus__inverse__ge__2,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8809_real__inv__sqrt__pow2,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 = ( inverse_inverse_real @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8810_arccos,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 13.25/7.71 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 13.25/7.71 & ( ( cos_real @ ( arccos @ Y3 ) )
% 13.25/7.71 = Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8811_arccos__minus__abs,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.71 => ( ( arccos @ ( uminus_uminus_real @ X ) )
% 13.25/7.71 = ( minus_minus_real @ pi @ ( arccos @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8812_tan__cot,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 13.25/7.71 = ( inverse_inverse_real @ ( tan_real @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8813_real__le__x__sinh,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( 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 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8814_real__le__abs__sinh,axiom,
% 13.25/7.71 ! [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 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8815_arccos__le__pi2,axiom,
% 13.25/7.71 ! [Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 13.25/7.71 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8816_arccos__cos__eq__abs__2pi,axiom,
% 13.25/7.71 ! [Theta: real] :
% 13.25/7.71 ~ ! [K3: int] :
% 13.25/7.71 ( ( arccos @ ( cos_real @ Theta ) )
% 13.25/7.71 != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K3 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8817_cot__less__zero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ zero_zero_real )
% 13.25/7.71 => ( ord_less_real @ ( cot_real @ X ) @ zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8818_i__even__power,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.71 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8819_sinh__real__less__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ ( sinh_real @ X ) @ ( sinh_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ X @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8820_sinh__real__le__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ ( sinh_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ X @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8821_sinh__real__neg__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( sinh_real @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_real @ X @ zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8822_sinh__real__pos__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X ) )
% 13.25/7.71 = ( ord_less_real @ zero_zero_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8823_sinh__real__nonpos__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( sinh_real @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8824_sinh__real__nonneg__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X ) )
% 13.25/7.71 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8825_norm__ii,axiom,
% 13.25/7.71 ( ( real_V1022390504157884413omplex @ imaginary_unit )
% 13.25/7.71 = one_one_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8826_complex__i__mult__minus,axiom,
% 13.25/7.71 ! [X: complex] :
% 13.25/7.71 ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
% 13.25/7.71 = ( uminus1482373934393186551omplex @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8827_divide__i,axiom,
% 13.25/7.71 ! [X: complex] :
% 13.25/7.71 ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
% 13.25/7.71 = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8828_i__squared,axiom,
% 13.25/7.71 ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
% 13.25/7.71 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8829_divide__numeral__i,axiom,
% 13.25/7.71 ! [Z: complex,N: num] :
% 13.25/7.71 ( ( divide1717551699836669952omplex @ Z @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
% 13.25/7.71 = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8830_cot__npi,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
% 13.25/7.71 = zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8831_cot__periodic,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 13.25/7.71 = ( cot_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8832_power2__i,axiom,
% 13.25/7.71 ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8833_divide__complex__def,axiom,
% 13.25/7.71 ( divide1717551699836669952omplex
% 13.25/7.71 = ( ^ [X2: complex,Y: complex] : ( times_times_complex @ X2 @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8834_sinh__less__cosh__real,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8835_sinh__le__cosh__real,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_eq_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8836_complex__i__not__numeral,axiom,
% 13.25/7.71 ! [W: num] :
% 13.25/7.71 ( imaginary_unit
% 13.25/7.71 != ( numera6690914467698888265omplex @ W ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8837_cosh__real__pos,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8838_cosh__real__nonpos__le__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ Y3 @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8839_cosh__real__nonneg__le__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8840_cosh__real__nonneg,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8841_cosh__real__ge__1,axiom,
% 13.25/7.71 ! [X: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8842_i__times__eq__iff,axiom,
% 13.25/7.71 ! [W: complex,Z: complex] :
% 13.25/7.71 ( ( ( times_times_complex @ imaginary_unit @ W )
% 13.25/7.71 = Z )
% 13.25/7.71 = ( W
% 13.25/7.71 = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8843_complex__i__not__neg__numeral,axiom,
% 13.25/7.71 ! [W: num] :
% 13.25/7.71 ( imaginary_unit
% 13.25/7.71 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8844_cosh__real__nonpos__less__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 13.25/7.71 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ Y3 @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8845_cosh__real__nonneg__less__iff,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8846_cosh__real__strict__mono,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_real @ ( cosh_real @ X ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8847_arcosh__cosh__real,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( arcosh_real @ ( cosh_real @ X ) )
% 13.25/7.71 = X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8848_imaginary__unit_Ocode,axiom,
% 13.25/7.71 ( imaginary_unit
% 13.25/7.71 = ( complex2 @ zero_zero_real @ one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8849_Complex__eq__i,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ( complex2 @ X @ Y3 )
% 13.25/7.71 = imaginary_unit )
% 13.25/7.71 = ( ( X = zero_zero_real )
% 13.25/7.71 & ( Y3 = one_one_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8850_i__mult__Complex,axiom,
% 13.25/7.71 ! [A: real,B: real] :
% 13.25/7.71 ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
% 13.25/7.71 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8851_Complex__mult__i,axiom,
% 13.25/7.71 ! [A: real,B: real] :
% 13.25/7.71 ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
% 13.25/7.71 = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8852_complex__inverse,axiom,
% 13.25/7.71 ! [A: real,B: real] :
% 13.25/7.71 ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
% 13.25/7.71 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8853_cosh__ln__real,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( cosh_real @ ( ln_ln_real @ X ) )
% 13.25/7.71 = ( divide_divide_real @ ( plus_plus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8854_cot__gt__zero,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8855_sinh__ln__real,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( sinh_real @ ( ln_ln_real @ X ) )
% 13.25/7.71 = ( divide_divide_real @ ( minus_minus_real @ X @ ( inverse_inverse_real @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8856_tan__cot_H,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) )
% 13.25/7.71 = ( cot_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8857_Arg__minus__ii,axiom,
% 13.25/7.71 ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
% 13.25/7.71 = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8858_Arg__ii,axiom,
% 13.25/7.71 ( ( arg @ imaginary_unit )
% 13.25/7.71 = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8859_cis__minus__pi__half,axiom,
% 13.25/7.71 ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8860_log__base__10__eq1,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 = ( 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 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8861_log__one,axiom,
% 13.25/7.71 ! [A: real] :
% 13.25/7.71 ( ( log @ A @ one_one_real )
% 13.25/7.71 = zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8862_norm__cis,axiom,
% 13.25/7.71 ! [A: real] :
% 13.25/7.71 ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
% 13.25/7.71 = one_one_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8863_zero__less__log__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X ) )
% 13.25/7.71 = ( ord_less_real @ one_one_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8864_log__less__zero__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ ( log @ A @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_real @ X @ one_one_real ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8865_one__less__log__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ ( log @ A @ X ) )
% 13.25/7.71 = ( ord_less_real @ A @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8866_log__less__one__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ ( log @ A @ X ) @ one_one_real )
% 13.25/7.71 = ( ord_less_real @ X @ A ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8867_log__less__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( ord_less_real @ ( log @ A @ X ) @ ( log @ A @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ X @ Y3 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8868_log__eq__one,axiom,
% 13.25/7.71 ! [A: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( log @ A @ A )
% 13.25/7.71 = one_one_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8869_zero__le__log__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X ) )
% 13.25/7.71 = ( ord_less_eq_real @ one_one_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8870_log__le__zero__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8871_one__le__log__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X ) )
% 13.25/7.71 = ( ord_less_eq_real @ A @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8872_log__le__one__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ one_one_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ A ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8873_log__le__cancel__iff,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( log @ A @ X ) @ ( log @ A @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ X @ Y3 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8874_log__pow__cancel,axiom,
% 13.25/7.71 ! [A: real,B: nat] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 13.25/7.71 = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8875_cis__pi__half,axiom,
% 13.25/7.71 ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 = imaginary_unit ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8876_cis__2pi,axiom,
% 13.25/7.71 ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 13.25/7.71 = one_one_complex ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8877_cis__divide,axiom,
% 13.25/7.71 ! [A: real,B: real] :
% 13.25/7.71 ( ( divide1717551699836669952omplex @ ( cis @ A ) @ ( cis @ B ) )
% 13.25/7.71 = ( cis @ ( minus_minus_real @ A @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8878_log__def,axiom,
% 13.25/7.71 ( log
% 13.25/7.71 = ( ^ [A3: real,X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ A3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8879_cis__mult,axiom,
% 13.25/7.71 ! [A: real,B: real] :
% 13.25/7.71 ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
% 13.25/7.71 = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8880_log__ln,axiom,
% 13.25/7.71 ( ln_ln_real
% 13.25/7.71 = ( log @ ( exp_real @ one_one_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8881_log__base__change,axiom,
% 13.25/7.71 ! [A: real,B: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( log @ B @ X )
% 13.25/7.71 = ( divide_divide_real @ ( log @ A @ X ) @ ( log @ A @ B ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8882_less__log__of__power,axiom,
% 13.25/7.71 ! [B: real,N: nat,M: real] :
% 13.25/7.71 ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8883_log__of__power__eq,axiom,
% 13.25/7.71 ! [M: nat,B: real,N: nat] :
% 13.25/7.71 ( ( ( semiri5074537144036343181t_real @ M )
% 13.25/7.71 = ( power_power_real @ B @ N ) )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( semiri5074537144036343181t_real @ N )
% 13.25/7.71 = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8884_DeMoivre,axiom,
% 13.25/7.71 ! [A: real,N: nat] :
% 13.25/7.71 ( ( power_power_complex @ ( cis @ A ) @ N )
% 13.25/7.71 = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8885_log__mult,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( log @ A @ ( times_times_real @ X @ Y3 ) )
% 13.25/7.71 = ( plus_plus_real @ ( log @ A @ X ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8886_log__divide,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( log @ A @ ( divide_divide_real @ X @ Y3 ) )
% 13.25/7.71 = ( minus_minus_real @ ( log @ A @ X ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8887_le__log__of__power,axiom,
% 13.25/7.71 ! [B: real,N: nat,M: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8888_log__base__pow,axiom,
% 13.25/7.71 ! [A: real,N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( log @ ( power_power_real @ A @ N ) @ X )
% 13.25/7.71 = ( divide_divide_real @ ( log @ A @ X ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8889_log__nat__power,axiom,
% 13.25/7.71 ! [X: real,B: real,N: nat] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( log @ B @ ( power_power_real @ X @ N ) )
% 13.25/7.71 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8890_log__inverse,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( log @ A @ ( inverse_inverse_real @ X ) )
% 13.25/7.71 = ( uminus_uminus_real @ ( log @ A @ X ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8891_log2__of__power__eq,axiom,
% 13.25/7.71 ! [M: nat,N: nat] :
% 13.25/7.71 ( ( M
% 13.25/7.71 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.71 => ( ( semiri5074537144036343181t_real @ N )
% 13.25/7.71 = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8892_log__of__power__less,axiom,
% 13.25/7.71 ! [M: nat,B: real,N: nat] :
% 13.25/7.71 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.71 => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8893_log__eq__div__ln__mult__log,axiom,
% 13.25/7.71 ! [A: real,B: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ B )
% 13.25/7.71 => ( ( B != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( log @ A @ X )
% 13.25/7.71 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8894_Arg__bounded,axiom,
% 13.25/7.71 ! [Z: complex] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 13.25/7.71 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8895_log__of__power__le,axiom,
% 13.25/7.71 ! [M: nat,B: real,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N ) )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.71 => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8896_less__log2__of__power,axiom,
% 13.25/7.71 ! [N: nat,M: nat] :
% 13.25/7.71 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 13.25/7.71 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8897_le__log2__of__power,axiom,
% 13.25/7.71 ! [N: nat,M: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 13.25/7.71 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8898_log2__of__power__less,axiom,
% 13.25/7.71 ! [M: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.71 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8899_log2__of__power__le,axiom,
% 13.25/7.71 ! [M: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.71 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8900_log__base__10__eq2,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
% 13.25/7.71 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8901_ceiling__log__nat__eq__powr__iff,axiom,
% 13.25/7.71 ! [B: nat,K: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.25/7.71 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 13.25/7.71 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 13.25/7.71 = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 13.25/7.71 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8902_bij__betw__roots__unity,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( bij_betw_nat_complex
% 13.25/7.71 @ ^ [K2: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 13.25/7.71 @ ( set_ord_lessThan_nat @ N )
% 13.25/7.71 @ ( collect_complex
% 13.25/7.71 @ ^ [Z2: complex] :
% 13.25/7.71 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.71 = one_one_complex ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8903_ceiling__log__nat__eq__if,axiom,
% 13.25/7.71 ! [B: nat,N: nat,K: nat] :
% 13.25/7.71 ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K )
% 13.25/7.71 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 13.25/7.71 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 13.25/7.71 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 13.25/7.71 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8904_ceiling__log2__div2,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.71 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 13.25/7.71 = ( 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 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8905_ceiling__divide__eq__div__numeral,axiom,
% 13.25/7.71 ! [A: num,B: num] :
% 13.25/7.71 ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 13.25/7.71 = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8906_ceiling__minus__divide__eq__div__numeral,axiom,
% 13.25/7.71 ! [A: num,B: num] :
% 13.25/7.71 ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 13.25/7.71 = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8907_ceiling__log__eq__powr__iff,axiom,
% 13.25/7.71 ! [X: real,B: real,K: nat] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X ) )
% 13.25/7.71 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 13.25/7.71 = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X )
% 13.25/7.71 & ( ord_less_eq_real @ X @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8908_floor__log__nat__eq__powr__iff,axiom,
% 13.25/7.71 ! [B: nat,K: nat,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 13.25/7.71 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 13.25/7.71 = ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.71 = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 13.25/7.71 & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8909_floor__log__nat__eq__if,axiom,
% 13.25/7.71 ! [B: nat,N: nat,K: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K )
% 13.25/7.71 => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 13.25/7.71 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 13.25/7.71 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 13.25/7.71 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8910_arctan__def,axiom,
% 13.25/7.71 ( arctan
% 13.25/7.71 = ( ^ [Y: real] :
% 13.25/7.71 ( the_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 13.25/7.71 & ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ( tan_real @ X2 )
% 13.25/7.71 = Y ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8911_powr__gt__zero,axiom,
% 13.25/7.71 ! [X: real,A: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X @ A ) )
% 13.25/7.71 = ( X != zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8912_powr__nonneg__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( powr_real @ A @ X ) @ zero_zero_real )
% 13.25/7.71 = ( A = zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8913_powr__less__cancel__iff,axiom,
% 13.25/7.71 ! [X: real,A: real,B: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 13.25/7.71 = ( ord_less_real @ A @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8914_powr__eq__one__iff,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ A )
% 13.25/7.71 => ( ( ( powr_real @ A @ X )
% 13.25/7.71 = one_one_real )
% 13.25/7.71 = ( X = zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8915_powr__one__gt__zero__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ( powr_real @ X @ one_one_real )
% 13.25/7.71 = X )
% 13.25/7.71 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8916_powr__one,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( powr_real @ X @ one_one_real )
% 13.25/7.71 = X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8917_powr__le__cancel__iff,axiom,
% 13.25/7.71 ! [X: real,A: real,B: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 13.25/7.71 = ( ord_less_eq_real @ A @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8918_numeral__powr__numeral__real,axiom,
% 13.25/7.71 ! [M: num,N: num] :
% 13.25/7.71 ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 13.25/7.71 = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8919_log__powr__cancel,axiom,
% 13.25/7.71 ! [A: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 13.25/7.71 = Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8920_powr__log__cancel,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( powr_real @ A @ ( log @ A @ X ) )
% 13.25/7.71 = X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8921_floor__divide__eq__div__numeral,axiom,
% 13.25/7.71 ! [A: num,B: num] :
% 13.25/7.71 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
% 13.25/7.71 = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8922_powr__numeral,axiom,
% 13.25/7.71 ! [X: real,N: num] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( powr_real @ X @ ( numeral_numeral_real @ N ) )
% 13.25/7.71 = ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8923_floor__one__divide__eq__div__numeral,axiom,
% 13.25/7.71 ! [B: num] :
% 13.25/7.71 ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
% 13.25/7.71 = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8924_floor__minus__divide__eq__div__numeral,axiom,
% 13.25/7.71 ! [A: num,B: num] :
% 13.25/7.71 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
% 13.25/7.71 = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8925_square__powr__half,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( powr_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 = ( abs_abs_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8926_floor__minus__one__divide__eq__div__numeral,axiom,
% 13.25/7.71 ! [B: num] :
% 13.25/7.71 ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
% 13.25/7.71 = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8927_powr__powr,axiom,
% 13.25/7.71 ! [X: real,A: real,B: real] :
% 13.25/7.71 ( ( powr_real @ ( powr_real @ X @ A ) @ B )
% 13.25/7.71 = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8928_powr__less__mono2__neg,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ A @ zero_zero_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8929_powr__non__neg,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ~ ( ord_less_real @ ( powr_real @ A @ X ) @ zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8930_powr__mono2,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8931_powr__ge__pzero,axiom,
% 13.25/7.71 ! [X: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8932_powr__less__mono,axiom,
% 13.25/7.71 ! [A: real,B: real,X: real] :
% 13.25/7.71 ( ( ord_less_real @ A @ B )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.71 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8933_powr__less__cancel,axiom,
% 13.25/7.71 ! [X: real,A: real,B: real] :
% 13.25/7.71 ( ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.71 => ( ord_less_real @ A @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8934_powr__mono,axiom,
% 13.25/7.71 ! [A: real,B: real,X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.71 => ( ( ord_less_eq_real @ one_one_real @ X )
% 13.25/7.71 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ X @ B ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8935_powr__mono2_H,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ A @ zero_zero_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X @ A ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8936_powr__less__mono2,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8937_gr__one__powr,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ord_less_real @ one_one_real @ ( powr_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8938_powr__inj,axiom,
% 13.25/7.71 ! [A: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( A != one_one_real )
% 13.25/7.71 => ( ( ( powr_real @ A @ X )
% 13.25/7.71 = ( powr_real @ A @ Y3 ) )
% 13.25/7.71 = ( X = Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8939_ge__one__powr__ge__zero,axiom,
% 13.25/7.71 ! [X: real,A: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ one_one_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X @ A ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8940_powr__mono__both,axiom,
% 13.25/7.71 ! [A: real,B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( ord_less_eq_real @ A @ B )
% 13.25/7.71 => ( ( ord_less_eq_real @ one_one_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ B ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8941_powr__le1,axiom,
% 13.25/7.71 ! [A: real,X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.71 => ( ord_less_eq_real @ ( powr_real @ X @ A ) @ one_one_real ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8942_powr__divide,axiom,
% 13.25/7.71 ! [X: real,Y3: real,A: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( powr_real @ ( divide_divide_real @ X @ Y3 ) @ A )
% 13.25/7.71 = ( divide_divide_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8943_powr__mult,axiom,
% 13.25/7.71 ! [X: real,Y3: real,A: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( powr_real @ ( times_times_real @ X @ Y3 ) @ A )
% 13.25/7.71 = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8944_inverse__powr,axiom,
% 13.25/7.71 ! [Y3: real,A: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.71 => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A )
% 13.25/7.71 = ( inverse_inverse_real @ ( powr_real @ Y3 @ A ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8945_divide__powr__uminus,axiom,
% 13.25/7.71 ! [A: real,B: real,C: real] :
% 13.25/7.71 ( ( divide_divide_real @ A @ ( powr_real @ B @ C ) )
% 13.25/7.71 = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8946_log__base__powr,axiom,
% 13.25/7.71 ! [A: real,B: real,X: real] :
% 13.25/7.71 ( ( A != zero_zero_real )
% 13.25/7.71 => ( ( log @ ( powr_real @ A @ B ) @ X )
% 13.25/7.71 = ( divide_divide_real @ ( log @ A @ X ) @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8947_ln__powr,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( X != zero_zero_real )
% 13.25/7.71 => ( ( ln_ln_real @ ( powr_real @ X @ Y3 ) )
% 13.25/7.71 = ( times_times_real @ Y3 @ ( ln_ln_real @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8948_log__powr,axiom,
% 13.25/7.71 ! [X: real,B: real,Y3: real] :
% 13.25/7.71 ( ( X != zero_zero_real )
% 13.25/7.71 => ( ( log @ B @ ( powr_real @ X @ Y3 ) )
% 13.25/7.71 = ( times_times_real @ Y3 @ ( log @ B @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8949_ln__neg__is__const,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.25/7.71 => ( ( ln_ln_real @ X )
% 13.25/7.71 = ( the_real
% 13.25/7.71 @ ^ [X2: real] : $false ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8950_floor__log__eq__powr__iff,axiom,
% 13.25/7.71 ! [X: real,B: real,K: int] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X ) )
% 13.25/7.71 = K )
% 13.25/7.71 = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X )
% 13.25/7.71 & ( ord_less_real @ X @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8951_powr__realpow,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( powr_real @ X @ ( semiri5074537144036343181t_real @ N ) )
% 13.25/7.71 = ( power_power_real @ X @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8952_powr__less__iff,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X )
% 13.25/7.71 = ( ord_less_real @ Y3 @ ( log @ B @ X ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8953_less__powr__iff,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( powr_real @ B @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ ( log @ B @ X ) @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8954_log__less__iff,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ ( log @ B @ X ) @ Y3 )
% 13.25/7.71 = ( ord_less_real @ X @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8955_less__log__iff,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ Y3 @ ( log @ B @ X ) )
% 13.25/7.71 = ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8956_floor__eq,axiom,
% 13.25/7.71 ! [N: int,X: real] :
% 13.25/7.71 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 13.25/7.71 => ( ( archim6058952711729229775r_real @ X )
% 13.25/7.71 = N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8957_real__of__int__floor__add__one__gt,axiom,
% 13.25/7.71 ! [R: real] : ( ord_less_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8958_real__of__int__floor__add__one__ge,axiom,
% 13.25/7.71 ! [R: real] : ( ord_less_eq_real @ R @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) @ one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8959_real__of__int__floor__gt__diff__one,axiom,
% 13.25/7.71 ! [R: real] : ( ord_less_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8960_real__of__int__floor__ge__diff__one,axiom,
% 13.25/7.71 ! [R: real] : ( ord_less_eq_real @ ( minus_minus_real @ R @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8961_powr__neg__one,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( powr_real @ X @ ( uminus_uminus_real @ one_one_real ) )
% 13.25/7.71 = ( divide_divide_real @ one_one_real @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8962_powr__mult__base,axiom,
% 13.25/7.71 ! [X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( times_times_real @ X @ ( powr_real @ X @ Y3 ) )
% 13.25/7.71 = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8963_le__log__iff,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ Y3 @ ( log @ B @ X ) )
% 13.25/7.71 = ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8964_log__le__iff,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( log @ B @ X ) @ Y3 )
% 13.25/7.71 = ( ord_less_eq_real @ X @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8965_le__powr__iff,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ ( powr_real @ B @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ ( log @ B @ X ) @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8966_powr__le__iff,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X )
% 13.25/7.71 = ( ord_less_eq_real @ Y3 @ ( log @ B @ X ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8967_floor__eq2,axiom,
% 13.25/7.71 ! [N: int,X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X )
% 13.25/7.71 => ( ( ord_less_real @ X @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 13.25/7.71 => ( ( archim6058952711729229775r_real @ X )
% 13.25/7.71 = N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8968_floor__divide__real__eq__div,axiom,
% 13.25/7.71 ! [B: int,A: real] :
% 13.25/7.71 ( ( ord_less_eq_int @ zero_zero_int @ B )
% 13.25/7.71 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
% 13.25/7.71 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8969_ln__powr__bound,axiom,
% 13.25/7.71 ! [X: real,A: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ one_one_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( divide_divide_real @ ( powr_real @ X @ A ) @ A ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8970_ln__powr__bound2,axiom,
% 13.25/7.71 ! [X: real,A: real] :
% 13.25/7.71 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.71 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8971_log__add__eq__powr,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ B )
% 13.25/7.71 => ( ( B != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( plus_plus_real @ ( log @ B @ X ) @ Y3 )
% 13.25/7.71 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8972_add__log__eq__powr,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ B )
% 13.25/7.71 => ( ( B != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( plus_plus_real @ Y3 @ ( log @ B @ X ) )
% 13.25/7.71 = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y3 ) @ X ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8973_minus__log__eq__powr,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ B )
% 13.25/7.71 => ( ( B != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( minus_minus_real @ Y3 @ ( log @ B @ X ) )
% 13.25/7.71 = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y3 ) @ X ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8974_arccos__def,axiom,
% 13.25/7.71 ( arccos
% 13.25/7.71 = ( ^ [Y: real] :
% 13.25/7.71 ( the_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 13.25/7.71 & ( ord_less_eq_real @ X2 @ pi )
% 13.25/7.71 & ( ( cos_real @ X2 )
% 13.25/7.71 = Y ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8975_log__minus__eq__powr,axiom,
% 13.25/7.71 ! [B: real,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ B )
% 13.25/7.71 => ( ( B != one_one_real )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( minus_minus_real @ ( log @ B @ X ) @ Y3 )
% 13.25/7.71 = ( log @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8976_powr__half__sqrt,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 = ( sqrt @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8977_powr__neg__numeral,axiom,
% 13.25/7.71 ! [X: real,N: num] :
% 13.25/7.71 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( powr_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 13.25/7.71 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8978_pi__half,axiom,
% 13.25/7.71 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.71 = ( the_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 13.25/7.71 & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.71 & ( ( cos_real @ X2 )
% 13.25/7.71 = zero_zero_real ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8979_pi__def,axiom,
% 13.25/7.71 ( pi
% 13.25/7.71 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 13.25/7.71 @ ( the_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 13.25/7.71 & ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 13.25/7.71 & ( ( cos_real @ X2 )
% 13.25/7.71 = zero_zero_real ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8980_floor__log2__div2,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.71 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 13.25/7.71 = ( 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 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8981_arcsin__def,axiom,
% 13.25/7.71 ( arcsin
% 13.25/7.71 = ( ^ [Y: real] :
% 13.25/7.71 ( the_real
% 13.25/7.71 @ ^ [X2: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 13.25/7.71 & ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.71 & ( ( sin_real @ X2 )
% 13.25/7.71 = Y ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8982_modulo__int__unfold,axiom,
% 13.25/7.71 ! [L2: int,K: int,N: nat,M: nat] :
% 13.25/7.71 ( ( ( ( ( sgn_sgn_int @ L2 )
% 13.25/7.71 = zero_zero_int )
% 13.25/7.71 | ( ( sgn_sgn_int @ K )
% 13.25/7.71 = zero_zero_int )
% 13.25/7.71 | ( N = zero_zero_nat ) )
% 13.25/7.71 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 13.25/7.71 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 13.25/7.71 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 13.25/7.71 = zero_zero_int )
% 13.25/7.71 | ( ( sgn_sgn_int @ K )
% 13.25/7.71 = zero_zero_int )
% 13.25/7.71 | ( N = zero_zero_nat ) )
% 13.25/7.71 => ( ( ( ( sgn_sgn_int @ K )
% 13.25/7.71 = ( sgn_sgn_int @ L2 ) )
% 13.25/7.71 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 13.25/7.71 = ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 13.25/7.71 & ( ( ( sgn_sgn_int @ K )
% 13.25/7.71 != ( sgn_sgn_int @ L2 ) )
% 13.25/7.71 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 13.25/7.71 = ( times_times_int @ ( sgn_sgn_int @ L2 )
% 13.25/7.71 @ ( minus_minus_int
% 13.25/7.71 @ ( semiri1314217659103216013at_int
% 13.25/7.71 @ ( times_times_nat @ N
% 13.25/7.71 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.71 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 13.25/7.71 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8983_sgn__mult__dvd__iff,axiom,
% 13.25/7.71 ! [R: int,L2: int,K: int] :
% 13.25/7.71 ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R ) @ L2 ) @ K )
% 13.25/7.71 = ( ( dvd_dvd_int @ L2 @ K )
% 13.25/7.71 & ( ( R = zero_zero_int )
% 13.25/7.71 => ( K = zero_zero_int ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8984_mult__sgn__dvd__iff,axiom,
% 13.25/7.71 ! [L2: int,R: int,K: int] :
% 13.25/7.71 ( ( dvd_dvd_int @ ( times_times_int @ L2 @ ( sgn_sgn_int @ R ) ) @ K )
% 13.25/7.71 = ( ( dvd_dvd_int @ L2 @ K )
% 13.25/7.71 & ( ( R = zero_zero_int )
% 13.25/7.71 => ( K = zero_zero_int ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8985_dvd__sgn__mult__iff,axiom,
% 13.25/7.71 ! [L2: int,R: int,K: int] :
% 13.25/7.71 ( ( dvd_dvd_int @ L2 @ ( times_times_int @ ( sgn_sgn_int @ R ) @ K ) )
% 13.25/7.71 = ( ( dvd_dvd_int @ L2 @ K )
% 13.25/7.71 | ( R = zero_zero_int ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8986_dvd__mult__sgn__iff,axiom,
% 13.25/7.71 ! [L2: int,K: int,R: int] :
% 13.25/7.71 ( ( dvd_dvd_int @ L2 @ ( times_times_int @ K @ ( sgn_sgn_int @ R ) ) )
% 13.25/7.71 = ( ( dvd_dvd_int @ L2 @ K )
% 13.25/7.71 | ( R = zero_zero_int ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8987_exp__pi__i_H,axiom,
% 13.25/7.71 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
% 13.25/7.71 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8988_exp__pi__i,axiom,
% 13.25/7.71 ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
% 13.25/7.71 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8989_exp__two__pi__i,axiom,
% 13.25/7.71 ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
% 13.25/7.71 = one_one_complex ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8990_exp__two__pi__i_H,axiom,
% 13.25/7.71 ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
% 13.25/7.71 = one_one_complex ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8991_complex__exp__exists,axiom,
% 13.25/7.71 ! [Z: complex] :
% 13.25/7.71 ? [A5: complex,R2: real] :
% 13.25/7.71 ( Z
% 13.25/7.71 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( exp_complex @ A5 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8992_int__sgnE,axiom,
% 13.25/7.71 ! [K: int] :
% 13.25/7.71 ~ ! [N4: nat,L4: int] :
% 13.25/7.71 ( K
% 13.25/7.71 != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N4 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8993_div__eq__sgn__abs,axiom,
% 13.25/7.71 ! [K: int,L2: int] :
% 13.25/7.71 ( ( ( sgn_sgn_int @ K )
% 13.25/7.71 = ( sgn_sgn_int @ L2 ) )
% 13.25/7.71 => ( ( divide_divide_int @ K @ L2 )
% 13.25/7.71 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8994_sgn__mod,axiom,
% 13.25/7.71 ! [L2: int,K: int] :
% 13.25/7.71 ( ( L2 != zero_zero_int )
% 13.25/7.71 => ( ~ ( dvd_dvd_int @ L2 @ K )
% 13.25/7.71 => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L2 ) )
% 13.25/7.71 = ( sgn_sgn_int @ L2 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8995_Complex__mult__complex__of__real,axiom,
% 13.25/7.71 ! [X: real,Y3: real,R: real] :
% 13.25/7.71 ( ( times_times_complex @ ( complex2 @ X @ Y3 ) @ ( real_V4546457046886955230omplex @ R ) )
% 13.25/7.71 = ( complex2 @ ( times_times_real @ X @ R ) @ ( times_times_real @ Y3 @ R ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8996_complex__of__real__mult__Complex,axiom,
% 13.25/7.71 ! [R: real,X: real,Y3: real] :
% 13.25/7.71 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y3 ) )
% 13.25/7.71 = ( complex2 @ ( times_times_real @ R @ X ) @ ( times_times_real @ R @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8997_Complex__add__complex__of__real,axiom,
% 13.25/7.71 ! [X: real,Y3: real,R: real] :
% 13.25/7.71 ( ( plus_plus_complex @ ( complex2 @ X @ Y3 ) @ ( real_V4546457046886955230omplex @ R ) )
% 13.25/7.71 = ( complex2 @ ( plus_plus_real @ X @ R ) @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8998_complex__of__real__add__Complex,axiom,
% 13.25/7.71 ! [R: real,X: real,Y3: real] :
% 13.25/7.71 ( ( plus_plus_complex @ ( real_V4546457046886955230omplex @ R ) @ ( complex2 @ X @ Y3 ) )
% 13.25/7.71 = ( complex2 @ ( plus_plus_real @ R @ X ) @ Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_8999_cis__conv__exp,axiom,
% 13.25/7.71 ( cis
% 13.25/7.71 = ( ^ [B4: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9000_zsgn__def,axiom,
% 13.25/7.71 ( sgn_sgn_int
% 13.25/7.71 = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9001_div__sgn__abs__cancel,axiom,
% 13.25/7.71 ! [V: int,K: int,L2: int] :
% 13.25/7.71 ( ( V != zero_zero_int )
% 13.25/7.71 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L2 ) ) )
% 13.25/7.71 = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9002_div__dvd__sgn__abs,axiom,
% 13.25/7.71 ! [L2: int,K: int] :
% 13.25/7.71 ( ( dvd_dvd_int @ L2 @ K )
% 13.25/7.71 => ( ( divide_divide_int @ K @ L2 )
% 13.25/7.71 = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L2 ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9003_i__complex__of__real,axiom,
% 13.25/7.71 ! [R: real] :
% 13.25/7.71 ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R ) )
% 13.25/7.71 = ( complex2 @ zero_zero_real @ R ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9004_complex__of__real__i,axiom,
% 13.25/7.71 ! [R: real] :
% 13.25/7.71 ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ imaginary_unit )
% 13.25/7.71 = ( complex2 @ zero_zero_real @ R ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9005_Complex__eq,axiom,
% 13.25/7.71 ( complex2
% 13.25/7.71 = ( ^ [A3: real,B4: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A3 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9006_complex__split__polar,axiom,
% 13.25/7.71 ! [Z: complex] :
% 13.25/7.71 ? [R2: real,A5: real] :
% 13.25/7.71 ( Z
% 13.25/7.71 = ( times_times_complex @ ( real_V4546457046886955230omplex @ R2 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A5 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9007_eucl__rel__int__remainderI,axiom,
% 13.25/7.71 ! [R: int,L2: int,K: int,Q2: int] :
% 13.25/7.71 ( ( ( sgn_sgn_int @ R )
% 13.25/7.71 = ( sgn_sgn_int @ L2 ) )
% 13.25/7.71 => ( ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L2 ) )
% 13.25/7.71 => ( ( K
% 13.25/7.71 = ( plus_plus_int @ ( times_times_int @ Q2 @ L2 ) @ R ) )
% 13.25/7.71 => ( eucl_rel_int @ K @ L2 @ ( product_Pair_int_int @ Q2 @ R ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9008_eucl__rel__int_Osimps,axiom,
% 13.25/7.71 ( eucl_rel_int
% 13.25/7.71 = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
% 13.25/7.71 ( ? [K2: int] :
% 13.25/7.71 ( ( A12 = K2 )
% 13.25/7.71 & ( A23 = zero_zero_int )
% 13.25/7.71 & ( A32
% 13.25/7.71 = ( product_Pair_int_int @ zero_zero_int @ K2 ) ) )
% 13.25/7.71 | ? [L: int,K2: int,Q5: int] :
% 13.25/7.71 ( ( A12 = K2 )
% 13.25/7.71 & ( A23 = L )
% 13.25/7.71 & ( A32
% 13.25/7.71 = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
% 13.25/7.71 & ( L != zero_zero_int )
% 13.25/7.71 & ( K2
% 13.25/7.71 = ( times_times_int @ Q5 @ L ) ) )
% 13.25/7.71 | ? [R5: int,L: int,K2: int,Q5: int] :
% 13.25/7.71 ( ( A12 = K2 )
% 13.25/7.71 & ( A23 = L )
% 13.25/7.71 & ( A32
% 13.25/7.71 = ( product_Pair_int_int @ Q5 @ R5 ) )
% 13.25/7.71 & ( ( sgn_sgn_int @ R5 )
% 13.25/7.71 = ( sgn_sgn_int @ L ) )
% 13.25/7.71 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L ) )
% 13.25/7.71 & ( K2
% 13.25/7.71 = ( plus_plus_int @ ( times_times_int @ Q5 @ L ) @ R5 ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9009_eucl__rel__int_Ocases,axiom,
% 13.25/7.71 ! [A1: int,A22: int,A33: product_prod_int_int] :
% 13.25/7.71 ( ( eucl_rel_int @ A1 @ A22 @ A33 )
% 13.25/7.71 => ( ( ( A22 = zero_zero_int )
% 13.25/7.71 => ( A33
% 13.25/7.71 != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
% 13.25/7.71 => ( ! [Q3: int] :
% 13.25/7.71 ( ( A33
% 13.25/7.71 = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
% 13.25/7.71 => ( ( A22 != zero_zero_int )
% 13.25/7.71 => ( A1
% 13.25/7.71 != ( times_times_int @ Q3 @ A22 ) ) ) )
% 13.25/7.71 => ~ ! [R2: int,Q3: int] :
% 13.25/7.71 ( ( A33
% 13.25/7.71 = ( product_Pair_int_int @ Q3 @ R2 ) )
% 13.25/7.71 => ( ( ( sgn_sgn_int @ R2 )
% 13.25/7.71 = ( sgn_sgn_int @ A22 ) )
% 13.25/7.71 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A22 ) )
% 13.25/7.71 => ( A1
% 13.25/7.71 != ( plus_plus_int @ ( times_times_int @ Q3 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9010_div__noneq__sgn__abs,axiom,
% 13.25/7.71 ! [L2: int,K: int] :
% 13.25/7.71 ( ( L2 != zero_zero_int )
% 13.25/7.71 => ( ( ( sgn_sgn_int @ K )
% 13.25/7.71 != ( sgn_sgn_int @ L2 ) )
% 13.25/7.71 => ( ( divide_divide_int @ K @ L2 )
% 13.25/7.71 = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) ) )
% 13.25/7.71 @ ( zero_n2684676970156552555ol_int
% 13.25/7.71 @ ~ ( dvd_dvd_int @ L2 @ K ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9011_cmod__unit__one,axiom,
% 13.25/7.71 ! [A: real] :
% 13.25/7.71 ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
% 13.25/7.71 = one_one_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9012_cmod__complex__polar,axiom,
% 13.25/7.71 ! [R: real,A: real] :
% 13.25/7.71 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
% 13.25/7.71 = ( abs_abs_real @ R ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9013_divide__int__unfold,axiom,
% 13.25/7.71 ! [L2: int,K: int,N: nat,M: nat] :
% 13.25/7.71 ( ( ( ( ( sgn_sgn_int @ L2 )
% 13.25/7.71 = zero_zero_int )
% 13.25/7.71 | ( ( sgn_sgn_int @ K )
% 13.25/7.71 = zero_zero_int )
% 13.25/7.71 | ( N = zero_zero_nat ) )
% 13.25/7.71 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 13.25/7.71 = zero_zero_int ) )
% 13.25/7.71 & ( ~ ( ( ( sgn_sgn_int @ L2 )
% 13.25/7.71 = zero_zero_int )
% 13.25/7.71 | ( ( sgn_sgn_int @ K )
% 13.25/7.71 = zero_zero_int )
% 13.25/7.71 | ( N = zero_zero_nat ) )
% 13.25/7.71 => ( ( ( ( sgn_sgn_int @ K )
% 13.25/7.71 = ( sgn_sgn_int @ L2 ) )
% 13.25/7.71 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 13.25/7.71 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 13.25/7.71 & ( ( ( sgn_sgn_int @ K )
% 13.25/7.71 != ( sgn_sgn_int @ L2 ) )
% 13.25/7.71 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 13.25/7.71 = ( uminus_uminus_int
% 13.25/7.71 @ ( semiri1314217659103216013at_int
% 13.25/7.71 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 13.25/7.71 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.71 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9014_csqrt__ii,axiom,
% 13.25/7.71 ( ( csqrt @ imaginary_unit )
% 13.25/7.71 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9015_bij__betw__nth__root__unity,axiom,
% 13.25/7.71 ! [C: complex,N: nat] :
% 13.25/7.71 ( ( C != zero_zero_complex )
% 13.25/7.71 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 13.25/7.71 @ ( collect_complex
% 13.25/7.71 @ ^ [Z2: complex] :
% 13.25/7.71 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.71 = one_one_complex ) )
% 13.25/7.71 @ ( collect_complex
% 13.25/7.71 @ ^ [Z2: complex] :
% 13.25/7.71 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.71 = C ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9016_modulo__int__def,axiom,
% 13.25/7.71 ( modulo_modulo_int
% 13.25/7.71 = ( ^ [K2: int,L: int] :
% 13.25/7.71 ( if_int @ ( L = zero_zero_int ) @ K2
% 13.25/7.71 @ ( if_int
% 13.25/7.71 @ ( ( sgn_sgn_int @ K2 )
% 13.25/7.71 = ( sgn_sgn_int @ L ) )
% 13.25/7.71 @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) )
% 13.25/7.71 @ ( times_times_int @ ( sgn_sgn_int @ L )
% 13.25/7.71 @ ( minus_minus_int
% 13.25/7.71 @ ( times_times_int @ ( abs_abs_int @ L )
% 13.25/7.71 @ ( zero_n2684676970156552555ol_int
% 13.25/7.71 @ ~ ( dvd_dvd_int @ L @ K2 ) ) )
% 13.25/7.71 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9017_divide__int__def,axiom,
% 13.25/7.71 ( divide_divide_int
% 13.25/7.71 = ( ^ [K2: int,L: int] :
% 13.25/7.71 ( if_int @ ( L = zero_zero_int ) @ zero_zero_int
% 13.25/7.71 @ ( if_int
% 13.25/7.71 @ ( ( sgn_sgn_int @ K2 )
% 13.25/7.71 = ( sgn_sgn_int @ L ) )
% 13.25/7.71 @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) )
% 13.25/7.71 @ ( uminus_uminus_int
% 13.25/7.71 @ ( semiri1314217659103216013at_int
% 13.25/7.71 @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) )
% 13.25/7.71 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.71 @ ~ ( dvd_dvd_int @ L @ K2 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9018_sgn__le__0__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9019_zero__le__sgn__iff,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X ) )
% 13.25/7.71 = ( ord_less_eq_real @ zero_zero_real @ X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9020_real__root__zero,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( root @ N @ zero_zero_real )
% 13.25/7.71 = zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9021_nat__numeral,axiom,
% 13.25/7.71 ! [K: num] :
% 13.25/7.71 ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 13.25/7.71 = ( numeral_numeral_nat @ K ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9022_real__root__Suc__0,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( root @ ( suc @ zero_zero_nat ) @ X )
% 13.25/7.71 = X ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9023_root__0,axiom,
% 13.25/7.71 ! [X: real] :
% 13.25/7.71 ( ( root @ zero_zero_nat @ X )
% 13.25/7.71 = zero_zero_real ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9024_real__root__eq__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ( root @ N @ X )
% 13.25/7.71 = ( root @ N @ Y3 ) )
% 13.25/7.71 = ( X = Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9025_real__root__eq__0__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ( root @ N @ X )
% 13.25/7.71 = zero_zero_real )
% 13.25/7.71 = ( X = zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9026_real__root__less__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ X @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9027_nat__0__iff,axiom,
% 13.25/7.71 ! [I: int] :
% 13.25/7.71 ( ( ( nat2 @ I )
% 13.25/7.71 = zero_zero_nat )
% 13.25/7.71 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9028_nat__le__0,axiom,
% 13.25/7.71 ! [Z: int] :
% 13.25/7.71 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 13.25/7.71 => ( ( nat2 @ Z )
% 13.25/7.71 = zero_zero_nat ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9029_real__root__le__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ X @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9030_zless__nat__conj,axiom,
% 13.25/7.71 ! [W: int,Z: int] :
% 13.25/7.71 ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 13.25/7.71 = ( ( ord_less_int @ zero_zero_int @ Z )
% 13.25/7.71 & ( ord_less_int @ W @ Z ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9031_real__root__one,axiom,
% 13.25/7.71 ! [N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( root @ N @ one_one_real )
% 13.25/7.71 = one_one_real ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9032_real__root__eq__1__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ( root @ N @ X )
% 13.25/7.71 = one_one_real )
% 13.25/7.71 = ( X = one_one_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9033_nat__neg__numeral,axiom,
% 13.25/7.71 ! [K: num] :
% 13.25/7.71 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.71 = zero_zero_nat ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9034_int__nat__eq,axiom,
% 13.25/7.71 ! [Z: int] :
% 13.25/7.71 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.71 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 13.25/7.71 = Z ) )
% 13.25/7.71 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.71 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 13.25/7.71 = zero_zero_int ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9035_real__root__lt__0__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9036_real__root__gt__0__iff,axiom,
% 13.25/7.71 ! [N: nat,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9037_real__root__le__0__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9038_real__root__ge__0__iff,axiom,
% 13.25/7.71 ! [N: nat,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9039_zero__less__nat__eq,axiom,
% 13.25/7.71 ! [Z: int] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 13.25/7.71 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9040_real__root__gt__1__iff,axiom,
% 13.25/7.71 ! [N: nat,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 13.25/7.71 = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9041_real__root__lt__1__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
% 13.25/7.71 = ( ord_less_real @ X @ one_one_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9042_real__root__le__1__iff,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
% 13.25/7.71 = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9043_real__root__ge__1__iff,axiom,
% 13.25/7.71 ! [N: nat,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 13.25/7.71 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9044_diff__nat__numeral,axiom,
% 13.25/7.71 ! [V: num,V3: num] :
% 13.25/7.71 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 13.25/7.71 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9045_nat__eq__numeral__power__cancel__iff,axiom,
% 13.25/7.71 ! [Y3: int,X: num,N: nat] :
% 13.25/7.71 ( ( ( nat2 @ Y3 )
% 13.25/7.71 = ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 13.25/7.71 = ( Y3
% 13.25/7.71 = ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9046_numeral__power__eq__nat__cancel__iff,axiom,
% 13.25/7.71 ! [X: num,N: nat,Y3: int] :
% 13.25/7.71 ( ( ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N )
% 13.25/7.71 = ( nat2 @ Y3 ) )
% 13.25/7.71 = ( ( power_power_int @ ( numeral_numeral_int @ X ) @ N )
% 13.25/7.71 = Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9047_power2__csqrt,axiom,
% 13.25/7.71 ! [Z: complex] :
% 13.25/7.71 ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.71 = Z ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9048_nat__ceiling__le__eq,axiom,
% 13.25/7.71 ! [X: real,A: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
% 13.25/7.71 = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9049_one__less__nat__eq,axiom,
% 13.25/7.71 ! [Z: int] :
% 13.25/7.71 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 13.25/7.71 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9050_real__root__pow__pos2,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 13.25/7.71 = X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9051_nat__numeral__diff__1,axiom,
% 13.25/7.71 ! [V: num] :
% 13.25/7.71 ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 13.25/7.71 = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9052_numeral__power__less__nat__cancel__iff,axiom,
% 13.25/7.71 ! [X: num,N: nat,A: int] :
% 13.25/7.71 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 13.25/7.71 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9053_nat__less__numeral__power__cancel__iff,axiom,
% 13.25/7.71 ! [A: int,X: num,N: nat] :
% 13.25/7.71 ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 13.25/7.71 = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9054_nat__le__numeral__power__cancel__iff,axiom,
% 13.25/7.71 ! [A: int,X: num,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
% 13.25/7.71 = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9055_numeral__power__le__nat__cancel__iff,axiom,
% 13.25/7.71 ! [X: num,N: nat,A: int] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
% 13.25/7.71 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9056_sgn__root,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( sgn_sgn_real @ ( root @ N @ X ) )
% 13.25/7.71 = ( sgn_sgn_real @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9057_real__root__commute,axiom,
% 13.25/7.71 ! [M: nat,N: nat,X: real] :
% 13.25/7.71 ( ( root @ M @ ( root @ N @ X ) )
% 13.25/7.71 = ( root @ N @ ( root @ M @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9058_real__root__mult__exp,axiom,
% 13.25/7.71 ! [M: nat,N: nat,X: real] :
% 13.25/7.71 ( ( root @ ( times_times_nat @ M @ N ) @ X )
% 13.25/7.71 = ( root @ M @ ( root @ N @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9059_real__root__mult,axiom,
% 13.25/7.71 ! [N: nat,X: real,Y3: real] :
% 13.25/7.71 ( ( root @ N @ ( times_times_real @ X @ Y3 ) )
% 13.25/7.71 = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9060_real__root__minus,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( root @ N @ ( uminus_uminus_real @ X ) )
% 13.25/7.71 = ( uminus_uminus_real @ ( root @ N @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9061_real__root__divide,axiom,
% 13.25/7.71 ! [N: nat,X: real,Y3: real] :
% 13.25/7.71 ( ( root @ N @ ( divide_divide_real @ X @ Y3 ) )
% 13.25/7.71 = ( divide_divide_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9062_real__root__inverse,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( root @ N @ ( inverse_inverse_real @ X ) )
% 13.25/7.71 = ( inverse_inverse_real @ ( root @ N @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9063_real__root__pos__pos__le,axiom,
% 13.25/7.71 ! [X: real,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.71 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9064_nat__zero__as__int,axiom,
% 13.25/7.71 ( zero_zero_nat
% 13.25/7.71 = ( nat2 @ zero_zero_int ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9065_nat__numeral__as__int,axiom,
% 13.25/7.71 ( numeral_numeral_nat
% 13.25/7.71 = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9066_real__sgn__eq,axiom,
% 13.25/7.71 ( sgn_sgn_real
% 13.25/7.71 = ( ^ [X2: real] : ( divide_divide_real @ X2 @ ( abs_abs_real @ X2 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9067_nat__mono,axiom,
% 13.25/7.71 ! [X: int,Y3: int] :
% 13.25/7.71 ( ( ord_less_eq_int @ X @ Y3 )
% 13.25/7.71 => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9068_eq__nat__nat__iff,axiom,
% 13.25/7.71 ! [Z: int,Z6: int] :
% 13.25/7.71 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.71 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 13.25/7.71 => ( ( ( nat2 @ Z )
% 13.25/7.71 = ( nat2 @ Z6 ) )
% 13.25/7.71 = ( Z = Z6 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9069_all__nat,axiom,
% 13.25/7.71 ( ( ^ [P2: nat > $o] :
% 13.25/7.71 ! [X4: nat] : ( P2 @ X4 ) )
% 13.25/7.71 = ( ^ [P3: nat > $o] :
% 13.25/7.71 ! [X2: int] :
% 13.25/7.71 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 13.25/7.71 => ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9070_ex__nat,axiom,
% 13.25/7.71 ( ( ^ [P2: nat > $o] :
% 13.25/7.71 ? [X4: nat] : ( P2 @ X4 ) )
% 13.25/7.71 = ( ^ [P3: nat > $o] :
% 13.25/7.71 ? [X2: int] :
% 13.25/7.71 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 13.25/7.71 & ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9071_nat__one__as__int,axiom,
% 13.25/7.71 ( one_one_nat
% 13.25/7.71 = ( nat2 @ one_one_int ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9072_root__sgn__power,axiom,
% 13.25/7.71 ! [N: nat,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 13.25/7.71 = Y3 ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9073_sgn__power__root,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
% 13.25/7.71 = X ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9074_sgn__eq,axiom,
% 13.25/7.71 ( sgn_sgn_complex
% 13.25/7.71 = ( ^ [Z2: complex] : ( divide1717551699836669952omplex @ Z2 @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9075_unset__bit__nat__def,axiom,
% 13.25/7.71 ( bit_se4205575877204974255it_nat
% 13.25/7.71 = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9076_split__root,axiom,
% 13.25/7.71 ! [P: real > $o,N: nat,X: real] :
% 13.25/7.71 ( ( P @ ( root @ N @ X ) )
% 13.25/7.71 = ( ( ( N = zero_zero_nat )
% 13.25/7.71 => ( P @ zero_zero_real ) )
% 13.25/7.71 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ! [Y: real] :
% 13.25/7.71 ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 13.25/7.71 = X )
% 13.25/7.71 => ( P @ Y ) ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9077_real__root__less__mono,axiom,
% 13.25/7.71 ! [N: nat,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9078_real__root__le__mono,axiom,
% 13.25/7.71 ! [N: nat,X: real,Y3: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( ord_less_eq_real @ X @ Y3 )
% 13.25/7.71 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y3 ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9079_real__root__power,axiom,
% 13.25/7.71 ! [N: nat,X: real,K: nat] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( root @ N @ ( power_power_real @ X @ K ) )
% 13.25/7.71 = ( power_power_real @ ( root @ N @ X ) @ K ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9080_real__root__abs,axiom,
% 13.25/7.71 ! [N: nat,X: real] :
% 13.25/7.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.71 => ( ( root @ N @ ( abs_abs_real @ X ) )
% 13.25/7.71 = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9081_nat__mono__iff,axiom,
% 13.25/7.71 ! [Z: int,W: int] :
% 13.25/7.71 ( ( ord_less_int @ zero_zero_int @ Z )
% 13.25/7.71 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 13.25/7.71 = ( ord_less_int @ W @ Z ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9082_zless__nat__eq__int__zless,axiom,
% 13.25/7.71 ! [M: nat,Z: int] :
% 13.25/7.71 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 13.25/7.71 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9083_nat__le__iff,axiom,
% 13.25/7.71 ! [X: int,N: nat] :
% 13.25/7.71 ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
% 13.25/7.71 = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9084_int__eq__iff,axiom,
% 13.25/7.71 ! [M: nat,Z: int] :
% 13.25/7.71 ( ( ( semiri1314217659103216013at_int @ M )
% 13.25/7.71 = Z )
% 13.25/7.71 = ( ( M
% 13.25/7.71 = ( nat2 @ Z ) )
% 13.25/7.71 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 13.25/7.71
% 13.25/7.71 thf(fact_9085_nat__0__le,axiom,
% 13.25/7.71 ! [Z: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.72 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 13.25/7.72 = Z ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9086_nat__int__add,axiom,
% 13.25/7.72 ! [A: nat,B: nat] :
% 13.25/7.72 ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 13.25/7.72 = ( plus_plus_nat @ A @ B ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9087_int__minus,axiom,
% 13.25/7.72 ! [N: nat,M: nat] :
% 13.25/7.72 ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M ) )
% 13.25/7.72 = ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9088_nat__abs__mult__distrib,axiom,
% 13.25/7.72 ! [W: int,Z: int] :
% 13.25/7.72 ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W @ Z ) ) )
% 13.25/7.72 = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W ) ) @ ( nat2 @ ( abs_abs_int @ Z ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9089_nat__plus__as__int,axiom,
% 13.25/7.72 ( plus_plus_nat
% 13.25/7.72 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9090_nat__times__as__int,axiom,
% 13.25/7.72 ( times_times_nat
% 13.25/7.72 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9091_real__nat__ceiling__ge,axiom,
% 13.25/7.72 ! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9092_nat__minus__as__int,axiom,
% 13.25/7.72 ( minus_minus_nat
% 13.25/7.72 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9093_nat__div__as__int,axiom,
% 13.25/7.72 ( divide_divide_nat
% 13.25/7.72 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9094_nat__mod__as__int,axiom,
% 13.25/7.72 ( modulo_modulo_nat
% 13.25/7.72 = ( ^ [A3: nat,B4: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9095_real__root__gt__zero,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9096_real__root__strict__decreasing,axiom,
% 13.25/7.72 ! [N: nat,N3: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_nat @ N @ N3 )
% 13.25/7.72 => ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.72 => ( ord_less_real @ ( root @ N3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9097_sqrt__def,axiom,
% 13.25/7.72 ( sqrt
% 13.25/7.72 = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9098_sgn__real__def,axiom,
% 13.25/7.72 ( sgn_sgn_real
% 13.25/7.72 = ( ^ [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 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9099_root__abs__power,axiom,
% 13.25/7.72 ! [N: nat,Y3: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 13.25/7.72 = ( abs_abs_real @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9100_nat__less__eq__zless,axiom,
% 13.25/7.72 ! [W: int,Z: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 13.25/7.72 => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 13.25/7.72 = ( ord_less_int @ W @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9101_nat__le__eq__zle,axiom,
% 13.25/7.72 ! [W: int,Z: int] :
% 13.25/7.72 ( ( ( ord_less_int @ zero_zero_int @ W )
% 13.25/7.72 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 13.25/7.72 => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
% 13.25/7.72 = ( ord_less_eq_int @ W @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9102_nat__eq__iff2,axiom,
% 13.25/7.72 ! [M: nat,W: int] :
% 13.25/7.72 ( ( M
% 13.25/7.72 = ( nat2 @ W ) )
% 13.25/7.72 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 13.25/7.72 => ( W
% 13.25/7.72 = ( semiri1314217659103216013at_int @ M ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 13.25/7.72 => ( M = zero_zero_nat ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9103_nat__eq__iff,axiom,
% 13.25/7.72 ! [W: int,M: nat] :
% 13.25/7.72 ( ( ( nat2 @ W )
% 13.25/7.72 = M )
% 13.25/7.72 = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
% 13.25/7.72 => ( W
% 13.25/7.72 = ( semiri1314217659103216013at_int @ M ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
% 13.25/7.72 => ( M = zero_zero_nat ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9104_split__nat,axiom,
% 13.25/7.72 ! [P: nat > $o,I: int] :
% 13.25/7.72 ( ( P @ ( nat2 @ I ) )
% 13.25/7.72 = ( ! [N2: nat] :
% 13.25/7.72 ( ( I
% 13.25/7.72 = ( semiri1314217659103216013at_int @ N2 ) )
% 13.25/7.72 => ( P @ N2 ) )
% 13.25/7.72 & ( ( ord_less_int @ I @ zero_zero_int )
% 13.25/7.72 => ( P @ zero_zero_nat ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9105_le__nat__iff,axiom,
% 13.25/7.72 ! [K: int,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.72 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 13.25/7.72 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9106_nat__add__distrib,axiom,
% 13.25/7.72 ! [Z: int,Z6: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.72 => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 13.25/7.72 => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
% 13.25/7.72 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9107_nat__mult__distrib,axiom,
% 13.25/7.72 ! [Z: int,Z6: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.72 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 13.25/7.72 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9108_Suc__as__int,axiom,
% 13.25/7.72 ( suc
% 13.25/7.72 = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9109_nat__diff__distrib,axiom,
% 13.25/7.72 ! [Z6: int,Z: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
% 13.25/7.72 => ( ( ord_less_eq_int @ Z6 @ Z )
% 13.25/7.72 => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
% 13.25/7.72 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9110_nat__diff__distrib_H,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ( nat2 @ ( minus_minus_int @ X @ Y3 ) )
% 13.25/7.72 = ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9111_nat__abs__triangle__ineq,axiom,
% 13.25/7.72 ! [K: int,L2: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L2 ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9112_nat__div__distrib_H,axiom,
% 13.25/7.72 ! [Y3: int,X: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ( nat2 @ ( divide_divide_int @ X @ Y3 ) )
% 13.25/7.72 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9113_nat__div__distrib,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( nat2 @ ( divide_divide_int @ X @ Y3 ) )
% 13.25/7.72 = ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9114_nat__power__eq,axiom,
% 13.25/7.72 ! [Z: int,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.72 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 13.25/7.72 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9115_nat__floor__neg,axiom,
% 13.25/7.72 ! [X: real] :
% 13.25/7.72 ( ( ord_less_eq_real @ X @ zero_zero_real )
% 13.25/7.72 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 13.25/7.72 = zero_zero_nat ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9116_nat__mod__distrib,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ( nat2 @ ( modulo_modulo_int @ X @ Y3 ) )
% 13.25/7.72 = ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9117_div__abs__eq__div__nat,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 13.25/7.72 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9118_floor__eq3,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 13.25/7.72 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 13.25/7.72 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 13.25/7.72 = N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9119_le__nat__floor,axiom,
% 13.25/7.72 ! [X: nat,A: real] :
% 13.25/7.72 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
% 13.25/7.72 => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9120_mod__abs__eq__div__nat,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L2 ) )
% 13.25/7.72 = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9121_nat__take__bit__eq,axiom,
% 13.25/7.72 ! [K: int,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.72 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 13.25/7.72 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9122_take__bit__nat__eq,axiom,
% 13.25/7.72 ! [K: int,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.72 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 13.25/7.72 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9123_bit__nat__iff,axiom,
% 13.25/7.72 ! [K: int,N: nat] :
% 13.25/7.72 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 13.25/7.72 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.72 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9124_of__real__sqrt,axiom,
% 13.25/7.72 ! [X: real] :
% 13.25/7.72 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X ) )
% 13.25/7.72 = ( csqrt @ ( real_V4546457046886955230omplex @ X ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9125_real__root__pos__pos,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9126_real__root__strict__increasing,axiom,
% 13.25/7.72 ! [N: nat,N3: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_nat @ N @ N3 )
% 13.25/7.72 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.72 => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9127_real__root__decreasing,axiom,
% 13.25/7.72 ! [N: nat,N3: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_eq_nat @ N @ N3 )
% 13.25/7.72 => ( ( ord_less_eq_real @ one_one_real @ X )
% 13.25/7.72 => ( ord_less_eq_real @ ( root @ N3 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9128_real__root__pow__pos,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 13.25/7.72 = X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9129_real__root__power__cancel,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 13.25/7.72 = X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9130_real__root__pos__unique,axiom,
% 13.25/7.72 ! [N: nat,Y3: real,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 13.25/7.72 => ( ( ( power_power_real @ Y3 @ N )
% 13.25/7.72 = X )
% 13.25/7.72 => ( ( root @ N @ X )
% 13.25/7.72 = Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9131_odd__real__root__power__cancel,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.72 => ( ( root @ N @ ( power_power_real @ X @ N ) )
% 13.25/7.72 = X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9132_odd__real__root__unique,axiom,
% 13.25/7.72 ! [N: nat,Y3: real,X: real] :
% 13.25/7.72 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.72 => ( ( ( power_power_real @ Y3 @ N )
% 13.25/7.72 = X )
% 13.25/7.72 => ( ( root @ N @ X )
% 13.25/7.72 = Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9133_odd__real__root__pow,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.72 => ( ( power_power_real @ ( root @ N @ X ) @ N )
% 13.25/7.72 = X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9134_nat__2,axiom,
% 13.25/7.72 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.72 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9135_sgn__power__injE,axiom,
% 13.25/7.72 ! [A: real,N: nat,X: real,B: real] :
% 13.25/7.72 ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
% 13.25/7.72 = X )
% 13.25/7.72 => ( ( X
% 13.25/7.72 = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
% 13.25/7.72 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( A = B ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9136_Suc__nat__eq__nat__zadd1,axiom,
% 13.25/7.72 ! [Z: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.72 => ( ( suc @ ( nat2 @ Z ) )
% 13.25/7.72 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9137_nat__less__iff,axiom,
% 13.25/7.72 ! [W: int,M: nat] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ W )
% 13.25/7.72 => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
% 13.25/7.72 = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9138_nat__mult__distrib__neg,axiom,
% 13.25/7.72 ! [Z: int,Z6: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 13.25/7.72 => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
% 13.25/7.72 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9139_nat__abs__int__diff,axiom,
% 13.25/7.72 ! [A: nat,B: nat] :
% 13.25/7.72 ( ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.72 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 13.25/7.72 = ( minus_minus_nat @ B @ A ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_nat @ A @ B )
% 13.25/7.72 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 13.25/7.72 = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9140_floor__eq4,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
% 13.25/7.72 => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 13.25/7.72 => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
% 13.25/7.72 = N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9141_diff__nat__eq__if,axiom,
% 13.25/7.72 ! [Z6: int,Z: int] :
% 13.25/7.72 ( ( ( ord_less_int @ Z6 @ zero_zero_int )
% 13.25/7.72 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 13.25/7.72 = ( nat2 @ Z ) ) )
% 13.25/7.72 & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
% 13.25/7.72 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
% 13.25/7.72 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9142_real__root__increasing,axiom,
% 13.25/7.72 ! [N: nat,N3: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_eq_nat @ N @ N3 )
% 13.25/7.72 => ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.72 => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N3 @ X ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9143_nat__dvd__iff,axiom,
% 13.25/7.72 ! [Z: int,M: nat] :
% 13.25/7.72 ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 13.25/7.72 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.72 => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 13.25/7.72 => ( M = zero_zero_nat ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9144_cis__Arg__unique,axiom,
% 13.25/7.72 ! [Z: complex,X: real] :
% 13.25/7.72 ( ( ( sgn_sgn_complex @ Z )
% 13.25/7.72 = ( cis @ X ) )
% 13.25/7.72 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X )
% 13.25/7.72 => ( ( ord_less_eq_real @ X @ pi )
% 13.25/7.72 => ( ( arg @ Z )
% 13.25/7.72 = X ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9145_ln__root,axiom,
% 13.25/7.72 ! [N: nat,B: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_real @ zero_zero_real @ B )
% 13.25/7.72 => ( ( ln_ln_real @ ( root @ N @ B ) )
% 13.25/7.72 = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9146_log__root,axiom,
% 13.25/7.72 ! [N: nat,A: real,B: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_real @ zero_zero_real @ A )
% 13.25/7.72 => ( ( log @ B @ ( root @ N @ A ) )
% 13.25/7.72 = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9147_log__base__root,axiom,
% 13.25/7.72 ! [N: nat,B: real,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_real @ zero_zero_real @ B )
% 13.25/7.72 => ( ( log @ ( root @ N @ B ) @ X )
% 13.25/7.72 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9148_Arg__correct,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( Z != zero_zero_complex )
% 13.25/7.72 => ( ( ( sgn_sgn_complex @ Z )
% 13.25/7.72 = ( cis @ ( arg @ Z ) ) )
% 13.25/7.72 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 13.25/7.72 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9149_root__powr__inverse,axiom,
% 13.25/7.72 ! [N: nat,X: real] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ( root @ N @ X )
% 13.25/7.72 = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9150_even__nat__iff,axiom,
% 13.25/7.72 ! [K: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.72 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 13.25/7.72 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9151_powr__real__of__int,axiom,
% 13.25/7.72 ! [X: real,N: int] :
% 13.25/7.72 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 13.25/7.72 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 13.25/7.72 = ( power_power_real @ X @ ( nat2 @ N ) ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 13.25/7.72 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 13.25/7.72 = ( inverse_inverse_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9152_arctan__inverse,axiom,
% 13.25/7.72 ! [X: real] :
% 13.25/7.72 ( ( X != zero_zero_real )
% 13.25/7.72 => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
% 13.25/7.72 = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9153_powr__int,axiom,
% 13.25/7.72 ! [X: real,I: int] :
% 13.25/7.72 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.72 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 13.25/7.72 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 13.25/7.72 = ( power_power_real @ X @ ( nat2 @ I ) ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 13.25/7.72 => ( ( powr_real @ X @ ( ring_1_of_int_real @ I ) )
% 13.25/7.72 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9154_num_Osize__gen_I3_J,axiom,
% 13.25/7.72 ! [X32: num] :
% 13.25/7.72 ( ( size_num @ ( bit1 @ X32 ) )
% 13.25/7.72 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9155_num_Osize__gen_I2_J,axiom,
% 13.25/7.72 ! [X22: num] :
% 13.25/7.72 ( ( size_num @ ( bit0 @ X22 ) )
% 13.25/7.72 = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9156_mask__nat__positive__iff,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 13.25/7.72 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9157_nat__mask__eq,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N ) )
% 13.25/7.72 = ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9158_less__eq__mask,axiom,
% 13.25/7.72 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9159_mask__nonnegative__int,axiom,
% 13.25/7.72 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9160_not__mask__negative__int,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9161_less__mask,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.72 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9162_take__bit__eq__mask__iff,axiom,
% 13.25/7.72 ! [N: nat,K: int] :
% 13.25/7.72 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 13.25/7.72 = ( bit_se2000444600071755411sk_int @ N ) )
% 13.25/7.72 = ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K @ one_one_int ) )
% 13.25/7.72 = zero_zero_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9163_num_Osize__gen_I1_J,axiom,
% 13.25/7.72 ( ( size_num @ one )
% 13.25/7.72 = zero_zero_nat ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9164_Suc__mask__eq__exp,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( suc @ ( bit_se2002935070580805687sk_nat @ N ) )
% 13.25/7.72 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9165_mask__nat__less__exp,axiom,
% 13.25/7.72 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9166_mask__nat__def,axiom,
% 13.25/7.72 ( bit_se2002935070580805687sk_nat
% 13.25/7.72 = ( ^ [N2: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9167_mask__half__int,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.72 = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9168_mask__int__def,axiom,
% 13.25/7.72 ( bit_se2000444600071755411sk_int
% 13.25/7.72 = ( ^ [N2: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9169_take__bit__eq__mask__iff__exp__dvd,axiom,
% 13.25/7.72 ! [N: nat,K: int] :
% 13.25/7.72 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 13.25/7.72 = ( bit_se2000444600071755411sk_int @ N ) )
% 13.25/7.72 = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9170_Arg__def,axiom,
% 13.25/7.72 ( arg
% 13.25/7.72 = ( ^ [Z2: complex] :
% 13.25/7.72 ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
% 13.25/7.72 @ ( fChoice_real
% 13.25/7.72 @ ^ [A3: real] :
% 13.25/7.72 ( ( ( sgn_sgn_complex @ Z2 )
% 13.25/7.72 = ( cis @ A3 ) )
% 13.25/7.72 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 13.25/7.72 & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9171_and__int__unfold,axiom,
% 13.25/7.72 ( bit_se725231765392027082nd_int
% 13.25/7.72 = ( ^ [K2: int,L: int] :
% 13.25/7.72 ( if_int
% 13.25/7.72 @ ( ( K2 = zero_zero_int )
% 13.25/7.72 | ( L = zero_zero_int ) )
% 13.25/7.72 @ zero_zero_int
% 13.25/7.72 @ ( if_int
% 13.25/7.72 @ ( K2
% 13.25/7.72 = ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.72 @ L
% 13.25/7.72 @ ( if_int
% 13.25/7.72 @ ( L
% 13.25/7.72 = ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.72 @ K2
% 13.25/7.72 @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9172_or__int__unfold,axiom,
% 13.25/7.72 ( bit_se1409905431419307370or_int
% 13.25/7.72 = ( ^ [K2: int,L: int] :
% 13.25/7.72 ( if_int
% 13.25/7.72 @ ( ( K2
% 13.25/7.72 = ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.72 | ( L
% 13.25/7.72 = ( uminus_uminus_int @ one_one_int ) ) )
% 13.25/7.72 @ ( uminus_uminus_int @ one_one_int )
% 13.25/7.72 @ ( if_int @ ( K2 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9173_and__nonnegative__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 13.25/7.72 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.72 | ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9174_and__negative__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ zero_zero_int )
% 13.25/7.72 = ( ( ord_less_int @ K @ zero_zero_int )
% 13.25/7.72 & ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9175_or__nonnegative__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) )
% 13.25/7.72 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.72 & ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9176_or__negative__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ zero_zero_int )
% 13.25/7.72 = ( ( ord_less_int @ K @ zero_zero_int )
% 13.25/7.72 | ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9177_and__minus__numerals_I6_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 13.25/7.72 = one_one_int ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9178_and__minus__numerals_I2_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = one_one_int ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9179_or__minus__numerals_I6_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ one_one_int )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9180_or__minus__numerals_I2_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9181_and__minus__numerals_I5_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 13.25/7.72 = zero_zero_int ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9182_and__minus__numerals_I1_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = zero_zero_int ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9183_plus__and__or,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ ( bit_se1409905431419307370or_int @ X @ Y3 ) )
% 13.25/7.72 = ( plus_plus_int @ X @ Y3 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9184_bit__or__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int,N: nat] :
% 13.25/7.72 ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L2 ) @ N )
% 13.25/7.72 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 13.25/7.72 | ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9185_bit__and__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int,N: nat] :
% 13.25/7.72 ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ N )
% 13.25/7.72 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 13.25/7.72 & ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9186_OR__lower,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X @ Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9187_or__greater__eq,axiom,
% 13.25/7.72 ! [L2: int,K: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ L2 )
% 13.25/7.72 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9188_AND__lower,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9189_AND__upper1,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9190_AND__upper2,axiom,
% 13.25/7.72 ! [Y3: int,X: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Y3 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9191_AND__upper1_H,axiom,
% 13.25/7.72 ! [Y3: int,Z: int,Ya: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ( ord_less_eq_int @ Y3 @ Z )
% 13.25/7.72 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9192_AND__upper2_H,axiom,
% 13.25/7.72 ! [Y3: int,Z: int,X: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ( ord_less_eq_int @ Y3 @ Z )
% 13.25/7.72 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9193_pow_Osimps_I1_J,axiom,
% 13.25/7.72 ! [X: num] :
% 13.25/7.72 ( ( pow @ X @ one )
% 13.25/7.72 = X ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9194_and__less__eq,axiom,
% 13.25/7.72 ! [L2: int,K: int] :
% 13.25/7.72 ( ( ord_less_int @ L2 @ zero_zero_int )
% 13.25/7.72 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L2 ) @ K ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9195_AND__upper1_H_H,axiom,
% 13.25/7.72 ! [Y3: int,Z: int,Ya: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ( ord_less_int @ Y3 @ Z )
% 13.25/7.72 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9196_AND__upper2_H_H,axiom,
% 13.25/7.72 ! [Y3: int,Z: int,X: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ( ord_less_int @ Y3 @ Z )
% 13.25/7.72 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X @ Y3 ) @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9197_even__and__iff__int,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L2 ) )
% 13.25/7.72 = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 13.25/7.72 | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9198_OR__upper,axiom,
% 13.25/7.72 ! [X: int,N: nat,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.72 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.72 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9199_or__int__rec,axiom,
% 13.25/7.72 ( bit_se1409905431419307370or_int
% 13.25/7.72 = ( ^ [K2: int,L: int] :
% 13.25/7.72 ( plus_plus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 13.25/7.72 | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 13.25/7.72 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9200_and__int__rec,axiom,
% 13.25/7.72 ( bit_se725231765392027082nd_int
% 13.25/7.72 = ( ^ [K2: int,L: int] :
% 13.25/7.72 ( plus_plus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 13.25/7.72 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9201_and__int_Oelims,axiom,
% 13.25/7.72 ! [X: int,Xa: int,Y3: int] :
% 13.25/7.72 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 13.25/7.72 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( uminus_uminus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 13.25/7.72 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 13.25/7.72 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( plus_plus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 13.25/7.72 @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9202_and__int_Osimps,axiom,
% 13.25/7.72 ( bit_se725231765392027082nd_int
% 13.25/7.72 = ( ^ [K2: int,L: int] :
% 13.25/7.72 ( if_int
% 13.25/7.72 @ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 13.25/7.72 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 13.25/7.72 @ ( uminus_uminus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 13.25/7.72 @ ( plus_plus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 13.25/7.72 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9203_or__minus__numerals_I1_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9204_or__minus__numerals_I5_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ one_one_int )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9205_and__nat__numerals_I3_J,axiom,
% 13.25/7.72 ! [X: num] :
% 13.25/7.72 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = zero_zero_nat ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9206_and__nat__numerals_I1_J,axiom,
% 13.25/7.72 ! [Y3: num] :
% 13.25/7.72 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 13.25/7.72 = zero_zero_nat ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9207_or__nat__numerals_I2_J,axiom,
% 13.25/7.72 ! [Y3: num] :
% 13.25/7.72 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 13.25/7.72 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9208_or__nat__numerals_I4_J,axiom,
% 13.25/7.72 ! [X: num] :
% 13.25/7.72 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9209_or__nat__numerals_I1_J,axiom,
% 13.25/7.72 ! [Y3: num] :
% 13.25/7.72 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 13.25/7.72 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9210_or__nat__numerals_I3_J,axiom,
% 13.25/7.72 ! [X: num] :
% 13.25/7.72 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9211_and__nat__numerals_I2_J,axiom,
% 13.25/7.72 ! [Y3: num] :
% 13.25/7.72 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 13.25/7.72 = one_one_nat ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9212_and__nat__numerals_I4_J,axiom,
% 13.25/7.72 ! [X: num] :
% 13.25/7.72 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = one_one_nat ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9213_or__minus__numerals_I4_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9214_or__minus__numerals_I8_J,axiom,
% 13.25/7.72 ! [N: num,M: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9215_or__minus__numerals_I3_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9216_or__minus__numerals_I7_J,axiom,
% 13.25/7.72 ! [N: num,M: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9217_Suc__0__and__eq,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.72 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9218_and__Suc__0__eq,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9219_or__not__num__neg_Osimps_I1_J,axiom,
% 13.25/7.72 ( ( bit_or_not_num_neg @ one @ one )
% 13.25/7.72 = one ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9220_or__not__num__neg_Osimps_I4_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one )
% 13.25/7.72 = ( bit0 @ one ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9221_or__not__num__neg_Osimps_I6_J,axiom,
% 13.25/7.72 ! [N: num,M: num] :
% 13.25/7.72 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
% 13.25/7.72 = ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9222_or__not__num__neg_Osimps_I7_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one )
% 13.25/7.72 = one ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9223_or__not__num__neg_Osimps_I3_J,axiom,
% 13.25/7.72 ! [M: num] :
% 13.25/7.72 ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
% 13.25/7.72 = ( bit1 @ M ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9224_or__not__num__neg_Osimps_I5_J,axiom,
% 13.25/7.72 ! [N: num,M: num] :
% 13.25/7.72 ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
% 13.25/7.72 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9225_or__not__num__neg_Osimps_I9_J,axiom,
% 13.25/7.72 ! [N: num,M: num] :
% 13.25/7.72 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
% 13.25/7.72 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9226_or__nat__def,axiom,
% 13.25/7.72 ( bit_se1412395901928357646or_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9227_and__nat__def,axiom,
% 13.25/7.72 ( bit_se727722235901077358nd_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9228_or__not__num__neg_Osimps_I2_J,axiom,
% 13.25/7.72 ! [M: num] :
% 13.25/7.72 ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
% 13.25/7.72 = ( bit1 @ M ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9229_or__not__num__neg_Osimps_I8_J,axiom,
% 13.25/7.72 ! [N: num,M: num] :
% 13.25/7.72 ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
% 13.25/7.72 = ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9230_atLeastAtMostPlus1__int__conv,axiom,
% 13.25/7.72 ! [M: int,N: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 13.25/7.72 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 13.25/7.72 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9231_simp__from__to,axiom,
% 13.25/7.72 ( set_or1266510415728281911st_int
% 13.25/7.72 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9232_or__not__num__neg_Oelims,axiom,
% 13.25/7.72 ! [X: num,Xa: num,Y3: num] :
% 13.25/7.72 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( ( X = one )
% 13.25/7.72 => ( ( Xa = one )
% 13.25/7.72 => ( Y3 != one ) ) )
% 13.25/7.72 => ( ( ( X = one )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit0 @ M4 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 != ( bit1 @ M4 ) ) ) )
% 13.25/7.72 => ( ( ( X = one )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit1 @ M4 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 != ( bit1 @ M4 ) ) ) )
% 13.25/7.72 => ( ( ? [N4: num] :
% 13.25/7.72 ( X
% 13.25/7.72 = ( bit0 @ N4 ) )
% 13.25/7.72 => ( ( Xa = one )
% 13.25/7.72 => ( Y3
% 13.25/7.72 != ( bit0 @ one ) ) ) )
% 13.25/7.72 => ( ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit0 @ N4 ) )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit0 @ M4 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 13.25/7.72 => ( ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit0 @ N4 ) )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit1 @ M4 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 != ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 13.25/7.72 => ( ( ? [N4: num] :
% 13.25/7.72 ( X
% 13.25/7.72 = ( bit1 @ N4 ) )
% 13.25/7.72 => ( ( Xa = one )
% 13.25/7.72 => ( Y3 != one ) ) )
% 13.25/7.72 => ( ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit1 @ N4 ) )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit0 @ M4 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) )
% 13.25/7.72 => ~ ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit1 @ N4 ) )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit1 @ M4 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 != ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9233_and__nat__unfold,axiom,
% 13.25/7.72 ( bit_se727722235901077358nd_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.72 ( if_nat
% 13.25/7.72 @ ( ( M3 = zero_zero_nat )
% 13.25/7.72 | ( N2 = zero_zero_nat ) )
% 13.25/7.72 @ zero_zero_nat
% 13.25/7.72 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9234_Suc__0__or__eq,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.72 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9235_or__Suc__0__eq,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9236_or__nat__rec,axiom,
% 13.25/7.72 ( bit_se1412395901928357646or_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.72 ( plus_plus_nat
% 13.25/7.72 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.72 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 13.25/7.72 | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 13.25/7.72 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9237_and__nat__rec,axiom,
% 13.25/7.72 ( bit_se727722235901077358nd_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.72 ( plus_plus_nat
% 13.25/7.72 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.72 @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 )
% 13.25/7.72 & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 13.25/7.72 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9238_or__nat__unfold,axiom,
% 13.25/7.72 ( bit_se1412395901928357646or_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9239_and__int_Opelims,axiom,
% 13.25/7.72 ! [X: int,Xa: int,Y3: int] :
% 13.25/7.72 ( ( ( bit_se725231765392027082nd_int @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 13.25/7.72 => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 13.25/7.72 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( uminus_uminus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
% 13.25/7.72 & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 13.25/7.72 & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( plus_plus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
% 13.25/7.72 @ ( 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 ) ) ) ) ) ) ) ) )
% 13.25/7.72 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9240_and__int_Opsimps,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L2 ) )
% 13.25/7.72 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 13.25/7.72 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 13.25/7.72 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 13.25/7.72 = ( uminus_uminus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) ) ) )
% 13.25/7.72 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 13.25/7.72 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 13.25/7.72 => ( ( bit_se725231765392027082nd_int @ K @ L2 )
% 13.25/7.72 = ( plus_plus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 13.25/7.72 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 13.25/7.72 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9241_cis__multiple__2pi,axiom,
% 13.25/7.72 ! [N: real] :
% 13.25/7.72 ( ( member_real @ N @ ring_1_Ints_real )
% 13.25/7.72 => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 13.25/7.72 = one_one_complex ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9242_set__encode__insert,axiom,
% 13.25/7.72 ! [A2: set_nat,N: nat] :
% 13.25/7.72 ( ( finite_finite_nat @ A2 )
% 13.25/7.72 => ( ~ ( member_nat @ N @ A2 )
% 13.25/7.72 => ( ( nat_set_encode @ ( insert_nat @ N @ A2 ) )
% 13.25/7.72 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9243_atLeastAtMost__insertL,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.72 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 13.25/7.72 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9244_atLeastAtMostSuc__conv,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 13.25/7.72 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 13.25/7.72 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9245_Icc__eq__insert__lb__nat,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.72 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 13.25/7.72 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9246_lessThan__nat__numeral,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 13.25/7.72 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9247_atMost__nat__numeral,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 13.25/7.72 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9248_sin__times__pi__eq__0,axiom,
% 13.25/7.72 ! [X: real] :
% 13.25/7.72 ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 = ( member_real @ X @ ring_1_Ints_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9249_atLeast1__atMost__eq__remove0,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.72 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9250_set__decode__plus__power__2,axiom,
% 13.25/7.72 ! [N: nat,Z: nat] :
% 13.25/7.72 ( ~ ( member_nat @ N @ ( nat_set_decode @ Z ) )
% 13.25/7.72 => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z ) )
% 13.25/7.72 = ( insert_nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9251_sin__integer__2pi,axiom,
% 13.25/7.72 ! [N: real] :
% 13.25/7.72 ( ( member_real @ N @ ring_1_Ints_real )
% 13.25/7.72 => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 13.25/7.72 = zero_zero_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9252_cos__integer__2pi,axiom,
% 13.25/7.72 ! [N: real] :
% 13.25/7.72 ( ( member_real @ N @ ring_1_Ints_real )
% 13.25/7.72 => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N ) )
% 13.25/7.72 = one_one_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9253_and__int_Opinduct,axiom,
% 13.25/7.72 ! [A0: int,A1: int,P: int > int > $o] :
% 13.25/7.72 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 13.25/7.72 => ( ! [K3: int,L4: int] :
% 13.25/7.72 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K3 @ L4 ) )
% 13.25/7.72 => ( ( ~ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 13.25/7.72 & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 13.25/7.72 => ( P @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 => ( P @ K3 @ L4 ) ) )
% 13.25/7.72 => ( P @ A0 @ A1 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9254_upto_Opinduct,axiom,
% 13.25/7.72 ! [A0: int,A1: int,P: int > int > $o] :
% 13.25/7.72 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
% 13.25/7.72 => ( ! [I2: int,J2: int] :
% 13.25/7.72 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
% 13.25/7.72 => ( ( ( ord_less_eq_int @ I2 @ J2 )
% 13.25/7.72 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
% 13.25/7.72 => ( P @ I2 @ J2 ) ) )
% 13.25/7.72 => ( P @ A0 @ A1 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9255_xor__Suc__0__eq,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.72 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.72 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9256_xor__nat__numerals_I1_J,axiom,
% 13.25/7.72 ! [Y3: num] :
% 13.25/7.72 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 13.25/7.72 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9257_xor__nat__numerals_I2_J,axiom,
% 13.25/7.72 ! [Y3: num] :
% 13.25/7.72 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 13.25/7.72 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9258_xor__nat__numerals_I3_J,axiom,
% 13.25/7.72 ! [X: num] :
% 13.25/7.72 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9259_xor__nat__numerals_I4_J,axiom,
% 13.25/7.72 ! [X: num] :
% 13.25/7.72 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9260_xor__nat__unfold,axiom,
% 13.25/7.72 ( bit_se6528837805403552850or_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M3 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9261_xor__nat__rec,axiom,
% 13.25/7.72 ( bit_se6528837805403552850or_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.72 ( plus_plus_nat
% 13.25/7.72 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.72 @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 13.25/7.72 != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 13.25/7.72 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9262_Suc__0__xor__eq,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.72 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.72 @ ( zero_n2687167440665602831ol_nat
% 13.25/7.72 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9263_horner__sum__of__bool__2__less,axiom,
% 13.25/7.72 ! [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 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9264_push__bit__nonnegative__int__iff,axiom,
% 13.25/7.72 ! [N: nat,K: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 13.25/7.72 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9265_push__bit__negative__int__iff,axiom,
% 13.25/7.72 ! [N: nat,K: int] :
% 13.25/7.72 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 13.25/7.72 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9266_concat__bit__of__zero__1,axiom,
% 13.25/7.72 ! [N: nat,L2: int] :
% 13.25/7.72 ( ( bit_concat_bit @ N @ zero_zero_int @ L2 )
% 13.25/7.72 = ( bit_se545348938243370406it_int @ N @ L2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9267_xor__nonnegative__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) )
% 13.25/7.72 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 13.25/7.72 = ( ord_less_eq_int @ zero_zero_int @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9268_xor__negative__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int] :
% 13.25/7.72 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ zero_zero_int )
% 13.25/7.72 = ( ( ord_less_int @ K @ zero_zero_int )
% 13.25/7.72 != ( ord_less_int @ L2 @ zero_zero_int ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9269_push__bit__of__Suc__0,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9270_bit__xor__int__iff,axiom,
% 13.25/7.72 ! [K: int,L2: int,N: nat] :
% 13.25/7.72 ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L2 ) @ N )
% 13.25/7.72 = ( ( bit_se1146084159140164899it_int @ K @ N )
% 13.25/7.72 != ( bit_se1146084159140164899it_int @ L2 @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9271_flip__bit__int__def,axiom,
% 13.25/7.72 ( bit_se2159334234014336723it_int
% 13.25/7.72 = ( ^ [N2: nat,K2: int] : ( bit_se6526347334894502574or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9272_push__bit__nat__eq,axiom,
% 13.25/7.72 ! [N: nat,K: int] :
% 13.25/7.72 ( ( bit_se547839408752420682it_nat @ N @ ( nat2 @ K ) )
% 13.25/7.72 = ( nat2 @ ( bit_se545348938243370406it_int @ N @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9273_XOR__lower,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X @ Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9274_set__bit__nat__def,axiom,
% 13.25/7.72 ( bit_se7882103937844011126it_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] : ( bit_se1412395901928357646or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9275_flip__bit__nat__def,axiom,
% 13.25/7.72 ( bit_se2161824704523386999it_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] : ( bit_se6528837805403552850or_nat @ N2 @ ( bit_se547839408752420682it_nat @ M3 @ one_one_nat ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9276_bit__push__bit__iff__int,axiom,
% 13.25/7.72 ! [M: nat,K: int,N: nat] :
% 13.25/7.72 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 13.25/7.72 = ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.72 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9277_xor__nat__def,axiom,
% 13.25/7.72 ( bit_se6528837805403552850or_nat
% 13.25/7.72 = ( ^ [M3: nat,N2: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9278_bit__push__bit__iff__nat,axiom,
% 13.25/7.72 ! [M: nat,Q2: nat,N: nat] :
% 13.25/7.72 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q2 ) @ N )
% 13.25/7.72 = ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.72 & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9279_concat__bit__eq,axiom,
% 13.25/7.72 ( bit_concat_bit
% 13.25/7.72 = ( ^ [N2: nat,K2: int,L: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9280_concat__bit__def,axiom,
% 13.25/7.72 ( bit_concat_bit
% 13.25/7.72 = ( ^ [N2: nat,K2: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N2 @ K2 ) @ ( bit_se545348938243370406it_int @ N2 @ L ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9281_set__bit__int__def,axiom,
% 13.25/7.72 ( bit_se7879613467334960850it_int
% 13.25/7.72 = ( ^ [N2: nat,K2: int] : ( bit_se1409905431419307370or_int @ K2 @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9282_push__bit__int__def,axiom,
% 13.25/7.72 ( bit_se545348938243370406it_int
% 13.25/7.72 = ( ^ [N2: nat,K2: int] : ( times_times_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9283_push__bit__nat__def,axiom,
% 13.25/7.72 ( bit_se547839408752420682it_nat
% 13.25/7.72 = ( ^ [N2: nat,M3: nat] : ( times_times_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9284_push__bit__minus__one,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se545348938243370406it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9285_XOR__upper,axiom,
% 13.25/7.72 ! [X: int,N: nat,Y3: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( ord_less_int @ X @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.72 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 13.25/7.72 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9286_xor__int__rec,axiom,
% 13.25/7.72 ( bit_se6526347334894502574or_int
% 13.25/7.72 = ( ^ [K2: int,L: int] :
% 13.25/7.72 ( plus_plus_int
% 13.25/7.72 @ ( zero_n2684676970156552555ol_int
% 13.25/7.72 @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) )
% 13.25/7.72 != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) )
% 13.25/7.72 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9287_xor__int__unfold,axiom,
% 13.25/7.72 ( bit_se6526347334894502574or_int
% 13.25/7.72 = ( ^ [K2: int,L: int] :
% 13.25/7.72 ( if_int
% 13.25/7.72 @ ( K2
% 13.25/7.72 = ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.72 @ ( bit_ri7919022796975470100ot_int @ L )
% 13.25/7.72 @ ( if_int
% 13.25/7.72 @ ( L
% 13.25/7.72 = ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.72 @ ( bit_ri7919022796975470100ot_int @ K2 )
% 13.25/7.72 @ ( if_int @ ( K2 = zero_zero_int ) @ L @ ( if_int @ ( L = zero_zero_int ) @ K2 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9288_not__nonnegative__int__iff,axiom,
% 13.25/7.72 ! [K: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 13.25/7.72 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9289_not__negative__int__iff,axiom,
% 13.25/7.72 ! [K: int] :
% 13.25/7.72 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 13.25/7.72 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9290_and__minus__minus__numerals,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.72 = ( 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 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9291_or__minus__minus__numerals,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.72 = ( 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 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9292_bit__not__int__iff,axiom,
% 13.25/7.72 ! [K: int,N: nat] :
% 13.25/7.72 ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N )
% 13.25/7.72 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9293_or__int__def,axiom,
% 13.25/7.72 ( bit_se1409905431419307370or_int
% 13.25/7.72 = ( ^ [K2: int,L: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( bit_ri7919022796975470100ot_int @ L ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9294_not__int__def,axiom,
% 13.25/7.72 ( bit_ri7919022796975470100ot_int
% 13.25/7.72 = ( ^ [K2: int] : ( minus_minus_int @ ( uminus_uminus_int @ K2 ) @ one_one_int ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9295_and__not__numerals_I1_J,axiom,
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 13.25/7.72 = zero_zero_int ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9296_or__not__numerals_I1_J,axiom,
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 13.25/7.72 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9297_unset__bit__int__def,axiom,
% 13.25/7.72 ( bit_se4203085406695923979it_int
% 13.25/7.72 = ( ^ [N2: nat,K2: int] : ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N2 @ one_one_int ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9298_xor__int__def,axiom,
% 13.25/7.72 ( bit_se6526347334894502574or_int
% 13.25/7.72 = ( ^ [K2: int,L: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K2 @ ( bit_ri7919022796975470100ot_int @ L ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ L ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9299_not__int__div__2,axiom,
% 13.25/7.72 ! [K: int] :
% 13.25/7.72 ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.72 = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9300_even__not__iff__int,axiom,
% 13.25/7.72 ! [K: int] :
% 13.25/7.72 ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
% 13.25/7.72 = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9301_and__not__numerals_I2_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = one_one_int ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9302_and__not__numerals_I4_J,axiom,
% 13.25/7.72 ! [M: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 13.25/7.72 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9303_or__not__numerals_I2_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9304_or__not__numerals_I4_J,axiom,
% 13.25/7.72 ! [M: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 13.25/7.72 = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9305_bit__minus__int__iff,axiom,
% 13.25/7.72 ! [K: int,N: nat] :
% 13.25/7.72 ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N )
% 13.25/7.72 = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9306_numeral__or__not__num__eq,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9307_int__numeral__not__or__num__neg,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9308_int__numeral__or__not__num__neg,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9309_and__not__numerals_I5_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9310_and__not__numerals_I7_J,axiom,
% 13.25/7.72 ! [M: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 13.25/7.72 = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9311_or__not__numerals_I3_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9312_and__not__numerals_I3_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = zero_zero_int ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9313_or__not__numerals_I7_J,axiom,
% 13.25/7.72 ! [M: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
% 13.25/7.72 = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9314_and__not__numerals_I9_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9315_and__not__numerals_I6_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9316_or__not__numerals_I6_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9317_or__not__numerals_I5_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9318_and__not__numerals_I8_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9319_or__not__numerals_I8_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9320_or__not__numerals_I9_J,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9321_not__int__rec,axiom,
% 13.25/7.72 ( bit_ri7919022796975470100ot_int
% 13.25/7.72 = ( ^ [K2: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9322_vebt__maxt_Opelims,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT,Y3: option_nat] :
% 13.25/7.72 ( ( ( vEBT_vebt_maxt @ X )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
% 13.25/7.72 => ( ! [A5: $o,B3: $o] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.72 => ( ( ( B3
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.72 & ( ~ B3
% 13.25/7.72 => ( ( A5
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( some_nat @ zero_zero_nat ) ) )
% 13.25/7.72 & ( ~ A5
% 13.25/7.72 => ( Y3 = none_nat ) ) ) ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A5 @ B3 ) ) ) )
% 13.25/7.72 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 13.25/7.72 => ( ( Y3 = none_nat )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 13.25/7.72 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( some_nat @ Ma2 ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9323_vebt__mint_Opelims,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT,Y3: option_nat] :
% 13.25/7.72 ( ( ( vEBT_vebt_mint @ X )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
% 13.25/7.72 => ( ! [A5: $o,B3: $o] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ A5 @ B3 ) )
% 13.25/7.72 => ( ( ( A5
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( some_nat @ zero_zero_nat ) ) )
% 13.25/7.72 & ( ~ A5
% 13.25/7.72 => ( ( B3
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( some_nat @ one_one_nat ) ) )
% 13.25/7.72 & ( ~ B3
% 13.25/7.72 => ( Y3 = none_nat ) ) ) ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A5 @ B3 ) ) ) )
% 13.25/7.72 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 13.25/7.72 => ( ( Y3 = none_nat )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 13.25/7.72 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( some_nat @ Mi2 ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9324_or__not__num__neg_Opelims,axiom,
% 13.25/7.72 ! [X: num,Xa: num,Y3: num] :
% 13.25/7.72 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 13.25/7.72 => ( ( ( X = one )
% 13.25/7.72 => ( ( Xa = one )
% 13.25/7.72 => ( ( Y3 = one )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 13.25/7.72 => ( ( ( X = one )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit0 @ M4 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( bit1 @ M4 ) )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M4 ) ) ) ) ) )
% 13.25/7.72 => ( ( ( X = one )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit1 @ M4 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( bit1 @ M4 ) )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M4 ) ) ) ) ) )
% 13.25/7.72 => ( ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit0 @ N4 ) )
% 13.25/7.72 => ( ( Xa = one )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( bit0 @ one ) )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ one ) ) ) ) )
% 13.25/7.72 => ( ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit0 @ N4 ) )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit0 @ M4 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 13.25/7.72 => ( ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit0 @ N4 ) )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit1 @ M4 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( bit0 @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) )
% 13.25/7.72 => ( ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit1 @ N4 ) )
% 13.25/7.72 => ( ( Xa = one )
% 13.25/7.72 => ( ( Y3 = one )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ one ) ) ) ) )
% 13.25/7.72 => ( ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit1 @ N4 ) )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit0 @ M4 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit0 @ M4 ) ) ) ) ) )
% 13.25/7.72 => ~ ! [N4: num] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( bit1 @ N4 ) )
% 13.25/7.72 => ! [M4: num] :
% 13.25/7.72 ( ( Xa
% 13.25/7.72 = ( bit1 @ M4 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( bitM @ ( bit_or_not_num_neg @ N4 @ M4 ) ) )
% 13.25/7.72 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N4 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9325_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT,Y3: $o] :
% 13.25/7.72 ( ( ( vEBT_VEBT_minNull @ X )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 13.25/7.72 => ( ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ $false @ $false ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 13.25/7.72 => ( ! [Uv2: $o] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 13.25/7.72 => ( ~ Y3
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 13.25/7.72 => ( ! [Uu2: $o] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 13.25/7.72 => ( ~ Y3
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 13.25/7.72 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 13.25/7.72 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 13.25/7.72 => ( ~ Y3
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9326_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT] :
% 13.25/7.72 ( ~ ( vEBT_VEBT_minNull @ X )
% 13.25/7.72 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 13.25/7.72 => ( ! [Uv2: $o] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 13.25/7.72 => ( ! [Uu2: $o] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 13.25/7.72 => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9327_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT] :
% 13.25/7.72 ( ( vEBT_VEBT_minNull @ X )
% 13.25/7.72 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
% 13.25/7.72 => ( ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ $false @ $false ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 13.25/7.72 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 13.25/7.72 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9328_int__ge__less__than2__def,axiom,
% 13.25/7.72 ( int_ge_less_than2
% 13.25/7.72 = ( ^ [D2: int] :
% 13.25/7.72 ( collec213857154873943460nt_int
% 13.25/7.72 @ ( produc4947309494688390418_int_o
% 13.25/7.72 @ ^ [Z7: int,Z2: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ D2 @ Z2 )
% 13.25/7.72 & ( ord_less_int @ Z7 @ Z2 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9329_int__ge__less__than__def,axiom,
% 13.25/7.72 ( int_ge_less_than
% 13.25/7.72 = ( ^ [D2: int] :
% 13.25/7.72 ( collec213857154873943460nt_int
% 13.25/7.72 @ ( produc4947309494688390418_int_o
% 13.25/7.72 @ ^ [Z7: int,Z2: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ D2 @ Z7 )
% 13.25/7.72 & ( ord_less_int @ Z7 @ Z2 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9330_Sum__Ico__nat,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( groups3542108847815614940at_nat
% 13.25/7.72 @ ^ [X2: nat] : X2
% 13.25/7.72 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 13.25/7.72 = ( 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 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9331_Cauchy__iff2,axiom,
% 13.25/7.72 ( topolo4055970368930404560y_real
% 13.25/7.72 = ( ^ [X6: nat > real] :
% 13.25/7.72 ! [J3: nat] :
% 13.25/7.72 ? [M8: nat] :
% 13.25/7.72 ! [M3: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ M8 @ M3 )
% 13.25/7.72 => ! [N2: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ M8 @ N2 )
% 13.25/7.72 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9332_all__nat__less__eq,axiom,
% 13.25/7.72 ! [N: nat,P: nat > $o] :
% 13.25/7.72 ( ( ! [M3: nat] :
% 13.25/7.72 ( ( ord_less_nat @ M3 @ N )
% 13.25/7.72 => ( P @ M3 ) ) )
% 13.25/7.72 = ( ! [X2: nat] :
% 13.25/7.72 ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 13.25/7.72 => ( P @ X2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9333_ex__nat__less__eq,axiom,
% 13.25/7.72 ! [N: nat,P: nat > $o] :
% 13.25/7.72 ( ( ? [M3: nat] :
% 13.25/7.72 ( ( ord_less_nat @ M3 @ N )
% 13.25/7.72 & ( P @ M3 ) ) )
% 13.25/7.72 = ( ? [X2: nat] :
% 13.25/7.72 ( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 13.25/7.72 & ( P @ X2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9334_subset__eq__atLeast0__lessThan__finite,axiom,
% 13.25/7.72 ! [N3: set_nat,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 13.25/7.72 => ( finite_finite_nat @ N3 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9335_atLeastLessThanSuc,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.72 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 13.25/7.72 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_nat @ M @ N )
% 13.25/7.72 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 13.25/7.72 = bot_bot_set_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9336_atLeastLessThan__nat__numeral,axiom,
% 13.25/7.72 ! [M: nat,K: num] :
% 13.25/7.72 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 13.25/7.72 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 13.25/7.72 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 13.25/7.72 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 13.25/7.72 = bot_bot_set_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9337_atLeast1__lessThan__eq__remove0,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 13.25/7.72 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9338_sum__power2,axiom,
% 13.25/7.72 ! [K: nat] :
% 13.25/7.72 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 13.25/7.72 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9339_Chebyshev__sum__upper__nat,axiom,
% 13.25/7.72 ! [N: nat,A: nat > nat,B: nat > nat] :
% 13.25/7.72 ( ! [I2: nat,J2: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ I2 @ J2 )
% 13.25/7.72 => ( ( ord_less_nat @ J2 @ N )
% 13.25/7.72 => ( ord_less_eq_nat @ ( A @ I2 ) @ ( A @ J2 ) ) ) )
% 13.25/7.72 => ( ! [I2: nat,J2: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ I2 @ J2 )
% 13.25/7.72 => ( ( ord_less_nat @ J2 @ N )
% 13.25/7.72 => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I2 ) ) ) )
% 13.25/7.72 => ( ord_less_eq_nat
% 13.25/7.72 @ ( times_times_nat @ N
% 13.25/7.72 @ ( groups3542108847815614940at_nat
% 13.25/7.72 @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
% 13.25/7.72 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 13.25/7.72 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9340_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 13.25/7.72 ! [L2: int,U: int] :
% 13.25/7.72 ( ( set_or4662586982721622107an_int @ L2 @ ( plus_plus_int @ U @ one_one_int ) )
% 13.25/7.72 = ( set_or1266510415728281911st_int @ L2 @ U ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9341_valid__eq2,axiom,
% 13.25/7.72 ! [T: vEBT_VEBT,D: nat] :
% 13.25/7.72 ( ( vEBT_VEBT_valid @ T @ D )
% 13.25/7.72 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9342_valid__eq1,axiom,
% 13.25/7.72 ! [T: vEBT_VEBT,D: nat] :
% 13.25/7.72 ( ( vEBT_invar_vebt @ T @ D )
% 13.25/7.72 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9343_valid__eq,axiom,
% 13.25/7.72 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9344_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 13.25/7.72 ! [Uu: $o,Uv: $o,D: nat] :
% 13.25/7.72 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 13.25/7.72 = ( D = one_one_nat ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9345_Code__Target__Int_Opositive__def,axiom,
% 13.25/7.72 code_Target_positive = numeral_numeral_int ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9346_divmod__step__integer__def,axiom,
% 13.25/7.72 ( unique4921790084139445826nteger
% 13.25/7.72 = ( ^ [L: num] :
% 13.25/7.72 ( produc6916734918728496179nteger
% 13.25/7.72 @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9347_divmod__integer_H__def,axiom,
% 13.25/7.72 ( unique3479559517661332726nteger
% 13.25/7.72 = ( ^ [M3: num,N2: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M3 ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9348_less__eq__integer__code_I1_J,axiom,
% 13.25/7.72 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9349_times__integer__code_I2_J,axiom,
% 13.25/7.72 ! [L2: code_integer] :
% 13.25/7.72 ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L2 )
% 13.25/7.72 = zero_z3403309356797280102nteger ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9350_times__integer__code_I1_J,axiom,
% 13.25/7.72 ! [K: code_integer] :
% 13.25/7.72 ( ( times_3573771949741848930nteger @ K @ zero_z3403309356797280102nteger )
% 13.25/7.72 = zero_z3403309356797280102nteger ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9351_plus__integer__code_I1_J,axiom,
% 13.25/7.72 ! [K: code_integer] :
% 13.25/7.72 ( ( plus_p5714425477246183910nteger @ K @ zero_z3403309356797280102nteger )
% 13.25/7.72 = K ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9352_plus__integer__code_I2_J,axiom,
% 13.25/7.72 ! [L2: code_integer] :
% 13.25/7.72 ( ( plus_p5714425477246183910nteger @ zero_z3403309356797280102nteger @ L2 )
% 13.25/7.72 = L2 ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9353_sgn__integer__code,axiom,
% 13.25/7.72 ( sgn_sgn_Code_integer
% 13.25/7.72 = ( ^ [K2: code_integer] : ( if_Code_integer @ ( K2 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9354_minus__integer__code_I1_J,axiom,
% 13.25/7.72 ! [K: code_integer] :
% 13.25/7.72 ( ( minus_8373710615458151222nteger @ K @ zero_z3403309356797280102nteger )
% 13.25/7.72 = K ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9355_minus__integer__code_I2_J,axiom,
% 13.25/7.72 ! [L2: code_integer] :
% 13.25/7.72 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ L2 )
% 13.25/7.72 = ( uminus1351360451143612070nteger @ L2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9356_one__natural_Orsp,axiom,
% 13.25/7.72 one_one_nat = one_one_nat ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9357_integer__of__int__code,axiom,
% 13.25/7.72 ( code_integer_of_int
% 13.25/7.72 = ( ^ [K2: int] :
% 13.25/7.72 ( if_Code_integer @ ( ord_less_int @ K2 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K2 ) ) )
% 13.25/7.72 @ ( if_Code_integer @ ( K2 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 13.25/7.72 @ ( if_Code_integer
% 13.25/7.72 @ ( ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 13.25/7.72 = zero_zero_int )
% 13.25/7.72 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9358_Code__Numeral_Opositive__def,axiom,
% 13.25/7.72 code_positive = numera6620942414471956472nteger ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9359_divide__integer_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: int,X: int] :
% 13.25/7.72 ( ( divide6298287555418463151nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 13.25/7.72 = ( code_integer_of_int @ ( divide_divide_int @ Xa @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9360_abs__integer__code,axiom,
% 13.25/7.72 ( abs_abs_Code_integer
% 13.25/7.72 = ( ^ [K2: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K2 ) @ K2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9361_less__integer_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: int,X: int] :
% 13.25/7.72 ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 13.25/7.72 = ( ord_less_int @ Xa @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9362_less__integer__code_I1_J,axiom,
% 13.25/7.72 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9363_plus__integer_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: int,X: int] :
% 13.25/7.72 ( ( plus_p5714425477246183910nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 13.25/7.72 = ( code_integer_of_int @ ( plus_plus_int @ Xa @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9364_times__integer_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: int,X: int] :
% 13.25/7.72 ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 13.25/7.72 = ( code_integer_of_int @ ( times_times_int @ Xa @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9365_less__eq__integer_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: int,X: int] :
% 13.25/7.72 ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 13.25/7.72 = ( ord_less_eq_int @ Xa @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9366_minus__integer_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: int,X: int] :
% 13.25/7.72 ( ( minus_8373710615458151222nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 13.25/7.72 = ( code_integer_of_int @ ( minus_minus_int @ Xa @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9367_integer__of__num_I3_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 13.25/7.72 = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ one_one_Code_integer ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9368_bit__cut__integer__def,axiom,
% 13.25/7.72 ( code_bit_cut_integer
% 13.25/7.72 = ( ^ [K2: code_integer] :
% 13.25/7.72 ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 13.25/7.72 @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9369_integer__of__num__def,axiom,
% 13.25/7.72 code_integer_of_num = numera6620942414471956472nteger ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9370_integer__of__num__triv_I1_J,axiom,
% 13.25/7.72 ( ( code_integer_of_num @ one )
% 13.25/7.72 = one_one_Code_integer ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9371_integer__of__num_I2_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 13.25/7.72 = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9372_integer__of__num__triv_I2_J,axiom,
% 13.25/7.72 ( ( code_integer_of_num @ ( bit0 @ one ) )
% 13.25/7.72 = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9373_bit__cut__integer__code,axiom,
% 13.25/7.72 ( code_bit_cut_integer
% 13.25/7.72 = ( ^ [K2: code_integer] :
% 13.25/7.72 ( if_Pro5737122678794959658eger_o @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 13.25/7.72 @ ( produc9125791028180074456eger_o
% 13.25/7.72 @ ^ [R5: code_integer,S6: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S6 ) ) @ ( S6 = one_one_Code_integer ) )
% 13.25/7.72 @ ( code_divmod_abs @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9374_divmod__integer__def,axiom,
% 13.25/7.72 ( code_divmod_integer
% 13.25/7.72 = ( ^ [K2: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K2 @ L ) @ ( modulo364778990260209775nteger @ K2 @ L ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9375_csqrt_Osimps_I1_J,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( re @ ( csqrt @ Z ) )
% 13.25/7.72 = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9376_complex__Re__numeral,axiom,
% 13.25/7.72 ! [V: num] :
% 13.25/7.72 ( ( re @ ( numera6690914467698888265omplex @ V ) )
% 13.25/7.72 = ( numeral_numeral_real @ V ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9377_Re__divide__of__nat,axiom,
% 13.25/7.72 ! [Z: complex,N: nat] :
% 13.25/7.72 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 13.25/7.72 = ( divide_divide_real @ ( re @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9378_Re__divide__of__real,axiom,
% 13.25/7.72 ! [Z: complex,R: real] :
% 13.25/7.72 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
% 13.25/7.72 = ( divide_divide_real @ ( re @ Z ) @ R ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9379_Re__sgn,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( re @ ( sgn_sgn_complex @ Z ) )
% 13.25/7.72 = ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9380_Re__divide__numeral,axiom,
% 13.25/7.72 ! [Z: complex,W: num] :
% 13.25/7.72 ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.72 = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9381_complex__Re__le__cmod,axiom,
% 13.25/7.72 ! [X: complex] : ( ord_less_eq_real @ ( re @ X ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9382_one__complex_Osimps_I1_J,axiom,
% 13.25/7.72 ( ( re @ one_one_complex )
% 13.25/7.72 = one_one_real ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9383_plus__complex_Osimps_I1_J,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( re @ ( plus_plus_complex @ X @ Y3 ) )
% 13.25/7.72 = ( plus_plus_real @ ( re @ X ) @ ( re @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9384_scaleR__complex_Osimps_I1_J,axiom,
% 13.25/7.72 ! [R: real,X: complex] :
% 13.25/7.72 ( ( re @ ( real_V2046097035970521341omplex @ R @ X ) )
% 13.25/7.72 = ( times_times_real @ R @ ( re @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9385_minus__complex_Osimps_I1_J,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( re @ ( minus_minus_complex @ X @ Y3 ) )
% 13.25/7.72 = ( minus_minus_real @ ( re @ X ) @ ( re @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9386_abs__Re__le__cmod,axiom,
% 13.25/7.72 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9387_Re__csqrt,axiom,
% 13.25/7.72 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9388_cmod__plus__Re__le__0__iff,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 13.25/7.72 = ( ( re @ Z )
% 13.25/7.72 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9389_cos__n__Re__cis__pow__n,axiom,
% 13.25/7.72 ! [N: nat,A: real] :
% 13.25/7.72 ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 13.25/7.72 = ( re @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9390_divmod__abs__def,axiom,
% 13.25/7.72 ( code_divmod_abs
% 13.25/7.72 = ( ^ [K2: code_integer,L: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K2 ) @ ( abs_abs_Code_integer @ L ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9391_divmod__integer__code,axiom,
% 13.25/7.72 ( code_divmod_integer
% 13.25/7.72 = ( ^ [K2: code_integer,L: code_integer] :
% 13.25/7.72 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 13.25/7.72 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L )
% 13.25/7.72 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K2 ) @ ( code_divmod_abs @ K2 @ L )
% 13.25/7.72 @ ( produc6916734918728496179nteger
% 13.25/7.72 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L @ S6 ) ) )
% 13.25/7.72 @ ( code_divmod_abs @ K2 @ L ) ) )
% 13.25/7.72 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 13.25/7.72 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 13.25/7.72 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K2 @ L )
% 13.25/7.72 @ ( produc6916734918728496179nteger
% 13.25/7.72 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L ) @ S6 ) ) )
% 13.25/7.72 @ ( code_divmod_abs @ K2 @ L ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9392_csqrt_Ocode,axiom,
% 13.25/7.72 ( csqrt
% 13.25/7.72 = ( ^ [Z2: complex] :
% 13.25/7.72 ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.72 @ ( times_times_real
% 13.25/7.72 @ ( if_real
% 13.25/7.72 @ ( ( im @ Z2 )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 @ one_one_real
% 13.25/7.72 @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
% 13.25/7.72 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9393_csqrt_Osimps_I2_J,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( im @ ( csqrt @ Z ) )
% 13.25/7.72 = ( times_times_real
% 13.25/7.72 @ ( if_real
% 13.25/7.72 @ ( ( im @ Z )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 @ one_one_real
% 13.25/7.72 @ ( sgn_sgn_real @ ( im @ Z ) ) )
% 13.25/7.72 @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9394_complex__Im__numeral,axiom,
% 13.25/7.72 ! [V: num] :
% 13.25/7.72 ( ( im @ ( numera6690914467698888265omplex @ V ) )
% 13.25/7.72 = zero_zero_real ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9395_Im__i__times,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( im @ ( times_times_complex @ imaginary_unit @ Z ) )
% 13.25/7.72 = ( re @ Z ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9396_Im__divide__of__real,axiom,
% 13.25/7.72 ! [Z: complex,R: real] :
% 13.25/7.72 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( real_V4546457046886955230omplex @ R ) ) )
% 13.25/7.72 = ( divide_divide_real @ ( im @ Z ) @ R ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9397_Im__sgn,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( im @ ( sgn_sgn_complex @ Z ) )
% 13.25/7.72 = ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9398_Re__power__real,axiom,
% 13.25/7.72 ! [X: complex,N: nat] :
% 13.25/7.72 ( ( ( im @ X )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 => ( ( re @ ( power_power_complex @ X @ N ) )
% 13.25/7.72 = ( power_power_real @ ( re @ X ) @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9399_Re__i__times,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( re @ ( times_times_complex @ imaginary_unit @ Z ) )
% 13.25/7.72 = ( uminus_uminus_real @ ( im @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9400_Im__divide__numeral,axiom,
% 13.25/7.72 ! [Z: complex,W: num] :
% 13.25/7.72 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.72 = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9401_Im__divide__of__nat,axiom,
% 13.25/7.72 ! [Z: complex,N: nat] :
% 13.25/7.72 ( ( im @ ( divide1717551699836669952omplex @ Z @ ( semiri8010041392384452111omplex @ N ) ) )
% 13.25/7.72 = ( divide_divide_real @ ( im @ Z ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9402_csqrt__of__real__nonneg,axiom,
% 13.25/7.72 ! [X: complex] :
% 13.25/7.72 ( ( ( im @ X )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) )
% 13.25/7.72 => ( ( csqrt @ X )
% 13.25/7.72 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9403_csqrt__minus,axiom,
% 13.25/7.72 ! [X: complex] :
% 13.25/7.72 ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
% 13.25/7.72 | ( ( ( im @ X )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
% 13.25/7.72 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
% 13.25/7.72 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9404_csqrt__of__real__nonpos,axiom,
% 13.25/7.72 ! [X: complex] :
% 13.25/7.72 ( ( ( im @ X )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
% 13.25/7.72 => ( ( csqrt @ X )
% 13.25/7.72 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9405_imaginary__unit_Osimps_I2_J,axiom,
% 13.25/7.72 ( ( im @ imaginary_unit )
% 13.25/7.72 = one_one_real ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9406_plus__complex_Osimps_I2_J,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( im @ ( plus_plus_complex @ X @ Y3 ) )
% 13.25/7.72 = ( plus_plus_real @ ( im @ X ) @ ( im @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9407_scaleR__complex_Osimps_I2_J,axiom,
% 13.25/7.72 ! [R: real,X: complex] :
% 13.25/7.72 ( ( im @ ( real_V2046097035970521341omplex @ R @ X ) )
% 13.25/7.72 = ( times_times_real @ R @ ( im @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9408_minus__complex_Osimps_I2_J,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( im @ ( minus_minus_complex @ X @ Y3 ) )
% 13.25/7.72 = ( minus_minus_real @ ( im @ X ) @ ( im @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9409_abs__Im__le__cmod,axiom,
% 13.25/7.72 ! [X: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( real_V1022390504157884413omplex @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9410_times__complex_Osimps_I2_J,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( im @ ( times_times_complex @ X @ Y3 ) )
% 13.25/7.72 = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y3 ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9411_cmod__Re__le__iff,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( ( im @ X )
% 13.25/7.72 = ( im @ Y3 ) )
% 13.25/7.72 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 13.25/7.72 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9412_cmod__Im__le__iff,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( ( re @ X )
% 13.25/7.72 = ( re @ Y3 ) )
% 13.25/7.72 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 13.25/7.72 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9413_times__complex_Osimps_I1_J,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( re @ ( times_times_complex @ X @ Y3 ) )
% 13.25/7.72 = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9414_plus__complex_Ocode,axiom,
% 13.25/7.72 ( plus_plus_complex
% 13.25/7.72 = ( ^ [X2: complex,Y: complex] : ( complex2 @ ( plus_plus_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( plus_plus_real @ ( im @ X2 ) @ ( im @ Y ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9415_scaleR__complex_Ocode,axiom,
% 13.25/7.72 ( real_V2046097035970521341omplex
% 13.25/7.72 = ( ^ [R5: real,X2: complex] : ( complex2 @ ( times_times_real @ R5 @ ( re @ X2 ) ) @ ( times_times_real @ R5 @ ( im @ X2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9416_minus__complex_Ocode,axiom,
% 13.25/7.72 ( minus_minus_complex
% 13.25/7.72 = ( ^ [X2: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( minus_minus_real @ ( im @ X2 ) @ ( im @ Y ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9417_csqrt__principal,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 13.25/7.72 | ( ( ( re @ ( csqrt @ Z ) )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9418_cmod__le,axiom,
% 13.25/7.72 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9419_sin__n__Im__cis__pow__n,axiom,
% 13.25/7.72 ! [N: nat,A: real] :
% 13.25/7.72 ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) )
% 13.25/7.72 = ( im @ ( power_power_complex @ ( cis @ A ) @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9420_Re__exp,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( re @ ( exp_complex @ Z ) )
% 13.25/7.72 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( cos_real @ ( im @ Z ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9421_Im__exp,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( im @ ( exp_complex @ Z ) )
% 13.25/7.72 = ( times_times_real @ ( exp_real @ ( re @ Z ) ) @ ( sin_real @ ( im @ Z ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9422_complex__eq,axiom,
% 13.25/7.72 ! [A: complex] :
% 13.25/7.72 ( A
% 13.25/7.72 = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9423_times__complex_Ocode,axiom,
% 13.25/7.72 ( times_times_complex
% 13.25/7.72 = ( ^ [X2: complex,Y: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9424_exp__eq__polar,axiom,
% 13.25/7.72 ( exp_complex
% 13.25/7.72 = ( ^ [Z2: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z2 ) ) ) @ ( cis @ ( im @ Z2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9425_cmod__power2,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.72 = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9426_Im__power2,axiom,
% 13.25/7.72 ! [X: complex] :
% 13.25/7.72 ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.72 = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9427_Re__power2,axiom,
% 13.25/7.72 ! [X: complex] :
% 13.25/7.72 ( ( re @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.72 = ( minus_minus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9428_complex__eq__0,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( Z = zero_zero_complex )
% 13.25/7.72 = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.72 = zero_zero_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9429_norm__complex__def,axiom,
% 13.25/7.72 ( real_V1022390504157884413omplex
% 13.25/7.72 = ( ^ [Z2: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9430_inverse__complex_Osimps_I1_J,axiom,
% 13.25/7.72 ! [X: complex] :
% 13.25/7.72 ( ( re @ ( invers8013647133539491842omplex @ X ) )
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9431_complex__neq__0,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( Z != zero_zero_complex )
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9432_Re__divide,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( re @ ( divide1717551699836669952omplex @ X @ Y3 ) )
% 13.25/7.72 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9433_csqrt__unique,axiom,
% 13.25/7.72 ! [W: complex,Z: complex] :
% 13.25/7.72 ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 13.25/7.72 = Z )
% 13.25/7.72 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
% 13.25/7.72 | ( ( ( re @ W )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
% 13.25/7.72 => ( ( csqrt @ Z )
% 13.25/7.72 = W ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9434_csqrt__square,axiom,
% 13.25/7.72 ! [B: complex] :
% 13.25/7.72 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
% 13.25/7.72 | ( ( ( re @ B )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
% 13.25/7.72 => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.72 = B ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9435_inverse__complex_Osimps_I2_J,axiom,
% 13.25/7.72 ! [X: complex] :
% 13.25/7.72 ( ( im @ ( invers8013647133539491842omplex @ X ) )
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9436_Im__divide,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( im @ ( divide1717551699836669952omplex @ X @ Y3 ) )
% 13.25/7.72 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y3 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9437_complex__abs__le__norm,axiom,
% 13.25/7.72 ! [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 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9438_complex__unit__circle,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( Z != zero_zero_complex )
% 13.25/7.72 => ( ( 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 ) ) ) )
% 13.25/7.72 = one_one_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9439_inverse__complex_Ocode,axiom,
% 13.25/7.72 ( invers8013647133539491842omplex
% 13.25/7.72 = ( ^ [X2: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X2 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X2 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9440_Complex__divide,axiom,
% 13.25/7.72 ( divide1717551699836669952omplex
% 13.25/7.72 = ( ^ [X2: complex,Y: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X2 ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X2 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9441_Im__Reals__divide,axiom,
% 13.25/7.72 ! [R: complex,Z: complex] :
% 13.25/7.72 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 13.25/7.72 => ( ( im @ ( divide1717551699836669952omplex @ R @ Z ) )
% 13.25/7.72 = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9442_Re__Reals__divide,axiom,
% 13.25/7.72 ! [R: complex,Z: complex] :
% 13.25/7.72 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 13.25/7.72 => ( ( re @ ( divide1717551699836669952omplex @ R @ Z ) )
% 13.25/7.72 = ( divide_divide_real @ ( times_times_real @ ( re @ R ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9443_complex__diff__cnj,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
% 13.25/7.72 = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9444_complex__cnj__mult,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( cnj @ ( times_times_complex @ X @ Y3 ) )
% 13.25/7.72 = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9445_complex__cnj__divide,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( cnj @ ( divide1717551699836669952omplex @ X @ Y3 ) )
% 13.25/7.72 = ( divide1717551699836669952omplex @ ( cnj @ X ) @ ( cnj @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9446_complex__cnj__add,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( cnj @ ( plus_plus_complex @ X @ Y3 ) )
% 13.25/7.72 = ( plus_plus_complex @ ( cnj @ X ) @ ( cnj @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9447_complex__cnj__numeral,axiom,
% 13.25/7.72 ! [W: num] :
% 13.25/7.72 ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
% 13.25/7.72 = ( numera6690914467698888265omplex @ W ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9448_complex__cnj__diff,axiom,
% 13.25/7.72 ! [X: complex,Y3: complex] :
% 13.25/7.72 ( ( cnj @ ( minus_minus_complex @ X @ Y3 ) )
% 13.25/7.72 = ( minus_minus_complex @ ( cnj @ X ) @ ( cnj @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9449_complex__cnj__neg__numeral,axiom,
% 13.25/7.72 ! [W: num] :
% 13.25/7.72 ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
% 13.25/7.72 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9450_complex__In__mult__cnj__zero,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( im @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 13.25/7.72 = zero_zero_real ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9451_Re__divide__Reals,axiom,
% 13.25/7.72 ! [R: complex,Z: complex] :
% 13.25/7.72 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 13.25/7.72 => ( ( re @ ( divide1717551699836669952omplex @ Z @ R ) )
% 13.25/7.72 = ( divide_divide_real @ ( re @ Z ) @ ( re @ R ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9452_real__eq__imaginary__iff,axiom,
% 13.25/7.72 ! [Y3: complex,X: complex] :
% 13.25/7.72 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 13.25/7.72 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 13.25/7.72 => ( ( X
% 13.25/7.72 = ( times_times_complex @ imaginary_unit @ Y3 ) )
% 13.25/7.72 = ( ( X = zero_zero_complex )
% 13.25/7.72 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9453_imaginary__eq__real__iff,axiom,
% 13.25/7.72 ! [Y3: complex,X: complex] :
% 13.25/7.72 ( ( member_complex @ Y3 @ real_V2521375963428798218omplex )
% 13.25/7.72 => ( ( member_complex @ X @ real_V2521375963428798218omplex )
% 13.25/7.72 => ( ( ( times_times_complex @ imaginary_unit @ Y3 )
% 13.25/7.72 = X )
% 13.25/7.72 = ( ( X = zero_zero_complex )
% 13.25/7.72 & ( Y3 = zero_zero_complex ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9454_Im__divide__Reals,axiom,
% 13.25/7.72 ! [R: complex,Z: complex] :
% 13.25/7.72 ( ( member_complex @ R @ real_V2521375963428798218omplex )
% 13.25/7.72 => ( ( im @ ( divide1717551699836669952omplex @ Z @ R ) )
% 13.25/7.72 = ( divide_divide_real @ ( im @ Z ) @ ( re @ R ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9455_Re__complex__div__eq__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 13.25/7.72 = zero_zero_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9456_Im__complex__div__eq__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
% 13.25/7.72 = zero_zero_real )
% 13.25/7.72 = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
% 13.25/7.72 = zero_zero_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9457_complex__mod__sqrt__Re__mult__cnj,axiom,
% 13.25/7.72 ( real_V1022390504157884413omplex
% 13.25/7.72 = ( ^ [Z2: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9458_Re__complex__div__lt__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 13.25/7.72 = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9459_Re__complex__div__gt__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 13.25/7.72 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9460_Re__complex__div__le__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 13.25/7.72 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9461_Re__complex__div__ge__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 13.25/7.72 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9462_Im__complex__div__lt__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 13.25/7.72 = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9463_Im__complex__div__gt__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 13.25/7.72 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9464_Im__complex__div__le__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
% 13.25/7.72 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9465_Im__complex__div__ge__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 13.25/7.72 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9466_complex__mod__mult__cnj,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
% 13.25/7.72 = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9467_complex__div__gt__0,axiom,
% 13.25/7.72 ! [A: complex,B: complex] :
% 13.25/7.72 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 13.25/7.72 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
% 13.25/7.72 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 13.25/7.72 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9468_complex__norm__square,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.72 = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9469_complex__add__cnj,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
% 13.25/7.72 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9470_complex__div__cnj,axiom,
% 13.25/7.72 ( divide1717551699836669952omplex
% 13.25/7.72 = ( ^ [A3: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9471_cnj__add__mult__eq__Re,axiom,
% 13.25/7.72 ! [Z: complex,W: complex] :
% 13.25/7.72 ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
% 13.25/7.72 = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9472_complex__mult__cnj,axiom,
% 13.25/7.72 ! [Z: complex] :
% 13.25/7.72 ( ( times_times_complex @ Z @ ( cnj @ Z ) )
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9473_card__Collect__less__nat,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( finite_card_nat
% 13.25/7.72 @ ( collect_nat
% 13.25/7.72 @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 13.25/7.72 = N ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9474_card__atLeastLessThan,axiom,
% 13.25/7.72 ! [L2: nat,U: nat] :
% 13.25/7.72 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L2 @ U ) )
% 13.25/7.72 = ( minus_minus_nat @ U @ L2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9475_card__Collect__le__nat,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( finite_card_nat
% 13.25/7.72 @ ( collect_nat
% 13.25/7.72 @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 13.25/7.72 = ( suc @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9476_card__atLeastAtMost,axiom,
% 13.25/7.72 ! [L2: nat,U: nat] :
% 13.25/7.72 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L2 @ U ) )
% 13.25/7.72 = ( minus_minus_nat @ ( suc @ U ) @ L2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9477_card__atLeastLessThan__int,axiom,
% 13.25/7.72 ! [L2: int,U: int] :
% 13.25/7.72 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L2 @ U ) )
% 13.25/7.72 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9478_card__atLeastAtMost__int,axiom,
% 13.25/7.72 ! [L2: int,U: int] :
% 13.25/7.72 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L2 @ U ) )
% 13.25/7.72 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L2 ) @ one_one_int ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9479_nat_Odisc__eq__case_I2_J,axiom,
% 13.25/7.72 ! [Nat: nat] :
% 13.25/7.72 ( ( Nat != zero_zero_nat )
% 13.25/7.72 = ( case_nat_o @ $false
% 13.25/7.72 @ ^ [Uu3: nat] : $true
% 13.25/7.72 @ Nat ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9480_nat_Odisc__eq__case_I1_J,axiom,
% 13.25/7.72 ! [Nat: nat] :
% 13.25/7.72 ( ( Nat = zero_zero_nat )
% 13.25/7.72 = ( case_nat_o @ $true
% 13.25/7.72 @ ^ [Uu3: nat] : $false
% 13.25/7.72 @ Nat ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9481_card__less__Suc2,axiom,
% 13.25/7.72 ! [M5: set_nat,I: nat] :
% 13.25/7.72 ( ~ ( member_nat @ zero_zero_nat @ M5 )
% 13.25/7.72 => ( ( finite_card_nat
% 13.25/7.72 @ ( collect_nat
% 13.25/7.72 @ ^ [K2: nat] :
% 13.25/7.72 ( ( member_nat @ ( suc @ K2 ) @ M5 )
% 13.25/7.72 & ( ord_less_nat @ K2 @ I ) ) ) )
% 13.25/7.72 = ( finite_card_nat
% 13.25/7.72 @ ( collect_nat
% 13.25/7.72 @ ^ [K2: nat] :
% 13.25/7.72 ( ( member_nat @ K2 @ M5 )
% 13.25/7.72 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9482_card__less__Suc,axiom,
% 13.25/7.72 ! [M5: set_nat,I: nat] :
% 13.25/7.72 ( ( member_nat @ zero_zero_nat @ M5 )
% 13.25/7.72 => ( ( suc
% 13.25/7.72 @ ( finite_card_nat
% 13.25/7.72 @ ( collect_nat
% 13.25/7.72 @ ^ [K2: nat] :
% 13.25/7.72 ( ( member_nat @ ( suc @ K2 ) @ M5 )
% 13.25/7.72 & ( ord_less_nat @ K2 @ I ) ) ) ) )
% 13.25/7.72 = ( finite_card_nat
% 13.25/7.72 @ ( collect_nat
% 13.25/7.72 @ ^ [K2: nat] :
% 13.25/7.72 ( ( member_nat @ K2 @ M5 )
% 13.25/7.72 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9483_card__less,axiom,
% 13.25/7.72 ! [M5: set_nat,I: nat] :
% 13.25/7.72 ( ( member_nat @ zero_zero_nat @ M5 )
% 13.25/7.72 => ( ( finite_card_nat
% 13.25/7.72 @ ( collect_nat
% 13.25/7.72 @ ^ [K2: nat] :
% 13.25/7.72 ( ( member_nat @ K2 @ M5 )
% 13.25/7.72 & ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
% 13.25/7.72 != zero_zero_nat ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9484_subset__card__intvl__is__intvl,axiom,
% 13.25/7.72 ! [A2: set_nat,K: nat] :
% 13.25/7.72 ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 13.25/7.72 => ( A2
% 13.25/7.72 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9485_less__eq__nat_Osimps_I2_J,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 13.25/7.72 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9486_max__Suc2,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 13.25/7.72 = ( case_nat_nat @ ( suc @ N )
% 13.25/7.72 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
% 13.25/7.72 @ M ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9487_max__Suc1,axiom,
% 13.25/7.72 ! [N: nat,M: nat] :
% 13.25/7.72 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 13.25/7.72 = ( case_nat_nat @ ( suc @ N )
% 13.25/7.72 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
% 13.25/7.72 @ M ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9488_subset__eq__atLeast0__lessThan__card,axiom,
% 13.25/7.72 ! [N3: set_nat,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_set_nat @ N3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 13.25/7.72 => ( ord_less_eq_nat @ ( finite_card_nat @ N3 ) @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9489_card__sum__le__nat__sum,axiom,
% 13.25/7.72 ! [S2: set_nat] :
% 13.25/7.72 ( ord_less_eq_nat
% 13.25/7.72 @ ( groups3542108847815614940at_nat
% 13.25/7.72 @ ^ [X2: nat] : X2
% 13.25/7.72 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
% 13.25/7.72 @ ( groups3542108847815614940at_nat
% 13.25/7.72 @ ^ [X2: nat] : X2
% 13.25/7.72 @ S2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9490_card__nth__roots,axiom,
% 13.25/7.72 ! [C: complex,N: nat] :
% 13.25/7.72 ( ( C != zero_zero_complex )
% 13.25/7.72 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( finite_card_complex
% 13.25/7.72 @ ( collect_complex
% 13.25/7.72 @ ^ [Z2: complex] :
% 13.25/7.72 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.72 = C ) ) )
% 13.25/7.72 = N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9491_card__roots__unity__eq,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( finite_card_complex
% 13.25/7.72 @ ( collect_complex
% 13.25/7.72 @ ^ [Z2: complex] :
% 13.25/7.72 ( ( power_power_complex @ Z2 @ N )
% 13.25/7.72 = one_one_complex ) ) )
% 13.25/7.72 = N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9492_diff__Suc,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 13.25/7.72 = ( case_nat_nat @ zero_zero_nat
% 13.25/7.72 @ ^ [K2: nat] : K2
% 13.25/7.72 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9493_floor__real__def,axiom,
% 13.25/7.72 ( archim6058952711729229775r_real
% 13.25/7.72 = ( ^ [X2: real] :
% 13.25/7.72 ( the_int
% 13.25/7.72 @ ^ [Z2: int] :
% 13.25/7.72 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X2 )
% 13.25/7.72 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9494_floor__rat__def,axiom,
% 13.25/7.72 ( archim3151403230148437115or_rat
% 13.25/7.72 = ( ^ [X2: rat] :
% 13.25/7.72 ( the_int
% 13.25/7.72 @ ^ [Z2: int] :
% 13.25/7.72 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X2 )
% 13.25/7.72 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9495_less__eq__rat__def,axiom,
% 13.25/7.72 ( ord_less_eq_rat
% 13.25/7.72 = ( ^ [X2: rat,Y: rat] :
% 13.25/7.72 ( ( ord_less_rat @ X2 @ Y )
% 13.25/7.72 | ( X2 = Y ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9496_obtain__pos__sum,axiom,
% 13.25/7.72 ! [R: rat] :
% 13.25/7.72 ( ( ord_less_rat @ zero_zero_rat @ R )
% 13.25/7.72 => ~ ! [S3: rat] :
% 13.25/7.72 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 13.25/7.72 => ! [T5: rat] :
% 13.25/7.72 ( ( ord_less_rat @ zero_zero_rat @ T5 )
% 13.25/7.72 => ( R
% 13.25/7.72 != ( plus_plus_rat @ S3 @ T5 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9497_sgn__rat__def,axiom,
% 13.25/7.72 ( sgn_sgn_rat
% 13.25/7.72 = ( ^ [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 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9498_abs__rat__def,axiom,
% 13.25/7.72 ( abs_abs_rat
% 13.25/7.72 = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9499_pred__def,axiom,
% 13.25/7.72 ( pred
% 13.25/7.72 = ( case_nat_nat @ zero_zero_nat
% 13.25/7.72 @ ^ [X24: nat] : X24 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9500_rat__inverse__code,axiom,
% 13.25/7.72 ! [P6: rat] :
% 13.25/7.72 ( ( quotient_of @ ( inverse_inverse_rat @ P6 ) )
% 13.25/7.72 = ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [A3: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B4 ) @ ( abs_abs_int @ A3 ) ) )
% 13.25/7.72 @ ( quotient_of @ P6 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9501_normalize__negative,axiom,
% 13.25/7.72 ! [Q2: int,P6: int] :
% 13.25/7.72 ( ( ord_less_int @ Q2 @ zero_zero_int )
% 13.25/7.72 => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q2 ) )
% 13.25/7.72 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9502_quotient__of__number_I3_J,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 13.25/7.72 = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9503_quotient__of__number_I5_J,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 13.25/7.72 = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9504_divide__rat__def,axiom,
% 13.25/7.72 ( divide_divide_rat
% 13.25/7.72 = ( ^ [Q5: rat,R5: rat] : ( times_times_rat @ Q5 @ ( inverse_inverse_rat @ R5 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9505_diff__rat__def,axiom,
% 13.25/7.72 ( minus_minus_rat
% 13.25/7.72 = ( ^ [Q5: rat,R5: rat] : ( plus_plus_rat @ Q5 @ ( uminus_uminus_rat @ R5 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9506_rat__divide__code,axiom,
% 13.25/7.72 ! [P6: rat,Q2: rat] :
% 13.25/7.72 ( ( quotient_of @ ( divide_divide_rat @ P6 @ Q2 ) )
% 13.25/7.72 = ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [A3: int,C3: int] :
% 13.25/7.72 ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B4 ) ) )
% 13.25/7.72 @ ( quotient_of @ Q2 ) )
% 13.25/7.72 @ ( quotient_of @ P6 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9507_rat__times__code,axiom,
% 13.25/7.72 ! [P6: rat,Q2: rat] :
% 13.25/7.72 ( ( quotient_of @ ( times_times_rat @ P6 @ Q2 ) )
% 13.25/7.72 = ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [A3: int,C3: int] :
% 13.25/7.72 ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B4 ) @ ( times_times_int @ C3 @ D2 ) ) )
% 13.25/7.72 @ ( quotient_of @ Q2 ) )
% 13.25/7.72 @ ( quotient_of @ P6 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9508_quotient__of__div,axiom,
% 13.25/7.72 ! [R: rat,N: int,D: int] :
% 13.25/7.72 ( ( ( quotient_of @ R )
% 13.25/7.72 = ( product_Pair_int_int @ N @ D ) )
% 13.25/7.72 => ( R
% 13.25/7.72 = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9509_rat__plus__code,axiom,
% 13.25/7.72 ! [P6: rat,Q2: rat] :
% 13.25/7.72 ( ( quotient_of @ ( plus_plus_rat @ P6 @ Q2 ) )
% 13.25/7.72 = ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [A3: int,C3: int] :
% 13.25/7.72 ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
% 13.25/7.72 @ ( quotient_of @ Q2 ) )
% 13.25/7.72 @ ( quotient_of @ P6 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9510_rat__minus__code,axiom,
% 13.25/7.72 ! [P6: rat,Q2: rat] :
% 13.25/7.72 ( ( quotient_of @ ( minus_minus_rat @ P6 @ Q2 ) )
% 13.25/7.72 = ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [A3: int,C3: int] :
% 13.25/7.72 ( produc4245557441103728435nt_int
% 13.25/7.72 @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
% 13.25/7.72 @ ( quotient_of @ Q2 ) )
% 13.25/7.72 @ ( quotient_of @ P6 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9511_quotient__of__denom__pos,axiom,
% 13.25/7.72 ! [R: rat,P6: int,Q2: int] :
% 13.25/7.72 ( ( ( quotient_of @ R )
% 13.25/7.72 = ( product_Pair_int_int @ P6 @ Q2 ) )
% 13.25/7.72 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9512_normalize__denom__pos,axiom,
% 13.25/7.72 ! [R: product_prod_int_int,P6: int,Q2: int] :
% 13.25/7.72 ( ( ( normalize @ R )
% 13.25/7.72 = ( product_Pair_int_int @ P6 @ Q2 ) )
% 13.25/7.72 => ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9513_normalize__crossproduct,axiom,
% 13.25/7.72 ! [Q2: int,S: int,P6: int,R: int] :
% 13.25/7.72 ( ( Q2 != zero_zero_int )
% 13.25/7.72 => ( ( S != zero_zero_int )
% 13.25/7.72 => ( ( ( normalize @ ( product_Pair_int_int @ P6 @ Q2 ) )
% 13.25/7.72 = ( normalize @ ( product_Pair_int_int @ R @ S ) ) )
% 13.25/7.72 => ( ( times_times_int @ P6 @ S )
% 13.25/7.72 = ( times_times_int @ R @ Q2 ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9514_rat__less__code,axiom,
% 13.25/7.72 ( ord_less_rat
% 13.25/7.72 = ( ^ [P4: rat,Q5: rat] :
% 13.25/7.72 ( produc4947309494688390418_int_o
% 13.25/7.72 @ ^ [A3: int,C3: int] :
% 13.25/7.72 ( produc4947309494688390418_int_o
% 13.25/7.72 @ ^ [B4: int,D2: int] : ( ord_less_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
% 13.25/7.72 @ ( quotient_of @ Q5 ) )
% 13.25/7.72 @ ( quotient_of @ P4 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9515_rat__less__eq__code,axiom,
% 13.25/7.72 ( ord_less_eq_rat
% 13.25/7.72 = ( ^ [P4: rat,Q5: rat] :
% 13.25/7.72 ( produc4947309494688390418_int_o
% 13.25/7.72 @ ^ [A3: int,C3: int] :
% 13.25/7.72 ( produc4947309494688390418_int_o
% 13.25/7.72 @ ^ [B4: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A3 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
% 13.25/7.72 @ ( quotient_of @ Q5 ) )
% 13.25/7.72 @ ( quotient_of @ P4 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9516_prod__decode__aux_Oelims,axiom,
% 13.25/7.72 ! [X: nat,Xa: nat,Y3: product_prod_nat_nat] :
% 13.25/7.72 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( ( ord_less_eq_nat @ Xa @ X )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9517_prod__decode__aux_Osimps,axiom,
% 13.25/7.72 ( nat_prod_decode_aux
% 13.25/7.72 = ( ^ [K2: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K2 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K2 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus_nat @ M3 @ ( suc @ K2 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9518_drop__bit__numeral__minus__bit1,axiom,
% 13.25/7.72 ! [L2: num,K: num] :
% 13.25/7.72 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 13.25/7.72 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9519_drop__bit__nonnegative__int__iff,axiom,
% 13.25/7.72 ! [N: nat,K: int] :
% 13.25/7.72 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 13.25/7.72 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9520_drop__bit__negative__int__iff,axiom,
% 13.25/7.72 ! [N: nat,K: int] :
% 13.25/7.72 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 13.25/7.72 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9521_drop__bit__minus__one,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ one_one_int ) )
% 13.25/7.72 = ( uminus_uminus_int @ one_one_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9522_drop__bit__Suc__minus__bit0,axiom,
% 13.25/7.72 ! [N: nat,K: num] :
% 13.25/7.72 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 13.25/7.72 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9523_drop__bit__numeral__minus__bit0,axiom,
% 13.25/7.72 ! [L2: num,K: num] :
% 13.25/7.72 ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 13.25/7.72 = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9524_drop__bit__Suc__minus__bit1,axiom,
% 13.25/7.72 ! [N: nat,K: num] :
% 13.25/7.72 ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 13.25/7.72 = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9525_drop__bit__push__bit__int,axiom,
% 13.25/7.72 ! [M: nat,N: nat,K: int] :
% 13.25/7.72 ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N @ K ) )
% 13.25/7.72 = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N @ M ) @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9526_drop__bit__int__def,axiom,
% 13.25/7.72 ( bit_se8568078237143864401it_int
% 13.25/7.72 = ( ^ [N2: nat,K2: int] : ( divide_divide_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9527_prod__decode__aux_Opelims,axiom,
% 13.25/7.72 ! [X: nat,Xa: nat,Y3: product_prod_nat_nat] :
% 13.25/7.72 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 13.25/7.72 => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
% 13.25/7.72 & ( ~ ( ord_less_eq_nat @ Xa @ X )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 13.25/7.72 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9528_Frct__code__post_I5_J,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 13.25/7.72 = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9529_Frct__code__post_I6_J,axiom,
% 13.25/7.72 ! [K: num,L2: num] :
% 13.25/7.72 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L2 ) ) )
% 13.25/7.72 = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9530_drop__bit__of__Suc__0,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 13.25/7.72 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9531_drop__bit__nat__eq,axiom,
% 13.25/7.72 ! [N: nat,K: int] :
% 13.25/7.72 ( ( bit_se8570568707652914677it_nat @ N @ ( nat2 @ K ) )
% 13.25/7.72 = ( nat2 @ ( bit_se8568078237143864401it_int @ N @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9532_drop__bit__nat__def,axiom,
% 13.25/7.72 ( bit_se8570568707652914677it_nat
% 13.25/7.72 = ( ^ [N2: nat,M3: nat] : ( divide_divide_nat @ M3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9533_Frct__code__post_I4_J,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 13.25/7.72 = ( numeral_numeral_rat @ K ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9534_Suc__0__div__numeral,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 13.25/7.72 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9535_Suc__0__mod__numeral,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 13.25/7.72 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9536_finite__enumerate,axiom,
% 13.25/7.72 ! [S2: set_nat] :
% 13.25/7.72 ( ( finite_finite_nat @ S2 )
% 13.25/7.72 => ? [R2: nat > nat] :
% 13.25/7.72 ( ( strict1292158309912662752at_nat @ R2 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
% 13.25/7.72 & ! [N7: nat] :
% 13.25/7.72 ( ( ord_less_nat @ N7 @ ( finite_card_nat @ S2 ) )
% 13.25/7.72 => ( member_nat @ ( R2 @ N7 ) @ S2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9537_fst__divmod__nat,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N ) )
% 13.25/7.72 = ( divide_divide_nat @ M @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9538_snd__divmod__nat,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N ) )
% 13.25/7.72 = ( modulo_modulo_nat @ M @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9539_fst__divmod__integer,axiom,
% 13.25/7.72 ! [K: code_integer,L2: code_integer] :
% 13.25/7.72 ( ( produc8508995932063986495nteger @ ( code_divmod_integer @ K @ L2 ) )
% 13.25/7.72 = ( divide6298287555418463151nteger @ K @ L2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9540_fst__divmod__abs,axiom,
% 13.25/7.72 ! [K: code_integer,L2: code_integer] :
% 13.25/7.72 ( ( produc8508995932063986495nteger @ ( code_divmod_abs @ K @ L2 ) )
% 13.25/7.72 = ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9541_quotient__of__denom__pos_H,axiom,
% 13.25/7.72 ! [R: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9542_minus__one__mod__numeral,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.72 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9543_one__mod__minus__numeral,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9544_numeral__mod__minus__numeral,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.72 = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9545_minus__numeral__mod__numeral,axiom,
% 13.25/7.72 ! [M: num,N: num] :
% 13.25/7.72 ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.72 = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9546_Divides_Oadjust__mod__def,axiom,
% 13.25/7.72 ( adjust_mod
% 13.25/7.72 = ( ^ [L: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L @ R5 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9547_bezw_Oelims,axiom,
% 13.25/7.72 ! [X: nat,Xa: nat,Y3: product_prod_int_int] :
% 13.25/7.72 ( ( ( bezw @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( ( Xa = zero_zero_nat )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 13.25/7.72 & ( ( Xa != zero_zero_nat )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9548_bezw_Osimps,axiom,
% 13.25/7.72 ( bezw
% 13.25/7.72 = ( ^ [X2: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X2 @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9549_bezw_Opelims,axiom,
% 13.25/7.72 ! [X: nat,Xa: nat,Y3: product_prod_int_int] :
% 13.25/7.72 ( ( ( bezw @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) )
% 13.25/7.72 => ~ ( ( ( ( Xa = zero_zero_nat )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 13.25/7.72 & ( ( Xa != zero_zero_nat )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( 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 ) ) ) ) ) ) ) )
% 13.25/7.72 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9550_bezw__non__0,axiom,
% 13.25/7.72 ! [Y3: nat,X: nat] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 13.25/7.72 => ( ( bezw @ X @ Y3 )
% 13.25/7.72 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9551_normalize__def,axiom,
% 13.25/7.72 ( normalize
% 13.25/7.72 = ( ^ [P4: product_prod_int_int] :
% 13.25/7.72 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P4 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) )
% 13.25/7.72 @ ( if_Pro3027730157355071871nt_int
% 13.25/7.72 @ ( ( product_snd_int_int @ P4 )
% 13.25/7.72 = zero_zero_int )
% 13.25/7.72 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 13.25/7.72 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P4 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P4 ) @ ( product_snd_int_int @ P4 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9552_divmod__integer__eq__cases,axiom,
% 13.25/7.72 ( code_divmod_integer
% 13.25/7.72 = ( ^ [K2: code_integer,L: code_integer] :
% 13.25/7.72 ( if_Pro6119634080678213985nteger @ ( K2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 13.25/7.72 @ ( if_Pro6119634080678213985nteger @ ( L = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K2 )
% 13.25/7.72 @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L
% 13.25/7.72 @ ( if_Pro6119634080678213985nteger
% 13.25/7.72 @ ( ( sgn_sgn_Code_integer @ K2 )
% 13.25/7.72 = ( sgn_sgn_Code_integer @ L ) )
% 13.25/7.72 @ ( code_divmod_abs @ K2 @ L )
% 13.25/7.72 @ ( produc6916734918728496179nteger
% 13.25/7.72 @ ^ [R5: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L ) @ S6 ) ) )
% 13.25/7.72 @ ( code_divmod_abs @ K2 @ L ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9553_gcd__pos__int,axiom,
% 13.25/7.72 ! [M: int,N: int] :
% 13.25/7.72 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 13.25/7.72 = ( ( M != zero_zero_int )
% 13.25/7.72 | ( N != zero_zero_int ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9554_gcd__neg__numeral__1__int,axiom,
% 13.25/7.72 ! [N: num,X: int] :
% 13.25/7.72 ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ X )
% 13.25/7.72 = ( gcd_gcd_int @ ( numeral_numeral_int @ N ) @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9555_gcd__neg__numeral__2__int,axiom,
% 13.25/7.72 ! [X: int,N: num] :
% 13.25/7.72 ( ( gcd_gcd_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.72 = ( gcd_gcd_int @ X @ ( numeral_numeral_int @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9556_gcd__ge__0__int,axiom,
% 13.25/7.72 ! [X: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X @ Y3 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9557_bezout__int,axiom,
% 13.25/7.72 ! [X: int,Y3: int] :
% 13.25/7.72 ? [U4: int,V2: int] :
% 13.25/7.72 ( ( plus_plus_int @ ( times_times_int @ U4 @ X ) @ ( times_times_int @ V2 @ Y3 ) )
% 13.25/7.72 = ( gcd_gcd_int @ X @ Y3 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9558_gcd__mult__distrib__int,axiom,
% 13.25/7.72 ! [K: int,M: int,N: int] :
% 13.25/7.72 ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N ) )
% 13.25/7.72 = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9559_gcd__le2__int,axiom,
% 13.25/7.72 ! [B: int,A: int] :
% 13.25/7.72 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.72 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9560_gcd__le1__int,axiom,
% 13.25/7.72 ! [A: int,B: int] :
% 13.25/7.72 ( ( ord_less_int @ zero_zero_int @ A )
% 13.25/7.72 => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9561_gcd__cases__int,axiom,
% 13.25/7.72 ! [X: int,Y3: int,P: int > $o] :
% 13.25/7.72 ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( P @ ( gcd_gcd_int @ X @ Y3 ) ) ) )
% 13.25/7.72 => ( ( ( ord_less_eq_int @ zero_zero_int @ X )
% 13.25/7.72 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 13.25/7.72 => ( P @ ( gcd_gcd_int @ X @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 13.25/7.72 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 13.25/7.72 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ Y3 ) ) ) )
% 13.25/7.72 => ( ( ( ord_less_eq_int @ X @ zero_zero_int )
% 13.25/7.72 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 13.25/7.72 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 13.25/7.72 => ( P @ ( gcd_gcd_int @ X @ Y3 ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9562_gcd__unique__int,axiom,
% 13.25/7.72 ! [D: int,A: int,B: int] :
% 13.25/7.72 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 13.25/7.72 & ( dvd_dvd_int @ D @ A )
% 13.25/7.72 & ( dvd_dvd_int @ D @ B )
% 13.25/7.72 & ! [E3: int] :
% 13.25/7.72 ( ( ( dvd_dvd_int @ E3 @ A )
% 13.25/7.72 & ( dvd_dvd_int @ E3 @ B ) )
% 13.25/7.72 => ( dvd_dvd_int @ E3 @ D ) ) )
% 13.25/7.72 = ( D
% 13.25/7.72 = ( gcd_gcd_int @ A @ B ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9563_gcd__non__0__int,axiom,
% 13.25/7.72 ! [Y3: int,X: int] :
% 13.25/7.72 ( ( ord_less_int @ zero_zero_int @ Y3 )
% 13.25/7.72 => ( ( gcd_gcd_int @ X @ Y3 )
% 13.25/7.72 = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X @ Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9564_nat__descend__induct,axiom,
% 13.25/7.72 ! [N: nat,P: nat > $o,M: nat] :
% 13.25/7.72 ( ! [K3: nat] :
% 13.25/7.72 ( ( ord_less_nat @ N @ K3 )
% 13.25/7.72 => ( P @ K3 ) )
% 13.25/7.72 => ( ! [K3: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ K3 @ N )
% 13.25/7.72 => ( ! [I3: nat] :
% 13.25/7.72 ( ( ord_less_nat @ K3 @ I3 )
% 13.25/7.72 => ( P @ I3 ) )
% 13.25/7.72 => ( P @ K3 ) ) )
% 13.25/7.72 => ( P @ M ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9565_card__greaterThanLessThan__int,axiom,
% 13.25/7.72 ! [L2: int,U: int] :
% 13.25/7.72 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L2 @ U ) )
% 13.25/7.72 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L2 @ one_one_int ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9566_gcd__1__nat,axiom,
% 13.25/7.72 ! [M: nat] :
% 13.25/7.72 ( ( gcd_gcd_nat @ M @ one_one_nat )
% 13.25/7.72 = one_one_nat ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9567_gcd__pos__nat,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 13.25/7.72 = ( ( M != zero_zero_nat )
% 13.25/7.72 | ( N != zero_zero_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9568_gcd__mult__distrib__nat,axiom,
% 13.25/7.72 ! [K: nat,M: nat,N: nat] :
% 13.25/7.72 ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N ) )
% 13.25/7.72 = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9569_gcd__le1__nat,axiom,
% 13.25/7.72 ! [A: nat,B: nat] :
% 13.25/7.72 ( ( A != zero_zero_nat )
% 13.25/7.72 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9570_gcd__le2__nat,axiom,
% 13.25/7.72 ! [B: nat,A: nat] :
% 13.25/7.72 ( ( B != zero_zero_nat )
% 13.25/7.72 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9571_gcd__diff2__nat,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.72 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 13.25/7.72 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9572_gcd__diff1__nat,axiom,
% 13.25/7.72 ! [N: nat,M: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ N @ M )
% 13.25/7.72 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 13.25/7.72 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9573_bezout__nat,axiom,
% 13.25/7.72 ! [A: nat,B: nat] :
% 13.25/7.72 ( ( A != zero_zero_nat )
% 13.25/7.72 => ? [X3: nat,Y4: nat] :
% 13.25/7.72 ( ( times_times_nat @ A @ X3 )
% 13.25/7.72 = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9574_bezout__gcd__nat_H,axiom,
% 13.25/7.72 ! [B: nat,A: nat] :
% 13.25/7.72 ? [X3: nat,Y4: nat] :
% 13.25/7.72 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X3 ) )
% 13.25/7.72 & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
% 13.25/7.72 = ( gcd_gcd_nat @ A @ B ) ) )
% 13.25/7.72 | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X3 ) )
% 13.25/7.72 & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
% 13.25/7.72 = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9575_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 13.25/7.72 ! [L2: int,U: int] :
% 13.25/7.72 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 13.25/7.72 = ( set_or5832277885323065728an_int @ L2 @ U ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9576_bezw__aux,axiom,
% 13.25/7.72 ! [X: nat,Y3: nat] :
% 13.25/7.72 ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y3 ) )
% 13.25/7.72 = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9577_Code__Numeral_Onegative__def,axiom,
% 13.25/7.72 ( code_negative
% 13.25/7.72 = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9578_Code__Target__Int_Onegative__def,axiom,
% 13.25/7.72 ( code_Target_negative
% 13.25/7.72 = ( comp_int_int_num @ uminus_uminus_int @ numeral_numeral_int ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9579_xor__minus__numerals_I1_J,axiom,
% 13.25/7.72 ! [N: num,K: int] :
% 13.25/7.72 ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K )
% 13.25/7.72 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9580_card__greaterThanLessThan,axiom,
% 13.25/7.72 ! [L2: nat,U: nat] :
% 13.25/7.72 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L2 @ U ) )
% 13.25/7.72 = ( minus_minus_nat @ U @ ( suc @ L2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9581_xor__minus__numerals_I2_J,axiom,
% 13.25/7.72 ! [K: int,N: num] :
% 13.25/7.72 ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.72 = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9582_tanh__real__bounds,axiom,
% 13.25/7.72 ! [X: real] : ( member_real @ ( tanh_real @ X ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9583_sub__BitM__One__eq,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( neg_numeral_sub_int @ ( bitM @ N ) @ one )
% 13.25/7.72 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N @ one ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9584_nat__of__integer__non__positive,axiom,
% 13.25/7.72 ! [K: code_integer] :
% 13.25/7.72 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 13.25/7.72 => ( ( code_nat_of_integer @ K )
% 13.25/7.72 = zero_zero_nat ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9585_Suc__funpow,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( compow_nat_nat @ N @ suc )
% 13.25/7.72 = ( plus_plus_nat @ N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9586_nat__of__integer__code__post_I3_J,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 13.25/7.72 = ( numeral_numeral_nat @ K ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9587_nat__of__integer__code__post_I2_J,axiom,
% 13.25/7.72 ( ( code_nat_of_integer @ one_one_Code_integer )
% 13.25/7.72 = one_one_nat ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9588_nat__of__integer__code,axiom,
% 13.25/7.72 ( code_nat_of_integer
% 13.25/7.72 = ( ^ [K2: code_integer] :
% 13.25/7.72 ( if_nat @ ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 13.25/7.72 @ ( produc1555791787009142072er_nat
% 13.25/7.72 @ ^ [L: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L ) @ ( code_nat_of_integer @ L ) ) @ one_one_nat ) )
% 13.25/7.72 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9589_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 13.25/7.72 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 13.25/7.72 @ ^ [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X2 )
% 13.25/7.72 @ ^ [X2: nat,Y: nat] : ( ord_less_nat @ Y @ X2 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9590_int__of__integer__code,axiom,
% 13.25/7.72 ( code_int_of_integer
% 13.25/7.72 = ( ^ [K2: code_integer] :
% 13.25/7.72 ( if_int @ ( ord_le6747313008572928689nteger @ K2 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K2 ) ) )
% 13.25/7.72 @ ( if_int @ ( K2 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 13.25/7.72 @ ( produc1553301316500091796er_int
% 13.25/7.72 @ ^ [L: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L ) ) @ one_one_int ) )
% 13.25/7.72 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9591_int__of__integer__numeral,axiom,
% 13.25/7.72 ! [K: num] :
% 13.25/7.72 ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K ) )
% 13.25/7.72 = ( numeral_numeral_int @ K ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9592_plus__integer_Orep__eq,axiom,
% 13.25/7.72 ! [X: code_integer,Xa: code_integer] :
% 13.25/7.72 ( ( code_int_of_integer @ ( plus_p5714425477246183910nteger @ X @ Xa ) )
% 13.25/7.72 = ( plus_plus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9593_times__integer_Orep__eq,axiom,
% 13.25/7.72 ! [X: code_integer,Xa: code_integer] :
% 13.25/7.72 ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa ) )
% 13.25/7.72 = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9594_minus__integer_Orep__eq,axiom,
% 13.25/7.72 ! [X: code_integer,Xa: code_integer] :
% 13.25/7.72 ( ( code_int_of_integer @ ( minus_8373710615458151222nteger @ X @ Xa ) )
% 13.25/7.72 = ( minus_minus_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9595_divide__integer_Orep__eq,axiom,
% 13.25/7.72 ! [X: code_integer,Xa: code_integer] :
% 13.25/7.72 ( ( code_int_of_integer @ ( divide6298287555418463151nteger @ X @ Xa ) )
% 13.25/7.72 = ( divide_divide_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9596_less__integer_Orep__eq,axiom,
% 13.25/7.72 ( ord_le6747313008572928689nteger
% 13.25/7.72 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9597_integer__less__iff,axiom,
% 13.25/7.72 ( ord_le6747313008572928689nteger
% 13.25/7.72 = ( ^ [K2: code_integer,L: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9598_integer__less__eq__iff,axiom,
% 13.25/7.72 ( ord_le3102999989581377725nteger
% 13.25/7.72 = ( ^ [K2: code_integer,L: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9599_less__eq__integer_Orep__eq,axiom,
% 13.25/7.72 ( ord_le3102999989581377725nteger
% 13.25/7.72 = ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9600_times__int_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 13.25/7.72 ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 13.25/7.72 = ( abs_Integ
% 13.25/7.72 @ ( produc27273713700761075at_nat
% 13.25/7.72 @ ^ [X2: nat,Y: nat] :
% 13.25/7.72 ( produc2626176000494625587at_nat
% 13.25/7.72 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) )
% 13.25/7.72 @ Xa
% 13.25/7.72 @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9601_nat_Oabs__eq,axiom,
% 13.25/7.72 ! [X: product_prod_nat_nat] :
% 13.25/7.72 ( ( nat2 @ ( abs_Integ @ X ) )
% 13.25/7.72 = ( produc6842872674320459806at_nat @ minus_minus_nat @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9602_one__int__def,axiom,
% 13.25/7.72 ( one_one_int
% 13.25/7.72 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9603_less__int_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 13.25/7.72 ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 13.25/7.72 = ( produc8739625826339149834_nat_o
% 13.25/7.72 @ ^ [X2: nat,Y: nat] :
% 13.25/7.72 ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) )
% 13.25/7.72 @ Xa
% 13.25/7.72 @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9604_less__eq__int_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 13.25/7.72 ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 13.25/7.72 = ( produc8739625826339149834_nat_o
% 13.25/7.72 @ ^ [X2: nat,Y: nat] :
% 13.25/7.72 ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) )
% 13.25/7.72 @ Xa
% 13.25/7.72 @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9605_plus__int_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 13.25/7.72 ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 13.25/7.72 = ( abs_Integ
% 13.25/7.72 @ ( produc27273713700761075at_nat
% 13.25/7.72 @ ^ [X2: nat,Y: nat] :
% 13.25/7.72 ( produc2626176000494625587at_nat
% 13.25/7.72 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 13.25/7.72 @ Xa
% 13.25/7.72 @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9606_minus__int_Oabs__eq,axiom,
% 13.25/7.72 ! [Xa: product_prod_nat_nat,X: product_prod_nat_nat] :
% 13.25/7.72 ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 13.25/7.72 = ( abs_Integ
% 13.25/7.72 @ ( produc27273713700761075at_nat
% 13.25/7.72 @ ^ [X2: nat,Y: nat] :
% 13.25/7.72 ( produc2626176000494625587at_nat
% 13.25/7.72 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) )
% 13.25/7.72 @ Xa
% 13.25/7.72 @ X ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9607_Gcd__remove0__nat,axiom,
% 13.25/7.72 ! [M5: set_nat] :
% 13.25/7.72 ( ( finite_finite_nat @ M5 )
% 13.25/7.72 => ( ( gcd_Gcd_nat @ M5 )
% 13.25/7.72 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9608_Gcd__nat__eq__one,axiom,
% 13.25/7.72 ! [N3: set_nat] :
% 13.25/7.72 ( ( member_nat @ one_one_nat @ N3 )
% 13.25/7.72 => ( ( gcd_Gcd_nat @ N3 )
% 13.25/7.72 = one_one_nat ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9609_less__eq__int_Orep__eq,axiom,
% 13.25/7.72 ( ord_less_eq_int
% 13.25/7.72 = ( ^ [X2: int,Xa4: int] :
% 13.25/7.72 ( produc8739625826339149834_nat_o
% 13.25/7.72 @ ^ [Y: nat,Z2: nat] :
% 13.25/7.72 ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U3 @ Z2 ) ) )
% 13.25/7.72 @ ( rep_Integ @ X2 )
% 13.25/7.72 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9610_less__int_Orep__eq,axiom,
% 13.25/7.72 ( ord_less_int
% 13.25/7.72 = ( ^ [X2: int,Xa4: int] :
% 13.25/7.72 ( produc8739625826339149834_nat_o
% 13.25/7.72 @ ^ [Y: nat,Z2: nat] :
% 13.25/7.72 ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U3 @ Z2 ) ) )
% 13.25/7.72 @ ( rep_Integ @ X2 )
% 13.25/7.72 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9611_prod__encode__def,axiom,
% 13.25/7.72 ( nat_prod_encode
% 13.25/7.72 = ( produc6842872674320459806at_nat
% 13.25/7.72 @ ^ [M3: nat,N2: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M3 @ N2 ) ) @ M3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9612_Gcd__int__greater__eq__0,axiom,
% 13.25/7.72 ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9613_le__prod__encode__1,axiom,
% 13.25/7.72 ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9614_le__prod__encode__2,axiom,
% 13.25/7.72 ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9615_nat_Orep__eq,axiom,
% 13.25/7.72 ( nat2
% 13.25/7.72 = ( ^ [X2: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X2 ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9616_prod__encode__prod__decode__aux,axiom,
% 13.25/7.72 ! [K: nat,M: nat] :
% 13.25/7.72 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 13.25/7.72 = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9617_num__of__nat_Osimps_I2_J,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( num_of_nat @ ( suc @ N ) )
% 13.25/7.72 = ( inc @ ( num_of_nat @ N ) ) ) )
% 13.25/7.72 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( num_of_nat @ ( suc @ N ) )
% 13.25/7.72 = one ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9618_num__of__nat__numeral__eq,axiom,
% 13.25/7.72 ! [Q2: num] :
% 13.25/7.72 ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
% 13.25/7.72 = Q2 ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9619_num__of__nat_Osimps_I1_J,axiom,
% 13.25/7.72 ( ( num_of_nat @ zero_zero_nat )
% 13.25/7.72 = one ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9620_numeral__num__of__nat,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 13.25/7.72 = N ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9621_num__of__nat__One,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 13.25/7.72 => ( ( num_of_nat @ N )
% 13.25/7.72 = one ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9622_num__of__nat__double,axiom,
% 13.25/7.72 ! [N: nat] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 13.25/7.72 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9623_num__of__nat__plus__distrib,axiom,
% 13.25/7.72 ! [M: nat,N: nat] :
% 13.25/7.72 ( ( ord_less_nat @ zero_zero_nat @ M )
% 13.25/7.72 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.72 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 13.25/7.72 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9624_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 13.25/7.72 ! [N: nat,J: nat,I: nat] :
% 13.25/7.72 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 13.25/7.72 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 13.25/7.72 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9625_pow_Osimps_I3_J,axiom,
% 13.25/7.72 ! [X: num,Y3: num] :
% 13.25/7.72 ( ( pow @ X @ ( bit1 @ Y3 ) )
% 13.25/7.72 = ( times_times_num @ ( sqr @ ( pow @ X @ Y3 ) ) @ X ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9626_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.72 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 13.25/7.72 => ( ( ? [Uu2: $o,Uv2: $o] :
% 13.25/7.72 ( X
% 13.25/7.72 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.72 => ( Xa = one_one_nat ) )
% 13.25/7.72 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.72 => ( ( Deg2 = Xa )
% 13.25/7.72 & ! [X3: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.72 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 & ( case_o184042715313410164at_nat
% 13.25/7.72 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 13.25/7.72 & ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 @ ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [Mi3: nat,Ma3: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.72 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.72 & ! [I4: nat] :
% 13.25/7.72 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 13.25/7.72 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 13.25/7.72 & ( ( Mi3 = Ma3 )
% 13.25/7.72 => ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 & ( ( Mi3 != Ma3 )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 13.25/7.72 & ! [X2: nat] :
% 13.25/7.72 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 13.25/7.72 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.72 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 13.25/7.72 @ Mima ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9627_sqr_Osimps_I2_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( sqr @ ( bit0 @ N ) )
% 13.25/7.72 = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9628_sqr_Osimps_I1_J,axiom,
% 13.25/7.72 ( ( sqr @ one )
% 13.25/7.72 = one ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9629_sqr__conv__mult,axiom,
% 13.25/7.72 ( sqr
% 13.25/7.72 = ( ^ [X2: num] : ( times_times_num @ X2 @ X2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9630_pow_Osimps_I2_J,axiom,
% 13.25/7.72 ! [X: num,Y3: num] :
% 13.25/7.72 ( ( pow @ X @ ( bit0 @ Y3 ) )
% 13.25/7.72 = ( sqr @ ( pow @ X @ Y3 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9631_sqr_Osimps_I3_J,axiom,
% 13.25/7.72 ! [N: num] :
% 13.25/7.72 ( ( sqr @ ( bit1 @ N ) )
% 13.25/7.72 = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9632_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 13.25/7.72 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
% 13.25/7.72 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
% 13.25/7.72 = ( ( Deg = Deg3 )
% 13.25/7.72 & ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.72 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 13.25/7.72 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 & ( case_o184042715313410164at_nat
% 13.25/7.72 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
% 13.25/7.72 & ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 @ ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [Mi3: nat,Ma3: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.72 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 13.25/7.72 & ! [I4: nat] :
% 13.25/7.72 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
% 13.25/7.72 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 13.25/7.72 & ( ( Mi3 = Ma3 )
% 13.25/7.72 => ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 & ( ( Mi3 != Ma3 )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 13.25/7.72 & ! [X2: nat] :
% 13.25/7.72 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X2 )
% 13.25/7.72 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.72 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 13.25/7.72 @ Mima2 ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9633_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 13.25/7.72 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( ? [Uu2: $o,Uv2: $o] :
% 13.25/7.72 ( X
% 13.25/7.72 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( Xa != one_one_nat ) ) )
% 13.25/7.72 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.72 => ( Y3
% 13.25/7.72 = ( ~ ( ( Deg2 = Xa )
% 13.25/7.72 & ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.72 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 & ( case_o184042715313410164at_nat
% 13.25/7.72 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 13.25/7.72 & ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 @ ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [Mi3: nat,Ma3: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.72 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.72 & ! [I4: nat] :
% 13.25/7.72 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 13.25/7.72 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 13.25/7.72 & ( ( Mi3 = Ma3 )
% 13.25/7.72 => ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 & ( ( Mi3 != Ma3 )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 13.25/7.72 & ! [X2: nat] :
% 13.25/7.72 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 13.25/7.72 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.72 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 13.25/7.72 @ Mima ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9634_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.72 ( ( vEBT_VEBT_valid @ X @ Xa )
% 13.25/7.72 => ( ( ? [Uu2: $o,Uv2: $o] :
% 13.25/7.72 ( X
% 13.25/7.72 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.72 => ( Xa != one_one_nat ) )
% 13.25/7.72 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.72 => ~ ( ( Deg2 = Xa )
% 13.25/7.72 & ! [X5: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.72 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 & ( case_o184042715313410164at_nat
% 13.25/7.72 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 13.25/7.72 & ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 @ ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [Mi3: nat,Ma3: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.72 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.72 & ! [I4: nat] :
% 13.25/7.72 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 13.25/7.72 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 13.25/7.72 & ( ( Mi3 = Ma3 )
% 13.25/7.72 => ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 & ( ( Mi3 != Ma3 )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 13.25/7.72 & ! [X2: nat] :
% 13.25/7.72 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 13.25/7.72 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.72 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 13.25/7.72 @ Mima ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9635_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 13.25/7.72 ! [X: vEBT_VEBT,Xa: nat,Y3: $o] :
% 13.25/7.72 ( ( ( vEBT_VEBT_valid @ X @ Xa )
% 13.25/7.72 = Y3 )
% 13.25/7.72 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.72 => ( ! [Uu2: $o,Uv2: $o] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( Xa = one_one_nat ) )
% 13.25/7.72 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
% 13.25/7.72 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.72 ( ( X
% 13.25/7.72 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.72 => ( ( Y3
% 13.25/7.72 = ( ( Deg2 = Xa )
% 13.25/7.72 & ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.72 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.72 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 & ( case_o184042715313410164at_nat
% 13.25/7.72 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 13.25/7.72 & ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 @ ( produc6081775807080527818_nat_o
% 13.25/7.72 @ ^ [Mi3: nat,Ma3: nat] :
% 13.25/7.72 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.72 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.72 & ! [I4: nat] :
% 13.25/7.72 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.72 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 13.25/7.72 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 13.25/7.72 & ( ( Mi3 = Ma3 )
% 13.25/7.72 => ! [X2: vEBT_VEBT] :
% 13.25/7.72 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.72 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.72 & ( ( Mi3 != Ma3 )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 13.25/7.72 & ! [X2: nat] :
% 13.25/7.72 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.72 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 13.25/7.72 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.72 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 13.25/7.72 @ Mima ) ) )
% 13.25/7.72 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% 13.25/7.72
% 13.25/7.72 thf(fact_9636_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 13.25/7.73 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.73 ( ( vEBT_VEBT_valid @ X @ Xa )
% 13.25/7.73 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.73 => ( ! [Uu2: $o,Uv2: $o] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.73 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 13.25/7.73 => ( Xa != one_one_nat ) ) )
% 13.25/7.73 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.73 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 13.25/7.73 => ~ ( ( Deg2 = Xa )
% 13.25/7.73 & ! [X5: vEBT_VEBT] :
% 13.25/7.73 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.73 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.73 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.73 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.73 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.73 & ( case_o184042715313410164at_nat
% 13.25/7.73 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 13.25/7.73 & ! [X2: vEBT_VEBT] :
% 13.25/7.73 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.73 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.73 @ ( produc6081775807080527818_nat_o
% 13.25/7.73 @ ^ [Mi3: nat,Ma3: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.73 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.73 & ! [I4: nat] :
% 13.25/7.73 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.73 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 13.25/7.73 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 13.25/7.73 & ( ( Mi3 = Ma3 )
% 13.25/7.73 => ! [X2: vEBT_VEBT] :
% 13.25/7.73 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.73 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.73 & ( ( Mi3 != Ma3 )
% 13.25/7.73 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 13.25/7.73 & ! [X2: nat] :
% 13.25/7.73 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.73 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 13.25/7.73 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.73 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 13.25/7.73 @ Mima ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9637_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 13.25/7.73 ! [X: vEBT_VEBT,Xa: nat] :
% 13.25/7.73 ( ~ ( vEBT_VEBT_valid @ X @ Xa )
% 13.25/7.73 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa ) )
% 13.25/7.73 => ( ! [Uu2: $o,Uv2: $o] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 13.25/7.73 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
% 13.25/7.73 => ( Xa = one_one_nat ) ) )
% 13.25/7.73 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 13.25/7.73 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa ) )
% 13.25/7.73 => ( ( Deg2 = Xa )
% 13.25/7.73 & ! [X3: vEBT_VEBT] :
% 13.25/7.73 ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.73 => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.73 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 13.25/7.73 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 13.25/7.73 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.73 & ( case_o184042715313410164at_nat
% 13.25/7.73 @ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
% 13.25/7.73 & ! [X2: vEBT_VEBT] :
% 13.25/7.73 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.73 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.73 @ ( produc6081775807080527818_nat_o
% 13.25/7.73 @ ^ [Mi3: nat,Ma3: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 13.25/7.73 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.73 & ! [I4: nat] :
% 13.25/7.73 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.73 => ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X6 ) )
% 13.25/7.73 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 13.25/7.73 & ( ( Mi3 = Ma3 )
% 13.25/7.73 => ! [X2: vEBT_VEBT] :
% 13.25/7.73 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 13.25/7.73 => ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X6 ) ) )
% 13.25/7.73 & ( ( Mi3 != Ma3 )
% 13.25/7.73 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 13.25/7.73 & ! [X2: nat] :
% 13.25/7.73 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 13.25/7.73 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X2 )
% 13.25/7.73 => ( ( ord_less_nat @ Mi3 @ X2 )
% 13.25/7.73 & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
% 13.25/7.73 @ Mima ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9638_Sup__int__def,axiom,
% 13.25/7.73 ( complete_Sup_Sup_int
% 13.25/7.73 = ( ^ [X6: set_int] :
% 13.25/7.73 ( the_int
% 13.25/7.73 @ ^ [X2: int] :
% 13.25/7.73 ( ( member_int @ X2 @ X6 )
% 13.25/7.73 & ! [Y: int] :
% 13.25/7.73 ( ( member_int @ Y @ X6 )
% 13.25/7.73 => ( ord_less_eq_int @ Y @ X2 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9639_take__bit__numeral__minus__numeral__int,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 = ( case_option_int_num @ zero_zero_int
% 13.25/7.73 @ ^ [Q5: 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 @ Q5 ) ) )
% 13.25/7.73 @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9640_and__minus__numerals_I7_J,axiom,
% 13.25/7.73 ! [N: num,M: num] :
% 13.25/7.73 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 13.25/7.73 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9641_take__bit__num__simps_I1_J,axiom,
% 13.25/7.73 ! [M: num] :
% 13.25/7.73 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 13.25/7.73 = none_num ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9642_take__bit__num__simps_I2_J,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( bit_take_bit_num @ ( suc @ N ) @ one )
% 13.25/7.73 = ( some_num @ one ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9643_take__bit__num__simps_I5_J,axiom,
% 13.25/7.73 ! [R: num] :
% 13.25/7.73 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ one )
% 13.25/7.73 = ( some_num @ one ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9644_take__bit__num__simps_I3_J,axiom,
% 13.25/7.73 ! [N: nat,M: num] :
% 13.25/7.73 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
% 13.25/7.73 = ( case_o6005452278849405969um_num @ none_num
% 13.25/7.73 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 13.25/7.73 @ ( bit_take_bit_num @ N @ M ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9645_take__bit__num__simps_I4_J,axiom,
% 13.25/7.73 ! [N: nat,M: num] :
% 13.25/7.73 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
% 13.25/7.73 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9646_take__bit__num__simps_I6_J,axiom,
% 13.25/7.73 ! [R: num,M: num] :
% 13.25/7.73 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit0 @ M ) )
% 13.25/7.73 = ( case_o6005452278849405969um_num @ none_num
% 13.25/7.73 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 13.25/7.73 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9647_take__bit__num__simps_I7_J,axiom,
% 13.25/7.73 ! [R: num,M: num] :
% 13.25/7.73 ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R ) @ ( bit1 @ M ) )
% 13.25/7.73 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9648_and__minus__numerals_I4_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
% 13.25/7.73 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9649_and__minus__numerals_I8_J,axiom,
% 13.25/7.73 ! [N: num,M: num] :
% 13.25/7.73 ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M ) )
% 13.25/7.73 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9650_and__minus__numerals_I3_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
% 13.25/7.73 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9651_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 13.25/7.73 ! [N: nat,M: num] :
% 13.25/7.73 ( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
% 13.25/7.73 = ( case_nat_option_num @ none_num
% 13.25/7.73 @ ^ [N2: nat] :
% 13.25/7.73 ( case_o6005452278849405969um_num @ none_num
% 13.25/7.73 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 13.25/7.73 @ ( bit_take_bit_num @ N2 @ M ) )
% 13.25/7.73 @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9652_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( bit_take_bit_num @ N @ one )
% 13.25/7.73 = ( case_nat_option_num @ none_num
% 13.25/7.73 @ ^ [N2: nat] : ( some_num @ one )
% 13.25/7.73 @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9653_and__not__num_Osimps_I1_J,axiom,
% 13.25/7.73 ( ( bit_and_not_num @ one @ one )
% 13.25/7.73 = none_num ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9654_and__not__num_Osimps_I4_J,axiom,
% 13.25/7.73 ! [M: num] :
% 13.25/7.73 ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
% 13.25/7.73 = ( some_num @ ( bit0 @ M ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9655_and__not__num_Osimps_I2_J,axiom,
% 13.25/7.73 ! [N: num] :
% 13.25/7.73 ( ( bit_and_not_num @ one @ ( bit0 @ N ) )
% 13.25/7.73 = ( some_num @ one ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9656_GreatestI__ex__nat,axiom,
% 13.25/7.73 ! [P: nat > $o,B: nat] :
% 13.25/7.73 ( ? [X_12: nat] : ( P @ X_12 )
% 13.25/7.73 => ( ! [Y4: nat] :
% 13.25/7.73 ( ( P @ Y4 )
% 13.25/7.73 => ( ord_less_eq_nat @ Y4 @ B ) )
% 13.25/7.73 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9657_Greatest__le__nat,axiom,
% 13.25/7.73 ! [P: nat > $o,K: nat,B: nat] :
% 13.25/7.73 ( ( P @ K )
% 13.25/7.73 => ( ! [Y4: nat] :
% 13.25/7.73 ( ( P @ Y4 )
% 13.25/7.73 => ( ord_less_eq_nat @ Y4 @ B ) )
% 13.25/7.73 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9658_GreatestI__nat,axiom,
% 13.25/7.73 ! [P: nat > $o,K: nat,B: nat] :
% 13.25/7.73 ( ( P @ K )
% 13.25/7.73 => ( ! [Y4: nat] :
% 13.25/7.73 ( ( P @ Y4 )
% 13.25/7.73 => ( ord_less_eq_nat @ Y4 @ B ) )
% 13.25/7.73 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9659_and__not__num_Osimps_I3_J,axiom,
% 13.25/7.73 ! [N: num] :
% 13.25/7.73 ( ( bit_and_not_num @ one @ ( bit1 @ N ) )
% 13.25/7.73 = none_num ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9660_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 13.25/7.73 ! [N: nat,M: num] :
% 13.25/7.73 ( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
% 13.25/7.73 = ( case_nat_option_num @ none_num
% 13.25/7.73 @ ^ [N2: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
% 13.25/7.73 @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9661_and__not__num_Osimps_I7_J,axiom,
% 13.25/7.73 ! [M: num] :
% 13.25/7.73 ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
% 13.25/7.73 = ( some_num @ ( bit0 @ M ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9662_and__not__num__eq__Some__iff,axiom,
% 13.25/7.73 ! [M: num,N: num,Q2: num] :
% 13.25/7.73 ( ( ( bit_and_not_num @ M @ N )
% 13.25/7.73 = ( some_num @ Q2 ) )
% 13.25/7.73 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 = ( numeral_numeral_int @ Q2 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9663_and__not__num_Osimps_I8_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.73 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 13.25/7.73 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 13.25/7.73 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9664_and__not__num__eq__None__iff,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( ( bit_and_not_num @ M @ N )
% 13.25/7.73 = none_num )
% 13.25/7.73 = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 = zero_zero_int ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9665_int__numeral__not__and__num,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9666_int__numeral__and__not__num,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9667_take__bit__num__def,axiom,
% 13.25/7.73 ( bit_take_bit_num
% 13.25/7.73 = ( ^ [N2: nat,M3: num] :
% 13.25/7.73 ( if_option_num
% 13.25/7.73 @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) )
% 13.25/7.73 = zero_zero_nat )
% 13.25/7.73 @ none_num
% 13.25/7.73 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M3 ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9668_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 13.25/7.73 ! [N: nat,J: nat,I: nat] :
% 13.25/7.73 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 13.25/7.73 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 13.25/7.73 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9669_card__greaterThanAtMost,axiom,
% 13.25/7.73 ! [L2: nat,U: nat] :
% 13.25/7.73 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L2 @ U ) )
% 13.25/7.73 = ( minus_minus_nat @ U @ L2 ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9670_and__not__num_Oelims,axiom,
% 13.25/7.73 ! [X: num,Xa: num,Y3: option_num] :
% 13.25/7.73 ( ( ( bit_and_not_num @ X @ Xa )
% 13.25/7.73 = Y3 )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3 != none_num ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ? [N4: num] :
% 13.25/7.73 ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ one ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ? [N4: num] :
% 13.25/7.73 ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3 != none_num ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 13.25/7.73 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 13.25/7.73 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) )
% 13.25/7.73 => ~ ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9671_Bit__Operations_Otake__bit__num__code,axiom,
% 13.25/7.73 ( bit_take_bit_num
% 13.25/7.73 = ( ^ [N2: nat,M3: num] :
% 13.25/7.73 ( produc478579273971653890on_num
% 13.25/7.73 @ ^ [A3: nat,X2: num] :
% 13.25/7.73 ( case_nat_option_num @ none_num
% 13.25/7.73 @ ^ [O: nat] :
% 13.25/7.73 ( case_num_option_num @ ( some_num @ one )
% 13.25/7.73 @ ^ [P4: num] :
% 13.25/7.73 ( case_o6005452278849405969um_num @ none_num
% 13.25/7.73 @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 13.25/7.73 @ ( bit_take_bit_num @ O @ P4 ) )
% 13.25/7.73 @ ^ [P4: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
% 13.25/7.73 @ X2 )
% 13.25/7.73 @ A3 )
% 13.25/7.73 @ ( product_Pair_nat_num @ N2 @ M3 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9672_xor__num_Osimps_I8_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.73 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9673_card__greaterThanAtMost__int,axiom,
% 13.25/7.73 ! [L2: int,U: int] :
% 13.25/7.73 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L2 @ U ) )
% 13.25/7.73 = ( nat2 @ ( minus_minus_int @ U @ L2 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9674_xor__num_Osimps_I5_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9675_xor__num_Osimps_I9_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9676_and__not__num_Osimps_I5_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9677_and__not__num_Osimps_I6_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9678_and__not__num_Osimps_I9_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9679_xor__num_Osimps_I1_J,axiom,
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ one @ one )
% 13.25/7.73 = none_num ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9680_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 13.25/7.73 ! [L2: int,U: int] :
% 13.25/7.73 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L2 @ one_one_int ) @ U )
% 13.25/7.73 = ( set_or6656581121297822940st_int @ L2 @ U ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9681_xor__num_Oelims,axiom,
% 13.25/7.73 ! [X: num,Xa: num,Y3: option_num] :
% 13.25/7.73 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 13.25/7.73 = Y3 )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3 != none_num ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ ( bit1 @ N4 ) ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ ( bit0 @ N4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ ( bit1 @ M4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ ( bit0 @ M4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ~ ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9682_xor__num_Osimps_I7_J,axiom,
% 13.25/7.73 ! [M: num] :
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
% 13.25/7.73 = ( some_num @ ( bit0 @ M ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9683_xor__num_Osimps_I4_J,axiom,
% 13.25/7.73 ! [M: num] :
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
% 13.25/7.73 = ( some_num @ ( bit1 @ M ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9684_xor__num_Osimps_I3_J,axiom,
% 13.25/7.73 ! [N: num] :
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
% 13.25/7.73 = ( some_num @ ( bit0 @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9685_xor__num_Osimps_I2_J,axiom,
% 13.25/7.73 ! [N: num] :
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
% 13.25/7.73 = ( some_num @ ( bit1 @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9686_xor__num_Osimps_I6_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.73 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9687_and__not__num_Opelims,axiom,
% 13.25/7.73 ! [X: num,Xa: num,Y3: option_num] :
% 13.25/7.73 ( ( ( bit_and_not_num @ X @ Xa )
% 13.25/7.73 = Y3 )
% 13.25/7.73 => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3 = none_num )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ one ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3 = none_num )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ ( bit0 @ M4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ ( bit0 @ M4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 13.25/7.73 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 13.25/7.73 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ~ ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9688_xor__num_Opelims,axiom,
% 13.25/7.73 ! [X: num,Xa: num,Y3: option_num] :
% 13.25/7.73 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 13.25/7.73 = Y3 )
% 13.25/7.73 => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3 = none_num )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ ( bit1 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ ( bit0 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ ( bit1 @ M4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ ( bit0 @ M4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ~ ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9689_and__num_Oelims,axiom,
% 13.25/7.73 ! [X: num,Xa: num,Y3: option_num] :
% 13.25/7.73 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 13.25/7.73 = Y3 )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ one ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ? [N4: num] :
% 13.25/7.73 ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3 != none_num ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ? [N4: num] :
% 13.25/7.73 ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ one ) ) ) )
% 13.25/7.73 => ( ( ? [M4: num] :
% 13.25/7.73 ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3 != none_num ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 13.25/7.73 => ( ( ? [M4: num] :
% 13.25/7.73 ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( some_num @ one ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) )
% 13.25/7.73 => ~ ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( Y3
% 13.25/7.73 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 13.25/7.73 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 13.25/7.73 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9690_xor__num__rel__dict,axiom,
% 13.25/7.73 bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9691_xor__num__dict,axiom,
% 13.25/7.73 bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9692_and__num_Osimps_I1_J,axiom,
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ one @ one )
% 13.25/7.73 = ( some_num @ one ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9693_and__num_Osimps_I5_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9694_and__num_Osimps_I3_J,axiom,
% 13.25/7.73 ! [N: num] :
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
% 13.25/7.73 = ( some_num @ one ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9695_and__num_Osimps_I7_J,axiom,
% 13.25/7.73 ! [M: num] :
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
% 13.25/7.73 = ( some_num @ one ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9696_and__num_Osimps_I4_J,axiom,
% 13.25/7.73 ! [M: num] :
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 13.25/7.73 = none_num ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9697_and__num_Osimps_I2_J,axiom,
% 13.25/7.73 ! [N: num] :
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
% 13.25/7.73 = none_num ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9698_and__num_Osimps_I6_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9699_and__num_Osimps_I8_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9700_and__num_Osimps_I9_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 13.25/7.73 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 13.25/7.73 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 13.25/7.73 @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9701_and__num_Opelims,axiom,
% 13.25/7.73 ! [X: num,Xa: num,Y3: option_num] :
% 13.25/7.73 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 13.25/7.73 = Y3 )
% 13.25/7.73 => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X @ Xa ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ one ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3 = none_num )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ( ( X = one )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ one ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3 = none_num )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit0 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ( ( Xa = one )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( some_num @ one ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) ) ) ) )
% 13.25/7.73 => ( ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit0 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N4 ) ) ) ) ) )
% 13.25/7.73 => ~ ! [M4: num] :
% 13.25/7.73 ( ( X
% 13.25/7.73 = ( bit1 @ M4 ) )
% 13.25/7.73 => ! [N4: num] :
% 13.25/7.73 ( ( Xa
% 13.25/7.73 = ( bit1 @ N4 ) )
% 13.25/7.73 => ( ( Y3
% 13.25/7.73 = ( case_o6005452278849405969um_num @ ( some_num @ one )
% 13.25/7.73 @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 13.25/7.73 @ ( bit_un7362597486090784418nd_num @ M4 @ N4 ) ) )
% 13.25/7.73 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9702_and__num__dict,axiom,
% 13.25/7.73 bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9703_and__num__rel__dict,axiom,
% 13.25/7.73 bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9704_rat__floor__lemma,axiom,
% 13.25/7.73 ! [A: int,B: int] :
% 13.25/7.73 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 13.25/7.73 & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9705_image__minus__const__atLeastLessThan__nat,axiom,
% 13.25/7.73 ! [C: nat,Y3: nat,X: nat] :
% 13.25/7.73 ( ( ( ord_less_nat @ C @ Y3 )
% 13.25/7.73 => ( ( image_nat_nat
% 13.25/7.73 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 13.25/7.73 @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
% 13.25/7.73 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_nat @ C @ Y3 )
% 13.25/7.73 => ( ( ( ord_less_nat @ X @ Y3 )
% 13.25/7.73 => ( ( image_nat_nat
% 13.25/7.73 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 13.25/7.73 @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
% 13.25/7.73 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 13.25/7.73 & ( ~ ( ord_less_nat @ X @ Y3 )
% 13.25/7.73 => ( ( image_nat_nat
% 13.25/7.73 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 13.25/7.73 @ ( set_or4665077453230672383an_nat @ X @ Y3 ) )
% 13.25/7.73 = bot_bot_set_nat ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9706_bij__betw__Suc,axiom,
% 13.25/7.73 ! [M5: set_nat,N3: set_nat] :
% 13.25/7.73 ( ( bij_betw_nat_nat @ suc @ M5 @ N3 )
% 13.25/7.73 = ( ( image_nat_nat @ suc @ M5 )
% 13.25/7.73 = N3 ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9707_mult__rat,axiom,
% 13.25/7.73 ! [A: int,B: int,C: int,D: int] :
% 13.25/7.73 ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 13.25/7.73 = ( fract @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9708_divide__rat,axiom,
% 13.25/7.73 ! [A: int,B: int,C: int,D: int] :
% 13.25/7.73 ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 13.25/7.73 = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9709_floor__Fract,axiom,
% 13.25/7.73 ! [A: int,B: int] :
% 13.25/7.73 ( ( archim3151403230148437115or_rat @ ( fract @ A @ B ) )
% 13.25/7.73 = ( divide_divide_int @ A @ B ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9710_less__rat,axiom,
% 13.25/7.73 ! [B: int,D: int,A: int,C: int] :
% 13.25/7.73 ( ( B != zero_zero_int )
% 13.25/7.73 => ( ( D != zero_zero_int )
% 13.25/7.73 => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 13.25/7.73 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9711_add__rat,axiom,
% 13.25/7.73 ! [B: int,D: int,A: int,C: int] :
% 13.25/7.73 ( ( B != zero_zero_int )
% 13.25/7.73 => ( ( D != zero_zero_int )
% 13.25/7.73 => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 13.25/7.73 = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9712_le__rat,axiom,
% 13.25/7.73 ! [B: int,D: int,A: int,C: int] :
% 13.25/7.73 ( ( B != zero_zero_int )
% 13.25/7.73 => ( ( D != zero_zero_int )
% 13.25/7.73 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 13.25/7.73 = ( 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 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9713_diff__rat,axiom,
% 13.25/7.73 ! [B: int,D: int,A: int,C: int] :
% 13.25/7.73 ( ( B != zero_zero_int )
% 13.25/7.73 => ( ( D != zero_zero_int )
% 13.25/7.73 => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 13.25/7.73 = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9714_sgn__rat,axiom,
% 13.25/7.73 ! [A: int,B: int] :
% 13.25/7.73 ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
% 13.25/7.73 = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9715_Rat__induct__pos,axiom,
% 13.25/7.73 ! [P: rat > $o,Q2: rat] :
% 13.25/7.73 ( ! [A5: int,B3: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B3 )
% 13.25/7.73 => ( P @ ( fract @ A5 @ B3 ) ) )
% 13.25/7.73 => ( P @ Q2 ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9716_mult__rat__cancel,axiom,
% 13.25/7.73 ! [C: int,A: int,B: int] :
% 13.25/7.73 ( ( C != zero_zero_int )
% 13.25/7.73 => ( ( fract @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 13.25/7.73 = ( fract @ A @ B ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9717_eq__rat_I1_J,axiom,
% 13.25/7.73 ! [B: int,D: int,A: int,C: int] :
% 13.25/7.73 ( ( B != zero_zero_int )
% 13.25/7.73 => ( ( D != zero_zero_int )
% 13.25/7.73 => ( ( ( fract @ A @ B )
% 13.25/7.73 = ( fract @ C @ D ) )
% 13.25/7.73 = ( ( times_times_int @ A @ D )
% 13.25/7.73 = ( times_times_int @ C @ B ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9718_Fract__coprime,axiom,
% 13.25/7.73 ! [A: int,B: int] :
% 13.25/7.73 ( ( fract @ ( divide_divide_int @ A @ ( gcd_gcd_int @ A @ B ) ) @ ( divide_divide_int @ B @ ( gcd_gcd_int @ A @ B ) ) )
% 13.25/7.73 = ( fract @ A @ B ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9719_Fract__of__int__quotient,axiom,
% 13.25/7.73 ( fract
% 13.25/7.73 = ( ^ [K2: int,L: int] : ( divide_divide_rat @ ( ring_1_of_int_rat @ K2 ) @ ( ring_1_of_int_rat @ L ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9720_rat__number__collapse_I3_J,axiom,
% 13.25/7.73 ! [W: num] :
% 13.25/7.73 ( ( fract @ ( numeral_numeral_int @ W ) @ one_one_int )
% 13.25/7.73 = ( numeral_numeral_rat @ W ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9721_rat__number__expand_I3_J,axiom,
% 13.25/7.73 ( numeral_numeral_rat
% 13.25/7.73 = ( ^ [K2: num] : ( fract @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9722_image__Suc__lessThan,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9723_image__Suc__atMost,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 13.25/7.73 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9724_zero__less__Fract__iff,axiom,
% 13.25/7.73 ! [B: int,A: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.73 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 13.25/7.73 = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9725_Fract__less__zero__iff,axiom,
% 13.25/7.73 ! [B: int,A: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.73 => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 13.25/7.73 = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9726_one__less__Fract__iff,axiom,
% 13.25/7.73 ! [B: int,A: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.73 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 13.25/7.73 = ( ord_less_int @ B @ A ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9727_Fract__less__one__iff,axiom,
% 13.25/7.73 ! [B: int,A: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.73 => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 13.25/7.73 = ( ord_less_int @ A @ B ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9728_Fract__add__one,axiom,
% 13.25/7.73 ! [N: int,M: int] :
% 13.25/7.73 ( ( N != zero_zero_int )
% 13.25/7.73 => ( ( fract @ ( plus_plus_int @ M @ N ) @ N )
% 13.25/7.73 = ( plus_plus_rat @ ( fract @ M @ N ) @ one_one_rat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9729_zero__le__Fract__iff,axiom,
% 13.25/7.73 ! [B: int,A: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.73 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 13.25/7.73 = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9730_Fract__le__zero__iff,axiom,
% 13.25/7.73 ! [B: int,A: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.73 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 13.25/7.73 = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9731_one__le__Fract__iff,axiom,
% 13.25/7.73 ! [B: int,A: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.73 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 13.25/7.73 = ( ord_less_eq_int @ B @ A ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9732_Fract__le__one__iff,axiom,
% 13.25/7.73 ! [B: int,A: int] :
% 13.25/7.73 ( ( ord_less_int @ zero_zero_int @ B )
% 13.25/7.73 => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 13.25/7.73 = ( ord_less_eq_int @ A @ B ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9733_rat__number__expand_I5_J,axiom,
% 13.25/7.73 ! [K: num] :
% 13.25/7.73 ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) )
% 13.25/7.73 = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9734_rat__number__collapse_I4_J,axiom,
% 13.25/7.73 ! [W: num] :
% 13.25/7.73 ( ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ one_one_int )
% 13.25/7.73 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9735_positive__rat,axiom,
% 13.25/7.73 ! [A: int,B: int] :
% 13.25/7.73 ( ( positive @ ( fract @ A @ B ) )
% 13.25/7.73 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9736_Inf__real__def,axiom,
% 13.25/7.73 ( comple4887499456419720421f_real
% 13.25/7.73 = ( ^ [X6: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X6 ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9737_Rat_Opositive__add,axiom,
% 13.25/7.73 ! [X: rat,Y3: rat] :
% 13.25/7.73 ( ( positive @ X )
% 13.25/7.73 => ( ( positive @ Y3 )
% 13.25/7.73 => ( positive @ ( plus_plus_rat @ X @ Y3 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9738_Rat_Opositive__mult,axiom,
% 13.25/7.73 ! [X: rat,Y3: rat] :
% 13.25/7.73 ( ( positive @ X )
% 13.25/7.73 => ( ( positive @ Y3 )
% 13.25/7.73 => ( positive @ ( times_times_rat @ X @ Y3 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9739_finite__int__iff__bounded,axiom,
% 13.25/7.73 ( finite_finite_int
% 13.25/7.73 = ( ^ [S5: set_int] :
% 13.25/7.73 ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K2 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9740_finite__int__iff__bounded__le,axiom,
% 13.25/7.73 ( finite_finite_int
% 13.25/7.73 = ( ^ [S5: set_int] :
% 13.25/7.73 ? [K2: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K2 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9741_suminf__eq__SUP__real,axiom,
% 13.25/7.73 ! [X7: nat > real] :
% 13.25/7.73 ( ( summable_real @ X7 )
% 13.25/7.73 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X7 @ I2 ) )
% 13.25/7.73 => ( ( suminf_real @ X7 )
% 13.25/7.73 = ( comple1385675409528146559p_real
% 13.25/7.73 @ ( image_nat_real
% 13.25/7.73 @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X7 @ ( set_ord_lessThan_nat @ I4 ) )
% 13.25/7.73 @ top_top_set_nat ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9742_less__rat__def,axiom,
% 13.25/7.73 ( ord_less_rat
% 13.25/7.73 = ( ^ [X2: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X2 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9743_image__add__int__atLeastLessThan,axiom,
% 13.25/7.73 ! [L2: int,U: int] :
% 13.25/7.73 ( ( image_int_int
% 13.25/7.73 @ ^ [X2: int] : ( plus_plus_int @ X2 @ L2 )
% 13.25/7.73 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L2 ) ) )
% 13.25/7.73 = ( set_or4662586982721622107an_int @ L2 @ U ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9744_range__mod,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( image_nat_nat
% 13.25/7.73 @ ^ [M3: nat] : ( modulo_modulo_nat @ M3 @ N )
% 13.25/7.73 @ top_top_set_nat )
% 13.25/7.73 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9745_image__atLeastZeroLessThan__int,axiom,
% 13.25/7.73 ! [U: int] :
% 13.25/7.73 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 13.25/7.73 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 13.25/7.73 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9746_Rat_Opositive_Orep__eq,axiom,
% 13.25/7.73 ( positive
% 13.25/7.73 = ( ^ [X2: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X2 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9747_card__UNIV__unit,axiom,
% 13.25/7.73 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 13.25/7.73 = one_one_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9748_card__UNIV__bool,axiom,
% 13.25/7.73 ( ( finite_card_o @ top_top_set_o )
% 13.25/7.73 = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9749_range__mult,axiom,
% 13.25/7.73 ! [A: real] :
% 13.25/7.73 ( ( ( A = zero_zero_real )
% 13.25/7.73 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 13.25/7.73 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 13.25/7.73 & ( ( A != zero_zero_real )
% 13.25/7.73 => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 13.25/7.73 = top_top_set_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9750_root__def,axiom,
% 13.25/7.73 ( root
% 13.25/7.73 = ( ^ [N2: nat,X2: real] :
% 13.25/7.73 ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 13.25/7.73 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 13.25/7.73 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 13.25/7.73 @ X2 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9751_card__UNIV__char,axiom,
% 13.25/7.73 ( ( finite_card_char @ top_top_set_char )
% 13.25/7.73 = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9752_UNIV__char__of__nat,axiom,
% 13.25/7.73 ( top_top_set_char
% 13.25/7.73 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9753_nat__of__char__less__256,axiom,
% 13.25/7.73 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9754_range__nat__of__char,axiom,
% 13.25/7.73 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 13.25/7.73 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9755_integer__of__char__code,axiom,
% 13.25/7.73 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 13.25/7.73 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 13.25/7.73 = ( 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 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9756_String_Ochar__of__ascii__of,axiom,
% 13.25/7.73 ! [C: char] :
% 13.25/7.73 ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
% 13.25/7.73 = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9757_DERIV__real__root__generic,axiom,
% 13.25/7.73 ! [N: nat,X: real,D4: real] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( X != zero_zero_real )
% 13.25/7.73 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.73 => ( D4
% 13.25/7.73 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 13.25/7.73 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.73 => ( ( ord_less_real @ X @ zero_zero_real )
% 13.25/7.73 => ( D4
% 13.25/7.73 = ( 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 ) ) ) ) ) ) ) ) )
% 13.25/7.73 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.73 => ( D4
% 13.25/7.73 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9758_DERIV__pos__imp__increasing,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_eq_real @ X3 @ B )
% 13.25/7.73 => ? [Y5: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 13.25/7.73 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9759_DERIV__neg__imp__decreasing,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_eq_real @ X3 @ B )
% 13.25/7.73 => ? [Y5: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 13.25/7.73 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9760_DERIV__pos__inc__left,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 13.25/7.73 => ? [D3: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.73 & ! [H4: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 13.25/7.73 => ( ( ord_less_real @ H4 @ D3 )
% 13.25/7.73 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9761_DERIV__neg__dec__left,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 13.25/7.73 => ? [D3: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.73 & ! [H4: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 13.25/7.73 => ( ( ord_less_real @ H4 @ D3 )
% 13.25/7.73 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9762_DERIV__pos__inc__right,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 13.25/7.73 => ? [D3: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.73 & ! [H4: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 13.25/7.73 => ( ( ord_less_real @ H4 @ D3 )
% 13.25/7.73 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9763_DERIV__neg__dec__right,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 13.25/7.73 => ? [D3: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.73 & ! [H4: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 13.25/7.73 => ( ( ord_less_real @ H4 @ D3 )
% 13.25/7.73 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9764_DERIV__isconst3,axiom,
% 13.25/7.73 ! [A: real,B: real,X: real,Y3: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 13.25/7.73 => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 13.25/7.73 => ( ( F @ X )
% 13.25/7.73 = ( F @ Y3 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9765_has__real__derivative__neg__dec__right,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real,S2: set_real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 13.25/7.73 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 13.25/7.73 => ? [D3: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.73 & ! [H4: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 13.25/7.73 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
% 13.25/7.73 => ( ( ord_less_real @ H4 @ D3 )
% 13.25/7.73 => ( ord_less_real @ ( F @ ( plus_plus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9766_has__real__derivative__pos__inc__right,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real,S2: set_real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 13.25/7.73 => ? [D3: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.73 & ! [H4: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 13.25/7.73 => ( ( member_real @ ( plus_plus_real @ X @ H4 ) @ S2 )
% 13.25/7.73 => ( ( ord_less_real @ H4 @ D3 )
% 13.25/7.73 => ( ord_less_real @ ( F @ X ) @ ( F @ ( plus_plus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9767_has__real__derivative__neg__dec__left,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real,S2: set_real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 13.25/7.73 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 13.25/7.73 => ? [D3: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.73 & ! [H4: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 13.25/7.73 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
% 13.25/7.73 => ( ( ord_less_real @ H4 @ D3 )
% 13.25/7.73 => ( ord_less_real @ ( F @ X ) @ ( F @ ( minus_minus_real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9768_has__real__derivative__pos__inc__left,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real,S2: set_real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ S2 ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 13.25/7.73 => ? [D3: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D3 )
% 13.25/7.73 & ! [H4: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 13.25/7.73 => ( ( member_real @ ( minus_minus_real @ X @ H4 ) @ S2 )
% 13.25/7.73 => ( ( ord_less_real @ H4 @ D3 )
% 13.25/7.73 => ( ord_less_real @ ( F @ ( minus_minus_real @ X @ H4 ) ) @ ( F @ X ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9769_DERIV__isconst__all,axiom,
% 13.25/7.73 ! [F: real > real,X: real,Y3: real] :
% 13.25/7.73 ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 => ( ( F @ X )
% 13.25/7.73 = ( F @ Y3 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9770_DERIV__mirror,axiom,
% 13.25/7.73 ! [F: real > real,Y3: real,X: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y3 @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ X ) @ top_top_set_real ) )
% 13.25/7.73 = ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X2: real] : ( F @ ( uminus_uminus_real @ X2 ) )
% 13.25/7.73 @ ( uminus_uminus_real @ Y3 )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9771_DERIV__const__ratio__const,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real,K: real] :
% 13.25/7.73 ( ( A != B )
% 13.25/7.73 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 13.25/7.73 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9772_DERIV__const__ratio__const2,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real,K: real] :
% 13.25/7.73 ( ( A != B )
% 13.25/7.73 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 => ( ( divide_divide_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( minus_minus_real @ B @ A ) )
% 13.25/7.73 = K ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9773_deriv__nonneg__imp__mono,axiom,
% 13.25/7.73 ! [A: real,B: real,G: real > real,G2: real > real] :
% 13.25/7.73 ( ! [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ ( set_or1222579329274155063t_real @ A @ B ) )
% 13.25/7.73 => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
% 13.25/7.73 => ( ( ord_less_eq_real @ A @ B )
% 13.25/7.73 => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9774_DERIV__nonpos__imp__nonincreasing,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_eq_real @ X3 @ B )
% 13.25/7.73 => ? [Y5: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 13.25/7.73 => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9775_DERIV__nonneg__imp__nondecreasing,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_eq_real @ X3 @ B )
% 13.25/7.73 => ? [Y5: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 13.25/7.73 => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9776_MVT2,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real,F3: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_eq_real @ X3 @ B )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ? [Z5: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ Z5 )
% 13.25/7.73 & ( ord_less_real @ Z5 @ B )
% 13.25/7.73 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 13.25/7.73 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F3 @ Z5 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9777_DERIV__local__const,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real,D: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ D )
% 13.25/7.73 => ( ! [Y4: real] :
% 13.25/7.73 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 13.25/7.73 => ( ( F @ X )
% 13.25/7.73 = ( F @ Y4 ) ) )
% 13.25/7.73 => ( L2 = zero_zero_real ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9778_DERIV__ln,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9779_DERIV__const__average,axiom,
% 13.25/7.73 ! [A: real,B: real,V: real > real,K: real] :
% 13.25/7.73 ( ( A != B )
% 13.25/7.73 => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 13.25/7.73 = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9780_DERIV__local__max,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real,D: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ D )
% 13.25/7.73 => ( ! [Y4: real] :
% 13.25/7.73 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 13.25/7.73 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X ) ) )
% 13.25/7.73 => ( L2 = zero_zero_real ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9781_DERIV__local__min,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,X: real,D: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ L2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ D )
% 13.25/7.73 => ( ! [Y4: real] :
% 13.25/7.73 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ Y4 ) ) @ D )
% 13.25/7.73 => ( ord_less_eq_real @ ( F @ X ) @ ( F @ Y4 ) ) )
% 13.25/7.73 => ( L2 = zero_zero_real ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9782_DERIV__ln__divide,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9783_DERIV__pow,axiom,
% 13.25/7.73 ! [N: nat,X: real,S: set_real] :
% 13.25/7.73 ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X2: real] : ( power_power_real @ X2 @ N )
% 13.25/7.73 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ S ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9784_DERIV__fun__pow,axiom,
% 13.25/7.73 ! [G: real > real,M: real,X: real,N: nat] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X2: real] : ( power_power_real @ ( G @ X2 ) @ N )
% 13.25/7.73 @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9785_has__real__derivative__powr,axiom,
% 13.25/7.73 ! [Z: real,R: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ Z )
% 13.25/7.73 => ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [Z2: real] : ( powr_real @ Z2 @ R )
% 13.25/7.73 @ ( times_times_real @ R @ ( powr_real @ Z @ ( minus_minus_real @ R @ one_one_real ) ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9786_DERIV__log,axiom,
% 13.25/7.73 ! [X: real,B: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.73 => ( 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 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9787_DERIV__fun__powr,axiom,
% 13.25/7.73 ! [G: real > real,M: real,X: real,R: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ R )
% 13.25/7.73 @ ( times_times_real @ ( times_times_real @ R @ ( powr_real @ ( G @ X ) @ ( minus_minus_real @ R @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9788_DERIV__powr,axiom,
% 13.25/7.73 ! [G: real > real,M: real,X: real,F: real > real,R: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ ( G @ X ) )
% 13.25/7.73 => ( ( has_fi5821293074295781190e_real @ F @ R @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X2: real] : ( powr_real @ ( G @ X2 ) @ ( F @ X2 ) )
% 13.25/7.73 @ ( times_times_real @ ( powr_real @ ( G @ X ) @ ( F @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R @ ( ln_ln_real @ ( G @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X ) ) @ ( G @ X ) ) ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9789_DERIV__real__sqrt,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9790_DERIV__series_H,axiom,
% 13.25/7.73 ! [F: real > nat > real,F3: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
% 13.25/7.73 ( ! [N4: nat] :
% 13.25/7.73 ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X2: real] : ( F @ X2 @ N4 )
% 13.25/7.73 @ ( F3 @ X0 @ N4 )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 13.25/7.73 => ( summable_real @ ( F @ X3 ) ) )
% 13.25/7.73 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 13.25/7.73 => ( ( summable_real @ ( F3 @ X0 ) )
% 13.25/7.73 => ( ( summable_real @ L5 )
% 13.25/7.73 => ( ! [N4: nat,X3: real,Y4: real] :
% 13.25/7.73 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 13.25/7.73 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 13.25/7.73 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X3 @ N4 ) @ ( F @ Y4 @ N4 ) ) ) @ ( times_times_real @ ( L5 @ N4 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y4 ) ) ) ) ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X2: real] : ( suminf_real @ ( F @ X2 ) )
% 13.25/7.73 @ ( suminf_real @ ( F3 @ X0 ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9791_DERIV__arctan,axiom,
% 13.25/7.73 ! [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 ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9792_arsinh__real__has__field__derivative,axiom,
% 13.25/7.73 ! [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 ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9793_DERIV__real__sqrt__generic,axiom,
% 13.25/7.73 ! [X: real,D4: real] :
% 13.25/7.73 ( ( X != zero_zero_real )
% 13.25/7.73 => ( ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.73 => ( D4
% 13.25/7.73 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 13.25/7.73 => ( ( ( ord_less_real @ X @ zero_zero_real )
% 13.25/7.73 => ( D4
% 13.25/7.73 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9794_arcosh__real__has__field__derivative,axiom,
% 13.25/7.73 ! [X: real,A2: set_real] :
% 13.25/7.73 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.73 => ( 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 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9795_artanh__real__has__field__derivative,axiom,
% 13.25/7.73 ! [X: real,A2: set_real] :
% 13.25/7.73 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.73 => ( 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 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9796_DERIV__power__series_H,axiom,
% 13.25/7.73 ! [R4: real,F: nat > real,X0: real] :
% 13.25/7.73 ( ! [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R4 ) @ R4 ) )
% 13.25/7.73 => ( summable_real
% 13.25/7.73 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X3 @ N2 ) ) ) )
% 13.25/7.73 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R4 ) @ R4 ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ R4 )
% 13.25/7.73 => ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( suminf_real
% 13.25/7.73 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ ( suc @ N2 ) ) ) )
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9797_DERIV__real__root,axiom,
% 13.25/7.73 ! [N: nat,X: real] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ X )
% 13.25/7.73 => ( 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 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9798_DERIV__arccos,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.73 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.73 => ( 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 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9799_DERIV__arcsin,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.73 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.73 => ( 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 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9800_Maclaurin__all__le,axiom,
% 13.25/7.73 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 13.25/7.73 ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 13.25/7.73 & ( ( F @ X )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9801_Maclaurin__all__le__objl,axiom,
% 13.25/7.73 ! [Diff: nat > real > real,F: real > real,X: real,N: nat] :
% 13.25/7.73 ( ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 & ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 13.25/7.73 & ( ( F @ X )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9802_DERIV__odd__real__root,axiom,
% 13.25/7.73 ! [N: nat,X: real] :
% 13.25/7.73 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.73 => ( ( X != zero_zero_real )
% 13.25/7.73 => ( 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 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9803_Maclaurin,axiom,
% 13.25/7.73 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H2 )
% 13.25/7.73 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ! [M4: nat,T5: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M4 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 13.25/7.73 & ( ord_less_eq_real @ T5 @ H2 ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ T5 )
% 13.25/7.73 & ( ord_less_real @ T5 @ H2 )
% 13.25/7.73 & ( ( F @ H2 )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9804_Maclaurin2,axiom,
% 13.25/7.73 ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ H2 )
% 13.25/7.73 => ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ! [M4: nat,T5: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M4 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 13.25/7.73 & ( ord_less_eq_real @ T5 @ H2 ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ T5 )
% 13.25/7.73 & ( ord_less_eq_real @ T5 @ H2 )
% 13.25/7.73 & ( ( F @ H2 )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9805_Maclaurin__minus,axiom,
% 13.25/7.73 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ H2 @ zero_zero_real )
% 13.25/7.73 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ! [M4: nat,T5: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M4 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ H2 @ T5 )
% 13.25/7.73 & ( ord_less_eq_real @ T5 @ zero_zero_real ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_real @ H2 @ T5 )
% 13.25/7.73 & ( ord_less_real @ T5 @ zero_zero_real )
% 13.25/7.73 & ( ( F @ H2 )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ H2 @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9806_Maclaurin__all__lt,axiom,
% 13.25/7.73 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 13.25/7.73 ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( X != zero_zero_real )
% 13.25/7.73 => ( ! [M4: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
% 13.25/7.73 & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 13.25/7.73 & ( ( F @ X )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9807_Maclaurin__bi__le,axiom,
% 13.25/7.73 ! [Diff: nat > real > real,F: real > real,N: nat,X: real] :
% 13.25/7.73 ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ! [M4: nat,T5: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M4 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X ) )
% 13.25/7.73 & ( ( F @ X )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ X @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9808_Taylor,axiom,
% 13.25/7.73 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X: real] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ! [M4: nat,T5: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M4 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ A @ T5 )
% 13.25/7.73 & ( ord_less_eq_real @ T5 @ B ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 13.25/7.73 => ( ( ord_less_eq_real @ A @ C )
% 13.25/7.73 => ( ( ord_less_eq_real @ C @ B )
% 13.25/7.73 => ( ( ord_less_eq_real @ A @ X )
% 13.25/7.73 => ( ( ord_less_eq_real @ X @ B )
% 13.25/7.73 => ( ( X != C )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ( ord_less_real @ X @ C )
% 13.25/7.73 => ( ( ord_less_real @ X @ T5 )
% 13.25/7.73 & ( ord_less_real @ T5 @ C ) ) )
% 13.25/7.73 & ( ~ ( ord_less_real @ X @ C )
% 13.25/7.73 => ( ( ord_less_real @ C @ T5 )
% 13.25/7.73 & ( ord_less_real @ T5 @ X ) ) )
% 13.25/7.73 & ( ( F @ X )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9809_Taylor__up,axiom,
% 13.25/7.73 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ! [M4: nat,T5: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M4 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ A @ T5 )
% 13.25/7.73 & ( ord_less_eq_real @ T5 @ B ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 13.25/7.73 => ( ( ord_less_eq_real @ A @ C )
% 13.25/7.73 => ( ( ord_less_real @ C @ B )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_real @ C @ T5 )
% 13.25/7.73 & ( ord_less_real @ T5 @ B )
% 13.25/7.73 & ( ( F @ B )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9810_Taylor__down,axiom,
% 13.25/7.73 ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( ( Diff @ zero_zero_nat )
% 13.25/7.73 = F )
% 13.25/7.73 => ( ! [M4: nat,T5: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M4 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ A @ T5 )
% 13.25/7.73 & ( ord_less_eq_real @ T5 @ B ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 13.25/7.73 => ( ( ord_less_real @ A @ C )
% 13.25/7.73 => ( ( ord_less_eq_real @ C @ B )
% 13.25/7.73 => ? [T5: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ T5 )
% 13.25/7.73 & ( ord_less_real @ T5 @ C )
% 13.25/7.73 & ( ( F @ A )
% 13.25/7.73 = ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [M3: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M3 @ C ) @ ( semiri2265585572941072030t_real @ M3 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M3 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ N ) )
% 13.25/7.73 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9811_Maclaurin__lemma2,axiom,
% 13.25/7.73 ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B5: real] :
% 13.25/7.73 ( ! [M4: nat,T5: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M4 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ zero_zero_real @ T5 )
% 13.25/7.73 & ( ord_less_eq_real @ T5 @ H2 ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
% 13.25/7.73 => ( ( N
% 13.25/7.73 = ( suc @ K ) )
% 13.25/7.73 => ! [M2: nat,T6: real] :
% 13.25/7.73 ( ( ( ord_less_nat @ M2 @ N )
% 13.25/7.73 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 13.25/7.73 & ( ord_less_eq_real @ T6 @ H2 ) )
% 13.25/7.73 => ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [U3: real] :
% 13.25/7.73 ( minus_minus_real @ ( Diff @ M2 @ U3 )
% 13.25/7.73 @ ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ U3 @ P4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M2 ) ) )
% 13.25/7.73 @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ U3 @ ( minus_minus_nat @ N @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) )
% 13.25/7.73 @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T6 )
% 13.25/7.73 @ ( plus_plus_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [P4: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P4 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P4 ) ) @ ( power_power_real @ T6 @ P4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) )
% 13.25/7.73 @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M2 ) ) ) ) ) ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9812_DERIV__arctan__series,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.73 => ( has_fi5821293074295781190e_real
% 13.25/7.73 @ ^ [X9: real] :
% 13.25/7.73 ( suminf_real
% 13.25/7.73 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [K2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( power_power_real @ X @ ( times_times_nat @ K2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9813_DERIV__even__real__root,axiom,
% 13.25/7.73 ! [N: nat,X: real] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.73 => ( ( ord_less_real @ X @ zero_zero_real )
% 13.25/7.73 => ( 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 ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9814_isCont__Lb__Ub,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 & ( ord_less_eq_real @ X3 @ B ) )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 13.25/7.73 => ? [L6: real,M9: real] :
% 13.25/7.73 ( ! [X5: real] :
% 13.25/7.73 ( ( ( ord_less_eq_real @ A @ X5 )
% 13.25/7.73 & ( ord_less_eq_real @ X5 @ B ) )
% 13.25/7.73 => ( ( ord_less_eq_real @ L6 @ ( F @ X5 ) )
% 13.25/7.73 & ( ord_less_eq_real @ ( F @ X5 ) @ M9 ) ) )
% 13.25/7.73 & ! [Y5: real] :
% 13.25/7.73 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 13.25/7.73 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 13.25/7.73 => ? [X3: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 & ( ord_less_eq_real @ X3 @ B )
% 13.25/7.73 & ( ( F @ X3 )
% 13.25/7.73 = Y5 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9815_LIM__fun__gt__zero,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,C: real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ L2 )
% 13.25/7.73 => ? [R2: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ R2 )
% 13.25/7.73 & ! [X5: real] :
% 13.25/7.73 ( ( ( X5 != C )
% 13.25/7.73 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 13.25/7.73 => ( ord_less_real @ zero_zero_real @ ( F @ X5 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9816_LIM__fun__not__zero,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,C: real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 13.25/7.73 => ( ( L2 != zero_zero_real )
% 13.25/7.73 => ? [R2: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ R2 )
% 13.25/7.73 & ! [X5: real] :
% 13.25/7.73 ( ( ( X5 != C )
% 13.25/7.73 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 13.25/7.73 => ( ( F @ X5 )
% 13.25/7.73 != zero_zero_real ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9817_LIM__fun__less__zero,axiom,
% 13.25/7.73 ! [F: real > real,L2: real,C: real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 13.25/7.73 => ( ( ord_less_real @ L2 @ zero_zero_real )
% 13.25/7.73 => ? [R2: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ R2 )
% 13.25/7.73 & ! [X5: real] :
% 13.25/7.73 ( ( ( X5 != C )
% 13.25/7.73 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X5 ) ) @ R2 ) )
% 13.25/7.73 => ( ord_less_real @ ( F @ X5 ) @ zero_zero_real ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9818_isCont__real__sqrt,axiom,
% 13.25/7.73 ! [X: real] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ sqrt ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9819_isCont__real__root,axiom,
% 13.25/7.73 ! [X: real,N: nat] : ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( root @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9820_continuous__frac,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ~ ( member_real @ X @ ring_1_Ints_real )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ archim2898591450579166408c_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9821_isCont__inverse__function2,axiom,
% 13.25/7.73 ! [A: real,X: real,B: real,G: real > real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X )
% 13.25/7.73 => ( ( ord_less_real @ X @ B )
% 13.25/7.73 => ( ! [Z5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ Z5 )
% 13.25/7.73 => ( ( ord_less_eq_real @ Z5 @ B )
% 13.25/7.73 => ( ( G @ ( F @ Z5 ) )
% 13.25/7.73 = Z5 ) ) )
% 13.25/7.73 => ( ! [Z5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ Z5 )
% 13.25/7.73 => ( ( ord_less_eq_real @ Z5 @ B )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9822_isCont__arcosh,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcosh_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9823_LIM__cos__div__sin,axiom,
% 13.25/7.73 ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( cos_real @ X2 ) @ ( sin_real @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9824_continuous__floor,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ~ ( member_real @ X @ ring_1_Ints_real )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ ( comp_int_real_real @ ring_1_of_int_real @ archim6058952711729229775r_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9825_DERIV__inverse__function,axiom,
% 13.25/7.73 ! [F: real > real,D4: real,G: real > real,X: real,A: real,B: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X ) @ top_top_set_real ) )
% 13.25/7.73 => ( ( D4 != zero_zero_real )
% 13.25/7.73 => ( ( ord_less_real @ A @ X )
% 13.25/7.73 => ( ( ord_less_real @ X @ B )
% 13.25/7.73 => ( ! [Y4: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ Y4 )
% 13.25/7.73 => ( ( ord_less_real @ Y4 @ B )
% 13.25/7.73 => ( ( F @ ( G @ Y4 ) )
% 13.25/7.73 = Y4 ) ) )
% 13.25/7.73 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ G )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9826_isCont__arccos,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.73 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arccos ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9827_isCont__arcsin,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.73 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ arcsin ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9828_LIM__less__bound,axiom,
% 13.25/7.73 ! [B: real,X: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ B @ X )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ B @ X ) )
% 13.25/7.73 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) )
% 13.25/7.73 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ F )
% 13.25/7.73 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9829_isCont__artanh,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X )
% 13.25/7.73 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) @ artanh_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9830_isCont__inverse__function,axiom,
% 13.25/7.73 ! [D: real,X: real,G: real > real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ D )
% 13.25/7.73 => ( ! [Z5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
% 13.25/7.73 => ( ( G @ ( F @ Z5 ) )
% 13.25/7.73 = Z5 ) )
% 13.25/7.73 => ( ! [Z5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z5 @ X ) ) @ D )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X ) @ top_top_set_real ) @ G ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9831_GMVT_H,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F3: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ! [Z5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ Z5 )
% 13.25/7.73 => ( ( ord_less_eq_real @ Z5 @ B )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ F ) ) )
% 13.25/7.73 => ( ! [Z5: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ Z5 )
% 13.25/7.73 => ( ( ord_less_eq_real @ Z5 @ B )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) @ G ) ) )
% 13.25/7.73 => ( ! [Z5: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ Z5 )
% 13.25/7.73 => ( ( ord_less_real @ Z5 @ B )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ( ! [Z5: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ Z5 )
% 13.25/7.73 => ( ( ord_less_real @ Z5 @ B )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ F @ ( F3 @ Z5 ) @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ? [C2: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ C2 )
% 13.25/7.73 & ( ord_less_real @ C2 @ B )
% 13.25/7.73 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
% 13.25/7.73 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F3 @ C2 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9832_summable__Leibniz_I3_J,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ( topolo6980174941875973593q_real @ A )
% 13.25/7.73 => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 13.25/7.73 => ! [N7: nat] :
% 13.25/7.73 ( member_real
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 13.25/7.73 @ ( set_or1222579329274155063t_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) )
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9833_summable__Leibniz_I2_J,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ( topolo6980174941875973593q_real @ A )
% 13.25/7.73 => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 13.25/7.73 => ! [N7: nat] :
% 13.25/7.73 ( member_real
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 13.25/7.73 @ ( set_or1222579329274155063t_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9834_mult__nat__left__at__top,axiom,
% 13.25/7.73 ! [C: nat] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ C )
% 13.25/7.73 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9835_mult__nat__right__at__top,axiom,
% 13.25/7.73 ! [C: nat] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ C )
% 13.25/7.73 => ( filterlim_nat_nat
% 13.25/7.73 @ ^ [X2: nat] : ( times_times_nat @ X2 @ C )
% 13.25/7.73 @ at_top_nat
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9836_monoseq__convergent,axiom,
% 13.25/7.73 ! [X7: nat > real,B5: real] :
% 13.25/7.73 ( ( topolo6980174941875973593q_real @ X7 )
% 13.25/7.73 => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X7 @ I2 ) ) @ B5 )
% 13.25/7.73 => ~ ! [L6: real] :
% 13.25/7.73 ~ ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9837_LIMSEQ__root,axiom,
% 13.25/7.73 ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( root @ N2 @ ( semiri5074537144036343181t_real @ N2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ one_one_real )
% 13.25/7.73 @ at_top_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9838_nested__sequence__unique,axiom,
% 13.25/7.73 ! [F: nat > real,G: nat > real] :
% 13.25/7.73 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N4 ) ) @ ( G @ N4 ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( G @ N4 ) )
% 13.25/7.73 => ( ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G @ N2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_nat )
% 13.25/7.73 => ? [L4: real] :
% 13.25/7.73 ( ! [N7: nat] : ( ord_less_eq_real @ ( F @ N7 ) @ L4 )
% 13.25/7.73 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 13.25/7.73 & ! [N7: nat] : ( ord_less_eq_real @ L4 @ ( G @ N7 ) )
% 13.25/7.73 & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9839_LIMSEQ__inverse__zero,axiom,
% 13.25/7.73 ! [X7: nat > real] :
% 13.25/7.73 ( ! [R2: real] :
% 13.25/7.73 ? [N8: nat] :
% 13.25/7.73 ! [N4: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ N8 @ N4 )
% 13.25/7.73 => ( ord_less_real @ R2 @ ( X7 @ N4 ) ) )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( inverse_inverse_real @ ( X7 @ N2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9840_lim__inverse__n_H,axiom,
% 13.25/7.73 ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9841_LIMSEQ__root__const,axiom,
% 13.25/7.73 ! [C: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ C )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( root @ N2 @ C )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ one_one_real )
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9842_LIMSEQ__inverse__real__of__nat__add,axiom,
% 13.25/7.73 ! [R: real] :
% 13.25/7.73 ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( plus_plus_real @ R @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ R )
% 13.25/7.73 @ at_top_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9843_increasing__LIMSEQ,axiom,
% 13.25/7.73 ! [F: nat > real,L2: real] :
% 13.25/7.73 ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ ( F @ N4 ) @ L2 )
% 13.25/7.73 => ( ! [E2: real] :
% 13.25/7.73 ( ( ord_less_real @ zero_zero_real @ E2 )
% 13.25/7.73 => ? [N7: nat] : ( ord_less_eq_real @ L2 @ ( plus_plus_real @ ( F @ N7 ) @ E2 ) ) )
% 13.25/7.73 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L2 ) @ at_top_nat ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9844_LIMSEQ__realpow__zero,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.73 => ( ( ord_less_real @ X @ one_one_real )
% 13.25/7.73 => ( filterlim_nat_real @ ( power_power_real @ X ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9845_LIMSEQ__divide__realpow__zero,axiom,
% 13.25/7.73 ! [X: real,A: real] :
% 13.25/7.73 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( divide_divide_real @ A @ ( power_power_real @ X @ N2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9846_LIMSEQ__abs__realpow__zero2,axiom,
% 13.25/7.73 ! [C: real] :
% 13.25/7.73 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 13.25/7.73 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9847_LIMSEQ__abs__realpow__zero,axiom,
% 13.25/7.73 ! [C: real] :
% 13.25/7.73 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 13.25/7.73 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9848_LIMSEQ__inverse__realpow__zero,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_real @ one_one_real @ X )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X @ N2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9849_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 13.25/7.73 ! [R: real] :
% 13.25/7.73 ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( plus_plus_real @ R @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ R )
% 13.25/7.73 @ at_top_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9850_tendsto__exp__limit__sequentially,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 13.25/7.73 @ at_top_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9851_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 13.25/7.73 ! [R: real] :
% 13.25/7.73 ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( times_times_real @ R @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ R )
% 13.25/7.73 @ at_top_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9852_summable__Leibniz_I1_J,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ( topolo6980174941875973593q_real @ A )
% 13.25/7.73 => ( summable_real
% 13.25/7.73 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9853_summable,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 13.25/7.73 => ( summable_real
% 13.25/7.73 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A @ N2 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9854_cos__diff__limit__1,axiom,
% 13.25/7.73 ! [Theta: nat > real,Theta2: real] :
% 13.25/7.73 ( ( filterlim_nat_real
% 13.25/7.73 @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ one_one_real )
% 13.25/7.73 @ at_top_nat )
% 13.25/7.73 => ~ ! [K3: nat > int] :
% 13.25/7.73 ~ ( filterlim_nat_real
% 13.25/7.73 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ Theta2 )
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9855_cos__limit__1,axiom,
% 13.25/7.73 ! [Theta: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real
% 13.25/7.73 @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ one_one_real )
% 13.25/7.73 @ at_top_nat )
% 13.25/7.73 => ? [K3: nat > int] :
% 13.25/7.73 ( filterlim_nat_real
% 13.25/7.73 @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K3 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9856_summable__Leibniz_I4_J,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ( topolo6980174941875973593q_real @ A )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] :
% 13.25/7.73 ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 13.25/7.73 @ at_top_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9857_zeroseq__arctan__series,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9858_summable__Leibniz_H_I2_J,axiom,
% 13.25/7.73 ! [A: nat > real,N: nat] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 13.25/7.73 => ( ord_less_eq_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9859_summable__Leibniz_H_I3_J,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] :
% 13.25/7.73 ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 13.25/7.73 @ at_top_nat ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9860_sums__alternating__upper__lower,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 13.25/7.73 => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ? [L4: real] :
% 13.25/7.73 ( ! [N7: nat] :
% 13.25/7.73 ( ord_less_eq_real
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) ) )
% 13.25/7.73 @ L4 )
% 13.25/7.73 & ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] :
% 13.25/7.73 ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ L4 )
% 13.25/7.73 @ at_top_nat )
% 13.25/7.73 & ! [N7: nat] :
% 13.25/7.73 ( ord_less_eq_real @ L4
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N7 ) @ one_one_nat ) ) ) )
% 13.25/7.73 & ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] :
% 13.25/7.73 ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ L4 )
% 13.25/7.73 @ at_top_nat ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9861_summable__Leibniz_I5_J,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ( topolo6980174941875973593q_real @ A )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] :
% 13.25/7.73 ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 13.25/7.73 @ at_top_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9862_summable__Leibniz_H_I5_J,axiom,
% 13.25/7.73 ! [A: nat > real] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 13.25/7.73 => ( filterlim_nat_real
% 13.25/7.73 @ ^ [N2: nat] :
% 13.25/7.73 ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) ) )
% 13.25/7.73 @ at_top_nat ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9863_summable__Leibniz_H_I4_J,axiom,
% 13.25/7.73 ! [A: nat > real,N: nat] :
% 13.25/7.73 ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N4 ) )
% 13.25/7.73 => ( ! [N4: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N4 ) ) @ ( A @ N4 ) )
% 13.25/7.73 => ( ord_less_eq_real
% 13.25/7.73 @ ( suminf_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) ) )
% 13.25/7.73 @ ( groups6591440286371151544t_real
% 13.25/7.73 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A @ I4 ) )
% 13.25/7.73 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9864_eventually__sequentially__seg,axiom,
% 13.25/7.73 ! [P: nat > $o,K: nat] :
% 13.25/7.73 ( ( eventually_nat
% 13.25/7.73 @ ^ [N2: nat] : ( P @ ( plus_plus_nat @ N2 @ K ) )
% 13.25/7.73 @ at_top_nat )
% 13.25/7.73 = ( eventually_nat @ P @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9865_le__sequentially,axiom,
% 13.25/7.73 ! [F4: filter_nat] :
% 13.25/7.73 ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
% 13.25/7.73 = ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F4 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9866_eventually__sequentiallyI,axiom,
% 13.25/7.73 ! [C: nat,P: nat > $o] :
% 13.25/7.73 ( ! [X3: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ C @ X3 )
% 13.25/7.73 => ( P @ X3 ) )
% 13.25/7.73 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9867_eventually__sequentially,axiom,
% 13.25/7.73 ! [P: nat > $o] :
% 13.25/7.73 ( ( eventually_nat @ P @ at_top_nat )
% 13.25/7.73 = ( ? [N5: nat] :
% 13.25/7.73 ! [N2: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ N5 @ N2 )
% 13.25/7.73 => ( P @ N2 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9868_sequentially__offset,axiom,
% 13.25/7.73 ! [P: nat > $o,K: nat] :
% 13.25/7.73 ( ( eventually_nat @ P @ at_top_nat )
% 13.25/7.73 => ( eventually_nat
% 13.25/7.73 @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K ) )
% 13.25/7.73 @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9869_real__bounded__linear,axiom,
% 13.25/7.73 ( real_V5970128139526366754l_real
% 13.25/7.73 = ( ^ [F5: real > real] :
% 13.25/7.73 ? [C3: real] :
% 13.25/7.73 ( F5
% 13.25/7.73 = ( ^ [X2: real] : ( times_times_real @ X2 @ C3 ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9870_sqrt__at__top,axiom,
% 13.25/7.73 filterlim_real_real @ sqrt @ at_top_real @ at_top_real ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9871_lhopital__left__at__top__at__top,axiom,
% 13.25/7.73 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ at_top_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ at_top_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9872_lhopital__at__top__at__top,axiom,
% 13.25/7.73 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ at_top_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ at_top_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9873_eventually__at__left__real,axiom,
% 13.25/7.73 ! [B: real,A: real] :
% 13.25/7.73 ( ( ord_less_real @ B @ A )
% 13.25/7.73 => ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ B @ A ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9874_lhopital__left__at__top,axiom,
% 13.25/7.73 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y3: real] :
% 13.25/7.73 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ Y3 )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ Y3 )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9875_lhopital__at__top,axiom,
% 13.25/7.73 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y3: real] :
% 13.25/7.73 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ Y3 )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ Y3 )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9876_lhospital__at__top__at__top,axiom,
% 13.25/7.73 ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 13.25/7.73 ( ( filterlim_real_real @ G @ at_top_real @ at_top_real )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ at_top_real )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ at_top_real )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ at_top_real )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ X )
% 13.25/7.73 @ at_top_real )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ X )
% 13.25/7.73 @ at_top_real ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9877_tanh__real__at__top,axiom,
% 13.25/7.73 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9878_artanh__real__at__left__1,axiom,
% 13.25/7.73 filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9879_ln__x__over__x__tendsto__0,axiom,
% 13.25/7.73 ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_real ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9880_tendsto__power__div__exp__0,axiom,
% 13.25/7.73 ! [K: nat] :
% 13.25/7.73 ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( power_power_real @ X2 @ K ) @ ( exp_real @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 13.25/7.73 @ at_top_real ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9881_lhopital__left,axiom,
% 13.25/7.73 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ F4
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ F4
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5984915006950818249n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9882_lhopital,axiom,
% 13.25/7.73 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ F4
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ F4
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9883_tendsto__exp__limit__at__top,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( filterlim_real_real
% 13.25/7.73 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ Y ) ) @ Y )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 13.25/7.73 @ at_top_real ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9884_filterlim__tan__at__left,axiom,
% 13.25/7.73 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 ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9885_DERIV__neg__imp__decreasing__at__top,axiom,
% 13.25/7.73 ! [B: real,F: real > real,Flim: real] :
% 13.25/7.73 ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ B @ X3 )
% 13.25/7.73 => ? [Y5: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 13.25/7.73 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9886_tendsto__arctan__at__top,axiom,
% 13.25/7.73 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9887_at__top__le__at__infinity,axiom,
% 13.25/7.73 ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9888_dist__complex__def,axiom,
% 13.25/7.73 ( real_V3694042436643373181omplex
% 13.25/7.73 = ( ^ [X2: complex,Y: complex] : ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9889_dist__real__def,axiom,
% 13.25/7.73 ( real_V975177566351809787t_real
% 13.25/7.73 = ( ^ [X2: real,Y: real] : ( abs_abs_real @ ( minus_minus_real @ X2 @ Y ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9890_filterlim__pow__at__bot__even,axiom,
% 13.25/7.73 ! [N: nat,F: real > real,F4: filter_real] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 13.25/7.73 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
% 13.25/7.73 @ at_top_real
% 13.25/7.73 @ F4 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9891_at__bot__le__at__infinity,axiom,
% 13.25/7.73 ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9892_tanh__real__at__bot,axiom,
% 13.25/7.73 filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9893_lhopital__at__top__at__bot,axiom,
% 13.25/7.73 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ at_bot_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ at_bot_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ top_top_set_real ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9894_DERIV__pos__imp__increasing__at__bot,axiom,
% 13.25/7.73 ! [B: real,F: real > real,Flim: real] :
% 13.25/7.73 ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ X3 @ B )
% 13.25/7.73 => ? [Y5: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 13.25/7.73 => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9895_filterlim__pow__at__bot__odd,axiom,
% 13.25/7.73 ! [N: nat,F: real > real,F4: filter_real] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
% 13.25/7.73 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( power_power_real @ ( F @ X2 ) @ N )
% 13.25/7.73 @ at_bot_real
% 13.25/7.73 @ F4 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9896_lhopital__left__at__top__at__bot,axiom,
% 13.25/7.73 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ at_bot_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ at_bot_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9897_tendsto__arctan__at__bot,axiom,
% 13.25/7.73 filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9898_Bseq__eq__bounded,axiom,
% 13.25/7.73 ! [F: nat > real,A: real,B: real] :
% 13.25/7.73 ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
% 13.25/7.73 => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9899_Bseq__realpow,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.73 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.73 => ( bfun_nat_real @ ( power_power_real @ X ) @ at_top_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9900_tendsto__exp__limit__at__right,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( filterlim_real_real
% 13.25/7.73 @ ^ [Y: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ one_one_real @ Y ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9901_filterlim__tan__at__right,axiom,
% 13.25/7.73 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 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9902_eventually__at__right__to__0,axiom,
% 13.25/7.73 ! [P: real > $o,A: real] :
% 13.25/7.73 ( ( eventually_real @ P @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 = ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( P @ ( plus_plus_real @ X2 @ A ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9903_eventually__at__right__real,axiom,
% 13.25/7.73 ! [A: real,B: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9904_tendsto__arcosh__at__left__1,axiom,
% 13.25/7.73 filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9905_artanh__real__at__right__1,axiom,
% 13.25/7.73 filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9906_lhopital__right__at__top__at__top,axiom,
% 13.25/7.73 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ at_top_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ at_top_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9907_lhopital__right,axiom,
% 13.25/7.73 ! [F: real > real,X: real,G: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ F4
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ F4
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9908_lhopital__right__0,axiom,
% 13.25/7.73 ! [F0: real > real,G0: real > real,G2: real > real,F3: real > real,F4: filter_real] :
% 13.25/7.73 ( ( filterlim_real_real @ F0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G0 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G0 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F0 @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G0 @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ F4
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
% 13.25/7.73 @ F4
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9909_lhopital__right__at__top__at__bot,axiom,
% 13.25/7.73 ! [F: real > real,A: real,G: real > real,F3: real > real,G2: real > real] :
% 13.25/7.73 ( ( filterlim_real_real @ F @ at_top_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( ( filterlim_real_real @ G @ at_bot_real @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ at_bot_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ at_bot_real
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9910_lhopital__right__0__at__top,axiom,
% 13.25/7.73 ! [G: real > real,G2: real > real,F: real > real,F3: real > real,X: real] :
% 13.25/7.73 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ X )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ X )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9911_lhopital__right__at__top,axiom,
% 13.25/7.73 ! [G: real > real,X: real,G2: real > real,F: real > real,F3: real > real,Y3: real] :
% 13.25/7.73 ( ( filterlim_real_real @ G @ at_top_real @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] :
% 13.25/7.73 ( ( G2 @ X2 )
% 13.25/7.73 != zero_zero_real )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( eventually_real
% 13.25/7.73 @ ^ [X2: real] : ( has_fi5821293074295781190e_real @ G @ ( G2 @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F3 @ X2 ) @ ( G2 @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ Y3 )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) )
% 13.25/7.73 => ( filterlim_real_real
% 13.25/7.73 @ ^ [X2: real] : ( divide_divide_real @ ( F @ X2 ) @ ( G @ X2 ) )
% 13.25/7.73 @ ( topolo2815343760600316023s_real @ Y3 )
% 13.25/7.73 @ ( topolo2177554685111907308n_real @ X @ ( set_or5849166863359141190n_real @ X ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9912_greaterThan__Suc,axiom,
% 13.25/7.73 ! [K: nat] :
% 13.25/7.73 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 13.25/7.73 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9913_atLeast__Suc,axiom,
% 13.25/7.73 ! [K: nat] :
% 13.25/7.73 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 13.25/7.73 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9914_decseq__bounded,axiom,
% 13.25/7.73 ! [X7: nat > real,B5: real] :
% 13.25/7.73 ( ( order_9091379641038594480t_real @ X7 )
% 13.25/7.73 => ( ! [I2: nat] : ( ord_less_eq_real @ B5 @ ( X7 @ I2 ) )
% 13.25/7.73 => ( bfun_nat_real @ X7 @ at_top_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9915_decseq__convergent,axiom,
% 13.25/7.73 ! [X7: nat > real,B5: real] :
% 13.25/7.73 ( ( order_9091379641038594480t_real @ X7 )
% 13.25/7.73 => ( ! [I2: nat] : ( ord_less_eq_real @ B5 @ ( X7 @ I2 ) )
% 13.25/7.73 => ~ ! [L6: real] :
% 13.25/7.73 ( ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 13.25/7.73 => ~ ! [I3: nat] : ( ord_less_eq_real @ L6 @ ( X7 @ I3 ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9916_GMVT,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real,G: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 & ( ord_less_eq_real @ X3 @ B ) )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F ) )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 & ( ord_less_real @ X3 @ B ) )
% 13.25/7.73 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ( ord_less_eq_real @ A @ X3 )
% 13.25/7.73 & ( ord_less_eq_real @ X3 @ B ) )
% 13.25/7.73 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G ) )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 & ( ord_less_real @ X3 @ B ) )
% 13.25/7.73 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
% 13.25/7.73 => ? [G_c: real,F_c: real,C2: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 13.25/7.73 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_real @ A @ C2 )
% 13.25/7.73 & ( ord_less_real @ C2 @ B )
% 13.25/7.73 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 13.25/7.73 = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9917_real__differentiable__def,axiom,
% 13.25/7.73 ! [F: real > real,X: real,S: set_real] :
% 13.25/7.73 ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X @ S ) )
% 13.25/7.73 = ( ? [D6: real] : ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9918_real__differentiableE,axiom,
% 13.25/7.73 ! [F: real > real,X: real,S: set_real] :
% 13.25/7.73 ( ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X @ S ) )
% 13.25/7.73 => ~ ! [Df: real] :
% 13.25/7.73 ~ ( has_fi5821293074295781190e_real @ F @ Df @ ( topolo2177554685111907308n_real @ X @ S ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9919_MVT,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_real @ X3 @ B )
% 13.25/7.73 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ? [L4: real,Z5: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ Z5 )
% 13.25/7.73 & ( ord_less_real @ Z5 @ B )
% 13.25/7.73 & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) )
% 13.25/7.73 & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 13.25/7.73 = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9920_continuous__on__arcosh_H,axiom,
% 13.25/7.73 ! [A2: set_real,F: real > real] :
% 13.25/7.73 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ A2 )
% 13.25/7.73 => ( ord_less_eq_real @ one_one_real @ ( F @ X3 ) ) )
% 13.25/7.73 => ( topolo5044208981011980120l_real @ A2
% 13.25/7.73 @ ^ [X2: real] : ( arcosh_real @ ( F @ X2 ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9921_continuous__image__closed__interval,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_eq_real @ A @ B )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ? [C2: real,D3: real] :
% 13.25/7.73 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
% 13.25/7.73 = ( set_or1222579329274155063t_real @ C2 @ D3 ) )
% 13.25/7.73 & ( ord_less_eq_real @ C2 @ D3 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9922_continuous__on__arcosh,axiom,
% 13.25/7.73 ! [A2: set_real] :
% 13.25/7.73 ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
% 13.25/7.73 => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9923_continuous__on__arccos_H,axiom,
% 13.25/7.73 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9924_continuous__on__arcsin_H,axiom,
% 13.25/7.73 topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9925_continuous__on__artanh_H,axiom,
% 13.25/7.73 ! [A2: set_real,F: real > real] :
% 13.25/7.73 ( ( topolo5044208981011980120l_real @ A2 @ F )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ A2 )
% 13.25/7.73 => ( member_real @ ( F @ X3 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
% 13.25/7.73 => ( topolo5044208981011980120l_real @ A2
% 13.25/7.73 @ ^ [X2: real] : ( artanh_real @ ( F @ X2 ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9926_Rolle__deriv,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ( ( F @ A )
% 13.25/7.73 = ( F @ B ) )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_real @ X3 @ B )
% 13.25/7.73 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ? [Z5: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ Z5 )
% 13.25/7.73 & ( ord_less_real @ Z5 @ B )
% 13.25/7.73 & ( ( F3 @ Z5 )
% 13.25/7.73 = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9927_mvt,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real,F3: real > real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_real @ X3 @ B )
% 13.25/7.73 => ( has_de1759254742604945161l_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ~ ! [Xi: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ Xi )
% 13.25/7.73 => ( ( ord_less_real @ Xi @ B )
% 13.25/7.73 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 13.25/7.73 != ( F3 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9928_DERIV__pos__imp__increasing__open,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_real @ X3 @ B )
% 13.25/7.73 => ? [Y5: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9929_DERIV__neg__imp__decreasing__open,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_real @ X3 @ B )
% 13.25/7.73 => ? [Y5: real] :
% 13.25/7.73 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9930_DERIV__isconst__end,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_real @ X3 @ B )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ( ( F @ B )
% 13.25/7.73 = ( F @ A ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9931_continuous__on__artanh,axiom,
% 13.25/7.73 ! [A2: set_real] :
% 13.25/7.73 ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 13.25/7.73 => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9932_DERIV__isconst2,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real,X: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_real @ X3 @ B )
% 13.25/7.73 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ( ( ord_less_eq_real @ A @ X )
% 13.25/7.73 => ( ( ord_less_eq_real @ X @ B )
% 13.25/7.73 => ( ( F @ X )
% 13.25/7.73 = ( F @ A ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9933_Rolle,axiom,
% 13.25/7.73 ! [A: real,B: real,F: real > real] :
% 13.25/7.73 ( ( ord_less_real @ A @ B )
% 13.25/7.73 => ( ( ( F @ A )
% 13.25/7.73 = ( F @ B ) )
% 13.25/7.73 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 13.25/7.73 => ( ! [X3: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ X3 )
% 13.25/7.73 => ( ( ord_less_real @ X3 @ B )
% 13.25/7.73 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
% 13.25/7.73 => ? [Z5: real] :
% 13.25/7.73 ( ( ord_less_real @ A @ Z5 )
% 13.25/7.73 & ( ord_less_real @ Z5 @ B )
% 13.25/7.73 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z5 @ top_top_set_real ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9934_uniformity__complex__def,axiom,
% 13.25/7.73 ( topolo896644834953643431omplex
% 13.25/7.73 = ( comple8358262395181532106omplex
% 13.25/7.73 @ ( image_5971271580939081552omplex
% 13.25/7.73 @ ^ [E3: real] :
% 13.25/7.73 ( princi3496590319149328850omplex
% 13.25/7.73 @ ( collec8663557070575231912omplex
% 13.25/7.73 @ ( produc6771430404735790350plex_o
% 13.25/7.73 @ ^ [X2: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X2 @ Y ) @ E3 ) ) ) )
% 13.25/7.73 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9935_uniformity__real__def,axiom,
% 13.25/7.73 ( topolo1511823702728130853y_real
% 13.25/7.73 = ( comple2936214249959783750l_real
% 13.25/7.73 @ ( image_2178119161166701260l_real
% 13.25/7.73 @ ^ [E3: real] :
% 13.25/7.73 ( princi6114159922880469582l_real
% 13.25/7.73 @ ( collec3799799289383736868l_real
% 13.25/7.73 @ ( produc5414030515140494994real_o
% 13.25/7.73 @ ^ [X2: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X2 @ Y ) @ E3 ) ) ) )
% 13.25/7.73 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9936_mono__Suc,axiom,
% 13.25/7.73 order_mono_nat_nat @ suc ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9937_mono__times__nat,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9938_incseq__bounded,axiom,
% 13.25/7.73 ! [X7: nat > real,B5: real] :
% 13.25/7.73 ( ( order_mono_nat_real @ X7 )
% 13.25/7.73 => ( ! [I2: nat] : ( ord_less_eq_real @ ( X7 @ I2 ) @ B5 )
% 13.25/7.73 => ( bfun_nat_real @ X7 @ at_top_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9939_incseq__convergent,axiom,
% 13.25/7.73 ! [X7: nat > real,B5: real] :
% 13.25/7.73 ( ( order_mono_nat_real @ X7 )
% 13.25/7.73 => ( ! [I2: nat] : ( ord_less_eq_real @ ( X7 @ I2 ) @ B5 )
% 13.25/7.73 => ~ ! [L6: real] :
% 13.25/7.73 ( ( filterlim_nat_real @ X7 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 13.25/7.73 => ~ ! [I3: nat] : ( ord_less_eq_real @ ( X7 @ I3 ) @ L6 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9940_mono__ge2__power__minus__self,axiom,
% 13.25/7.73 ! [K: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 13.25/7.73 => ( order_mono_nat_nat
% 13.25/7.73 @ ^ [M3: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M3 ) @ M3 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9941_inj__sgn__power,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( inj_on_real_real
% 13.25/7.73 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 13.25/7.73 @ top_top_set_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9942_log__inj,axiom,
% 13.25/7.73 ! [B: real] :
% 13.25/7.73 ( ( ord_less_real @ one_one_real @ B )
% 13.25/7.73 => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9943_inj__on__diff__nat,axiom,
% 13.25/7.73 ! [N3: set_nat,K: nat] :
% 13.25/7.73 ( ! [N4: nat] :
% 13.25/7.73 ( ( member_nat @ N4 @ N3 )
% 13.25/7.73 => ( ord_less_eq_nat @ K @ N4 ) )
% 13.25/7.73 => ( inj_on_nat_nat
% 13.25/7.73 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 13.25/7.73 @ N3 ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9944_inj__Suc,axiom,
% 13.25/7.73 ! [N3: set_nat] : ( inj_on_nat_nat @ suc @ N3 ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9945_summable__reindex,axiom,
% 13.25/7.73 ! [F: nat > real,G: nat > nat] :
% 13.25/7.73 ( ( summable_real @ F )
% 13.25/7.73 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 13.25/7.73 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 13.25/7.73 => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9946_suminf__reindex__mono,axiom,
% 13.25/7.73 ! [F: nat > real,G: nat > nat] :
% 13.25/7.73 ( ( summable_real @ F )
% 13.25/7.73 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 13.25/7.73 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 13.25/7.73 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9947_inj__on__char__of__nat,axiom,
% 13.25/7.73 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9948_suminf__reindex,axiom,
% 13.25/7.73 ! [F: nat > real,G: nat > nat] :
% 13.25/7.73 ( ( summable_real @ F )
% 13.25/7.73 => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
% 13.25/7.73 => ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 13.25/7.73 => ( ! [X3: nat] :
% 13.25/7.73 ( ~ ( member_nat @ X3 @ ( image_nat_nat @ G @ top_top_set_nat ) )
% 13.25/7.73 => ( ( F @ X3 )
% 13.25/7.73 = zero_zero_real ) )
% 13.25/7.73 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
% 13.25/7.73 = ( suminf_real @ F ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9949_nonneg__incseq__Bseq__subseq__iff,axiom,
% 13.25/7.73 ! [F: nat > real,G: nat > nat] :
% 13.25/7.73 ( ! [X3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) )
% 13.25/7.73 => ( ( order_mono_nat_real @ F )
% 13.25/7.73 => ( ( order_5726023648592871131at_nat @ G )
% 13.25/7.73 => ( ( bfun_nat_real
% 13.25/7.73 @ ^ [X2: nat] : ( F @ ( G @ X2 ) )
% 13.25/7.73 @ at_top_nat )
% 13.25/7.73 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9950_strict__mono__imp__increasing,axiom,
% 13.25/7.73 ! [F: nat > nat,N: nat] :
% 13.25/7.73 ( ( order_5726023648592871131at_nat @ F )
% 13.25/7.73 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9951_powr__real__of__int_H,axiom,
% 13.25/7.73 ! [X: real,N: int] :
% 13.25/7.73 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.73 => ( ( ( X != zero_zero_real )
% 13.25/7.73 | ( ord_less_int @ zero_zero_int @ N ) )
% 13.25/7.73 => ( ( powr_real @ X @ ( ring_1_of_int_real @ N ) )
% 13.25/7.73 = ( power_int_real @ X @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9952_Rats__eq__int__div__nat,axiom,
% 13.25/7.73 ( field_5140801741446780682s_real
% 13.25/7.73 = ( collect_real
% 13.25/7.73 @ ^ [Uu3: real] :
% 13.25/7.73 ? [I4: int,N2: nat] :
% 13.25/7.73 ( ( Uu3
% 13.25/7.73 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 13.25/7.73 & ( N2 != zero_zero_nat ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9953_Rats__abs__iff,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ( ( member_real @ ( abs_abs_real @ X ) @ field_5140801741446780682s_real )
% 13.25/7.73 = ( member_real @ X @ field_5140801741446780682s_real ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9954_Rats__no__top__le,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ? [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 13.25/7.73 & ( ord_less_eq_real @ X @ X3 ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9955_Rats__dense__in__real,axiom,
% 13.25/7.73 ! [X: real,Y3: real] :
% 13.25/7.73 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.73 => ? [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 13.25/7.73 & ( ord_less_real @ X @ X3 )
% 13.25/7.73 & ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9956_Rats__no__bot__less,axiom,
% 13.25/7.73 ! [X: real] :
% 13.25/7.73 ? [X3: real] :
% 13.25/7.73 ( ( member_real @ X3 @ field_5140801741446780682s_real )
% 13.25/7.73 & ( ord_less_real @ X3 @ X ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9957_atLeastLessThan__add__Un,axiom,
% 13.25/7.73 ! [I: nat,J: nat,K: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.73 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 13.25/7.73 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9958_Rats__eq__int__div__int,axiom,
% 13.25/7.73 ( field_5140801741446780682s_real
% 13.25/7.73 = ( collect_real
% 13.25/7.73 @ ^ [Uu3: real] :
% 13.25/7.73 ? [I4: int,J3: int] :
% 13.25/7.73 ( ( Uu3
% 13.25/7.73 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( ring_1_of_int_real @ J3 ) ) )
% 13.25/7.73 & ( J3 != zero_zero_int ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9959_pos__deriv__imp__strict__mono,axiom,
% 13.25/7.73 ! [F: real > real,F3: real > real] :
% 13.25/7.73 ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F @ ( F3 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 13.25/7.73 => ( ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( F3 @ X3 ) )
% 13.25/7.73 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9960_sup__enat__def,axiom,
% 13.25/7.73 sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9961_sup__nat__def,axiom,
% 13.25/7.73 sup_sup_nat = ord_max_nat ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9962_num__of__integer__code,axiom,
% 13.25/7.73 ( code_num_of_integer
% 13.25/7.73 = ( ^ [K2: code_integer] :
% 13.25/7.73 ( if_num @ ( ord_le3102999989581377725nteger @ K2 @ one_one_Code_integer ) @ one
% 13.25/7.73 @ ( produc7336495610019696514er_num
% 13.25/7.73 @ ^ [L: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L ) @ ( code_num_of_integer @ L ) ) @ one ) )
% 13.25/7.73 @ ( code_divmod_integer @ K2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9963_min__Suc__Suc,axiom,
% 13.25/7.73 ! [M: nat,N: nat] :
% 13.25/7.73 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 13.25/7.73 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9964_min__0R,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( ord_min_nat @ N @ zero_zero_nat )
% 13.25/7.73 = zero_zero_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9965_min__0L,axiom,
% 13.25/7.73 ! [N: nat] :
% 13.25/7.73 ( ( ord_min_nat @ zero_zero_nat @ N )
% 13.25/7.73 = zero_zero_nat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9966_min__Suc__numeral,axiom,
% 13.25/7.73 ! [N: nat,K: num] :
% 13.25/7.73 ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 13.25/7.73 = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9967_min__numeral__Suc,axiom,
% 13.25/7.73 ! [K: num,N: nat] :
% 13.25/7.73 ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 13.25/7.73 = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9968_min__diff,axiom,
% 13.25/7.73 ! [M: nat,I: nat,N: nat] :
% 13.25/7.73 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 13.25/7.73 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9969_nat__mult__min__left,axiom,
% 13.25/7.73 ! [M: nat,N: nat,Q2: nat] :
% 13.25/7.73 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q2 )
% 13.25/7.73 = ( ord_min_nat @ ( times_times_nat @ M @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9970_nat__mult__min__right,axiom,
% 13.25/7.73 ! [M: nat,N: nat,Q2: nat] :
% 13.25/7.73 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q2 ) )
% 13.25/7.73 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q2 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9971_concat__bit__assoc__sym,axiom,
% 13.25/7.73 ! [M: nat,N: nat,K: int,L2: int,R: int] :
% 13.25/7.73 ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N @ K @ L2 ) @ R )
% 13.25/7.73 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N ) @ L2 @ R ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9972_inf__nat__def,axiom,
% 13.25/7.73 inf_inf_nat = ord_min_nat ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9973_take__bit__concat__bit__eq,axiom,
% 13.25/7.73 ! [M: nat,N: nat,K: int,L2: int] :
% 13.25/7.73 ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N @ K @ L2 ) )
% 13.25/7.73 = ( bit_concat_bit @ ( ord_min_nat @ M @ N ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N ) @ L2 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9974_min__Suc1,axiom,
% 13.25/7.73 ! [N: nat,M: nat] :
% 13.25/7.73 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 13.25/7.73 = ( case_nat_nat @ zero_zero_nat
% 13.25/7.73 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
% 13.25/7.73 @ M ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9975_min__Suc2,axiom,
% 13.25/7.73 ! [M: nat,N: nat] :
% 13.25/7.73 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 13.25/7.73 = ( case_nat_nat @ zero_zero_nat
% 13.25/7.73 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
% 13.25/7.73 @ M ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9976_min__enat__simps_I2_J,axiom,
% 13.25/7.73 ! [Q2: extended_enat] :
% 13.25/7.73 ( ( ord_mi8085742599997312461d_enat @ Q2 @ zero_z5237406670263579293d_enat )
% 13.25/7.73 = zero_z5237406670263579293d_enat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9977_min__enat__simps_I3_J,axiom,
% 13.25/7.73 ! [Q2: extended_enat] :
% 13.25/7.73 ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q2 )
% 13.25/7.73 = zero_z5237406670263579293d_enat ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9978_inf__enat__def,axiom,
% 13.25/7.73 inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9979_card__le__Suc__Max,axiom,
% 13.25/7.73 ! [S2: set_nat] :
% 13.25/7.73 ( ( finite_finite_nat @ S2 )
% 13.25/7.73 => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9980_divide__nat__def,axiom,
% 13.25/7.73 ( divide_divide_nat
% 13.25/7.73 = ( ^ [M3: nat,N2: nat] :
% 13.25/7.73 ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 13.25/7.73 @ ( lattic8265883725875713057ax_nat
% 13.25/7.73 @ ( collect_nat
% 13.25/7.73 @ ^ [K2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K2 @ N2 ) @ M3 ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9981_gcd__is__Max__divisors__nat,axiom,
% 13.25/7.73 ! [N: nat,M: nat] :
% 13.25/7.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.73 => ( ( gcd_gcd_nat @ M @ N )
% 13.25/7.73 = ( lattic8265883725875713057ax_nat
% 13.25/7.73 @ ( collect_nat
% 13.25/7.73 @ ^ [D2: nat] :
% 13.25/7.73 ( ( dvd_dvd_nat @ D2 @ M )
% 13.25/7.73 & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9982_sorted__list__of__set__greaterThanAtMost,axiom,
% 13.25/7.73 ! [I: nat,J: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 13.25/7.73 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 13.25/7.73 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9983_sorted__list__of__set__greaterThanLessThan,axiom,
% 13.25/7.73 ! [I: nat,J: nat] :
% 13.25/7.73 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 13.25/7.73 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 13.25/7.73 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9984_upto__aux__rec,axiom,
% 13.25/7.73 ( upto_aux
% 13.25/7.73 = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9985_upto_Opelims,axiom,
% 13.25/7.73 ! [X: int,Xa: int,Y3: list_int] :
% 13.25/7.73 ( ( ( upto @ X @ Xa )
% 13.25/7.73 = Y3 )
% 13.25/7.73 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
% 13.25/7.73 => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
% 13.25/7.73 => ( Y3
% 13.25/7.73 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 13.25/7.73 => ( Y3 = nil_int ) ) )
% 13.25/7.73 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9986_upto_Opsimps,axiom,
% 13.25/7.73 ! [I: int,J: int] :
% 13.25/7.73 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 13.25/7.73 => ( ( ( ord_less_eq_int @ I @ J )
% 13.25/7.73 => ( ( upto @ I @ J )
% 13.25/7.73 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_eq_int @ I @ J )
% 13.25/7.73 => ( ( upto @ I @ J )
% 13.25/7.73 = nil_int ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9987_upto__Nil,axiom,
% 13.25/7.73 ! [I: int,J: int] :
% 13.25/7.73 ( ( ( upto @ I @ J )
% 13.25/7.73 = nil_int )
% 13.25/7.73 = ( ord_less_int @ J @ I ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9988_upto__Nil2,axiom,
% 13.25/7.73 ! [I: int,J: int] :
% 13.25/7.73 ( ( nil_int
% 13.25/7.73 = ( upto @ I @ J ) )
% 13.25/7.73 = ( ord_less_int @ J @ I ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9989_upto__empty,axiom,
% 13.25/7.73 ! [J: int,I: int] :
% 13.25/7.73 ( ( ord_less_int @ J @ I )
% 13.25/7.73 => ( ( upto @ I @ J )
% 13.25/7.73 = nil_int ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9990_nth__upto,axiom,
% 13.25/7.73 ! [I: int,K: nat,J: int] :
% 13.25/7.73 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 13.25/7.73 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 13.25/7.73 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9991_length__upto,axiom,
% 13.25/7.73 ! [I: int,J: int] :
% 13.25/7.73 ( ( size_size_list_int @ ( upto @ I @ J ) )
% 13.25/7.73 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9992_upto__rec__numeral_I1_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 = nil_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9993_upto__rec__numeral_I2_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 = nil_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9994_upto__rec__numeral_I3_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 = ( 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 ) ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 13.25/7.73 = nil_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9995_upto__rec__numeral_I4_J,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 = ( 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 ) ) ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 13.25/7.73 = nil_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9996_upto__split2,axiom,
% 13.25/7.73 ! [I: int,J: int,K: int] :
% 13.25/7.73 ( ( ord_less_eq_int @ I @ J )
% 13.25/7.73 => ( ( ord_less_eq_int @ J @ K )
% 13.25/7.73 => ( ( upto @ I @ K )
% 13.25/7.73 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9997_upto__split1,axiom,
% 13.25/7.73 ! [I: int,J: int,K: int] :
% 13.25/7.73 ( ( ord_less_eq_int @ I @ J )
% 13.25/7.73 => ( ( ord_less_eq_int @ J @ K )
% 13.25/7.73 => ( ( upto @ I @ K )
% 13.25/7.73 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9998_atLeastLessThan__upto,axiom,
% 13.25/7.73 ( set_or4662586982721622107an_int
% 13.25/7.73 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_9999_greaterThanAtMost__upto,axiom,
% 13.25/7.73 ( set_or6656581121297822940st_int
% 13.25/7.73 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10000_upto_Osimps,axiom,
% 13.25/7.73 ( upto
% 13.25/7.73 = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10001_upto_Oelims,axiom,
% 13.25/7.73 ! [X: int,Xa: int,Y3: list_int] :
% 13.25/7.73 ( ( ( upto @ X @ Xa )
% 13.25/7.73 = Y3 )
% 13.25/7.73 => ( ( ( ord_less_eq_int @ X @ Xa )
% 13.25/7.73 => ( Y3
% 13.25/7.73 = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_eq_int @ X @ Xa )
% 13.25/7.73 => ( Y3 = nil_int ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10002_upto__rec1,axiom,
% 13.25/7.73 ! [I: int,J: int] :
% 13.25/7.73 ( ( ord_less_eq_int @ I @ J )
% 13.25/7.73 => ( ( upto @ I @ J )
% 13.25/7.73 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10003_upto__rec2,axiom,
% 13.25/7.73 ! [I: int,J: int] :
% 13.25/7.73 ( ( ord_less_eq_int @ I @ J )
% 13.25/7.73 => ( ( upto @ I @ J )
% 13.25/7.73 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10004_greaterThanLessThan__upto,axiom,
% 13.25/7.73 ( set_or5832277885323065728an_int
% 13.25/7.73 = ( ^ [I4: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10005_upto__split3,axiom,
% 13.25/7.73 ! [I: int,J: int,K: int] :
% 13.25/7.73 ( ( ord_less_eq_int @ I @ J )
% 13.25/7.73 => ( ( ord_less_eq_int @ J @ K )
% 13.25/7.73 => ( ( upto @ I @ K )
% 13.25/7.73 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10006_hd__upt,axiom,
% 13.25/7.73 ! [I: nat,J: nat] :
% 13.25/7.73 ( ( ord_less_nat @ I @ J )
% 13.25/7.73 => ( ( hd_nat @ ( upt @ I @ J ) )
% 13.25/7.73 = I ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10007_upt__conv__Nil,axiom,
% 13.25/7.73 ! [J: nat,I: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ J @ I )
% 13.25/7.73 => ( ( upt @ I @ J )
% 13.25/7.73 = nil_nat ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10008_drop__upt,axiom,
% 13.25/7.73 ! [M: nat,I: nat,J: nat] :
% 13.25/7.73 ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 13.25/7.73 = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10009_length__upt,axiom,
% 13.25/7.73 ! [I: nat,J: nat] :
% 13.25/7.73 ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 13.25/7.73 = ( minus_minus_nat @ J @ I ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10010_upt__eq__Nil__conv,axiom,
% 13.25/7.73 ! [I: nat,J: nat] :
% 13.25/7.73 ( ( ( upt @ I @ J )
% 13.25/7.73 = nil_nat )
% 13.25/7.73 = ( ( J = zero_zero_nat )
% 13.25/7.73 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10011_take__upt,axiom,
% 13.25/7.73 ! [I: nat,M: nat,N: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 13.25/7.73 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 13.25/7.73 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10012_nth__upt,axiom,
% 13.25/7.73 ! [I: nat,K: nat,J: nat] :
% 13.25/7.73 ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 13.25/7.73 => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 13.25/7.73 = ( plus_plus_nat @ I @ K ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10013_upt__rec__numeral,axiom,
% 13.25/7.73 ! [M: num,N: num] :
% 13.25/7.73 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.73 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.73 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.73 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 13.25/7.73 = nil_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10014_map__add__upt,axiom,
% 13.25/7.73 ! [N: nat,M: nat] :
% 13.25/7.73 ( ( map_nat_nat
% 13.25/7.73 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 13.25/7.73 @ ( upt @ zero_zero_nat @ M ) )
% 13.25/7.73 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10015_upt__add__eq__append,axiom,
% 13.25/7.73 ! [I: nat,J: nat,K: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.73 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 13.25/7.73 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10016_upt__conv__Cons,axiom,
% 13.25/7.73 ! [I: nat,J: nat] :
% 13.25/7.73 ( ( ord_less_nat @ I @ J )
% 13.25/7.73 => ( ( upt @ I @ J )
% 13.25/7.73 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10017_upt__rec,axiom,
% 13.25/7.73 ( upt
% 13.25/7.73 = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10018_upt__Suc,axiom,
% 13.25/7.73 ! [I: nat,J: nat] :
% 13.25/7.73 ( ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.73 => ( ( upt @ I @ ( suc @ J ) )
% 13.25/7.73 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 13.25/7.73 & ( ~ ( ord_less_eq_nat @ I @ J )
% 13.25/7.73 => ( ( upt @ I @ ( suc @ J ) )
% 13.25/7.73 = nil_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10019_upt__Suc__append,axiom,
% 13.25/7.73 ! [I: nat,J: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ I @ J )
% 13.25/7.73 => ( ( upt @ I @ ( suc @ J ) )
% 13.25/7.73 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10020_map__decr__upt,axiom,
% 13.25/7.73 ! [M: nat,N: nat] :
% 13.25/7.73 ( ( map_nat_nat
% 13.25/7.73 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 13.25/7.73 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 13.25/7.73 = ( upt @ M @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10021_upt__eq__Cons__conv,axiom,
% 13.25/7.73 ! [I: nat,J: nat,X: nat,Xs2: list_nat] :
% 13.25/7.73 ( ( ( upt @ I @ J )
% 13.25/7.73 = ( cons_nat @ X @ Xs2 ) )
% 13.25/7.73 = ( ( ord_less_nat @ I @ J )
% 13.25/7.73 & ( I = X )
% 13.25/7.73 & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 13.25/7.73 = Xs2 ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10022_sum__list__upt,axiom,
% 13.25/7.73 ! [M: nat,N: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ M @ N )
% 13.25/7.73 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 13.25/7.73 = ( groups3542108847815614940at_nat
% 13.25/7.73 @ ^ [X2: nat] : X2
% 13.25/7.73 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10023_card__length__sum__list__rec,axiom,
% 13.25/7.73 ! [M: nat,N3: nat] :
% 13.25/7.73 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 13.25/7.73 => ( ( finite_card_list_nat
% 13.25/7.73 @ ( collect_list_nat
% 13.25/7.73 @ ^ [L: list_nat] :
% 13.25/7.73 ( ( ( size_size_list_nat @ L )
% 13.25/7.73 = M )
% 13.25/7.73 & ( ( groups4561878855575611511st_nat @ L )
% 13.25/7.73 = N3 ) ) ) )
% 13.25/7.73 = ( plus_plus_nat
% 13.25/7.73 @ ( finite_card_list_nat
% 13.25/7.73 @ ( collect_list_nat
% 13.25/7.73 @ ^ [L: list_nat] :
% 13.25/7.73 ( ( ( size_size_list_nat @ L )
% 13.25/7.73 = ( minus_minus_nat @ M @ one_one_nat ) )
% 13.25/7.73 & ( ( groups4561878855575611511st_nat @ L )
% 13.25/7.73 = N3 ) ) ) )
% 13.25/7.73 @ ( finite_card_list_nat
% 13.25/7.73 @ ( collect_list_nat
% 13.25/7.73 @ ^ [L: list_nat] :
% 13.25/7.73 ( ( ( size_size_list_nat @ L )
% 13.25/7.73 = M )
% 13.25/7.73 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L ) @ one_one_nat )
% 13.25/7.73 = N3 ) ) ) ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10024_card__length__sum__list,axiom,
% 13.25/7.73 ! [M: nat,N3: nat] :
% 13.25/7.73 ( ( finite_card_list_nat
% 13.25/7.73 @ ( collect_list_nat
% 13.25/7.73 @ ^ [L: list_nat] :
% 13.25/7.73 ( ( ( size_size_list_nat @ L )
% 13.25/7.73 = M )
% 13.25/7.73 & ( ( groups4561878855575611511st_nat @ L )
% 13.25/7.73 = N3 ) ) ) )
% 13.25/7.73 = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M ) @ one_one_nat ) @ N3 ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10025_sorted__upt,axiom,
% 13.25/7.73 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10026_sorted__wrt__upt,axiom,
% 13.25/7.73 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10027_sorted__wrt__less__idx,axiom,
% 13.25/7.73 ! [Ns: list_nat,I: nat] :
% 13.25/7.73 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 13.25/7.73 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 13.25/7.73 => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10028_VEBT_Osize_I3_J,axiom,
% 13.25/7.73 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 13.25/7.73 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 13.25/7.73 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10029_VEBT_Osize__gen_I1_J,axiom,
% 13.25/7.73 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 13.25/7.73 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 13.25/7.73 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10030_sorted__upto,axiom,
% 13.25/7.73 ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10031_sorted__wrt__upto,axiom,
% 13.25/7.73 ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 13.25/7.73
% 13.25/7.73 thf(fact_10032_quotient__of__def,axiom,
% 13.25/7.73 ( quotient_of
% 13.25/7.73 = ( ^ [X2: rat] :
% 13.25/7.73 ( the_Pr4378521158711661632nt_int
% 13.25/7.73 @ ^ [Pair: product_prod_int_int] :
% 13.25/7.73 ( ( X2
% 13.25/7.73 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 13.25/7.73 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 13.25/7.73 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10033_coprime__abs__right__iff,axiom,
% 13.25/7.74 ! [K: int,L2: int] :
% 13.25/7.74 ( ( algebr932160517623751201me_int @ K @ ( abs_abs_int @ L2 ) )
% 13.25/7.74 = ( algebr932160517623751201me_int @ K @ L2 ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10034_coprime__abs__left__iff,axiom,
% 13.25/7.74 ! [K: int,L2: int] :
% 13.25/7.74 ( ( algebr932160517623751201me_int @ ( abs_abs_int @ K ) @ L2 )
% 13.25/7.74 = ( algebr932160517623751201me_int @ K @ L2 ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10035_normalize__stable,axiom,
% 13.25/7.74 ! [Q2: int,P6: int] :
% 13.25/7.74 ( ( ord_less_int @ zero_zero_int @ Q2 )
% 13.25/7.74 => ( ( algebr932160517623751201me_int @ P6 @ Q2 )
% 13.25/7.74 => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q2 ) )
% 13.25/7.74 = ( product_Pair_int_int @ P6 @ Q2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10036_coprime__common__divisor__int,axiom,
% 13.25/7.74 ! [A: int,B: int,X: int] :
% 13.25/7.74 ( ( algebr932160517623751201me_int @ A @ B )
% 13.25/7.74 => ( ( dvd_dvd_int @ X @ A )
% 13.25/7.74 => ( ( dvd_dvd_int @ X @ B )
% 13.25/7.74 => ( ( abs_abs_int @ X )
% 13.25/7.74 = one_one_int ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10037_coprime__crossproduct__int,axiom,
% 13.25/7.74 ! [A: int,D: int,B: int,C: int] :
% 13.25/7.74 ( ( algebr932160517623751201me_int @ A @ D )
% 13.25/7.74 => ( ( algebr932160517623751201me_int @ B @ C )
% 13.25/7.74 => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C ) )
% 13.25/7.74 = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
% 13.25/7.74 = ( ( ( abs_abs_int @ A )
% 13.25/7.74 = ( abs_abs_int @ B ) )
% 13.25/7.74 & ( ( abs_abs_int @ C )
% 13.25/7.74 = ( abs_abs_int @ D ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10038_Rat__induct,axiom,
% 13.25/7.74 ! [P: rat > $o,Q2: rat] :
% 13.25/7.74 ( ! [A5: int,B3: int] :
% 13.25/7.74 ( ( ord_less_int @ zero_zero_int @ B3 )
% 13.25/7.74 => ( ( algebr932160517623751201me_int @ A5 @ B3 )
% 13.25/7.74 => ( P @ ( fract @ A5 @ B3 ) ) ) )
% 13.25/7.74 => ( P @ Q2 ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10039_Rat__cases,axiom,
% 13.25/7.74 ! [Q2: rat] :
% 13.25/7.74 ~ ! [A5: int,B3: int] :
% 13.25/7.74 ( ( Q2
% 13.25/7.74 = ( fract @ A5 @ B3 ) )
% 13.25/7.74 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 13.25/7.74 => ~ ( algebr932160517623751201me_int @ A5 @ B3 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10040_Rat__cases__nonzero,axiom,
% 13.25/7.74 ! [Q2: rat] :
% 13.25/7.74 ( ! [A5: int,B3: int] :
% 13.25/7.74 ( ( Q2
% 13.25/7.74 = ( fract @ A5 @ B3 ) )
% 13.25/7.74 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 13.25/7.74 => ( ( A5 != zero_zero_int )
% 13.25/7.74 => ~ ( algebr932160517623751201me_int @ A5 @ B3 ) ) ) )
% 13.25/7.74 => ( Q2 = zero_zero_rat ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10041_quotient__of__unique,axiom,
% 13.25/7.74 ! [R: rat] :
% 13.25/7.74 ? [X3: product_prod_int_int] :
% 13.25/7.74 ( ( R
% 13.25/7.74 = ( fract @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) ) )
% 13.25/7.74 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X3 ) )
% 13.25/7.74 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X3 ) @ ( product_snd_int_int @ X3 ) )
% 13.25/7.74 & ! [Y5: product_prod_int_int] :
% 13.25/7.74 ( ( ( R
% 13.25/7.74 = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 13.25/7.74 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 13.25/7.74 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 13.25/7.74 => ( Y5 = X3 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10042_vanishes__mult__bounded,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ? [A8: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ A8 )
% 13.25/7.74 & ! [N4: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N4 ) ) @ A8 ) )
% 13.25/7.74 => ( ( vanishes @ Y7 )
% 13.25/7.74 => ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10043_coprime__int__iff,axiom,
% 13.25/7.74 ! [M: nat,N: nat] :
% 13.25/7.74 ( ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 13.25/7.74 = ( algebr934650988132801477me_nat @ M @ N ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10044_vanishes__const,axiom,
% 13.25/7.74 ! [C: rat] :
% 13.25/7.74 ( ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : C )
% 13.25/7.74 = ( C = zero_zero_rat ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10045_coprime__nat__abs__right__iff,axiom,
% 13.25/7.74 ! [N: nat,K: int] :
% 13.25/7.74 ( ( algebr934650988132801477me_nat @ N @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 13.25/7.74 = ( algebr932160517623751201me_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10046_coprime__nat__abs__left__iff,axiom,
% 13.25/7.74 ! [K: int,N: nat] :
% 13.25/7.74 ( ( algebr934650988132801477me_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N )
% 13.25/7.74 = ( algebr932160517623751201me_int @ K @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10047_coprime__Suc__0__left,axiom,
% 13.25/7.74 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10048_coprime__Suc__0__right,axiom,
% 13.25/7.74 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10049_coprime__crossproduct__nat,axiom,
% 13.25/7.74 ! [A: nat,D: nat,B: nat,C: nat] :
% 13.25/7.74 ( ( algebr934650988132801477me_nat @ A @ D )
% 13.25/7.74 => ( ( algebr934650988132801477me_nat @ B @ C )
% 13.25/7.74 => ( ( ( times_times_nat @ A @ C )
% 13.25/7.74 = ( times_times_nat @ B @ D ) )
% 13.25/7.74 = ( ( A = B )
% 13.25/7.74 & ( C = D ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10050_coprime__common__divisor__nat,axiom,
% 13.25/7.74 ! [A: nat,B: nat,X: nat] :
% 13.25/7.74 ( ( algebr934650988132801477me_nat @ A @ B )
% 13.25/7.74 => ( ( dvd_dvd_nat @ X @ A )
% 13.25/7.74 => ( ( dvd_dvd_nat @ X @ B )
% 13.25/7.74 => ( X = one_one_nat ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10051_vanishes__diff,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( vanishes @ X7 )
% 13.25/7.74 => ( ( vanishes @ Y7 )
% 13.25/7.74 => ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10052_vanishes__minus,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( vanishes @ X7 )
% 13.25/7.74 => ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X7 @ N2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10053_vanishes__add,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( vanishes @ X7 )
% 13.25/7.74 => ( ( vanishes @ Y7 )
% 13.25/7.74 => ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10054_coprime__diff__one__left__nat,axiom,
% 13.25/7.74 ! [N: nat] :
% 13.25/7.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.74 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10055_coprime__diff__one__right__nat,axiom,
% 13.25/7.74 ! [N: nat] :
% 13.25/7.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 13.25/7.74 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10056_Rats__abs__nat__div__natE,axiom,
% 13.25/7.74 ! [X: real] :
% 13.25/7.74 ( ( member_real @ X @ field_5140801741446780682s_real )
% 13.25/7.74 => ~ ! [M4: nat,N4: nat] :
% 13.25/7.74 ( ( N4 != zero_zero_nat )
% 13.25/7.74 => ( ( ( abs_abs_real @ X )
% 13.25/7.74 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
% 13.25/7.74 => ~ ( algebr934650988132801477me_nat @ M4 @ N4 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10057_vanishes__def,axiom,
% 13.25/7.74 ( vanishes
% 13.25/7.74 = ( ^ [X6: nat > rat] :
% 13.25/7.74 ! [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 => ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10058_vanishesI,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ! [R2: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 13.25/7.74 => ? [K4: nat] :
% 13.25/7.74 ! [N4: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K4 @ N4 )
% 13.25/7.74 => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N4 ) ) @ R2 ) ) )
% 13.25/7.74 => ( vanishes @ X7 ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10059_vanishesD,axiom,
% 13.25/7.74 ! [X7: nat > rat,R: rat] :
% 13.25/7.74 ( ( vanishes @ X7 )
% 13.25/7.74 => ( ( ord_less_rat @ zero_zero_rat @ R )
% 13.25/7.74 => ? [K3: nat] :
% 13.25/7.74 ! [N7: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K3 @ N7 )
% 13.25/7.74 => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N7 ) ) @ R ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10060_rcis__inverse,axiom,
% 13.25/7.74 ! [R: real,A: real] :
% 13.25/7.74 ( ( invers8013647133539491842omplex @ ( rcis @ R @ A ) )
% 13.25/7.74 = ( rcis @ ( divide_divide_real @ one_one_real @ R ) @ ( uminus_uminus_real @ A ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10061_Re__rcis,axiom,
% 13.25/7.74 ! [R: real,A: real] :
% 13.25/7.74 ( ( re @ ( rcis @ R @ A ) )
% 13.25/7.74 = ( times_times_real @ R @ ( cos_real @ A ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10062_Im__rcis,axiom,
% 13.25/7.74 ! [R: real,A: real] :
% 13.25/7.74 ( ( im @ ( rcis @ R @ A ) )
% 13.25/7.74 = ( times_times_real @ R @ ( sin_real @ A ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10063_cis__rcis__eq,axiom,
% 13.25/7.74 ( cis
% 13.25/7.74 = ( rcis @ one_one_real ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10064_rcis__mult,axiom,
% 13.25/7.74 ! [R1: real,A: real,R22: real,B: real] :
% 13.25/7.74 ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 13.25/7.74 = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10065_rcis__divide,axiom,
% 13.25/7.74 ! [R1: real,A: real,R22: real,B: real] :
% 13.25/7.74 ( ( divide1717551699836669952omplex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
% 13.25/7.74 = ( rcis @ ( divide_divide_real @ R1 @ R22 ) @ ( minus_minus_real @ A @ B ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10066_rcis__def,axiom,
% 13.25/7.74 ( rcis
% 13.25/7.74 = ( ^ [R5: real,A3: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R5 ) @ ( cis @ A3 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10067_DeMoivre2,axiom,
% 13.25/7.74 ! [R: real,A: real,N: nat] :
% 13.25/7.74 ( ( power_power_complex @ ( rcis @ R @ A ) @ N )
% 13.25/7.74 = ( rcis @ ( power_power_real @ R @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10068_cauchy__def,axiom,
% 13.25/7.74 ( cauchy
% 13.25/7.74 = ( ^ [X6: nat > rat] :
% 13.25/7.74 ! [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 => ? [K2: nat] :
% 13.25/7.74 ! [M3: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ M3 )
% 13.25/7.74 => ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M3 ) @ ( X6 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10069_cauchyI,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ! [R2: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 13.25/7.74 => ? [K4: nat] :
% 13.25/7.74 ! [M4: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K4 @ M4 )
% 13.25/7.74 => ! [N4: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K4 @ N4 )
% 13.25/7.74 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) ) @ R2 ) ) ) )
% 13.25/7.74 => ( cauchy @ X7 ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10070_cauchy__mult,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( cauchy
% 13.25/7.74 @ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10071_cauchy__diff,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( cauchy
% 13.25/7.74 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10072_cauchy__const,axiom,
% 13.25/7.74 ! [X: rat] :
% 13.25/7.74 ( cauchy
% 13.25/7.74 @ ^ [N2: nat] : X ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10073_cauchy__add,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( cauchy
% 13.25/7.74 @ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10074_cauchy__minus,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( cauchy
% 13.25/7.74 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X7 @ N2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10075_cauchy__inverse,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ~ ( vanishes @ X7 )
% 13.25/7.74 => ( cauchy
% 13.25/7.74 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X7 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10076_cauchy__imp__bounded,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ? [B3: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 13.25/7.74 & ! [N7: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N7 ) ) @ B3 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10077_vanishes__diff__inverse,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ~ ( vanishes @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ~ ( vanishes @ Y7 )
% 13.25/7.74 => ( ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) )
% 13.25/7.74 => ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( minus_minus_rat @ ( inverse_inverse_rat @ ( X7 @ N2 ) ) @ ( inverse_inverse_rat @ ( Y7 @ N2 ) ) ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10078_cauchy__not__vanishes__cases,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ~ ( vanishes @ X7 )
% 13.25/7.74 => ? [B3: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 13.25/7.74 & ? [K3: nat] :
% 13.25/7.74 ( ! [N7: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K3 @ N7 )
% 13.25/7.74 => ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X7 @ N7 ) ) ) )
% 13.25/7.74 | ! [N7: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K3 @ N7 )
% 13.25/7.74 => ( ord_less_rat @ B3 @ ( X7 @ N7 ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10079_cauchy__not__vanishes,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ~ ( vanishes @ X7 )
% 13.25/7.74 => ? [B3: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 13.25/7.74 & ? [K3: nat] :
% 13.25/7.74 ! [N7: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K3 @ N7 )
% 13.25/7.74 => ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X7 @ N7 ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10080_cauchyD,axiom,
% 13.25/7.74 ! [X7: nat > rat,R: rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( ord_less_rat @ zero_zero_rat @ R )
% 13.25/7.74 => ? [K3: nat] :
% 13.25/7.74 ! [M2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K3 @ M2 )
% 13.25/7.74 => ! [N7: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K3 @ N7 )
% 13.25/7.74 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M2 ) @ ( X7 @ N7 ) ) ) @ R ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10081_le__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ( ord_less_eq_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
% 13.25/7.74 = ( ! [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 => ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_eq_rat @ ( X7 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10082_inverse__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( ( vanishes @ X7 )
% 13.25/7.74 => ( ( inverse_inverse_real @ ( real2 @ X7 ) )
% 13.25/7.74 = zero_zero_real ) )
% 13.25/7.74 & ( ~ ( vanishes @ X7 )
% 13.25/7.74 => ( ( inverse_inverse_real @ ( real2 @ X7 ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X7 @ N2 ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10083_Real__induct,axiom,
% 13.25/7.74 ! [P: real > $o,X: real] :
% 13.25/7.74 ( ! [X10: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X10 )
% 13.25/7.74 => ( P @ ( real2 @ X10 ) ) )
% 13.25/7.74 => ( P @ X ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10084_of__int__Real,axiom,
% 13.25/7.74 ( ring_1_of_int_real
% 13.25/7.74 = ( ^ [X2: int] :
% 13.25/7.74 ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( ring_1_of_int_rat @ X2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10085_one__real__def,axiom,
% 13.25/7.74 ( one_one_real
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : one_one_rat ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10086_zero__real__def,axiom,
% 13.25/7.74 ( zero_zero_real
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10087_of__nat__Real,axiom,
% 13.25/7.74 ( semiri5074537144036343181t_real
% 13.25/7.74 = ( ^ [X2: nat] :
% 13.25/7.74 ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( semiri681578069525770553at_rat @ X2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10088_minus__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( uminus_uminus_real @ ( real2 @ X7 ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X7 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10089_add__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ( plus_plus_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( plus_plus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10090_mult__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ( times_times_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( times_times_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10091_diff__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ( minus_minus_real @ ( real2 @ X7 ) @ ( real2 @ Y7 ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10092_eq__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ( ( real2 @ X7 )
% 13.25/7.74 = ( real2 @ Y7 ) )
% 13.25/7.74 = ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10093_not__positive__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( ~ ( positive2 @ ( real2 @ X7 ) ) )
% 13.25/7.74 = ( ! [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 => ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_eq_rat @ ( X7 @ N2 ) @ R5 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10094_positive__Real,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( positive2 @ ( real2 @ X7 ) )
% 13.25/7.74 = ( ? [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 & ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_rat @ R5 @ ( X7 @ N2 ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10095_Real_Opositive__minus,axiom,
% 13.25/7.74 ! [X: real] :
% 13.25/7.74 ( ~ ( positive2 @ X )
% 13.25/7.74 => ( ( X != zero_zero_real )
% 13.25/7.74 => ( positive2 @ ( uminus_uminus_real @ X ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10096_Real_Opositive__mult,axiom,
% 13.25/7.74 ! [X: real,Y3: real] :
% 13.25/7.74 ( ( positive2 @ X )
% 13.25/7.74 => ( ( positive2 @ Y3 )
% 13.25/7.74 => ( positive2 @ ( times_times_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10097_Real_Opositive__add,axiom,
% 13.25/7.74 ! [X: real,Y3: real] :
% 13.25/7.74 ( ( positive2 @ X )
% 13.25/7.74 => ( ( positive2 @ Y3 )
% 13.25/7.74 => ( positive2 @ ( plus_plus_real @ X @ Y3 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10098_Real_Opositive__zero,axiom,
% 13.25/7.74 ~ ( positive2 @ zero_zero_real ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10099_less__real__def,axiom,
% 13.25/7.74 ( ord_less_real
% 13.25/7.74 = ( ^ [X2: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10100_Real_Opositive_Orep__eq,axiom,
% 13.25/7.74 ( positive2
% 13.25/7.74 = ( ^ [X2: real] :
% 13.25/7.74 ? [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 & ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_rat @ R5 @ ( rep_real @ X2 @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10101_inverse__real_Oabs__eq,axiom,
% 13.25/7.74 ! [X: nat > rat] :
% 13.25/7.74 ( ( realrel @ X @ X )
% 13.25/7.74 => ( ( inverse_inverse_real @ ( real2 @ X ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ( if_nat_rat @ ( vanishes @ X )
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat
% 13.25/7.74 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10102_realrel__refl,axiom,
% 13.25/7.74 ! [X7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( realrel @ X7 @ X7 ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10103_zero__real_Orsp,axiom,
% 13.25/7.74 ( realrel
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10104_one__real_Orsp,axiom,
% 13.25/7.74 ( realrel
% 13.25/7.74 @ ^ [N2: nat] : one_one_rat
% 13.25/7.74 @ ^ [N2: nat] : one_one_rat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10105_real_Oabs__induct,axiom,
% 13.25/7.74 ! [P: real > $o,X: real] :
% 13.25/7.74 ( ! [Y4: nat > rat] :
% 13.25/7.74 ( ( realrel @ Y4 @ Y4 )
% 13.25/7.74 => ( P @ ( real2 @ Y4 ) ) )
% 13.25/7.74 => ( P @ X ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10106_uminus__real_Oabs__eq,axiom,
% 13.25/7.74 ! [X: nat > rat] :
% 13.25/7.74 ( ( realrel @ X @ X )
% 13.25/7.74 => ( ( uminus_uminus_real @ ( real2 @ X ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( uminus_uminus_rat @ ( X @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10107_plus__real_Oabs__eq,axiom,
% 13.25/7.74 ! [Xa: nat > rat,X: nat > rat] :
% 13.25/7.74 ( ( realrel @ Xa @ Xa )
% 13.25/7.74 => ( ( realrel @ X @ X )
% 13.25/7.74 => ( ( plus_plus_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( plus_plus_rat @ ( Xa @ N2 ) @ ( X @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10108_times__real_Oabs__eq,axiom,
% 13.25/7.74 ! [Xa: nat > rat,X: nat > rat] :
% 13.25/7.74 ( ( realrel @ Xa @ Xa )
% 13.25/7.74 => ( ( realrel @ X @ X )
% 13.25/7.74 => ( ( times_times_real @ ( real2 @ Xa ) @ ( real2 @ X ) )
% 13.25/7.74 = ( real2
% 13.25/7.74 @ ^ [N2: nat] : ( times_times_rat @ ( Xa @ N2 ) @ ( X @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10109_realrelI,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y7: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( minus_minus_rat @ ( X7 @ N2 ) @ ( Y7 @ N2 ) ) )
% 13.25/7.74 => ( realrel @ X7 @ Y7 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10110_realrel__def,axiom,
% 13.25/7.74 ( realrel
% 13.25/7.74 = ( ^ [X6: nat > rat,Y8: nat > rat] :
% 13.25/7.74 ( ( cauchy @ X6 )
% 13.25/7.74 & ( cauchy @ Y8 )
% 13.25/7.74 & ( vanishes
% 13.25/7.74 @ ^ [N2: nat] : ( minus_minus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10111_Real_Opositive_Oabs__eq,axiom,
% 13.25/7.74 ! [X: nat > rat] :
% 13.25/7.74 ( ( realrel @ X @ X )
% 13.25/7.74 => ( ( positive2 @ ( real2 @ X ) )
% 13.25/7.74 = ( ? [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 & ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_rat @ R5 @ ( X @ N2 ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10112_inverse__real__def,axiom,
% 13.25/7.74 ( inverse_inverse_real
% 13.25/7.74 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 13.25/7.74 @ ^ [X6: nat > rat] :
% 13.25/7.74 ( if_nat_rat @ ( vanishes @ X6 )
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat
% 13.25/7.74 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10113_cr__real__def,axiom,
% 13.25/7.74 ( cr_real
% 13.25/7.74 = ( ^ [X2: nat > rat,Y: real] :
% 13.25/7.74 ( ( realrel @ X2 @ X2 )
% 13.25/7.74 & ( ( real2 @ X2 )
% 13.25/7.74 = Y ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10114_uminus__real__def,axiom,
% 13.25/7.74 ( uminus_uminus_real
% 13.25/7.74 = ( map_fu7146612038024189824t_real @ rep_real @ real2
% 13.25/7.74 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10115_times__real__def,axiom,
% 13.25/7.74 ( times_times_real
% 13.25/7.74 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 13.25/7.74 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10116_plus__real__def,axiom,
% 13.25/7.74 ( plus_plus_real
% 13.25/7.74 = ( map_fu1532550112467129777l_real @ rep_real @ ( map_fu7146612038024189824t_real @ rep_real @ real2 )
% 13.25/7.74 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10117_Real_Opositive_Orsp,axiom,
% 13.25/7.74 ( bNF_re728719798268516973at_o_o @ realrel
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [X6: nat > rat] :
% 13.25/7.74 ? [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 & ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 13.25/7.74 @ ^ [X6: nat > rat] :
% 13.25/7.74 ? [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 & ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10118_uminus__real_Orsp,axiom,
% 13.25/7.74 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 13.25/7.74 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
% 13.25/7.74 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10119_plus__real_Orsp,axiom,
% 13.25/7.74 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 13.25/7.74 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 13.25/7.74 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10120_times__real_Orsp,axiom,
% 13.25/7.74 ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
% 13.25/7.74 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 13.25/7.74 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10121_less__natural_Orsp,axiom,
% 13.25/7.74 ( bNF_re578469030762574527_nat_o
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re4705727531993890431at_o_o
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ord_less_nat
% 13.25/7.74 @ ord_less_nat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10122_less__integer_Orsp,axiom,
% 13.25/7.74 ( bNF_re3403563459893282935_int_o
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re5089333283451836215nt_o_o
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ord_less_int
% 13.25/7.74 @ ord_less_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10123_sub_Orsp,axiom,
% 13.25/7.74 ( bNF_re8402795839162346335um_int
% 13.25/7.74 @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re1822329894187522285nt_int
% 13.25/7.74 @ ^ [Y6: num,Z4: num] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) )
% 13.25/7.74 @ ^ [M3: num,N2: num] : ( minus_minus_int @ ( numeral_numeral_int @ M3 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10124_divide__natural_Orsp,axiom,
% 13.25/7.74 ( bNF_re1345281282404953727at_nat
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re5653821019739307937at_nat
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.74 @ divide_divide_nat
% 13.25/7.74 @ divide_divide_nat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10125_divide__integer_Orsp,axiom,
% 13.25/7.74 ( bNF_re711492959462206631nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re4712519889275205905nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.74 @ divide_divide_int
% 13.25/7.74 @ divide_divide_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10126_minus__integer_Orsp,axiom,
% 13.25/7.74 ( bNF_re711492959462206631nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re4712519889275205905nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.74 @ minus_minus_int
% 13.25/7.74 @ minus_minus_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10127_minus__natural_Orsp,axiom,
% 13.25/7.74 ( bNF_re1345281282404953727at_nat
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re5653821019739307937at_nat
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.74 @ minus_minus_nat
% 13.25/7.74 @ minus_minus_nat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10128_times__natural_Orsp,axiom,
% 13.25/7.74 ( bNF_re1345281282404953727at_nat
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re5653821019739307937at_nat
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.74 @ times_times_nat
% 13.25/7.74 @ times_times_nat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10129_times__integer_Orsp,axiom,
% 13.25/7.74 ( bNF_re711492959462206631nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re4712519889275205905nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.74 @ times_times_int
% 13.25/7.74 @ times_times_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10130_dup_Orsp,axiom,
% 13.25/7.74 ( bNF_re4712519889275205905nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 )
% 13.25/7.74 @ ^ [K2: int] : ( plus_plus_int @ K2 @ K2 ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10131_plus__integer_Orsp,axiom,
% 13.25/7.74 ( bNF_re711492959462206631nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re4712519889275205905nt_int
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
% 13.25/7.74 @ plus_plus_int
% 13.25/7.74 @ plus_plus_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10132_plus__natural_Orsp,axiom,
% 13.25/7.74 ( bNF_re1345281282404953727at_nat
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re5653821019739307937at_nat
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
% 13.25/7.74 @ plus_plus_nat
% 13.25/7.74 @ plus_plus_nat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10133_less__eq__natural_Orsp,axiom,
% 13.25/7.74 ( bNF_re578469030762574527_nat_o
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re4705727531993890431at_o_o
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ord_less_eq_nat
% 13.25/7.74 @ ord_less_eq_nat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10134_less__eq__integer_Orsp,axiom,
% 13.25/7.74 ( bNF_re3403563459893282935_int_o
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ( bNF_re5089333283451836215nt_o_o
% 13.25/7.74 @ ^ [Y6: int,Z4: int] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ord_less_eq_int
% 13.25/7.74 @ ord_less_eq_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10135_inverse__real_Orsp,axiom,
% 13.25/7.74 ( bNF_re895249473297799549at_rat @ realrel @ realrel
% 13.25/7.74 @ ^ [X6: nat > rat] :
% 13.25/7.74 ( if_nat_rat @ ( vanishes @ X6 )
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat
% 13.25/7.74 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
% 13.25/7.74 @ ^ [X6: nat > rat] :
% 13.25/7.74 ( if_nat_rat @ ( vanishes @ X6 )
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat
% 13.25/7.74 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10136_Real_Opositive_Otransfer,axiom,
% 13.25/7.74 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [X6: nat > rat] :
% 13.25/7.74 ? [R5: rat] :
% 13.25/7.74 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 13.25/7.74 & ? [K2: nat] :
% 13.25/7.74 ! [N2: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ K2 @ N2 )
% 13.25/7.74 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) )
% 13.25/7.74 @ positive2 ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10137_plus__rat_Otransfer,axiom,
% 13.25/7.74 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 13.25/7.74 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
% 13.25/7.74 @ plus_plus_rat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10138_real_Orel__eq__transfer,axiom,
% 13.25/7.74 ( bNF_re4521903465945308077real_o @ pcr_real
% 13.25/7.74 @ ( bNF_re4297313714947099218al_o_o @ pcr_real
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ realrel
% 13.25/7.74 @ ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10139_zero__real_Otransfer,axiom,
% 13.25/7.74 ( pcr_real
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat
% 13.25/7.74 @ zero_zero_real ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10140_real_Opcr__cr__eq,axiom,
% 13.25/7.74 pcr_real = cr_real ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10141_one__real_Otransfer,axiom,
% 13.25/7.74 ( pcr_real
% 13.25/7.74 @ ^ [N2: nat] : one_one_rat
% 13.25/7.74 @ one_one_real ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10142_cr__real__eq,axiom,
% 13.25/7.74 ( pcr_real
% 13.25/7.74 = ( ^ [X2: nat > rat,Y: real] :
% 13.25/7.74 ( ( cauchy @ X2 )
% 13.25/7.74 & ( ( real2 @ X2 )
% 13.25/7.74 = Y ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10143_uminus__real_Otransfer,axiom,
% 13.25/7.74 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 13.25/7.74 @ ^ [X6: nat > rat,N2: nat] : ( uminus_uminus_rat @ ( X6 @ N2 ) )
% 13.25/7.74 @ uminus_uminus_real ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10144_plus__real_Otransfer,axiom,
% 13.25/7.74 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 13.25/7.74 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( plus_plus_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 13.25/7.74 @ plus_plus_real ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10145_times__real_Otransfer,axiom,
% 13.25/7.74 ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
% 13.25/7.74 @ ^ [X6: nat > rat,Y8: nat > rat,N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y8 @ N2 ) )
% 13.25/7.74 @ times_times_real ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10146_inverse__real_Otransfer,axiom,
% 13.25/7.74 ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real
% 13.25/7.74 @ ^ [X6: nat > rat] :
% 13.25/7.74 ( if_nat_rat @ ( vanishes @ X6 )
% 13.25/7.74 @ ^ [N2: nat] : zero_zero_rat
% 13.25/7.74 @ ^ [N2: nat] : ( inverse_inverse_rat @ ( X6 @ N2 ) ) )
% 13.25/7.74 @ inverse_inverse_real ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10147_times__rat_Otransfer,axiom,
% 13.25/7.74 ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
% 13.25/7.74 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
% 13.25/7.74 @ times_times_rat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10148_Rat_Opositive_Otransfer,axiom,
% 13.25/7.74 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
% 13.25/7.74 @ positive ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10149_times__int_Otransfer,axiom,
% 13.25/7.74 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) )
% 13.25/7.74 @ times_times_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10150_nat_Otransfer,axiom,
% 13.25/7.74 ( bNF_re4555766996558763186at_nat @ pcr_int
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 13.25/7.74 @ nat2 ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10151_one__int_Otransfer,axiom,
% 13.25/7.74 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10152_less__int_Otransfer,axiom,
% 13.25/7.74 ( bNF_re717283939379294677_int_o @ pcr_int
% 13.25/7.74 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ( produc8739625826339149834_nat_o
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc6081775807080527818_nat_o
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
% 13.25/7.74 @ ord_less_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10153_less__eq__int_Otransfer,axiom,
% 13.25/7.74 ( bNF_re717283939379294677_int_o @ pcr_int
% 13.25/7.74 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ( produc8739625826339149834_nat_o
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc6081775807080527818_nat_o
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
% 13.25/7.74 @ ord_less_eq_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10154_plus__int_Otransfer,axiom,
% 13.25/7.74 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 13.25/7.74 @ plus_plus_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10155_minus__int_Otransfer,axiom,
% 13.25/7.74 ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) )
% 13.25/7.74 @ minus_minus_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10156_times__int_Orsp,axiom,
% 13.25/7.74 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X2 @ U3 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X2 @ V4 ) @ ( times_times_nat @ Y @ U3 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10157_intrel__iff,axiom,
% 13.25/7.74 ! [X: nat,Y3: nat,U: nat,V: nat] :
% 13.25/7.74 ( ( intrel @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( product_Pair_nat_nat @ U @ V ) )
% 13.25/7.74 = ( ( plus_plus_nat @ X @ V )
% 13.25/7.74 = ( plus_plus_nat @ U @ Y3 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10158_nat_Orsp,axiom,
% 13.25/7.74 ( bNF_re8246922863344978751at_nat @ intrel
% 13.25/7.74 @ ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
% 13.25/7.74 @ ( produc6842872674320459806at_nat @ minus_minus_nat )
% 13.25/7.74 @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10159_le__RealI,axiom,
% 13.25/7.74 ! [Y7: nat > rat,X: real] :
% 13.25/7.74 ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ! [N4: nat] : ( ord_less_eq_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) )
% 13.25/7.74 => ( ord_less_eq_real @ X @ ( real2 @ Y7 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10160_Real__leI,axiom,
% 13.25/7.74 ! [X7: nat > rat,Y3: real] :
% 13.25/7.74 ( ( cauchy @ X7 )
% 13.25/7.74 => ( ! [N4: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X7 @ N4 ) ) @ Y3 )
% 13.25/7.74 => ( ord_less_eq_real @ ( real2 @ X7 ) @ Y3 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10161_less__RealD,axiom,
% 13.25/7.74 ! [Y7: nat > rat,X: real] :
% 13.25/7.74 ( ( cauchy @ Y7 )
% 13.25/7.74 => ( ( ord_less_real @ X @ ( real2 @ Y7 ) )
% 13.25/7.74 => ? [N4: nat] : ( ord_less_real @ X @ ( field_7254667332652039916t_real @ ( Y7 @ N4 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10162_of__rat__dense,axiom,
% 13.25/7.74 ! [X: real,Y3: real] :
% 13.25/7.74 ( ( ord_less_real @ X @ Y3 )
% 13.25/7.74 => ? [Q3: rat] :
% 13.25/7.74 ( ( ord_less_real @ X @ ( field_7254667332652039916t_real @ Q3 ) )
% 13.25/7.74 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q3 ) @ Y3 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10163_of__rat__Real,axiom,
% 13.25/7.74 ( field_7254667332652039916t_real
% 13.25/7.74 = ( ^ [X2: rat] :
% 13.25/7.74 ( real2
% 13.25/7.74 @ ^ [N2: nat] : X2 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10164_one__int_Orsp,axiom,
% 13.25/7.74 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10165_intrel__def,axiom,
% 13.25/7.74 ( intrel
% 13.25/7.74 = ( produc8739625826339149834_nat_o
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc6081775807080527818_nat_o
% 13.25/7.74 @ ^ [U3: nat,V4: nat] :
% 13.25/7.74 ( ( plus_plus_nat @ X2 @ V4 )
% 13.25/7.74 = ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10166_less__int_Orsp,axiom,
% 13.25/7.74 ( bNF_re4202695980764964119_nat_o @ intrel
% 13.25/7.74 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ( produc8739625826339149834_nat_o
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc6081775807080527818_nat_o
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
% 13.25/7.74 @ ( produc8739625826339149834_nat_o
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc6081775807080527818_nat_o
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10167_less__eq__int_Orsp,axiom,
% 13.25/7.74 ( bNF_re4202695980764964119_nat_o @ intrel
% 13.25/7.74 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 ) )
% 13.25/7.74 @ ( produc8739625826339149834_nat_o
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc6081775807080527818_nat_o
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) )
% 13.25/7.74 @ ( produc8739625826339149834_nat_o
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc6081775807080527818_nat_o
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ U3 @ Y ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10168_plus__int_Orsp,axiom,
% 13.25/7.74 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ U3 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10169_minus__int_Orsp,axiom,
% 13.25/7.74 ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) )
% 13.25/7.74 @ ( produc27273713700761075at_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] :
% 13.25/7.74 ( produc2626176000494625587at_nat
% 13.25/7.74 @ ^ [U3: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X2 @ V4 ) @ ( plus_plus_nat @ Y @ U3 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10170_le__enumerate,axiom,
% 13.25/7.74 ! [S2: set_nat,N: nat] :
% 13.25/7.74 ( ~ ( finite_finite_nat @ S2 )
% 13.25/7.74 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10171_finite__le__enumerate,axiom,
% 13.25/7.74 ! [S2: set_nat,N: nat] :
% 13.25/7.74 ( ( finite_finite_nat @ S2 )
% 13.25/7.74 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
% 13.25/7.74 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10172_Least__eq__0,axiom,
% 13.25/7.74 ! [P: nat > $o] :
% 13.25/7.74 ( ( P @ zero_zero_nat )
% 13.25/7.74 => ( ( ord_Least_nat @ P )
% 13.25/7.74 = zero_zero_nat ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10173_Least__Suc,axiom,
% 13.25/7.74 ! [P: nat > $o,N: nat] :
% 13.25/7.74 ( ( P @ N )
% 13.25/7.74 => ( ~ ( P @ zero_zero_nat )
% 13.25/7.74 => ( ( ord_Least_nat @ P )
% 13.25/7.74 = ( suc
% 13.25/7.74 @ ( ord_Least_nat
% 13.25/7.74 @ ^ [M3: nat] : ( P @ ( suc @ M3 ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10174_Least__Suc2,axiom,
% 13.25/7.74 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 13.25/7.74 ( ( P @ N )
% 13.25/7.74 => ( ( Q @ M )
% 13.25/7.74 => ( ~ ( P @ zero_zero_nat )
% 13.25/7.74 => ( ! [K3: nat] :
% 13.25/7.74 ( ( P @ ( suc @ K3 ) )
% 13.25/7.74 = ( Q @ K3 ) )
% 13.25/7.74 => ( ( ord_Least_nat @ P )
% 13.25/7.74 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10175_Sup__real__def,axiom,
% 13.25/7.74 ( comple1385675409528146559p_real
% 13.25/7.74 = ( ^ [X6: set_real] :
% 13.25/7.74 ( ord_Least_real
% 13.25/7.74 @ ^ [Z2: real] :
% 13.25/7.74 ! [X2: real] :
% 13.25/7.74 ( ( member_real @ X2 @ X6 )
% 13.25/7.74 => ( ord_less_eq_real @ X2 @ Z2 ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10176_has__field__derivative__iff__has__vector__derivative,axiom,
% 13.25/7.74 has_fi5821293074295781190e_real = has_ve631408500373753343e_real ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10177_has__vector__derivative__id,axiom,
% 13.25/7.74 ! [Net: filter_real] :
% 13.25/7.74 ( has_ve631408500373753343e_real
% 13.25/7.74 @ ^ [X2: real] : X2
% 13.25/7.74 @ one_one_real
% 13.25/7.74 @ Net ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10178_plus__rat_Orsp,axiom,
% 13.25/7.74 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 13.25/7.74 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
% 13.25/7.74 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) @ ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10179_ratrel__iff,axiom,
% 13.25/7.74 ( ratrel
% 13.25/7.74 = ( ^ [X2: product_prod_int_int,Y: product_prod_int_int] :
% 13.25/7.74 ( ( ( product_snd_int_int @ X2 )
% 13.25/7.74 != zero_zero_int )
% 13.25/7.74 & ( ( product_snd_int_int @ Y )
% 13.25/7.74 != zero_zero_int )
% 13.25/7.74 & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) )
% 13.25/7.74 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10180_ratrel__def,axiom,
% 13.25/7.74 ( ratrel
% 13.25/7.74 = ( ^ [X2: product_prod_int_int,Y: product_prod_int_int] :
% 13.25/7.74 ( ( ( product_snd_int_int @ X2 )
% 13.25/7.74 != zero_zero_int )
% 13.25/7.74 & ( ( product_snd_int_int @ Y )
% 13.25/7.74 != zero_zero_int )
% 13.25/7.74 & ( ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ Y ) )
% 13.25/7.74 = ( times_times_int @ ( product_fst_int_int @ Y ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10181_times__rat_Orsp,axiom,
% 13.25/7.74 ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
% 13.25/7.74 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) )
% 13.25/7.74 @ ^ [X2: product_prod_int_int,Y: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_fst_int_int @ Y ) ) @ ( times_times_int @ ( product_snd_int_int @ X2 ) @ ( product_snd_int_int @ Y ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10182_Rat_Opositive_Orsp,axiom,
% 13.25/7.74 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 13.25/7.74 @ ^ [Y6: $o,Z4: $o] : ( Y6 = Z4 )
% 13.25/7.74 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) )
% 13.25/7.74 @ ^ [X2: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10183_plus__rat_Oabs__eq,axiom,
% 13.25/7.74 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 13.25/7.74 ( ( ratrel @ Xa @ Xa )
% 13.25/7.74 => ( ( ratrel @ X @ X )
% 13.25/7.74 => ( ( plus_plus_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 13.25/7.74 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10184_Rat_Opositive_Oabs__eq,axiom,
% 13.25/7.74 ! [X: product_prod_int_int] :
% 13.25/7.74 ( ( ratrel @ X @ X )
% 13.25/7.74 => ( ( positive @ ( abs_Rat @ X ) )
% 13.25/7.74 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10185_times__rat_Oabs__eq,axiom,
% 13.25/7.74 ! [Xa: product_prod_int_int,X: product_prod_int_int] :
% 13.25/7.74 ( ( ratrel @ Xa @ Xa )
% 13.25/7.74 => ( ( ratrel @ X @ X )
% 13.25/7.74 => ( ( times_times_rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 13.25/7.74 = ( 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 ) ) ) ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10186_last__upt,axiom,
% 13.25/7.74 ! [I: nat,J: nat] :
% 13.25/7.74 ( ( ord_less_nat @ I @ J )
% 13.25/7.74 => ( ( last_nat @ ( upt @ I @ J ) )
% 13.25/7.74 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10187_set__decode__div__2,axiom,
% 13.25/7.74 ! [X: nat] :
% 13.25/7.74 ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 13.25/7.74 = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10188_set__encode__vimage__Suc,axiom,
% 13.25/7.74 ! [A2: set_nat] :
% 13.25/7.74 ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
% 13.25/7.74 = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10189_Bseq__monoseq__convergent_H__dec,axiom,
% 13.25/7.74 ! [F: nat > real,M5: nat] :
% 13.25/7.74 ( ( bfun_nat_real
% 13.25/7.74 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
% 13.25/7.74 @ at_top_nat )
% 13.25/7.74 => ( ! [M4: nat,N4: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ M5 @ M4 )
% 13.25/7.74 => ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.74 => ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ M4 ) ) ) )
% 13.25/7.74 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10190_Bseq__mono__convergent,axiom,
% 13.25/7.74 ! [X7: nat > real] :
% 13.25/7.74 ( ( bfun_nat_real @ X7 @ at_top_nat )
% 13.25/7.74 => ( ! [M4: nat,N4: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.74 => ( ord_less_eq_real @ ( X7 @ M4 ) @ ( X7 @ N4 ) ) )
% 13.25/7.74 => ( topolo7531315842566124627t_real @ X7 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10191_convergent__realpow,axiom,
% 13.25/7.74 ! [X: real] :
% 13.25/7.74 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 13.25/7.74 => ( ( ord_less_eq_real @ X @ one_one_real )
% 13.25/7.74 => ( topolo7531315842566124627t_real @ ( power_power_real @ X ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10192_Bseq__monoseq__convergent_H__inc,axiom,
% 13.25/7.74 ! [F: nat > real,M5: nat] :
% 13.25/7.74 ( ( bfun_nat_real
% 13.25/7.74 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
% 13.25/7.74 @ at_top_nat )
% 13.25/7.74 => ( ! [M4: nat,N4: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ M5 @ M4 )
% 13.25/7.74 => ( ( ord_less_eq_nat @ M4 @ N4 )
% 13.25/7.74 => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N4 ) ) ) )
% 13.25/7.74 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10193_filtermap__at__right__shift,axiom,
% 13.25/7.74 ! [D: real,A: real] :
% 13.25/7.74 ( ( filtermap_real_real
% 13.25/7.74 @ ^ [X2: real] : ( minus_minus_real @ X2 @ D )
% 13.25/7.74 @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) )
% 13.25/7.74 = ( topolo2177554685111907308n_real @ ( minus_minus_real @ A @ D ) @ ( set_or5849166863359141190n_real @ ( minus_minus_real @ A @ D ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10194_at__right__to__0,axiom,
% 13.25/7.74 ! [A: real] :
% 13.25/7.74 ( ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) )
% 13.25/7.74 = ( filtermap_real_real
% 13.25/7.74 @ ^ [X2: real] : ( plus_plus_real @ X2 @ A )
% 13.25/7.74 @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10195_pair__lessI2,axiom,
% 13.25/7.74 ! [A: nat,B: nat,S: nat,T: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.74 => ( ( ord_less_nat @ S @ T )
% 13.25/7.74 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10196_pair__less__iff1,axiom,
% 13.25/7.74 ! [X: nat,Y3: nat,Z: nat] :
% 13.25/7.74 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
% 13.25/7.74 = ( ord_less_nat @ Y3 @ Z ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10197_pair__lessI1,axiom,
% 13.25/7.74 ! [A: nat,B: nat,S: nat,T: nat] :
% 13.25/7.74 ( ( ord_less_nat @ A @ B )
% 13.25/7.74 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10198_bot__nat__0_Oordering__top__axioms,axiom,
% 13.25/7.74 ( ordering_top_nat
% 13.25/7.74 @ ^ [X2: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X2 )
% 13.25/7.74 @ ^ [X2: nat,Y: nat] : ( ord_less_nat @ Y @ X2 )
% 13.25/7.74 @ zero_zero_nat ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10199_pair__leqI2,axiom,
% 13.25/7.74 ! [A: nat,B: nat,S: nat,T: nat] :
% 13.25/7.74 ( ( ord_less_eq_nat @ A @ B )
% 13.25/7.74 => ( ( ord_less_eq_nat @ S @ T )
% 13.25/7.74 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10200_pair__leqI1,axiom,
% 13.25/7.74 ! [A: nat,B: nat,S: nat,T: nat] :
% 13.25/7.74 ( ( ord_less_nat @ A @ B )
% 13.25/7.74 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10201_euclidean__size__int__def,axiom,
% 13.25/7.74 ( euclid4774559944035922753ze_int
% 13.25/7.74 = ( comp_int_nat_int @ nat2 @ abs_abs_int ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10202_abs__division__segment,axiom,
% 13.25/7.74 ! [K: int] :
% 13.25/7.74 ( ( abs_abs_int @ ( euclid3395696857347342551nt_int @ K ) )
% 13.25/7.74 = one_one_int ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10203_division__segment__nat__def,axiom,
% 13.25/7.74 ( euclid3398187327856392827nt_nat
% 13.25/7.74 = ( ^ [N2: nat] : one_one_nat ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10204_division__segment__eq__sgn,axiom,
% 13.25/7.74 ! [K: int] :
% 13.25/7.74 ( ( K != zero_zero_int )
% 13.25/7.74 => ( ( euclid3395696857347342551nt_int @ K )
% 13.25/7.74 = ( sgn_sgn_int @ K ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10205_division__segment__int__def,axiom,
% 13.25/7.74 ( euclid3395696857347342551nt_int
% 13.25/7.74 = ( ^ [K2: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10206_less__eq__enat__def,axiom,
% 13.25/7.74 ( ord_le2932123472753598470d_enat
% 13.25/7.74 = ( ^ [M3: extended_enat] :
% 13.25/7.74 ( extended_case_enat_o
% 13.25/7.74 @ ^ [N1: nat] :
% 13.25/7.74 ( extended_case_enat_o
% 13.25/7.74 @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
% 13.25/7.74 @ $false
% 13.25/7.74 @ M3 )
% 13.25/7.74 @ $true ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10207_less__enat__def,axiom,
% 13.25/7.74 ( ord_le72135733267957522d_enat
% 13.25/7.74 = ( ^ [M3: extended_enat,N2: extended_enat] :
% 13.25/7.74 ( extended_case_enat_o
% 13.25/7.74 @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
% 13.25/7.74 @ $false
% 13.25/7.74 @ M3 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(fact_10208_transp__realrel,axiom,
% 13.25/7.74 transp_nat_rat @ realrel ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 13.25/7.74 ! [X: int,Y3: int] :
% 13.25/7.74 ( ( if_int @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 13.25/7.74 ! [X: int,Y3: int] :
% 13.25/7.74 ( ( if_int @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 13.25/7.74 ! [X: nat,Y3: nat] :
% 13.25/7.74 ( ( if_nat @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 13.25/7.74 ! [X: nat,Y3: nat] :
% 13.25/7.74 ( ( if_nat @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 13.25/7.74 ! [X: num,Y3: num] :
% 13.25/7.74 ( ( if_num @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 13.25/7.74 ! [X: num,Y3: num] :
% 13.25/7.74 ( ( if_num @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 13.25/7.74 ! [X: rat,Y3: rat] :
% 13.25/7.74 ( ( if_rat @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 13.25/7.74 ! [X: rat,Y3: rat] :
% 13.25/7.74 ( ( if_rat @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 13.25/7.74 ! [X: real,Y3: real] :
% 13.25/7.74 ( ( if_real @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 13.25/7.74 ! [X: real,Y3: real] :
% 13.25/7.74 ( ( if_real @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 13.25/7.74 ! [P: real > $o] :
% 13.25/7.74 ( ( P @ ( fChoice_real @ P ) )
% 13.25/7.74 = ( ? [X6: real] : ( P @ X6 ) ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 13.25/7.74 ! [X: complex,Y3: complex] :
% 13.25/7.74 ( ( if_complex @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 13.25/7.74 ! [X: complex,Y3: complex] :
% 13.25/7.74 ( ( if_complex @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 13.25/7.74 ! [X: extended_enat,Y3: extended_enat] :
% 13.25/7.74 ( ( if_Extended_enat @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 13.25/7.74 ! [X: extended_enat,Y3: extended_enat] :
% 13.25/7.74 ( ( if_Extended_enat @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 13.25/7.74 ! [X: code_integer,Y3: code_integer] :
% 13.25/7.74 ( ( if_Code_integer @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 13.25/7.74 ! [X: code_integer,Y3: code_integer] :
% 13.25/7.74 ( ( if_Code_integer @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 13.25/7.74 ! [X: set_int,Y3: set_int] :
% 13.25/7.74 ( ( if_set_int @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 13.25/7.74 ! [X: set_int,Y3: set_int] :
% 13.25/7.74 ( ( if_set_int @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 13.25/7.74 ! [X: vEBT_VEBT,Y3: vEBT_VEBT] :
% 13.25/7.74 ( ( if_VEBT_VEBT @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 13.25/7.74 ! [X: vEBT_VEBT,Y3: vEBT_VEBT] :
% 13.25/7.74 ( ( if_VEBT_VEBT @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 13.25/7.74 ! [X: list_int,Y3: list_int] :
% 13.25/7.74 ( ( if_list_int @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 13.25/7.74 ! [X: list_int,Y3: list_int] :
% 13.25/7.74 ( ( if_list_int @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 13.25/7.74 ! [X: list_nat,Y3: list_nat] :
% 13.25/7.74 ( ( if_list_nat @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 13.25/7.74 ! [X: list_nat,Y3: list_nat] :
% 13.25/7.74 ( ( if_list_nat @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 13.25/7.74 ! [X: int > int,Y3: int > int] :
% 13.25/7.74 ( ( if_int_int @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 13.25/7.74 ! [X: int > int,Y3: int > int] :
% 13.25/7.74 ( ( if_int_int @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 13.25/7.74 ! [X: nat > rat,Y3: nat > rat] :
% 13.25/7.74 ( ( if_nat_rat @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001_062_It__Nat__Onat_Mt__Rat__Orat_J_T,axiom,
% 13.25/7.74 ! [X: nat > rat,Y3: nat > rat] :
% 13.25/7.74 ( ( if_nat_rat @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 13.25/7.74 ! [X: option_nat,Y3: option_nat] :
% 13.25/7.74 ( ( if_option_nat @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 13.25/7.74 ! [X: option_nat,Y3: option_nat] :
% 13.25/7.74 ( ( if_option_nat @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 13.25/7.74 ! [X: option_num,Y3: option_num] :
% 13.25/7.74 ( ( if_option_num @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 13.25/7.74 ! [X: option_num,Y3: option_num] :
% 13.25/7.74 ( ( if_option_num @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 13.25/7.74 ! [X: product_prod_int_int,Y3: product_prod_int_int] :
% 13.25/7.74 ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 13.25/7.74 ! [X: product_prod_int_int,Y3: product_prod_int_int] :
% 13.25/7.74 ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 13.25/7.74 ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 13.25/7.74 ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 13.25/7.74 ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 13.25/7.74 ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 13.25/7.74 ! [X: nat > int > int,Y3: nat > int > int] :
% 13.25/7.74 ( ( if_nat_int_int @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
% 13.25/7.74 ! [X: nat > int > int,Y3: nat > int > int] :
% 13.25/7.74 ( ( if_nat_int_int @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 13.25/7.74 ! [X: nat > nat > nat,Y3: nat > nat > nat] :
% 13.25/7.74 ( ( if_nat_nat_nat @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
% 13.25/7.74 ! [X: nat > nat > nat,Y3: nat > nat > nat] :
% 13.25/7.74 ( ( if_nat_nat_nat @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 13.25/7.74 ! [X: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 13.25/7.74 ( ( if_Pro5737122678794959658eger_o @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 13.25/7.74 ! [X: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 13.25/7.74 ( ( if_Pro5737122678794959658eger_o @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 13.25/7.74 ! [P: $o] :
% 13.25/7.74 ( ( P = $true )
% 13.25/7.74 | ( P = $false ) ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 13.25/7.74 ! [X: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 13.25/7.74 ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y3 )
% 13.25/7.74 = Y3 ) ).
% 13.25/7.74
% 13.25/7.74 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 13.25/7.74 ! [X: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 13.25/7.74 ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y3 )
% 13.25/7.74 = X ) ).
% 13.25/7.74
% 13.25/7.74 thf(conj_0,conjecture,
% 13.25/7.74 vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ summary ) @ ( vEBT_VEBT_high @ xa @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ sc ).
% 13.25/7.74 ------------------------
% 13.92/8.49 % (31486)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31377 for (2992ds/183Mi)
% 13.92/8.49 % (31487)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_31377 for (2992ds/4Mi)
% 13.92/8.49 % (31488)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_31377 for (2992ds/27Mi)
% 13.92/8.49 % (31489)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31377 for (2992ds/2Mi)
% 13.92/8.49 % (31490)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_31377 for (2992ds/2Mi)
% 13.92/8.49 % (31491)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_31377 for (2992ds/275Mi)
% 13.92/8.49 % (31492)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31377 for (2992ds/18Mi)
% 13.92/8.49 % (31489)Instruction limit reached!
% 13.92/8.49 % (31489)------------------------------
% 13.92/8.49 % (31489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.92/8.49 % (31489)Termination reason: Unknown
% 13.92/8.49 % (31489)Termination phase: shuffling
% 13.92/8.49
% 13.92/8.49 % (31489)Memory used [KB]: 12920
% 13.92/8.49 % (31489)Time elapsed: 0.003 s
% 13.92/8.49 % (31489)Instructions burned: 2 (million)
% 13.92/8.49 % (31489)------------------------------
% 13.92/8.49 % (31489)------------------------------
% 13.92/8.49 % (31490)Instruction limit reached!
% 13.92/8.49 % (31490)------------------------------
% 13.92/8.49 % (31490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.92/8.49 % (31490)Termination reason: Unknown
% 13.92/8.49 % (31490)Termination phase: shuffling
% 13.92/8.49
% 13.92/8.49 % (31490)Memory used [KB]: 12920
% 13.92/8.49 % (31490)Time elapsed: 0.003 s
% 13.92/8.49 % (31490)Instructions burned: 2 (million)
% 13.92/8.49 % (31490)------------------------------
% 13.92/8.49 % (31490)------------------------------
% 13.92/8.49 % (31487)Instruction limit reached!
% 13.92/8.49 % (31487)------------------------------
% 13.92/8.49 % (31487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.92/8.49 % (31487)Termination reason: Unknown
% 13.92/8.49 % (31487)Termination phase: shuffling
% 13.92/8.49
% 13.92/8.49 % (31487)Memory used [KB]: 13176
% 13.92/8.49 % (31487)Time elapsed: 0.006 s
% 13.92/8.49 % (31487)Instructions burned: 4 (million)
% 13.92/8.49 % (31487)------------------------------
% 13.92/8.49 % (31487)------------------------------
% 14.05/8.51 % (31492)Instruction limit reached!
% 14.05/8.51 % (31492)------------------------------
% 14.05/8.51 % (31492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.51 % (31492)Termination reason: Unknown
% 14.05/8.51 % (31492)Termination phase: shuffling
% 14.05/8.51
% 14.05/8.51 % (31492)Memory used [KB]: 13816
% 14.05/8.51 % (31492)Time elapsed: 0.019 s
% 14.05/8.51 % (31492)Instructions burned: 18 (million)
% 14.05/8.51 % (31492)------------------------------
% 14.05/8.51 % (31492)------------------------------
% 14.05/8.51 % (31494)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_31377 for (2992ds/37Mi)
% 14.05/8.51 % (31493)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31377 for (2992ds/3Mi)
% 14.05/8.51 % (31495)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_31377 for (2992ds/15Mi)
% 14.05/8.51 % (31493)Instruction limit reached!
% 14.05/8.51 % (31493)------------------------------
% 14.05/8.51 % (31493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.51 % (31493)Termination reason: Unknown
% 14.05/8.51 % (31493)Termination phase: shuffling
% 14.05/8.51
% 14.05/8.51 % (31493)Memory used [KB]: 13048
% 14.05/8.51 % (31493)Time elapsed: 0.004 s
% 14.05/8.51 % (31493)Instructions burned: 3 (million)
% 14.05/8.51 % (31493)------------------------------
% 14.05/8.51 % (31493)------------------------------
% 14.05/8.51 % (31488)Instruction limit reached!
% 14.05/8.51 % (31488)------------------------------
% 14.05/8.51 % (31488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.51 % (31488)Termination reason: Unknown
% 14.05/8.51 % (31488)Termination phase: shuffling
% 14.05/8.51
% 14.05/8.51 % (31488)Memory used [KB]: 13816
% 14.05/8.51 % (31488)Time elapsed: 0.025 s
% 14.05/8.51 % (31488)Instructions burned: 28 (million)
% 14.05/8.51 % (31488)------------------------------
% 14.05/8.51 % (31488)------------------------------
% 14.05/8.53 % (31496)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_31377 for (2992ds/3Mi)
% 14.05/8.53 % (31495)Instruction limit reached!
% 14.05/8.53 % (31495)------------------------------
% 14.05/8.53 % (31495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.53 % (31495)Termination reason: Unknown
% 14.05/8.53 % (31495)Termination phase: shuffling
% 14.05/8.53
% 14.05/8.53 % (31495)Memory used [KB]: 13816
% 14.05/8.53 % (31495)Time elapsed: 0.018 s
% 14.05/8.53 % (31495)Instructions burned: 16 (million)
% 14.05/8.53 % (31495)------------------------------
% 14.05/8.53 % (31495)------------------------------
% 14.05/8.53 % (31497)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_31377 for (2992ds/1041Mi)
% 14.05/8.53 % (31496)Instruction limit reached!
% 14.05/8.53 % (31496)------------------------------
% 14.05/8.53 % (31496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.53 % (31496)Termination reason: Unknown
% 14.05/8.53 % (31496)Termination phase: shuffling
% 14.05/8.53
% 14.05/8.53 % (31496)Memory used [KB]: 13048
% 14.05/8.53 % (31496)Time elapsed: 0.005 s
% 14.05/8.53 % (31496)Instructions burned: 3 (million)
% 14.05/8.53 % (31496)------------------------------
% 14.05/8.53 % (31496)------------------------------
% 14.05/8.53 % (31498)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_31377 for (2992ds/7Mi)
% 14.05/8.54 % (31494)Instruction limit reached!
% 14.05/8.54 % (31494)------------------------------
% 14.05/8.54 % (31494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.54 % (31494)Termination reason: Unknown
% 14.05/8.54 % (31494)Termination phase: shuffling
% 14.05/8.54
% 14.05/8.54 % (31494)Memory used [KB]: 14072
% 14.05/8.54 % (31494)Time elapsed: 0.029 s
% 14.05/8.54 % (31494)Instructions burned: 38 (million)
% 14.05/8.54 % (31494)------------------------------
% 14.05/8.54 % (31494)------------------------------
% 14.05/8.54 % (31498)Instruction limit reached!
% 14.05/8.54 % (31498)------------------------------
% 14.05/8.54 % (31498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.54 % (31498)Termination reason: Unknown
% 14.05/8.54 % (31498)Termination phase: shuffling
% 14.05/8.54
% 14.05/8.54 % (31498)Memory used [KB]: 13304
% 14.05/8.54 % (31498)Time elapsed: 0.009 s
% 14.05/8.54 % (31498)Instructions burned: 7 (million)
% 14.05/8.54 % (31498)------------------------------
% 14.05/8.54 % (31498)------------------------------
% 14.05/8.55 % (31499)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_31377 for (2992ds/16Mi)
% 14.05/8.55 % (31500)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_31377 for (2992ds/3Mi)
% 14.05/8.55 % (31500)Instruction limit reached!
% 14.05/8.55 % (31500)------------------------------
% 14.05/8.55 % (31500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.55 % (31500)Termination reason: Unknown
% 14.05/8.55 % (31500)Termination phase: shuffling
% 14.05/8.55
% 14.05/8.55 % (31500)Memory used [KB]: 13048
% 14.05/8.55 % (31500)Time elapsed: 0.004 s
% 14.05/8.55 % (31500)Instructions burned: 3 (million)
% 14.05/8.55 % (31500)------------------------------
% 14.05/8.55 % (31500)------------------------------
% 14.05/8.55 % (31501)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_31377 for (2992ds/3Mi)
% 14.05/8.56 % (31502)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_31377 for (2992ds/7Mi)
% 14.05/8.56 % (31501)Instruction limit reached!
% 14.05/8.56 % (31501)------------------------------
% 14.05/8.56 % (31501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.56 % (31501)Termination reason: Unknown
% 14.05/8.56 % (31501)Termination phase: shuffling
% 14.05/8.56
% 14.05/8.56 % (31501)Memory used [KB]: 13048
% 14.05/8.56 % (31501)Time elapsed: 0.005 s
% 14.05/8.56 % (31501)Instructions burned: 3 (million)
% 14.05/8.56 % (31501)------------------------------
% 14.05/8.56 % (31501)------------------------------
% 14.05/8.56 % (31499)Instruction limit reached!
% 14.05/8.56 % (31499)------------------------------
% 14.05/8.56 % (31499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.56 % (31499)Termination reason: Unknown
% 14.05/8.56 % (31499)Termination phase: shuffling
% 14.05/8.56
% 14.05/8.56 % (31499)Memory used [KB]: 13816
% 14.05/8.56 % (31499)Time elapsed: 0.019 s
% 14.05/8.56 % (31499)Instructions burned: 16 (million)
% 14.05/8.56 % (31499)------------------------------
% 14.05/8.56 % (31499)------------------------------
% 14.05/8.56 % (31502)Instruction limit reached!
% 14.05/8.56 % (31502)------------------------------
% 14.05/8.56 % (31502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.56 % (31502)Termination reason: Unknown
% 14.05/8.56 % (31502)Termination phase: shuffling
% 14.05/8.56
% 14.05/8.56 % (31502)Memory used [KB]: 13304
% 14.05/8.56 % (31502)Time elapsed: 0.009 s
% 14.05/8.56 % (31502)Instructions burned: 7 (million)
% 14.05/8.56 % (31502)------------------------------
% 14.05/8.56 % (31502)------------------------------
% 14.05/8.57 % (31503)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_31377 for (2991ds/3Mi)
% 14.05/8.57 % (31503)Instruction limit reached!
% 14.05/8.57 % (31503)------------------------------
% 14.05/8.57 % (31503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.57 % (31503)Termination reason: Unknown
% 14.05/8.57 % (31503)Termination phase: shuffling
% 14.05/8.57
% 14.05/8.57 % (31503)Memory used [KB]: 13048
% 14.05/8.57 % (31503)Time elapsed: 0.005 s
% 14.05/8.57 % (31503)Instructions burned: 3 (million)
% 14.05/8.57 % (31503)------------------------------
% 14.05/8.57 % (31503)------------------------------
% 14.05/8.57 % (31504)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_31377 for (2991ds/4Mi)
% 14.05/8.58 % (31504)Instruction limit reached!
% 14.05/8.58 % (31504)------------------------------
% 14.05/8.58 % (31504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.58 % (31504)Termination reason: Unknown
% 14.05/8.58 % (31504)Termination phase: shuffling
% 14.05/8.58
% 14.05/8.58 % (31504)Memory used [KB]: 13176
% 14.05/8.58 % (31504)Time elapsed: 0.006 s
% 14.05/8.58 % (31504)Instructions burned: 4 (million)
% 14.05/8.58 % (31504)------------------------------
% 14.05/8.58 % (31504)------------------------------
% 14.05/8.58 % (31505)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_31377 for (2991ds/18Mi)
% 14.05/8.58 % (31506)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_31377 for (2991ds/710Mi)
% 14.05/8.59 % (31507)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31377 for (2991ds/6Mi)
% 14.05/8.59 % (31507)Instruction limit reached!
% 14.05/8.59 % (31507)------------------------------
% 14.05/8.59 % (31507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.59 % (31507)Termination reason: Unknown
% 14.05/8.59 % (31507)Termination phase: shuffling
% 14.05/8.59
% 14.05/8.59 % (31507)Memory used [KB]: 13304
% 14.05/8.59 % (31507)Time elapsed: 0.008 s
% 14.05/8.59 % (31507)Instructions burned: 6 (million)
% 14.05/8.59 % (31507)------------------------------
% 14.05/8.59 % (31507)------------------------------
% 14.05/8.59 % (31486)Instruction limit reached!
% 14.05/8.59 % (31486)------------------------------
% 14.05/8.59 % (31486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.05/8.59 % (31486)Termination reason: Unknown
% 14.05/8.59 % (31486)Termination phase: shuffling
% 14.05/8.59
% 14.05/8.59 % (31486)Memory used [KB]: 16119
% 14.05/8.59 % (31486)Time elapsed: 0.104 s
% 14.05/8.59 % (31486)Instructions burned: 183 (million)
% 14.05/8.59 % (31486)------------------------------
% 14.05/8.59 % (31486)------------------------------
% 14.05/8.60 % (31508)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_31377 for (2991ds/902Mi)
% 14.54/8.60 % (31505)Instruction limit reached!
% 14.54/8.60 % (31505)------------------------------
% 14.54/8.60 % (31505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.54/8.60 % (31505)Termination reason: Unknown
% 14.54/8.60 % (31505)Termination phase: shuffling
% 14.54/8.60
% 14.54/8.60 % (31505)Memory used [KB]: 13816
% 14.54/8.60 % (31505)Time elapsed: 0.020 s
% 14.54/8.60 % (31505)Instructions burned: 18 (million)
% 14.54/8.60 % (31505)------------------------------
% 14.54/8.60 % (31505)------------------------------
% 14.54/8.61 % (31509)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_31377 for (2991ds/21Mi)
% 14.54/8.61 % (31510)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_31377 for (2991ds/5Mi)
% 14.54/8.62 % (31510)Instruction limit reached!
% 14.54/8.62 % (31510)------------------------------
% 14.54/8.62 % (31510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.54/8.62 % (31510)Termination reason: Unknown
% 14.54/8.62 % (31510)Termination phase: shuffling
% 14.54/8.62
% 14.54/8.62 % (31510)Memory used [KB]: 13176
% 14.54/8.62 % (31510)Time elapsed: 0.007 s
% 14.54/8.62 % (31510)Instructions burned: 5 (million)
% 14.54/8.62 % (31510)------------------------------
% 14.54/8.62 % (31510)------------------------------
% 14.54/8.62 % (31511)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_31377 for (2991ds/6Mi)
% 14.54/8.62 % (31511)Instruction limit reached!
% 14.54/8.62 % (31511)------------------------------
% 14.54/8.62 % (31511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.54/8.62 % (31511)Termination reason: Unknown
% 14.54/8.62 % (31511)Termination phase: shuffling
% 14.54/8.62
% 14.54/8.62 % (31511)Memory used [KB]: 13304
% 14.54/8.62 % (31511)Time elapsed: 0.008 s
% 14.54/8.62 % (31511)Instructions burned: 6 (million)
% 14.54/8.62 % (31511)------------------------------
% 14.54/8.62 % (31511)------------------------------
% 14.54/8.63 % (31509)Instruction limit reached!
% 14.54/8.63 % (31509)------------------------------
% 14.54/8.63 % (31509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.54/8.63 % (31509)Termination reason: Unknown
% 14.54/8.63 % (31509)Termination phase: shuffling
% 14.54/8.63
% 14.54/8.63 % (31509)Memory used [KB]: 13816
% 14.54/8.63 % (31509)Time elapsed: 0.022 s
% 14.54/8.63 % (31509)Instructions burned: 21 (million)
% 14.54/8.63 % (31509)------------------------------
% 14.54/8.63 % (31509)------------------------------
% 14.54/8.63 % (31512)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_31377 for (2991ds/377Mi)
% 14.54/8.64 % (31513)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_31377 for (2991ds/779Mi)
% 14.54/8.64 % (31491)Instruction limit reached!
% 14.54/8.64 % (31491)------------------------------
% 14.54/8.64 % (31491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.54/8.64 % (31491)Termination reason: Unknown
% 14.54/8.64 % (31491)Termination phase: shuffling
% 14.54/8.64
% 14.54/8.64 % (31491)Memory used [KB]: 17526
% 14.54/8.64 % (31491)Time elapsed: 0.153 s
% 14.54/8.64 % (31491)Instructions burned: 276 (million)
% 14.54/8.64 % (31491)------------------------------
% 14.54/8.64 % (31491)------------------------------
% 14.54/8.65 % (31514)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_31377 for (2991ds/19Mi)
% 14.54/8.66 % (31515)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_31377 for (2990ds/879Mi)
% 14.54/8.66 % (31514)Instruction limit reached!
% 14.54/8.66 % (31514)------------------------------
% 14.54/8.66 % (31514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.54/8.66 % (31514)Termination reason: Unknown
% 14.54/8.66 % (31514)Termination phase: shuffling
% 14.54/8.66
% 14.54/8.66 % (31514)Memory used [KB]: 13816
% 14.54/8.66 % (31514)Time elapsed: 0.020 s
% 14.54/8.66 % (31514)Instructions burned: 19 (million)
% 14.54/8.66 % (31514)------------------------------
% 14.54/8.66 % (31514)------------------------------
% 14.54/8.68 % (31516)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_31377 for (2990ds/17Mi)
% 14.87/8.70 % (31516)Instruction limit reached!
% 14.87/8.70 % (31516)------------------------------
% 14.87/8.70 % (31516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.87/8.70 % (31516)Termination reason: Unknown
% 14.87/8.70 % (31516)Termination phase: shuffling
% 14.87/8.70
% 14.87/8.70 % (31516)Memory used [KB]: 13816
% 14.87/8.70 % (31516)Time elapsed: 0.020 s
% 14.87/8.70 % (31516)Instructions burned: 18 (million)
% 14.87/8.70 % (31516)------------------------------
% 14.87/8.70 % (31516)------------------------------
% 14.87/8.72 % (31517)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_31377 for (2990ds/3Mi)
% 14.87/8.72 % (31517)Instruction limit reached!
% 14.87/8.72 % (31517)------------------------------
% 14.87/8.72 % (31517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.87/8.72 % (31517)Termination reason: Unknown
% 14.87/8.72 % (31517)Termination phase: shuffling
% 14.87/8.72
% 14.87/8.72 % (31517)Memory used [KB]: 13048
% 14.87/8.72 % (31517)Time elapsed: 0.005 s
% 14.87/8.72 % (31517)Instructions burned: 3 (million)
% 14.87/8.72 % (31517)------------------------------
% 14.87/8.72 % (31517)------------------------------
% 14.87/8.74 % (31518)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_31377 for (2990ds/30Mi)
% 14.87/8.76 % (31518)Instruction limit reached!
% 14.87/8.76 % (31518)------------------------------
% 14.87/8.76 % (31518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.87/8.76 % (31518)Termination reason: Unknown
% 14.87/8.76 % (31518)Termination phase: shuffling
% 14.87/8.76
% 14.87/8.76 % (31518)Memory used [KB]: 13944
% 14.87/8.76 % (31518)Time elapsed: 0.026 s
% 14.87/8.76 % (31518)Instructions burned: 30 (million)
% 14.87/8.76 % (31518)------------------------------
% 14.87/8.76 % (31518)------------------------------
% 14.87/8.78 % (31519)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_31377 for (2989ds/127Mi)
% 14.98/8.83 % (31512)Instruction limit reached!
% 14.98/8.83 % (31512)------------------------------
% 14.98/8.83 % (31512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.98/8.84 % (31512)Termination reason: Unknown
% 14.98/8.84 % (31512)Termination phase: shuffling
% 14.98/8.84
% 14.98/8.84 % (31512)Memory used [KB]: 18677
% 14.98/8.84 % (31512)Time elapsed: 0.204 s
% 14.98/8.84 % (31512)Instructions burned: 377 (million)
% 14.98/8.84 % (31512)------------------------------
% 14.98/8.84 % (31512)------------------------------
% 14.98/8.85 % (31520)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_31377 for (2989ds/100Mi)
% 14.98/8.86 % (31519)Instruction limit reached!
% 14.98/8.86 % (31519)------------------------------
% 14.98/8.86 % (31519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.98/8.86 % (31519)Termination reason: Unknown
% 14.98/8.86 % (31519)Termination phase: shuffling
% 14.98/8.86
% 14.98/8.86 % (31519)Memory used [KB]: 15351
% 14.98/8.86 % (31519)Time elapsed: 0.076 s
% 14.98/8.86 % (31519)Instructions burned: 128 (million)
% 14.98/8.86 % (31519)------------------------------
% 14.98/8.86 % (31519)------------------------------
% 14.98/8.87 % (31521)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_31377 for (2988ds/3Mi)
% 14.98/8.88 % (31521)Instruction limit reached!
% 14.98/8.88 % (31521)------------------------------
% 14.98/8.88 % (31521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.98/8.88 % (31521)Termination reason: Unknown
% 14.98/8.88 % (31521)Termination phase: shuffling
% 14.98/8.88
% 14.98/8.88 % (31521)Memory used [KB]: 13048
% 14.98/8.88 % (31521)Time elapsed: 0.005 s
% 14.98/8.88 % (31521)Instructions burned: 3 (million)
% 14.98/8.88 % (31521)------------------------------
% 14.98/8.88 % (31521)------------------------------
% 14.98/8.89 % (31522)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_31377 for (2988ds/20Mi)
% 15.30/8.91 % (31520)Instruction limit reached!
% 15.30/8.91 % (31520)------------------------------
% 15.30/8.91 % (31520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.30/8.91 % (31520)Termination reason: Unknown
% 15.30/8.91 % (31520)Termination phase: shuffling
% 15.30/8.91
% 15.30/8.91 % (31520)Memory used [KB]: 14967
% 15.30/8.91 % (31520)Time elapsed: 0.058 s
% 15.30/8.91 % (31520)Instructions burned: 101 (million)
% 15.30/8.91 % (31520)------------------------------
% 15.30/8.91 % (31520)------------------------------
% 15.30/8.91 % (31522)Instruction limit reached!
% 15.30/8.91 % (31522)------------------------------
% 15.30/8.91 % (31522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.30/8.91 % (31522)Termination reason: Unknown
% 15.30/8.91 % (31522)Termination phase: shuffling
% 15.30/8.91
% 15.30/8.91 % (31522)Memory used [KB]: 13816
% 15.30/8.91 % (31522)Time elapsed: 0.021 s
% 15.30/8.91 % (31522)Instructions burned: 20 (million)
% 15.30/8.91 % (31522)------------------------------
% 15.30/8.91 % (31522)------------------------------
% 15.30/8.93 % (31523)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_31377 for (2988ds/86Mi)
% 15.30/8.93 % (31524)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_31377 for (2988ds/107Mi)
% 15.30/8.96 % (31506)Instruction limit reached!
% 15.30/8.96 % (31506)------------------------------
% 15.30/8.96 % (31506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.30/8.96 % (31506)Termination reason: Unknown
% 15.30/8.96 % (31506)Termination phase: shuffling
% 15.30/8.96
% 15.30/8.96 % (31506)Memory used [KB]: 25330
% 15.30/8.96 % (31506)Time elapsed: 0.380 s
% 15.30/8.96 % (31506)Instructions burned: 710 (million)
% 15.30/8.96 % (31506)------------------------------
% 15.30/8.96 % (31506)------------------------------
% 15.30/8.98 % (31525)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_31377 for (2987ds/39Mi)
% 15.30/8.98 % (31523)Instruction limit reached!
% 15.30/8.98 % (31523)------------------------------
% 15.30/8.98 % (31523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.30/8.98 % (31523)Termination reason: Unknown
% 15.30/8.98 % (31523)Termination phase: shuffling
% 15.30/8.98
% 15.30/8.98 % (31523)Memory used [KB]: 14839
% 15.30/8.98 % (31523)Time elapsed: 0.054 s
% 15.30/8.98 % (31523)Instructions burned: 87 (million)
% 15.30/8.98 % (31523)------------------------------
% 15.30/8.98 % (31523)------------------------------
% 15.30/8.99 % (31524)Instruction limit reached!
% 15.30/8.99 % (31524)------------------------------
% 15.30/8.99 % (31524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.30/8.99 % (31524)Termination reason: Unknown
% 15.30/8.99 % (31524)Termination phase: shuffling
% 15.30/8.99
% 15.30/8.99 % (31524)Memory used [KB]: 15095
% 15.30/8.99 % (31524)Time elapsed: 0.065 s
% 15.30/8.99 % (31524)Instructions burned: 108 (million)
% 15.30/8.99 % (31524)------------------------------
% 15.30/8.99 % (31524)------------------------------
% 15.30/9.00 % (31526)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_31377 for (2987ds/448Mi)
% 15.30/9.01 % (31525)Instruction limit reached!
% 15.30/9.01 % (31525)------------------------------
% 15.30/9.01 % (31525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.30/9.01 % (31525)Termination reason: Unknown
% 15.30/9.01 % (31525)Termination phase: shuffling
% 15.30/9.01
% 15.30/9.01 % (31525)Memory used [KB]: 14072
% 15.30/9.01 % (31525)Time elapsed: 0.030 s
% 15.30/9.01 % (31525)Instructions burned: 40 (million)
% 15.30/9.01 % (31525)------------------------------
% 15.30/9.01 % (31525)------------------------------
% 15.30/9.01 % (31527)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_31377 for (2987ds/46Mi)
% 15.30/9.02 % (31513)Instruction limit reached!
% 15.30/9.02 % (31513)------------------------------
% 15.30/9.02 % (31513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.30/9.02 % (31513)Termination reason: Unknown
% 15.30/9.02 % (31513)Termination phase: shuffling
% 15.30/9.02
% 15.30/9.02 % (31513)Memory used [KB]: 26481
% 15.30/9.02 % (31513)Time elapsed: 0.383 s
% 15.30/9.02 % (31513)Instructions burned: 780 (million)
% 15.30/9.02 % (31513)------------------------------
% 15.30/9.02 % (31513)------------------------------
% 15.30/9.02 % (31528)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_31377 for (2987ds/98Mi)
% 17.77/9.04 % (31497)Instruction limit reached!
% 17.77/9.04 % (31497)------------------------------
% 17.77/9.04 % (31497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.77/9.04 % (31497)Termination reason: Unknown
% 17.77/9.04 % (31497)Termination phase: shuffling
% 17.77/9.04
% 17.77/9.04 % (31497)Memory used [KB]: 34029
% 17.77/9.04 % (31497)Time elapsed: 0.512 s
% 17.77/9.04 % (31497)Instructions burned: 1042 (million)
% 17.77/9.04 % (31497)------------------------------
% 17.77/9.04 % (31497)------------------------------
% 17.77/9.04 % (31529)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_31377 for (2987ds/507Mi)
% 17.77/9.04 % (31527)Instruction limit reached!
% 17.77/9.04 % (31527)------------------------------
% 17.77/9.04 % (31527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.77/9.04 % (31527)Termination reason: Unknown
% 17.77/9.04 % (31527)Termination phase: shuffling
% 17.77/9.04
% 17.77/9.04 % (31527)Memory used [KB]: 14200
% 17.77/9.04 % (31527)Time elapsed: 0.034 s
% 17.77/9.04 % (31527)Instructions burned: 46 (million)
% 17.77/9.04 % (31527)------------------------------
% 17.77/9.04 % (31527)------------------------------
% 17.77/9.06 % (31530)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_31377 for (2987ds/149Mi)
% 17.77/9.06 % (31531)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_31377 for (2987ds/39Mi)
% 17.77/9.07 % (31508)Instruction limit reached!
% 17.77/9.07 % (31508)------------------------------
% 17.77/9.07 % (31508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.77/9.07 % (31508)Termination reason: Unknown
% 17.77/9.07 % (31508)Termination phase: Property scanning
% 17.77/9.07
% 17.77/9.07 % (31508)Memory used [KB]: 26609
% 17.77/9.07 % (31508)Time elapsed: 0.476 s
% 17.77/9.07 % (31508)Instructions burned: 902 (million)
% 17.77/9.07 % (31508)------------------------------
% 17.77/9.07 % (31508)------------------------------
% 17.77/9.08 % (31528)Instruction limit reached!
% 17.77/9.08 % (31528)------------------------------
% 17.77/9.08 % (31528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.77/9.08 % (31528)Termination reason: Unknown
% 17.77/9.08 % (31528)Termination phase: shuffling
% 17.77/9.08
% 17.77/9.08 % (31528)Memory used [KB]: 14967
% 17.77/9.08 % (31528)Time elapsed: 0.057 s
% 17.77/9.08 % (31528)Instructions burned: 99 (million)
% 17.77/9.08 % (31528)------------------------------
% 17.77/9.08 % (31528)------------------------------
% 17.77/9.09 % (31532)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_31377 for (2986ds/6Mi)
% 17.77/9.09 % (31531)Instruction limit reached!
% 17.77/9.09 % (31531)------------------------------
% 17.77/9.09 % (31531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.77/9.09 % (31531)Termination reason: Unknown
% 17.77/9.09 % (31531)Termination phase: shuffling
% 17.77/9.09
% 17.77/9.09 % (31531)Memory used [KB]: 14072
% 17.77/9.09 % (31531)Time elapsed: 0.031 s
% 17.77/9.09 % (31531)Instructions burned: 40 (million)
% 17.77/9.09 % (31531)------------------------------
% 17.77/9.09 % (31531)------------------------------
% 17.77/9.10 % (31532)Instruction limit reached!
% 17.77/9.10 % (31532)------------------------------
% 17.77/9.10 % (31532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.77/9.10 % (31532)Termination reason: Unknown
% 17.77/9.10 % (31532)Termination phase: shuffling
% 17.77/9.10
% 17.77/9.10 % (31532)Memory used [KB]: 13304
% 17.77/9.10 % (31532)Time elapsed: 0.008 s
% 17.77/9.10 % (31532)Instructions burned: 6 (million)
% 17.77/9.10 % (31532)------------------------------
% 17.77/9.10 % (31532)------------------------------
% 17.77/9.10 % (31533)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_31377 for (2986ds/1087Mi)
% 17.77/9.11 % (31534)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_31377 for (2986ds/47Mi)
% 17.77/9.11 % (31535)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_31377 for (2986ds/44Mi)
% 17.77/9.12 % (31515)Instruction limit reached!
% 17.77/9.12 % (31515)------------------------------
% 17.77/9.12 % (31515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.77/9.12 % (31515)Termination reason: Unknown
% 17.77/9.12 % (31515)Termination phase: Property scanning
% 17.77/9.12
% 17.77/9.12 % (31515)Memory used [KB]: 26481
% 17.77/9.12 % (31515)Time elapsed: 0.462 s
% 17.77/9.12 % (31515)Instructions burned: 879 (million)
% 17.77/9.12 % (31515)------------------------------
% 17.77/9.12 % (31515)------------------------------
% 18.25/9.14 % (31536)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_31377 for (2986ds/432Mi)
% 18.25/9.14 % (31534)Instruction limit reached!
% 18.25/9.14 % (31534)------------------------------
% 18.25/9.14 % (31534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.25/9.14 % (31534)Termination reason: Unknown
% 18.25/9.14 % (31534)Termination phase: shuffling
% 18.25/9.14
% 18.25/9.14 % (31534)Memory used [KB]: 14200
% 18.25/9.14 % (31534)Time elapsed: 0.035 s
% 18.25/9.14 % (31534)Instructions burned: 47 (million)
% 18.25/9.14 % (31534)------------------------------
% 18.25/9.14 % (31534)------------------------------
% 18.25/9.14 % (31535)Instruction limit reached!
% 18.25/9.14 % (31535)------------------------------
% 18.25/9.14 % (31535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.25/9.14 % (31535)Termination reason: Unknown
% 18.25/9.14 % (31535)Termination phase: shuffling
% 18.25/9.14
% 18.25/9.14 % (31535)Memory used [KB]: 14200
% 18.25/9.14 % (31535)Time elapsed: 0.032 s
% 18.25/9.14 % (31535)Instructions burned: 45 (million)
% 18.25/9.14 % (31535)------------------------------
% 18.25/9.14 % (31535)------------------------------
% 18.25/9.15 % (31530)Instruction limit reached!
% 18.25/9.15 % (31530)------------------------------
% 18.25/9.15 % (31530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.25/9.15 % (31530)Termination reason: Unknown
% 18.25/9.15 % (31530)Termination phase: shuffling
% 18.25/9.15
% 18.25/9.15 % (31530)Memory used [KB]: 15607
% 18.25/9.15 % (31530)Time elapsed: 0.088 s
% 18.25/9.15 % (31530)Instructions burned: 149 (million)
% 18.25/9.15 % (31530)------------------------------
% 18.25/9.15 % (31530)------------------------------
% 18.25/9.16 % (31537)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_31377 for (2985ds/62Mi)
% 18.25/9.16 % (31538)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_31377 for (2985ds/31Mi)
% 18.25/9.16 % (31539)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_31377 for (2985ds/96Mi)
% 18.25/9.19 % (31538)Instruction limit reached!
% 18.25/9.19 % (31538)------------------------------
% 18.25/9.19 % (31538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.25/9.19 % (31538)Termination reason: Unknown
% 18.25/9.19 % (31538)Termination phase: shuffling
% 18.25/9.19
% 18.25/9.19 % (31538)Memory used [KB]: 13944
% 18.25/9.19 % (31538)Time elapsed: 0.027 s
% 18.25/9.19 % (31538)Instructions burned: 32 (million)
% 18.25/9.19 % (31538)------------------------------
% 18.25/9.19 % (31538)------------------------------
% 18.57/9.20 % (31537)Instruction limit reached!
% 18.57/9.20 % (31537)------------------------------
% 18.57/9.20 % (31537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.57/9.20 % (31537)Termination reason: Unknown
% 18.57/9.20 % (31537)Termination phase: shuffling
% 18.57/9.20
% 18.57/9.20 % (31537)Memory used [KB]: 14456
% 18.57/9.20 % (31537)Time elapsed: 0.041 s
% 18.57/9.20 % (31537)Instructions burned: 63 (million)
% 18.57/9.20 % (31537)------------------------------
% 18.57/9.20 % (31537)------------------------------
% 18.57/9.22 % (31540)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_31377 for (2985ds/453Mi)
% 18.57/9.22 % (31541)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_31377 for (2985ds/71Mi)
% 18.57/9.22 % (31539)Instruction limit reached!
% 18.57/9.22 % (31539)------------------------------
% 18.57/9.22 % (31539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.57/9.22 % (31539)Termination reason: Unknown
% 18.57/9.22 % (31539)Termination phase: shuffling
% 18.57/9.22 % (31526)Instruction limit reached!
% 18.57/9.22 % (31526)------------------------------
% 18.57/9.22 % (31526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.57/9.22 % (31526)Termination reason: Unknown
% 18.57/9.22 % (31526)Termination phase: shuffling
% 18.57/9.22
% 18.57/9.22 % (31526)Memory used [KB]: 21875
% 18.57/9.22 % (31526)Time elapsed: 0.212 s
% 18.57/9.22 % (31526)Instructions burned: 448 (million)
% 18.57/9.22 % (31526)------------------------------
% 18.57/9.22 % (31526)------------------------------
% 18.57/9.22
% 18.57/9.22 % (31539)Memory used [KB]: 14839
% 18.57/9.22 % (31539)Time elapsed: 0.058 s
% 18.57/9.22 % (31539)Instructions burned: 96 (million)
% 18.57/9.22 % (31539)------------------------------
% 18.57/9.22 % (31539)------------------------------
% 18.57/9.24 % (31542)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_31377 for (2985ds/83Mi)
% 18.57/9.24 % (31543)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_31377 for (2985ds/11Mi)
% 18.57/9.25 % (31543)Instruction limit reached!
% 18.57/9.25 % (31543)------------------------------
% 18.57/9.25 % (31543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.57/9.25 % (31543)Termination reason: Unknown
% 18.57/9.25 % (31543)Termination phase: shuffling
% 18.57/9.25
% 18.57/9.25 % (31543)Memory used [KB]: 13688
% 18.57/9.25 % (31543)Time elapsed: 0.014 s
% 18.57/9.25 % (31543)Instructions burned: 11 (million)
% 18.57/9.25 % (31543)------------------------------
% 18.57/9.25 % (31543)------------------------------
% 18.57/9.26 % (31541)Instruction limit reached!
% 18.57/9.26 % (31541)------------------------------
% 18.57/9.26 % (31541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.57/9.26 % (31541)Termination reason: Unknown
% 18.57/9.26 % (31541)Termination phase: shuffling
% 18.57/9.26
% 18.57/9.26 % (31541)Memory used [KB]: 14456
% 18.57/9.26 % (31541)Time elapsed: 0.047 s
% 18.57/9.26 % (31541)Instructions burned: 72 (million)
% 18.57/9.26 % (31541)------------------------------
% 18.57/9.26 % (31541)------------------------------
% 18.57/9.27 % (31544)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_31377 for (2984ds/41Mi)
% 18.57/9.28 % (31545)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_31377 for (2984ds/6Mi)
% 18.57/9.29 % (31545)Instruction limit reached!
% 18.57/9.29 % (31545)------------------------------
% 18.57/9.29 % (31545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.57/9.29 % (31545)Termination reason: Unknown
% 18.57/9.29 % (31545)Termination phase: shuffling
% 18.57/9.29
% 18.57/9.29 % (31545)Memory used [KB]: 13304
% 18.57/9.29 % (31545)Time elapsed: 0.008 s
% 18.57/9.29 % (31545)Instructions burned: 6 (million)
% 18.57/9.29 % (31545)------------------------------
% 18.57/9.29 % (31545)------------------------------
% 18.57/9.29 % (31542)Instruction limit reached!
% 18.57/9.29 % (31542)------------------------------
% 18.57/9.29 % (31542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.57/9.29 % (31542)Termination reason: Unknown
% 18.57/9.29 % (31542)Termination phase: shuffling
% 18.57/9.29
% 18.57/9.29 % (31542)Memory used [KB]: 14711
% 18.57/9.29 % (31542)Time elapsed: 0.054 s
% 18.57/9.29 % (31542)Instructions burned: 84 (million)
% 18.57/9.29 % (31542)------------------------------
% 18.57/9.29 % (31542)------------------------------
% 18.91/9.30 % (31544)Instruction limit reached!
% 18.91/9.30 % (31544)------------------------------
% 18.91/9.30 % (31544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.91/9.30 % (31544)Termination reason: Unknown
% 18.91/9.30 % (31544)Termination phase: shuffling
% 18.91/9.30
% 18.91/9.30 % (31544)Memory used [KB]: 14072
% 18.91/9.30 % (31544)Time elapsed: 0.030 s
% 18.91/9.30 % (31544)Instructions burned: 42 (million)
% 18.91/9.30 % (31544)------------------------------
% 18.91/9.30 % (31544)------------------------------
% 18.91/9.31 % (31546)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_31377 for (2984ds/150Mi)
% 18.91/9.31 % (31529)Instruction limit reached!
% 18.91/9.31 % (31529)------------------------------
% 18.91/9.31 % (31529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.91/9.31 % (31529)Termination reason: Unknown
% 18.91/9.31 % (31529)Termination phase: shuffling
% 18.91/9.31
% 18.91/9.31 % (31529)Memory used [KB]: 20212
% 18.91/9.31 % (31529)Time elapsed: 0.269 s
% 18.91/9.31 % (31529)Instructions burned: 507 (million)
% 18.91/9.31 % (31529)------------------------------
% 18.91/9.31 % (31529)------------------------------
% 18.91/9.31 % (31547)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_31377 for (2984ds/343Mi)
% 18.91/9.32 % (31548)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_31377 for (2984ds/109Mi)
% 18.91/9.33 % (31549)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_31377 for (2984ds/253Mi)
% 18.91/9.36 % (31536)Instruction limit reached!
% 18.91/9.36 % (31536)------------------------------
% 18.91/9.36 % (31536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.91/9.36 % (31536)Termination reason: Unknown
% 18.91/9.36 % (31536)Termination phase: shuffling
% 18.91/9.36
% 18.91/9.36 % (31536)Memory used [KB]: 21748
% 18.91/9.36 % (31536)Time elapsed: 0.216 s
% 18.91/9.36 % (31536)Instructions burned: 432 (million)
% 18.91/9.36 % (31536)------------------------------
% 18.91/9.36 % (31536)------------------------------
% 18.91/9.37 % (31550)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_31377 for (2983ds/140Mi)
% 18.91/9.38 % (31548)Instruction limit reached!
% 18.91/9.38 % (31548)------------------------------
% 18.91/9.38 % (31548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.91/9.38 % (31548)Termination reason: Unknown
% 18.91/9.38 % (31548)Termination phase: shuffling
% 18.91/9.38
% 18.91/9.38 % (31548)Memory used [KB]: 15095
% 18.91/9.38 % (31548)Time elapsed: 0.067 s
% 18.91/9.38 % (31548)Instructions burned: 110 (million)
% 18.91/9.38 % (31548)------------------------------
% 18.91/9.38 % (31548)------------------------------
% 18.91/9.39 % (31546)Instruction limit reached!
% 18.91/9.39 % (31546)------------------------------
% 18.91/9.39 % (31546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.91/9.39 % (31546)Termination reason: Unknown
% 18.91/9.39 % (31546)Termination phase: shuffling
% 18.91/9.39
% 18.91/9.39 % (31546)Memory used [KB]: 15607
% 18.91/9.39 % (31546)Time elapsed: 0.087 s
% 18.91/9.39 % (31546)Instructions burned: 150 (million)
% 18.91/9.39 % (31546)------------------------------
% 18.91/9.39 % (31546)------------------------------
% 19.68/9.40 % (31551)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_31377 for (2983ds/614Mi)
% 19.68/9.41 % (31552)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_31377 for (2983ds/202Mi)
% 19.68/9.43 % (31540)Instruction limit reached!
% 19.68/9.43 % (31540)------------------------------
% 19.68/9.43 % (31540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.68/9.43 % (31540)Termination reason: Unknown
% 19.68/9.43 % (31540)Termination phase: shuffling
% 19.68/9.43
% 19.68/9.43 % (31540)Memory used [KB]: 22003
% 19.68/9.43 % (31540)Time elapsed: 0.217 s
% 19.68/9.43 % (31540)Instructions burned: 453 (million)
% 19.68/9.43 % (31540)------------------------------
% 19.68/9.43 % (31540)------------------------------
% 19.68/9.45 % (31553)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_31377 for (2983ds/339Mi)
% 19.68/9.46 % (31550)Instruction limit reached!
% 19.68/9.46 % (31550)------------------------------
% 19.68/9.46 % (31550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.68/9.46 % (31550)Termination reason: Unknown
% 19.68/9.46 % (31550)Termination phase: shuffling
% 19.68/9.46
% 19.68/9.46 % (31550)Memory used [KB]: 15479
% 19.68/9.46 % (31550)Time elapsed: 0.082 s
% 19.68/9.46 % (31550)Instructions burned: 140 (million)
% 19.68/9.46 % (31550)------------------------------
% 19.68/9.46 % (31550)------------------------------
% 19.68/9.47 % (31549)Instruction limit reached!
% 19.68/9.47 % (31549)------------------------------
% 19.68/9.47 % (31549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.68/9.47 % (31549)Termination reason: Unknown
% 19.68/9.47 % (31549)Termination phase: shuffling
% 19.68/9.47
% 19.68/9.47 % (31549)Memory used [KB]: 17142
% 19.68/9.47 % (31549)Time elapsed: 0.141 s
% 19.68/9.47 % (31549)Instructions burned: 254 (million)
% 19.68/9.47 % (31549)------------------------------
% 19.68/9.47 % (31549)------------------------------
% 19.68/9.47 % (31554)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_31377 for (2982ds/283Mi)
% 19.68/9.49 % (31555)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_31377 for (2982ds/134Mi)
% 19.68/9.49 % (31547)Instruction limit reached!
% 19.68/9.49 % (31547)------------------------------
% 19.68/9.49 % (31547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.68/9.49 % (31547)Termination reason: Unknown
% 19.68/9.49 % (31547)Termination phase: shuffling
% 19.68/9.49
% 19.68/9.49 % (31547)Memory used [KB]: 18421
% 19.68/9.49 % (31547)Time elapsed: 0.186 s
% 19.68/9.49 % (31547)Instructions burned: 344 (million)
% 19.68/9.49 % (31547)------------------------------
% 19.68/9.49 % (31547)------------------------------
% 20.36/9.51 % (31556)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_31377 for (2982ds/298Mi)
% 20.36/9.52 % (31552)Instruction limit reached!
% 20.36/9.52 % (31552)------------------------------
% 20.36/9.52 % (31552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.36/9.52 % (31552)Termination reason: Unknown
% 20.36/9.52 % (31552)Termination phase: shuffling
% 20.36/9.52
% 20.36/9.52 % (31552)Memory used [KB]: 16502
% 20.36/9.52 % (31552)Time elapsed: 0.115 s
% 20.36/9.52 % (31552)Instructions burned: 202 (million)
% 20.36/9.52 % (31552)------------------------------
% 20.36/9.52 % (31552)------------------------------
% 20.36/9.54 % (31557)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_31377 for (2982ds/192Mi)
% 20.36/9.56 % (31555)Instruction limit reached!
% 20.36/9.56 % (31555)------------------------------
% 20.36/9.56 % (31555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.36/9.56 % (31555)Termination reason: Unknown
% 20.36/9.56 % (31555)Termination phase: shuffling
% 20.36/9.56
% 20.36/9.56 % (31555)Memory used [KB]: 15479
% 20.36/9.56 % (31555)Time elapsed: 0.079 s
% 20.36/9.56 % (31555)Instructions burned: 134 (million)
% 20.36/9.56 % (31555)------------------------------
% 20.36/9.56 % (31555)------------------------------
% 20.36/9.58 % (31558)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_31377 for (2981ds/258Mi)
% 20.58/9.62 % (31554)Instruction limit reached!
% 20.58/9.62 % (31554)------------------------------
% 20.58/9.62 % (31554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/9.62 % (31554)Termination reason: Unknown
% 20.58/9.62 % (31554)Termination phase: shuffling
% 20.58/9.62
% 20.58/9.62 % (31554)Memory used [KB]: 19573
% 20.58/9.62 % (31554)Time elapsed: 0.145 s
% 20.58/9.62 % (31554)Instructions burned: 284 (million)
% 20.58/9.62 % (31554)------------------------------
% 20.58/9.62 % (31554)------------------------------
% 20.58/9.64 % (31553)Instruction limit reached!
% 20.58/9.64 % (31553)------------------------------
% 20.58/9.64 % (31553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/9.64 % (31553)Termination reason: Unknown
% 20.58/9.64 % (31553)Termination phase: shuffling
% 20.58/9.64
% 20.58/9.64 % (31553)Memory used [KB]: 18293
% 20.58/9.64 % (31553)Time elapsed: 0.184 s
% 20.58/9.64 % (31553)Instructions burned: 339 (million)
% 20.58/9.64 % (31553)------------------------------
% 20.58/9.64 % (31553)------------------------------
% 20.58/9.64 % (31559)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_31377 for (2981ds/90Mi)
% 20.58/9.65 % (31557)Instruction limit reached!
% 20.58/9.65 % (31557)------------------------------
% 20.58/9.65 % (31557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/9.65 % (31557)Termination reason: Unknown
% 20.58/9.65 % (31557)Termination phase: shuffling
% 20.58/9.65
% 20.58/9.65 % (31557)Memory used [KB]: 16247
% 20.58/9.65 % (31557)Time elapsed: 0.108 s
% 20.58/9.65 % (31557)Instructions burned: 193 (million)
% 20.58/9.65 % (31557)------------------------------
% 20.58/9.65 % (31557)------------------------------
% 20.58/9.66 % (31560)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_31377 for (2981ds/231Mi)
% 20.58/9.66 % (31533)Instruction limit reached!
% 20.58/9.66 % (31533)------------------------------
% 20.58/9.66 % (31533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/9.66 % (31533)Termination reason: Unknown
% 20.58/9.66 % (31533)Termination phase: Property scanning
% 20.58/9.66
% 20.58/9.66 % (31533)Memory used [KB]: 26609
% 20.58/9.66 % (31533)Time elapsed: 0.560 s
% 20.58/9.66 % (31533)Instructions burned: 1089 (million)
% 20.58/9.66 % (31533)------------------------------
% 20.58/9.66 % (31533)------------------------------
% 20.58/9.66 % (31556)Instruction limit reached!
% 20.58/9.66 % (31556)------------------------------
% 20.58/9.66 % (31556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/9.66 % (31556)Termination reason: Unknown
% 20.58/9.66 % (31556)Termination phase: shuffling
% 20.58/9.66
% 20.58/9.66 % (31556)Memory used [KB]: 19829
% 20.58/9.66 % (31556)Time elapsed: 0.152 s
% 20.58/9.66 % (31556)Instructions burned: 299 (million)
% 20.58/9.66 % (31556)------------------------------
% 20.58/9.66 % (31556)------------------------------
% 20.58/9.67 % (31561)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_31377 for (2981ds/458Mi)
% 20.58/9.68 % (31562)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_31377 for (2980ds/31Mi)
% 20.58/9.68 % (31563)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_31377 for (2980ds/297Mi)
% 20.58/9.69 % (31559)Instruction limit reached!
% 20.58/9.69 % (31559)------------------------------
% 20.58/9.69 % (31559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/9.69 % (31559)Termination reason: Unknown
% 20.58/9.69 % (31559)Termination phase: shuffling
% 20.58/9.69
% 20.58/9.69 % (31559)Memory used [KB]: 14839
% 20.58/9.69 % (31559)Time elapsed: 0.057 s
% 20.58/9.69 % (31559)Instructions burned: 91 (million)
% 20.58/9.69 % (31559)------------------------------
% 20.58/9.69 % (31559)------------------------------
% 21.90/9.70 % (31562)Instruction limit reached!
% 21.90/9.70 % (31562)------------------------------
% 21.90/9.70 % (31562)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.90/9.70 % (31562)Termination reason: Unknown
% 21.90/9.70 % (31562)Termination phase: shuffling
% 21.90/9.70
% 21.90/9.70 % (31562)Memory used [KB]: 13944
% 21.90/9.70 % (31562)Time elapsed: 0.027 s
% 21.90/9.70 % (31562)Instructions burned: 31 (million)
% 21.90/9.70 % (31562)------------------------------
% 21.90/9.70 % (31562)------------------------------
% 21.90/9.71 % (31564)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_31377 for (2980ds/987Mi)
% 21.90/9.72 % (31565)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_31377 for (2980ds/182Mi)
% 21.90/9.72 % (31558)Instruction limit reached!
% 21.90/9.72 % (31558)------------------------------
% 21.90/9.72 % (31558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.90/9.72 % (31558)Termination reason: Unknown
% 21.90/9.72 % (31558)Termination phase: shuffling
% 21.90/9.72
% 21.90/9.72 % (31558)Memory used [KB]: 17142
% 21.90/9.72 % (31558)Time elapsed: 0.143 s
% 21.90/9.72 % (31558)Instructions burned: 258 (million)
% 21.90/9.72 % (31558)------------------------------
% 21.90/9.72 % (31558)------------------------------
% 21.90/9.73 % (31551)Instruction limit reached!
% 21.90/9.73 % (31551)------------------------------
% 21.90/9.73 % (31551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.90/9.73 % (31551)Termination reason: Unknown
% 21.90/9.73 % (31551)Termination phase: shuffling
% 21.90/9.73
% 21.90/9.73 % (31551)Memory used [KB]: 23922
% 21.90/9.73 % (31551)Time elapsed: 0.330 s
% 21.90/9.73 % (31551)Instructions burned: 615 (million)
% 21.90/9.73 % (31551)------------------------------
% 21.90/9.73 % (31551)------------------------------
% 21.90/9.74 % (31566)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_31377 for (2980ds/182Mi)
% 21.90/9.75 % (31567)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_31377 for (2980ds/154Mi)
% 21.90/9.77 % (31560)Instruction limit reached!
% 21.90/9.77 % (31560)------------------------------
% 21.90/9.77 % (31560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.90/9.77 % (31560)Termination reason: Unknown
% 21.90/9.77 % (31560)Termination phase: shuffling
% 21.90/9.77
% 21.90/9.77 % (31560)Memory used [KB]: 16886
% 21.90/9.77 % (31560)Time elapsed: 0.120 s
% 21.90/9.77 % (31560)Instructions burned: 231 (million)
% 21.90/9.77 % (31560)------------------------------
% 21.90/9.77 % (31560)------------------------------
% 21.90/9.79 % (31568)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_31377 for (2979ds/79Mi)
% 22.59/9.82 % (31565)Instruction limit reached!
% 22.59/9.82 % (31565)------------------------------
% 22.59/9.82 % (31565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.59/9.82 % (31565)Termination reason: Unknown
% 22.59/9.82 % (31565)Termination phase: shuffling
% 22.59/9.82
% 22.59/9.82 % (31565)Memory used [KB]: 16119
% 22.59/9.82 % (31565)Time elapsed: 0.104 s
% 22.59/9.82 % (31565)Instructions burned: 182 (million)
% 22.59/9.82 % (31565)------------------------------
% 22.59/9.82 % (31565)------------------------------
% 22.59/9.83 % (31567)Instruction limit reached!
% 22.59/9.83 % (31567)------------------------------
% 22.59/9.83 % (31567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.59/9.83 % (31567)Termination reason: Unknown
% 22.59/9.83 % (31567)Termination phase: shuffling
% 22.59/9.83
% 22.59/9.83 % (31567)Memory used [KB]: 15735
% 22.59/9.83 % (31567)Time elapsed: 0.083 s
% 22.59/9.83 % (31567)Instructions burned: 154 (million)
% 22.59/9.83 % (31567)------------------------------
% 22.59/9.83 % (31567)------------------------------
% 22.59/9.83 % (31563)Instruction limit reached!
% 22.59/9.83 % (31563)------------------------------
% 22.59/9.83 % (31563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.59/9.83 % (31563)Termination reason: Unknown
% 22.59/9.83 % (31563)Termination phase: shuffling
% 22.59/9.83
% 22.59/9.83 % (31563)Memory used [KB]: 19829
% 22.59/9.83 % (31563)Time elapsed: 0.152 s
% 22.59/9.83 % (31563)Instructions burned: 297 (million)
% 22.59/9.83 % (31563)------------------------------
% 22.59/9.83 % (31563)------------------------------
% 22.59/9.84 % (31569)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_31377 for (2979ds/216Mi)
% 22.59/9.84 % (31568)Instruction limit reached!
% 22.59/9.84 % (31568)------------------------------
% 22.59/9.84 % (31568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.59/9.84 % (31568)Termination reason: Unknown
% 22.59/9.84 % (31568)Termination phase: shuffling
% 22.59/9.84
% 22.59/9.84 % (31568)Memory used [KB]: 14711
% 22.59/9.84 % (31568)Time elapsed: 0.051 s
% 22.59/9.84 % (31568)Instructions burned: 80 (million)
% 22.59/9.84 % (31568)------------------------------
% 22.59/9.84 % (31568)------------------------------
% 22.59/9.85 % (31566)Instruction limit reached!
% 22.59/9.85 % (31566)------------------------------
% 22.59/9.85 % (31566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.59/9.85 % (31566)Termination reason: Unknown
% 22.59/9.85 % (31566)Termination phase: shuffling
% 22.59/9.85
% 22.59/9.85 % (31566)Memory used [KB]: 16119
% 22.59/9.85 % (31566)Time elapsed: 0.104 s
% 22.59/9.85 % (31566)Instructions burned: 182 (million)
% 22.59/9.85 % (31566)------------------------------
% 22.59/9.85 % (31566)------------------------------
% 22.59/9.85 % (31570)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_31377 for (2979ds/114Mi)
% 22.59/9.85 % (31571)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_31377 for (2979ds/922Mi)
% 22.59/9.86 % (31572)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_31377 for (2978ds/182Mi)
% 22.59/9.86 % (31573)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_31377 for (2978ds/164Mi)
% 23.16/9.91 % (31561)Instruction limit reached!
% 23.16/9.91 % (31561)------------------------------
% 23.16/9.91 % (31561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.16/9.91 % (31561)Termination reason: Unknown
% 23.16/9.91 % (31561)Termination phase: shuffling
% 23.16/9.91
% 23.16/9.91 % (31561)Memory used [KB]: 19829
% 23.16/9.91 % (31561)Time elapsed: 0.246 s
% 23.16/9.91 % (31561)Instructions burned: 458 (million)
% 23.16/9.91 % (31561)------------------------------
% 23.16/9.91 % (31561)------------------------------
% 23.16/9.92 % (31570)Instruction limit reached!
% 23.16/9.92 % (31570)------------------------------
% 23.16/9.92 % (31570)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.16/9.92 % (31570)Termination reason: Unknown
% 23.16/9.92 % (31570)Termination phase: shuffling
% 23.16/9.92
% 23.16/9.92 % (31570)Memory used [KB]: 15223
% 23.16/9.92 % (31570)Time elapsed: 0.069 s
% 23.16/9.92 % (31570)Instructions burned: 114 (million)
% 23.16/9.92 % (31570)------------------------------
% 23.16/9.92 % (31570)------------------------------
% 23.16/9.93 % (31574)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_31377 for (2978ds/50Mi)
% 23.16/9.93 % (31575)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_31377 for (2978ds/368Mi)
% 23.16/9.95 % (31573)Instruction limit reached!
% 23.16/9.95 % (31573)------------------------------
% 23.16/9.95 % (31573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.16/9.95 % (31573)Termination reason: Unknown
% 23.16/9.95 % (31573)Termination phase: shuffling
% 23.16/9.95
% 23.16/9.95 % (31573)Memory used [KB]: 15863
% 23.16/9.95 % (31573)Time elapsed: 0.088 s
% 23.16/9.95 % (31573)Instructions burned: 164 (million)
% 23.16/9.95 % (31573)------------------------------
% 23.16/9.95 % (31573)------------------------------
% 23.16/9.95 % (31569)Instruction limit reached!
% 23.16/9.95 % (31569)------------------------------
% 23.16/9.95 % (31569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.16/9.95 % (31569)Termination reason: Unknown
% 23.16/9.95 % (31569)Termination phase: shuffling
% 23.16/9.95
% 23.16/9.95 % (31569)Memory used [KB]: 16630
% 23.16/9.95 % (31569)Time elapsed: 0.112 s
% 23.16/9.95 % (31569)Instructions burned: 216 (million)
% 23.16/9.95 % (31569)------------------------------
% 23.16/9.95 % (31569)------------------------------
% 23.16/9.96 % (31572)Instruction limit reached!
% 23.16/9.96 % (31572)------------------------------
% 23.16/9.96 % (31572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.16/9.96 % (31572)Termination reason: Unknown
% 23.16/9.96 % (31572)Termination phase: shuffling
% 23.16/9.96
% 23.16/9.96 % (31572)Memory used [KB]: 16119
% 23.16/9.96 % (31572)Time elapsed: 0.097 s
% 23.16/9.96 % (31572)Instructions burned: 183 (million)
% 23.16/9.96 % (31572)------------------------------
% 23.16/9.96 % (31572)------------------------------
% 23.16/9.96 % (31574)Instruction limit reached!
% 23.16/9.96 % (31574)------------------------------
% 23.16/9.96 % (31574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.16/9.96 % (31574)Termination reason: Unknown
% 23.16/9.96 % (31574)Termination phase: shuffling
% 23.16/9.96
% 23.16/9.96 % (31574)Memory used [KB]: 14200
% 23.16/9.96 % (31574)Time elapsed: 0.034 s
% 23.16/9.96 % (31574)Instructions burned: 50 (million)
% 23.16/9.96 % (31574)------------------------------
% 23.16/9.96 % (31574)------------------------------
% 23.16/9.97 % (31576)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_31377 for (2977ds/1021Mi)
% 23.16/9.97 % (31577)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_31377 for (2977ds/267Mi)
% 23.16/9.97 % (31578)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_31377 for (2977ds/204Mi)
% 23.16/9.98 % (31579)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_31377 for (2977ds/26Mi)
% 23.16/10.00 % (31579)Instruction limit reached!
% 23.16/10.00 % (31579)------------------------------
% 23.16/10.00 % (31579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.16/10.00 % (31579)Termination reason: Unknown
% 23.16/10.00 % (31579)Termination phase: shuffling
% 23.16/10.00
% 23.16/10.00 % (31579)Memory used [KB]: 13944
% 23.16/10.00 % (31579)Time elapsed: 0.024 s
% 23.16/10.00 % (31579)Instructions burned: 26 (million)
% 23.16/10.00 % (31579)------------------------------
% 23.16/10.00 % (31579)------------------------------
% 23.16/10.02 % (31580)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_31377 for (2977ds/53Mi)
% 24.71/10.05 % (31580)Instruction limit reached!
% 24.71/10.05 % (31580)------------------------------
% 24.71/10.05 % (31580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.71/10.05 % (31580)Termination reason: Unknown
% 24.71/10.05 % (31580)Termination phase: shuffling
% 24.71/10.05
% 24.71/10.05 % (31580)Memory used [KB]: 14328
% 24.71/10.05 % (31580)Time elapsed: 0.036 s
% 24.71/10.05 % (31580)Instructions burned: 53 (million)
% 24.71/10.05 % (31580)------------------------------
% 24.71/10.05 % (31580)------------------------------
% 24.71/10.07 % (31581)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_31377 for (2976ds/4436Mi)
% 24.71/10.09 % (31578)Instruction limit reached!
% 24.71/10.09 % (31578)------------------------------
% 24.71/10.09 % (31578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.71/10.09 % (31578)Termination reason: Unknown
% 24.71/10.09 % (31578)Termination phase: shuffling
% 24.71/10.09
% 24.71/10.09 % (31578)Memory used [KB]: 16375
% 24.71/10.09 % (31578)Time elapsed: 0.116 s
% 24.71/10.09 % (31578)Instructions burned: 204 (million)
% 24.71/10.09 % (31578)------------------------------
% 24.71/10.09 % (31578)------------------------------
% 24.71/10.11 % (31577)Instruction limit reached!
% 24.71/10.11 % (31577)------------------------------
% 24.71/10.11 % (31577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.71/10.11 % (31577)Termination reason: Unknown
% 24.71/10.11 % (31577)Termination phase: shuffling
% 24.71/10.11
% 24.71/10.11 % (31577)Memory used [KB]: 19445
% 24.71/10.11 % (31577)Time elapsed: 0.138 s
% 24.71/10.11 % (31577)Instructions burned: 268 (million)
% 24.71/10.11 % (31577)------------------------------
% 24.71/10.11 % (31577)------------------------------
% 24.71/10.11 % (31582)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_31377 for (2976ds/1539Mi)
% 24.71/10.12 % (31583)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_31377 for (2976ds/408Mi)
% 25.35/10.13 % (31575)Instruction limit reached!
% 25.35/10.13 % (31575)------------------------------
% 25.35/10.13 % (31575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.35/10.13 % (31575)Termination reason: Unknown
% 25.35/10.13 % (31575)Termination phase: shuffling
% 25.35/10.13
% 25.35/10.13 % (31575)Memory used [KB]: 18677
% 25.35/10.13 % (31575)Time elapsed: 0.199 s
% 25.35/10.13 % (31575)Instructions burned: 369 (million)
% 25.35/10.13 % (31575)------------------------------
% 25.35/10.13 % (31575)------------------------------
% 25.35/10.15 % (31584)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_31377 for (2976ds/5205Mi)
% 25.58/10.20 % (31564)Instruction limit reached!
% 25.58/10.20 % (31564)------------------------------
% 25.58/10.20 % (31564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.58/10.20 % (31564)Termination reason: Unknown
% 25.58/10.20 % (31564)Termination phase: shuffling
% 25.58/10.20
% 25.58/10.20 % (31564)Memory used [KB]: 33261
% 25.58/10.20 % (31564)Time elapsed: 0.486 s
% 25.58/10.20 % (31564)Instructions burned: 987 (million)
% 25.58/10.20 % (31564)------------------------------
% 25.58/10.20 % (31564)------------------------------
% 25.58/10.23 % (31585)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_31377 for (2975ds/1913Mi)
% 25.81/10.33 % (31571)Instruction limit reached!
% 25.81/10.33 % (31571)------------------------------
% 25.81/10.33 % (31571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.81/10.33 % (31571)Termination reason: Unknown
% 25.81/10.33 % (31571)Termination phase: Property scanning
% 25.81/10.33
% 25.81/10.33 % (31571)Memory used [KB]: 26609
% 25.81/10.33 % (31571)Time elapsed: 0.483 s
% 25.81/10.33 % (31571)Instructions burned: 922 (million)
% 25.81/10.33 % (31571)------------------------------
% 25.81/10.33 % (31571)------------------------------
% 25.81/10.34 % (31583)Instruction limit reached!
% 25.81/10.34 % (31583)------------------------------
% 25.81/10.34 % (31583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.81/10.34 % (31583)Termination reason: Unknown
% 25.81/10.34 % (31583)Termination phase: shuffling
% 25.81/10.34
% 25.81/10.34 % (31583)Memory used [KB]: 19317
% 25.81/10.34 % (31583)Time elapsed: 0.222 s
% 25.81/10.34 % (31583)Instructions burned: 409 (million)
% 25.81/10.34 % (31583)------------------------------
% 25.81/10.34 % (31583)------------------------------
% 25.81/10.35 % (31586)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_31377 for (2974ds/4350Mi)
% 25.81/10.36 % (31587)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_31377 for (2974ds/68Mi)
% 26.18/10.41 % (31587)Instruction limit reached!
% 26.18/10.41 % (31587)------------------------------
% 26.18/10.41 % (31587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.18/10.41 % (31587)Termination reason: Unknown
% 26.18/10.41 % (31587)Termination phase: shuffling
% 26.18/10.41
% 26.18/10.41 % (31587)Memory used [KB]: 14456
% 26.18/10.41 % (31587)Time elapsed: 0.045 s
% 26.18/10.41 % (31587)Instructions burned: 69 (million)
% 26.18/10.41 % (31587)------------------------------
% 26.18/10.41 % (31587)------------------------------
% 26.18/10.43 % (31588)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_31377 for (2973ds/358Mi)
% 26.18/10.50 % (31576)Instruction limit reached!
% 26.18/10.50 % (31576)------------------------------
% 26.18/10.50 % (31576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.18/10.50 % (31576)Termination reason: Unknown
% 26.18/10.50 % (31576)Termination phase: Property scanning
% 26.18/10.50
% 26.18/10.50 % (31576)Memory used [KB]: 26609
% 26.18/10.50 % (31576)Time elapsed: 0.528 s
% 26.18/10.50 % (31576)Instructions burned: 1021 (million)
% 26.18/10.50 % (31576)------------------------------
% 26.18/10.50 % (31576)------------------------------
% 25.81/10.52 % (31589)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_31377 for (2972ds/31Mi)
% 25.81/10.54 % (31589)Instruction limit reached!
% 25.81/10.54 % (31589)------------------------------
% 25.81/10.54 % (31589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 25.81/10.54 % (31589)Termination reason: Unknown
% 25.81/10.54 % (31589)Termination phase: shuffling
% 25.81/10.54
% 25.81/10.54 % (31589)Memory used [KB]: 13944
% 25.81/10.54 % (31589)Time elapsed: 0.026 s
% 25.81/10.54 % (31589)Instructions burned: 31 (million)
% 25.81/10.54 % (31589)------------------------------
% 25.81/10.54 % (31589)------------------------------
% 25.81/10.56 % (31590)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_31377 for (2972ds/279Mi)
% 26.41/10.62 % (31588)Instruction limit reached!
% 26.41/10.62 % (31588)------------------------------
% 26.41/10.62 % (31588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.41/10.62 % (31588)Termination reason: Unknown
% 26.41/10.62 % (31588)Termination phase: shuffling
% 26.41/10.62
% 26.41/10.62 % (31588)Memory used [KB]: 18549
% 26.41/10.62 % (31588)Time elapsed: 0.194 s
% 26.41/10.62 % (31588)Instructions burned: 358 (million)
% 26.41/10.62 % (31588)------------------------------
% 26.41/10.62 % (31588)------------------------------
% 26.41/10.64 % (31591)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_31377 for (2971ds/258Mi)
% 26.62/10.70 % (31590)Instruction limit reached!
% 26.62/10.70 % (31590)------------------------------
% 26.62/10.70 % (31590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.62/10.70 % (31590)Termination reason: Unknown
% 26.62/10.70 % (31590)Termination phase: shuffling
% 26.62/10.70
% 26.62/10.70 % (31590)Memory used [KB]: 19573
% 26.62/10.70 % (31590)Time elapsed: 0.144 s
% 26.62/10.70 % (31590)Instructions burned: 279 (million)
% 26.62/10.70 % (31590)------------------------------
% 26.62/10.70 % (31590)------------------------------
% 26.62/10.72 % (31592)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_31377 for (2970ds/26Mi)
% 26.62/10.74 % (31592)Instruction limit reached!
% 26.62/10.74 % (31592)------------------------------
% 26.62/10.74 % (31592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.62/10.74 % (31592)Termination reason: Unknown
% 26.62/10.74 % (31592)Termination phase: shuffling
% 26.62/10.74
% 26.62/10.74 % (31592)Memory used [KB]: 13944
% 26.62/10.74 % (31592)Time elapsed: 0.023 s
% 26.62/10.74 % (31592)Instructions burned: 27 (million)
% 26.62/10.74 % (31592)------------------------------
% 26.62/10.74 % (31592)------------------------------
% 26.62/10.76 % (31593)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_31377 for (2969ds/995Mi)
% 26.62/10.77 % (31591)Instruction limit reached!
% 26.62/10.77 % (31591)------------------------------
% 26.62/10.77 % (31591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.62/10.77 % (31591)Termination reason: Unknown
% 26.62/10.77 % (31591)Termination phase: shuffling
% 26.62/10.77
% 26.62/10.77 % (31591)Memory used [KB]: 17142
% 26.62/10.77 % (31591)Time elapsed: 0.135 s
% 26.62/10.77 % (31591)Instructions burned: 258 (million)
% 26.62/10.77 % (31591)------------------------------
% 26.62/10.77 % (31591)------------------------------
% 26.62/10.79 % (31594)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_31377 for (2969ds/373Mi)
% 26.98/10.88 % (31582)Instruction limit reached!
% 26.98/10.88 % (31582)------------------------------
% 26.98/10.88 % (31582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.98/10.88 % (31582)Termination reason: Unknown
% 26.98/10.88 % (31582)Termination phase: SInE selection
% 26.98/10.88
% 26.98/10.88 % (31582)Memory used [KB]: 26481
% 26.98/10.88 % (31582)Time elapsed: 0.771 s
% 26.98/10.88 % (31582)Instructions burned: 1539 (million)
% 26.98/10.88 % (31582)------------------------------
% 26.98/10.88 % (31582)------------------------------
% 27.77/10.90 % (31595)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_31377 for (2968ds/253Mi)
% 27.77/10.99 % (31594)Instruction limit reached!
% 27.77/10.99 % (31594)------------------------------
% 27.77/10.99 % (31594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.77/10.99 % (31594)Termination reason: Unknown
% 27.77/10.99 % (31594)Termination phase: shuffling
% 27.77/10.99
% 27.77/10.99 % (31594)Memory used [KB]: 18677
% 27.77/10.99 % (31594)Time elapsed: 0.201 s
% 27.77/10.99 % (31594)Instructions burned: 373 (million)
% 27.77/10.99 % (31594)------------------------------
% 27.77/10.99 % (31594)------------------------------
% 27.77/11.01 % (31596)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_31377 for (2967ds/832Mi)
% 27.77/11.02 % (31595)Instruction limit reached!
% 27.77/11.02 % (31595)------------------------------
% 27.77/11.02 % (31595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.77/11.02 % (31595)Termination reason: Unknown
% 27.77/11.02 % (31595)Termination phase: shuffling
% 27.77/11.02
% 27.77/11.02 % (31595)Memory used [KB]: 17142
% 27.77/11.02 % (31595)Time elapsed: 0.129 s
% 27.77/11.02 % (31595)Instructions burned: 253 (million)
% 27.77/11.03 % (31595)------------------------------
% 27.77/11.03 % (31595)------------------------------
% 31.61/11.04 % (31597)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_31377 for (2967ds/668Mi)
% 32.35/11.18 % (31585)Instruction limit reached!
% 32.35/11.18 % (31585)------------------------------
% 32.35/11.18 % (31585)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.35/11.18 % (31585)Termination reason: Unknown
% 32.35/11.18 % (31585)Termination phase: SInE selection
% 32.35/11.18
% 32.35/11.18 % (31585)Memory used [KB]: 27249
% 32.35/11.18 % (31585)Time elapsed: 0.956 s
% 32.35/11.18 % (31585)Instructions burned: 1914 (million)
% 32.35/11.18 % (31585)------------------------------
% 32.35/11.18 % (31585)------------------------------
% 32.76/11.20 % (31598)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_31377 for (2965ds/40Mi)
% 32.76/11.23 % (31598)Instruction limit reached!
% 32.76/11.23 % (31598)------------------------------
% 32.76/11.23 % (31598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.76/11.23 % (31598)Termination reason: Unknown
% 32.76/11.23 % (31598)Termination phase: shuffling
% 32.76/11.23
% 32.76/11.23 % (31598)Memory used [KB]: 14072
% 32.76/11.23 % (31598)Time elapsed: 0.031 s
% 32.76/11.23 % (31598)Instructions burned: 41 (million)
% 32.76/11.23 % (31598)------------------------------
% 32.76/11.23 % (31598)------------------------------
% 32.76/11.24 % (31593)Instruction limit reached!
% 32.76/11.24 % (31593)------------------------------
% 32.76/11.24 % (31593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.76/11.24 % (31593)Termination reason: Unknown
% 32.76/11.24 % (31593)Termination phase: shuffling
% 32.76/11.24
% 32.76/11.24 % (31593)Memory used [KB]: 33389
% 32.76/11.24 % (31593)Time elapsed: 0.478 s
% 32.76/11.24 % (31593)Instructions burned: 996 (million)
% 32.76/11.24 % (31593)------------------------------
% 32.76/11.24 % (31593)------------------------------
% 32.76/11.25 % (31599)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_31377 for (2965ds/294Mi)
% 32.76/11.25 % (31600)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_31377 for (2964ds/187Mi)
% 33.25/11.35 % (31600)Instruction limit reached!
% 33.25/11.35 % (31600)------------------------------
% 33.25/11.35 % (31600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.25/11.35 % (31600)Termination reason: Unknown
% 33.25/11.35 % (31600)Termination phase: shuffling
% 33.25/11.35
% 33.25/11.35 % (31600)Memory used [KB]: 16247
% 33.25/11.35 % (31600)Time elapsed: 0.096 s
% 33.25/11.35 % (31600)Instructions burned: 189 (million)
% 33.25/11.35 % (31600)------------------------------
% 33.25/11.35 % (31600)------------------------------
% 33.25/11.37 % (31601)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_31377 for (2963ds/371Mi)
% 33.25/11.37 % (31597)Instruction limit reached!
% 33.25/11.37 % (31597)------------------------------
% 33.25/11.37 % (31597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.25/11.37 % (31597)Termination reason: Unknown
% 33.25/11.37 % (31597)Termination phase: shuffling
% 33.25/11.37
% 33.25/11.37 % (31597)Memory used [KB]: 24946
% 33.25/11.37 % (31597)Time elapsed: 0.329 s
% 33.25/11.37 % (31597)Instructions burned: 668 (million)
% 33.25/11.37 % (31597)------------------------------
% 33.25/11.37 % (31597)------------------------------
% 33.25/11.39 % (31602)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_31377 for (2963ds/1827Mi)
% 34.12/11.40 % (31599)Instruction limit reached!
% 34.12/11.40 % (31599)------------------------------
% 34.12/11.40 % (31599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.12/11.40 % (31599)Termination reason: Unknown
% 34.12/11.40 % (31599)Termination phase: shuffling
% 34.12/11.40
% 34.12/11.40 % (31599)Memory used [KB]: 19829
% 34.12/11.40 % (31599)Time elapsed: 0.151 s
% 34.12/11.40 % (31599)Instructions burned: 295 (million)
% 34.12/11.40 % (31599)------------------------------
% 34.12/11.40 % (31599)------------------------------
% 34.12/11.42 % (31603)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_31377 for (2963ds/70Mi)
% 34.12/11.45 % (31596)Instruction limit reached!
% 34.12/11.45 % (31596)------------------------------
% 34.12/11.45 % (31596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.12/11.45 % (31596)Termination reason: Unknown
% 34.12/11.45 % (31596)Termination phase: Property scanning
% 34.12/11.45
% 34.12/11.45 % (31596)Memory used [KB]: 26609
% 34.12/11.45 % (31596)Time elapsed: 0.445 s
% 34.12/11.45 % (31596)Instructions burned: 833 (million)
% 34.12/11.45 % (31596)------------------------------
% 34.12/11.45 % (31596)------------------------------
% 34.12/11.46 % (31603)Instruction limit reached!
% 34.12/11.46 % (31603)------------------------------
% 34.12/11.46 % (31603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.12/11.46 % (31603)Termination reason: Unknown
% 34.12/11.46 % (31603)Termination phase: shuffling
% 34.12/11.46
% 34.12/11.46 % (31603)Memory used [KB]: 14583
% 34.12/11.46 % (31603)Time elapsed: 0.044 s
% 34.12/11.46 % (31603)Instructions burned: 71 (million)
% 34.12/11.46 % (31603)------------------------------
% 34.12/11.46 % (31603)------------------------------
% 34.12/11.47 % (31604)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_31377 for (2962ds/316Mi)
% 34.12/11.48 % (31605)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_31377 for (2962ds/932Mi)
% 34.73/11.57 % (31601)Instruction limit reached!
% 34.73/11.57 % (31601)------------------------------
% 34.73/11.57 % (31601)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.73/11.57 % (31601)Termination reason: Unknown
% 34.73/11.57 % (31601)Termination phase: shuffling
% 34.73/11.57
% 34.73/11.57 % (31601)Memory used [KB]: 18677
% 34.73/11.57 % (31601)Time elapsed: 0.203 s
% 34.73/11.57 % (31601)Instructions burned: 372 (million)
% 34.73/11.57 % (31601)------------------------------
% 34.73/11.57 % (31601)------------------------------
% 34.73/11.59 % (31606)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_31377 for (2961ds/4714Mi)
% 35.25/11.63 % (31604)Instruction limit reached!
% 35.25/11.63 % (31604)------------------------------
% 35.25/11.63 % (31604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.25/11.63 % (31604)Termination reason: Unknown
% 35.25/11.63 % (31604)Termination phase: shuffling
% 35.25/11.63
% 35.25/11.63 % (31604)Memory used [KB]: 20084
% 35.25/11.63 % (31604)Time elapsed: 0.162 s
% 35.25/11.63 % (31604)Instructions burned: 316 (million)
% 35.25/11.63 % (31604)------------------------------
% 35.25/11.63 % (31604)------------------------------
% 35.25/11.65 % (31607)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_31377 for (2961ds/8250Mi)
% 36.33/11.94 % (31605)Instruction limit reached!
% 36.33/11.94 % (31605)------------------------------
% 36.33/11.94 % (31605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.33/11.94 % (31605)Termination reason: Unknown
% 36.33/11.94 % (31605)Termination phase: shuffling
% 36.33/11.94
% 36.33/11.94 % (31605)Memory used [KB]: 32622
% 36.33/11.94 % (31605)Time elapsed: 0.462 s
% 36.33/11.94 % (31605)Instructions burned: 933 (million)
% 36.33/11.94 % (31605)------------------------------
% 36.33/11.94 % (31605)------------------------------
% 36.33/11.97 % (31608)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_31377 for (2958ds/770Mi)
% 39.90/12.22 % (31608)Instruction limit reached!
% 39.90/12.22 % (31608)------------------------------
% 39.90/12.22 % (31608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.90/12.22 % (31608)Termination reason: Unknown
% 39.90/12.22 % (31608)Termination phase: shuffling
% 39.90/12.22
% 39.90/12.22 % (31608)Memory used [KB]: 25713
% 39.90/12.22 % (31608)Time elapsed: 0.259 s
% 39.90/12.22 % (31608)Instructions burned: 770 (million)
% 39.90/12.22 % (31608)------------------------------
% 39.90/12.22 % (31608)------------------------------
% 39.90/12.24 % (31609)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_31377 for (2955ds/794Mi)
% 40.56/12.32 % (31602)Instruction limit reached!
% 40.56/12.32 % (31602)------------------------------
% 40.56/12.32 % (31602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.56/12.32 % (31602)Termination reason: Unknown
% 40.56/12.32 % (31602)Termination phase: Preprocessing 3
% 40.56/12.32
% 40.56/12.32 % (31602)Memory used [KB]: 34413
% 40.56/12.32 % (31602)Time elapsed: 0.932 s
% 40.56/12.32 % (31602)Instructions burned: 1828 (million)
% 40.56/12.32 % (31602)------------------------------
% 40.56/12.32 % (31602)------------------------------
% 40.56/12.34 % (31610)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_31377 for (2954ds/464Mi)
% 40.56/12.35 % (31581)Instruction limit reached!
% 40.56/12.35 % (31581)------------------------------
% 40.56/12.35 % (31581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.56/12.35 % (31581)Termination reason: Unknown
% 40.56/12.35 % (31581)Termination phase: Saturation
% 40.56/12.35
% 40.56/12.35 % (31581)Memory used [KB]: 34924
% 40.56/12.35 % (31581)Time elapsed: 2.274 s
% 40.56/12.35 % (31581)Instructions burned: 4436 (million)
% 40.56/12.35 % (31581)------------------------------
% 40.56/12.35 % (31581)------------------------------
% 40.56/12.37 % (31611)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_31377 for (2954ds/1376Mi)
% 41.04/12.44 % (31584)Instruction limit reached!
% 41.04/12.44 % (31584)------------------------------
% 41.04/12.44 % (31584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.04/12.44 % (31584)Termination reason: Unknown
% 41.04/12.44 % (31584)Termination phase: Function definition elimination
% 41.04/12.44
% 41.04/12.44 % (31584)Memory used [KB]: 56928
% 41.04/12.44 % (31584)Time elapsed: 2.292 s
% 41.04/12.44 % (31584)Instructions burned: 5207 (million)
% 41.04/12.44 % (31584)------------------------------
% 41.04/12.44 % (31584)------------------------------
% 41.04/12.46 % (31612)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_31377 for (2952ds/906Mi)
% 41.41/12.51 % (31609)Instruction limit reached!
% 41.41/12.51 % (31609)------------------------------
% 41.41/12.51 % (31609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.41/12.51 % (31609)Termination reason: Unknown
% 41.41/12.51 % (31609)Termination phase: shuffling
% 41.41/12.51
% 41.41/12.51 % (31609)Memory used [KB]: 26225
% 41.41/12.51 % (31609)Time elapsed: 0.272 s
% 41.41/12.51 % (31609)Instructions burned: 794 (million)
% 41.41/12.51 % (31609)------------------------------
% 41.41/12.51 % (31609)------------------------------
% 41.41/12.52 % (31613)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_31377 for (2952ds/434Mi)
% 41.41/12.53 % (31586)Instruction limit reached!
% 41.41/12.53 % (31586)------------------------------
% 41.41/12.53 % (31586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.41/12.53 % (31586)Termination reason: Unknown
% 41.41/12.53 % (31586)Termination phase: Saturation
% 41.41/12.53
% 41.41/12.53 % (31586)Memory used [KB]: 33901
% 41.41/12.53 % (31586)Time elapsed: 2.179 s
% 41.41/12.53 % (31586)Instructions burned: 4350 (million)
% 41.41/12.53 % (31586)------------------------------
% 41.41/12.53 % (31586)------------------------------
% 41.41/12.55 % (31615)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_31377 for (2952ds/865Mi)
% 41.41/12.59 % (31610)Instruction limit reached!
% 41.41/12.59 % (31610)------------------------------
% 41.41/12.59 % (31610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.41/12.59 % (31610)Termination reason: Unknown
% 41.41/12.59 % (31610)Termination phase: shuffling
% 41.41/12.59
% 41.41/12.59 % (31610)Memory used [KB]: 19829
% 41.41/12.59 % (31610)Time elapsed: 0.249 s
% 41.41/12.59 % (31610)Instructions burned: 464 (million)
% 41.41/12.59 % (31610)------------------------------
% 41.41/12.59 % (31610)------------------------------
% 42.21/12.62 % (31616)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_31377 for (2951ds/593Mi)
% 42.42/12.73 % (31613)Instruction limit reached!
% 42.42/12.73 % (31613)------------------------------
% 42.42/12.73 % (31613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.42/12.73 % (31613)Termination reason: Unknown
% 42.42/12.73 % (31613)Termination phase: shuffling
% 42.42/12.73
% 42.42/12.73 % (31613)Memory used [KB]: 19573
% 42.42/12.73 % (31613)Time elapsed: 0.171 s
% 42.42/12.73 % (31613)Instructions burned: 435 (million)
% 42.42/12.73 % (31613)------------------------------
% 42.42/12.73 % (31613)------------------------------
% 42.42/12.75 % (31617)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_31377 for (2950ds/443Mi)
% 42.80/12.83 % (31616)Instruction limit reached!
% 42.80/12.83 % (31616)------------------------------
% 42.80/12.83 % (31616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.80/12.83 % (31616)Termination reason: Unknown
% 42.80/12.83 % (31616)Termination phase: shuffling
% 42.80/12.83
% 42.80/12.83 % (31616)Memory used [KB]: 23922
% 42.80/12.83 % (31616)Time elapsed: 0.178 s
% 42.80/12.83 % (31616)Instructions burned: 595 (million)
% 42.80/12.83 % (31616)------------------------------
% 42.80/12.83 % (31616)------------------------------
% 42.80/12.84 % (31618)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_31377 for (2949ds/594Mi)
% 43.18/12.94 % (31612)Instruction limit reached!
% 43.18/12.94 % (31612)------------------------------
% 43.18/12.94 % (31612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.18/12.94 % (31612)Termination reason: Unknown
% 43.18/12.94 % (31612)Termination phase: Property scanning
% 43.18/12.94
% 43.18/12.94 % (31612)Memory used [KB]: 26481
% 43.18/12.94 % (31612)Time elapsed: 0.475 s
% 43.18/12.94 % (31612)Instructions burned: 907 (million)
% 43.18/12.94 % (31612)------------------------------
% 43.18/12.94 % (31612)------------------------------
% 43.18/12.96 % (31619)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_31377 for (2948ds/327Mi)
% 43.18/12.97 % (31615)Instruction limit reached!
% 43.18/12.97 % (31615)------------------------------
% 43.18/12.97 % (31615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.18/12.97 % (31615)Termination reason: Unknown
% 43.18/12.97 % (31615)Termination phase: shuffling
% 43.18/12.97
% 43.18/12.97 % (31615)Memory used [KB]: 27632
% 43.18/12.97 % (31615)Time elapsed: 0.422 s
% 43.18/12.97 % (31615)Instructions burned: 865 (million)
% 43.18/12.97 % (31615)------------------------------
% 43.18/12.97 % (31615)------------------------------
% 43.18/12.98 % (31617)Instruction limit reached!
% 43.18/12.98 % (31617)------------------------------
% 43.18/12.98 % (31617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.18/12.98 % (31617)Termination reason: Unknown
% 43.18/12.98 % (31617)Termination phase: shuffling
% 43.18/12.98
% 43.18/12.98 % (31617)Memory used [KB]: 19701
% 43.18/12.98 % (31617)Time elapsed: 0.236 s
% 43.18/12.98 % (31617)Instructions burned: 443 (million)
% 43.18/12.98 % (31617)------------------------------
% 43.18/12.98 % (31617)------------------------------
% 43.18/13.00 % (31620)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_31377 for (2947ds/5411Mi)
% 43.18/13.01 % (31621)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_31377 for (2947ds/1762Mi)
% 45.48/13.04 % (31611)Instruction limit reached!
% 45.48/13.04 % (31611)------------------------------
% 45.48/13.04 % (31611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.48/13.04 % (31611)Termination reason: Unknown
% 45.48/13.04 % (31611)Termination phase: shuffling
% 45.48/13.04
% 45.48/13.04 % (31611)Memory used [KB]: 38506
% 45.48/13.04 % (31611)Time elapsed: 0.671 s
% 45.48/13.04 % (31611)Instructions burned: 1376 (million)
% 45.48/13.04 % (31611)------------------------------
% 45.48/13.04 % (31611)------------------------------
% 45.48/13.06 % (31622)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_31377 for (2947ds/96Mi)
% 45.48/13.09 % (31619)Instruction limit reached!
% 45.48/13.09 % (31619)------------------------------
% 45.48/13.09 % (31619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.48/13.09 % (31619)Termination reason: Unknown
% 45.48/13.09 % (31619)Termination phase: shuffling
% 45.48/13.09
% 45.48/13.09 % (31619)Memory used [KB]: 18166
% 45.48/13.09 % (31619)Time elapsed: 0.119 s
% 45.48/13.09 % (31619)Instructions burned: 327 (million)
% 45.48/13.09 % (31619)------------------------------
% 45.48/13.09 % (31619)------------------------------
% 45.48/13.10 % (31618)Instruction limit reached!
% 45.48/13.10 % (31618)------------------------------
% 45.48/13.10 % (31618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 45.48/13.10 % (31618)Termination reason: Unknown
% 45.48/13.10 % (31618)Termination phase: shuffling
% 45.48/13.10
% 45.48/13.10 % (31618)Memory used [KB]: 21620
% 45.48/13.10 % (31618)Time elapsed: 0.256 s
% 45.48/13.10 % (31618)Instructions burned: 595 (million)
% 45.48/13.10 % (31618)------------------------------
% 45.48/13.10 % (31618)------------------------------
% 45.48/13.10 % (31623)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_31377 for (2946ds/503Mi)
% 46.09/13.12 % (31624)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_31377 for (2946ds/90Mi)
% 46.09/13.12 % (31622)Instruction limit reached!
% 46.09/13.12 % (31622)------------------------------
% 46.09/13.12 % (31622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.09/13.12 % (31622)Termination reason: Unknown
% 46.09/13.12 % (31622)Termination phase: shuffling
% 46.09/13.12
% 46.09/13.12 % (31622)Memory used [KB]: 14967
% 46.09/13.12 % (31622)Time elapsed: 0.055 s
% 46.09/13.12 % (31622)Instructions burned: 97 (million)
% 46.09/13.12 % (31622)------------------------------
% 46.09/13.12 % (31622)------------------------------
% 46.09/13.14 % (31625)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_31377 for (2946ds/593Mi)
% 46.09/13.18 % (31624)Instruction limit reached!
% 46.09/13.18 % (31624)------------------------------
% 46.09/13.18 % (31624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.09/13.18 % (31624)Termination reason: Unknown
% 46.09/13.18 % (31624)Termination phase: shuffling
% 46.09/13.18
% 46.09/13.18 % (31624)Memory used [KB]: 14839
% 46.09/13.18 % (31624)Time elapsed: 0.057 s
% 46.09/13.18 % (31624)Instructions burned: 91 (million)
% 46.09/13.18 % (31624)------------------------------
% 46.09/13.18 % (31624)------------------------------
% 46.09/13.20 % (31626)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_31377 for (2945ds/1512Mi)
% 46.56/13.28 % (31623)Instruction limit reached!
% 46.56/13.28 % (31623)------------------------------
% 46.56/13.28 % (31623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.56/13.28 % (31623)Termination reason: Unknown
% 46.56/13.28 % (31623)Termination phase: shuffling
% 46.56/13.28
% 46.56/13.28 % (31623)Memory used [KB]: 20340
% 46.56/13.28 % (31623)Time elapsed: 0.174 s
% 46.56/13.28 % (31623)Instructions burned: 503 (million)
% 46.56/13.28 % (31623)------------------------------
% 46.56/13.28 % (31623)------------------------------
% 46.56/13.29 % (31627)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_31377 for (2944ds/569Mi)
% 47.13/13.34 % (31606)Instruction limit reached!
% 47.13/13.34 % (31606)------------------------------
% 47.13/13.34 % (31606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.13/13.34 % (31606)Termination reason: Unknown
% 47.13/13.34 % (31606)Termination phase: Function definition elimination
% 47.13/13.34
% 47.13/13.34 % (31606)Memory used [KB]: 55905
% 47.13/13.34 % (31606)Time elapsed: 1.739 s
% 47.13/13.34 % (31606)Instructions burned: 4716 (million)
% 47.13/13.34 % (31606)------------------------------
% 47.13/13.34 % (31606)------------------------------
% 47.13/13.37 % (31628)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_31377 for (2943ds/501Mi)
% 47.50/13.46 % (31627)Instruction limit reached!
% 47.50/13.46 % (31627)------------------------------
% 47.50/13.46 % (31627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.50/13.46 % (31627)Termination reason: Unknown
% 47.50/13.46 % (31627)Termination phase: shuffling
% 47.50/13.46
% 47.50/13.46 % (31627)Memory used [KB]: 23666
% 47.50/13.46 % (31627)Time elapsed: 0.169 s
% 47.50/13.46 % (31627)Instructions burned: 570 (million)
% 47.50/13.46 % (31627)------------------------------
% 47.50/13.46 % (31627)------------------------------
% 47.50/13.46 % (31625)Instruction limit reached!
% 47.50/13.46 % (31625)------------------------------
% 47.50/13.46 % (31625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.50/13.46 % (31625)Termination reason: Unknown
% 47.50/13.46 % (31625)Termination phase: shuffling
% 47.50/13.46
% 47.50/13.46 % (31625)Memory used [KB]: 23666
% 47.50/13.46 % (31625)Time elapsed: 0.317 s
% 47.50/13.46 % (31625)Instructions burned: 593 (million)
% 47.50/13.46 % (31625)------------------------------
% 47.50/13.46 % (31625)------------------------------
% 47.50/13.47 % (31629)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_31377 for (2942ds/562Mi)
% 47.50/13.48 % (31630)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_31377 for (2942ds/406Mi)
% 48.62/13.63 % (31628)Instruction limit reached!
% 48.62/13.63 % (31628)------------------------------
% 48.62/13.63 % (31628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.62/13.63 % (31628)Termination reason: Unknown
% 48.62/13.63 % (31628)Termination phase: shuffling
% 48.62/13.63
% 48.62/13.63 % (31628)Memory used [KB]: 20340
% 48.62/13.63 % (31628)Time elapsed: 0.269 s
% 48.62/13.63 % (31628)Instructions burned: 501 (million)
% 48.62/13.63 % (31628)------------------------------
% 48.62/13.63 % (31628)------------------------------
% 48.62/13.66 % (31631)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_31377 for (2941ds/582Mi)
% 49.21/13.70 % (31629)Instruction limit reached!
% 49.21/13.70 % (31629)------------------------------
% 49.21/13.70 % (31629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.21/13.70 % (31629)Termination reason: Unknown
% 49.21/13.70 % (31629)Termination phase: shuffling
% 49.21/13.70
% 49.21/13.70 % (31629)Memory used [KB]: 23539
% 49.21/13.70 % (31629)Time elapsed: 0.172 s
% 49.21/13.70 % (31629)Instructions burned: 564 (million)
% 49.21/13.70 % (31629)------------------------------
% 49.21/13.70 % (31629)------------------------------
% 49.21/13.71 % (31630)Instruction limit reached!
% 49.21/13.71 % (31630)------------------------------
% 49.21/13.71 % (31630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.21/13.71 % (31630)Termination reason: Unknown
% 49.21/13.71 % (31630)Termination phase: shuffling
% 49.21/13.71
% 49.21/13.71 % (31630)Memory used [KB]: 19061
% 49.21/13.71 % (31630)Time elapsed: 0.204 s
% 49.21/13.71 % (31630)Instructions burned: 407 (million)
% 49.21/13.71 % (31630)------------------------------
% 49.21/13.71 % (31630)------------------------------
% 49.21/13.72 % (31632)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_31377 for (2940ds/456Mi)
% 49.21/13.73 % (31633)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_31377 for (2940ds/393Mi)
% 49.89/13.87 % (31621)Instruction limit reached!
% 49.89/13.87 % (31621)------------------------------
% 49.89/13.87 % (31621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.89/13.87 % (31621)Termination reason: Unknown
% 49.89/13.87 % (31621)Termination phase: Property scanning
% 49.89/13.87
% 49.89/13.87 % (31621)Memory used [KB]: 41321
% 49.89/13.87 % (31621)Time elapsed: 0.854 s
% 49.89/13.87 % (31621)Instructions burned: 1763 (million)
% 49.89/13.87 % (31621)------------------------------
% 49.89/13.87 % (31621)------------------------------
% 49.89/13.89 % (31631)Instruction limit reached!
% 49.89/13.89 % (31631)------------------------------
% 49.89/13.89 % (31631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.89/13.89 % (31631)Termination reason: Unknown
% 49.89/13.89 % (31631)Termination phase: shuffling
% 49.89/13.89
% 49.89/13.89 % (31631)Memory used [KB]: 21748
% 49.89/13.89 % (31631)Time elapsed: 0.204 s
% 49.89/13.89 % (31631)Instructions burned: 584 (million)
% 49.89/13.89 % (31631)------------------------------
% 49.89/13.89 % (31631)------------------------------
% 49.89/13.89 % (31634)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_31377 for (2938ds/1335Mi)
% 50.39/13.90 % (31635)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_31377 for (2938ds/3373Mi)
% 50.39/13.93 % (31633)Instruction limit reached!
% 50.39/13.93 % (31633)------------------------------
% 50.39/13.93 % (31633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 50.39/13.93 % (31633)Termination reason: Unknown
% 50.39/13.93 % (31633)Termination phase: shuffling
% 50.39/13.93
% 50.39/13.93 % (31633)Memory used [KB]: 21236
% 50.39/13.93 % (31633)Time elapsed: 0.198 s
% 50.39/13.93 % (31633)Instructions burned: 394 (million)
% 50.39/13.93 % (31633)------------------------------
% 50.39/13.93 % (31633)------------------------------
% 50.39/13.95 % (31636)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_31377 for (2938ds/474Mi)
% 50.39/13.95 % (31626)Instruction limit reached!
% 50.39/13.95 % (31626)------------------------------
% 50.39/13.95 % (31626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 50.39/13.95 % (31626)Termination reason: Unknown
% 50.39/13.95 % (31626)Termination phase: Property scanning
% 50.39/13.95
% 50.39/13.95 % (31626)Memory used [KB]: 29040
% 50.39/13.95 % (31626)Time elapsed: 0.757 s
% 50.39/13.95 % (31626)Instructions burned: 1512 (million)
% 50.39/13.95 % (31626)------------------------------
% 50.39/13.95 % (31626)------------------------------
% 50.39/13.96 % (31632)Instruction limit reached!
% 50.39/13.96 % (31632)------------------------------
% 50.39/13.96 % (31632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 50.39/13.96 % (31632)Termination reason: Unknown
% 50.39/13.96 % (31632)Termination phase: shuffling
% 50.39/13.96
% 50.39/13.96 % (31632)Memory used [KB]: 19957
% 50.39/13.96 % (31632)Time elapsed: 0.245 s
% 50.39/13.96 % (31632)Instructions burned: 457 (million)
% 50.39/13.96 % (31632)------------------------------
% 50.39/13.96 % (31632)------------------------------
% 50.39/13.98 % (31637)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_31377 for (2937ds/157Mi)
% 50.39/13.98 % (31638)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_31377 for (2937ds/590Mi)
% 52.30/14.07 % (31637)Instruction limit reached!
% 52.30/14.07 % (31637)------------------------------
% 52.30/14.07 % (31637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.30/14.07 % (31637)Termination reason: Unknown
% 52.30/14.07 % (31637)Termination phase: shuffling
% 52.30/14.07
% 52.30/14.07 % (31637)Memory used [KB]: 15735
% 52.30/14.07 % (31637)Time elapsed: 0.073 s
% 52.30/14.07 % (31637)Instructions burned: 158 (million)
% 52.30/14.07 % (31637)------------------------------
% 52.30/14.07 % (31637)------------------------------
% 52.30/14.09 % (31639)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_31377 for (2936ds/104Mi)
% 52.98/14.15 % (31639)Instruction limit reached!
% 52.98/14.15 % (31639)------------------------------
% 52.98/14.15 % (31639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.98/14.15 % (31639)Termination reason: Unknown
% 52.98/14.15 % (31639)Termination phase: shuffling
% 52.98/14.15
% 52.98/14.15 % (31639)Memory used [KB]: 14967
% 52.98/14.15 % (31639)Time elapsed: 0.063 s
% 52.98/14.15 % (31639)Instructions burned: 104 (million)
% 52.98/14.15 % (31639)------------------------------
% 52.98/14.15 % (31639)------------------------------
% 52.98/14.17 % (31640)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_31377 for (2935ds/414Mi)
% 52.98/14.18 % (31636)Instruction limit reached!
% 52.98/14.18 % (31636)------------------------------
% 52.98/14.18 % (31636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 52.98/14.18 % (31636)Termination reason: Unknown
% 52.98/14.18 % (31636)Termination phase: shuffling
% 52.98/14.18
% 52.98/14.18 % (31636)Memory used [KB]: 22259
% 52.98/14.18 % (31636)Time elapsed: 0.235 s
% 52.98/14.18 % (31636)Instructions burned: 474 (million)
% 52.98/14.18 % (31636)------------------------------
% 52.98/14.18 % (31636)------------------------------
% 53.48/14.20 % (31641)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_31377 for (2935ds/715Mi)
% 53.48/14.29 % (31638)Instruction limit reached!
% 53.48/14.29 % (31638)------------------------------
% 53.48/14.29 % (31638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.48/14.29 % (31638)Termination reason: Unknown
% 53.48/14.29 % (31638)Termination phase: shuffling
% 53.48/14.29
% 53.48/14.29 % (31638)Memory used [KB]: 21620
% 53.48/14.29 % (31638)Time elapsed: 0.314 s
% 53.48/14.29 % (31638)Instructions burned: 590 (million)
% 53.48/14.29 % (31638)------------------------------
% 53.48/14.29 % (31638)------------------------------
% 53.97/14.31 % (31643)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_31377 for (2934ds/961Mi)
% 53.97/14.38 % (31640)Instruction limit reached!
% 53.97/14.38 % (31640)------------------------------
% 53.97/14.38 % (31640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.97/14.38 % (31640)Termination reason: Unknown
% 53.97/14.38 % (31640)Termination phase: shuffling
% 53.97/14.38
% 53.97/14.38 % (31640)Memory used [KB]: 21492
% 53.97/14.38 % (31640)Time elapsed: 0.207 s
% 53.97/14.38 % (31640)Instructions burned: 414 (million)
% 53.97/14.38 % (31640)------------------------------
% 53.97/14.38 % (31640)------------------------------
% 54.59/14.40 % (31644)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_31377 for (2933ds/1213Mi)
% 54.99/14.54 % (31634)Instruction limit reached!
% 54.99/14.54 % (31634)------------------------------
% 54.99/14.54 % (31634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.99/14.54 % (31634)Termination reason: Unknown
% 54.99/14.54 % (31634)Termination phase: shuffling
% 54.99/14.54
% 54.99/14.54 % (31634)Memory used [KB]: 37867
% 54.99/14.54 % (31634)Time elapsed: 0.651 s
% 54.99/14.54 % (31634)Instructions burned: 1335 (million)
% 54.99/14.54 % (31634)------------------------------
% 54.99/14.54 % (31634)------------------------------
% 54.99/14.55 % (31641)Instruction limit reached!
% 54.99/14.55 % (31641)------------------------------
% 54.99/14.55 % (31641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.99/14.55 % (31641)Termination reason: Unknown
% 54.99/14.55 % (31641)Termination phase: shuffling
% 54.99/14.55
% 54.99/14.55 % (31641)Memory used [KB]: 25330
% 54.99/14.55 % (31641)Time elapsed: 0.354 s
% 54.99/14.55 % (31641)Instructions burned: 715 (million)
% 54.99/14.55 % (31641)------------------------------
% 54.99/14.55 % (31641)------------------------------
% 54.99/14.57 % (31645)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_31377 for (2932ds/1979Mi)
% 54.99/14.57 % (31646)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_31377 for (2931ds/481Mi)
% 56.02/14.72 % (31620)Refutation not found, incomplete strategy
% 56.02/14.72 % (31620)------------------------------
% 56.02/14.72 % (31620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.02/14.72 % (31620)Termination reason: Refutation not found, incomplete strategy
% 56.02/14.72
% 56.02/14.72
% 56.02/14.72 % (31620)Memory used [KB]: 35820
% 56.02/14.72 % (31620)Time elapsed: 1.736 s
% 56.02/14.72 % (31620)Instructions burned: 3710 (million)
% 56.02/14.72 % (31620)------------------------------
% 56.02/14.72 % (31620)------------------------------
% 56.02/14.73 % (31647)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_31377 for (2930ds/783Mi)
% 56.02/14.79 % (31646)Instruction limit reached!
% 56.02/14.79 % (31646)------------------------------
% 56.02/14.79 % (31646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.02/14.79 % (31646)Termination reason: Unknown
% 56.02/14.79 % (31646)Termination phase: shuffling
% 56.02/14.79
% 56.02/14.79 % (31646)Memory used [KB]: 22387
% 56.02/14.79 % (31646)Time elapsed: 0.222 s
% 56.02/14.79 % (31646)Instructions burned: 482 (million)
% 56.02/14.79 % (31646)------------------------------
% 56.02/14.79 % (31646)------------------------------
% 56.50/14.81 % (31643)Instruction limit reached!
% 56.50/14.81 % (31643)------------------------------
% 56.50/14.81 % (31643)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.50/14.81 % (31643)Termination reason: Unknown
% 56.50/14.81 % (31643)Termination phase: Property scanning
% 56.50/14.81
% 56.50/14.81 % (31643)Memory used [KB]: 26481
% 56.50/14.81 % (31643)Time elapsed: 0.500 s
% 56.50/14.81 % (31643)Instructions burned: 962 (million)
% 56.50/14.81 % (31643)------------------------------
% 56.50/14.81 % (31643)------------------------------
% 56.50/14.81 % (31648)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_31377 for (2929ds/432Mi)
% 56.50/14.84 % (31649)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_31377 for (2929ds/1670Mi)
% 57.21/14.97 % (31635)Instruction limit reached!
% 57.21/14.97 % (31635)------------------------------
% 57.21/14.97 % (31635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.21/14.97 % (31635)Termination reason: Unknown
% 57.21/14.97 % (31635)Termination phase: Function definition elimination
% 57.21/14.97
% 57.21/14.97 % (31635)Memory used [KB]: 39146
% 57.21/14.97 % (31635)Time elapsed: 1.037 s
% 57.21/14.97 % (31635)Instructions burned: 3375 (million)
% 57.21/14.97 % (31635)------------------------------
% 57.21/14.97 % (31635)------------------------------
% 57.21/15.01 % (31650)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_31377 for (2927ds/90Mi)
% 57.21/15.01 % (31648)Instruction limit reached!
% 57.21/15.01 % (31648)------------------------------
% 57.21/15.01 % (31648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.21/15.01 % (31648)Termination reason: Unknown
% 57.21/15.01 % (31648)Termination phase: shuffling
% 57.21/15.01
% 57.21/15.01 % (31648)Memory used [KB]: 19573
% 57.21/15.01 % (31648)Time elapsed: 0.206 s
% 57.21/15.01 % (31648)Instructions burned: 433 (million)
% 57.21/15.01 % (31648)------------------------------
% 57.21/15.01 % (31648)------------------------------
% 57.21/15.02 % (31644)Instruction limit reached!
% 57.21/15.02 % (31644)------------------------------
% 57.21/15.02 % (31644)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.21/15.02 % (31644)Termination reason: Unknown
% 57.21/15.02 % (31644)Termination phase: Property scanning
% 57.21/15.02
% 57.21/15.02 % (31644)Memory used [KB]: 26609
% 57.21/15.02 % (31644)Time elapsed: 0.623 s
% 57.21/15.02 % (31644)Instructions burned: 1215 (million)
% 57.21/15.02 % (31644)------------------------------
% 57.21/15.02 % (31644)------------------------------
% 59.21/15.03 % (31651)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_31377 for (2927ds/346Mi)
% 59.21/15.04 % (31647)Instruction limit reached!
% 59.21/15.04 % (31647)------------------------------
% 59.21/15.04 % (31647)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.21/15.04 % (31647)Termination reason: Unknown
% 59.21/15.04 % (31647)Termination phase: shuffling
% 59.21/15.04
% 59.21/15.04 % (31647)Memory used [KB]: 25841
% 59.21/15.04 % (31647)Time elapsed: 0.295 s
% 59.21/15.04 % (31647)Instructions burned: 784 (million)
% 59.21/15.04 % (31647)------------------------------
% 59.21/15.04 % (31647)------------------------------
% 59.21/15.04 % (31652)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_31377 for (2927ds/401Mi)
% 59.21/15.05 % (31650)Instruction limit reached!
% 59.21/15.05 % (31650)------------------------------
% 59.21/15.05 % (31650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.21/15.05 % (31650)Termination reason: Unknown
% 59.21/15.05 % (31650)Termination phase: shuffling
% 59.21/15.05
% 59.21/15.05 % (31650)Memory used [KB]: 14839
% 59.21/15.05 % (31650)Time elapsed: 0.043 s
% 59.21/15.05 % (31650)Instructions burned: 91 (million)
% 59.21/15.05 % (31650)------------------------------
% 59.21/15.05 % (31650)------------------------------
% 59.21/15.06 % (31653)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_31377 for (2926ds/6454Mi)
% 59.21/15.07 % (31654)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_31377 for (2926ds/283Mi)
% 59.80/15.20 % (31654)Instruction limit reached!
% 59.80/15.20 % (31654)------------------------------
% 59.80/15.20 % (31654)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.80/15.20 % (31654)Termination reason: Unknown
% 59.80/15.20 % (31654)Termination phase: shuffling
% 59.80/15.20
% 59.80/15.20 % (31654)Memory used [KB]: 17526
% 59.80/15.20 % (31654)Time elapsed: 0.129 s
% 59.80/15.20 % (31654)Instructions burned: 285 (million)
% 59.80/15.20 % (31654)------------------------------
% 59.80/15.20 % (31654)------------------------------
% 59.80/15.21 % (31655)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_31377 for (2925ds/865Mi)
% 59.80/15.21 % (31651)Instruction limit reached!
% 59.80/15.21 % (31651)------------------------------
% 59.80/15.21 % (31651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.80/15.21 % (31651)Termination reason: Unknown
% 59.80/15.21 % (31651)Termination phase: shuffling
% 59.80/15.21
% 59.80/15.21 % (31651)Memory used [KB]: 20468
% 59.80/15.21 % (31651)Time elapsed: 0.168 s
% 59.80/15.21 % (31651)Instructions burned: 347 (million)
% 59.80/15.21 % (31651)------------------------------
% 59.80/15.21 % (31651)------------------------------
% 59.80/15.22 % (31652)Instruction limit reached!
% 59.80/15.22 % (31652)------------------------------
% 59.80/15.22 % (31652)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 59.80/15.22 % (31652)Termination reason: Unknown
% 59.80/15.22 % (31652)Termination phase: shuffling
% 59.80/15.22
% 59.80/15.22 % (31652)Memory used [KB]: 21236
% 59.80/15.22 % (31652)Time elapsed: 0.161 s
% 59.80/15.22 % (31652)Instructions burned: 402 (million)
% 59.80/15.22 % (31652)------------------------------
% 59.80/15.22 % (31652)------------------------------
% 59.80/15.23 % (31656)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_31377 for (2925ds/4528Mi)
% 59.80/15.24 % (31657)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_31377 for (2925ds/1001Mi)
% 60.60/15.37 % (31645)Instruction limit reached!
% 60.60/15.37 % (31645)------------------------------
% 60.60/15.37 % (31645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.60/15.37 % (31645)Termination reason: Unknown
% 60.60/15.37 % (31645)Termination phase: Property scanning
% 60.60/15.37
% 60.60/15.37 % (31645)Memory used [KB]: 37099
% 60.60/15.37 % (31645)Time elapsed: 0.808 s
% 60.60/15.37 % (31645)Instructions burned: 1980 (million)
% 60.60/15.37 % (31645)------------------------------
% 60.60/15.37 % (31645)------------------------------
% 60.60/15.39 % (31658)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_31377 for (2923ds/102Mi)
% 60.91/15.40 % (31649)Instruction limit reached!
% 60.91/15.40 % (31649)------------------------------
% 60.91/15.40 % (31649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.91/15.40 % (31649)Termination reason: Unknown
% 60.91/15.40 % (31649)Termination phase: Property scanning
% 60.91/15.40
% 60.91/15.40 % (31649)Memory used [KB]: 41321
% 60.91/15.40 % (31649)Time elapsed: 0.555 s
% 60.91/15.40 % (31649)Instructions burned: 1670 (million)
% 60.91/15.40 % (31649)------------------------------
% 60.91/15.40 % (31649)------------------------------
% 60.91/15.41 % (31659)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_31377 for (2923ds/952Mi)
% 60.91/15.42 % (31658)Instruction limit reached!
% 60.91/15.42 % (31658)------------------------------
% 60.91/15.42 % (31658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.91/15.42 % (31658)Termination reason: Unknown
% 60.91/15.42 % (31658)Termination phase: shuffling
% 60.91/15.42
% 60.91/15.42 % (31658)Memory used [KB]: 15095
% 60.91/15.42 % (31658)Time elapsed: 0.038 s
% 60.91/15.42 % (31658)Instructions burned: 103 (million)
% 60.91/15.42 % (31658)------------------------------
% 60.91/15.42 % (31658)------------------------------
% 60.91/15.43 % (31660)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_31377 for (2923ds/578Mi)
% 61.16/15.61 % (31660)Instruction limit reached!
% 61.16/15.61 % (31660)------------------------------
% 61.16/15.61 % (31660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.16/15.61 % (31660)Termination reason: Unknown
% 61.16/15.61 % (31660)Termination phase: shuffling
% 61.16/15.61
% 61.16/15.61 % (31660)Memory used [KB]: 23794
% 61.16/15.61 % (31660)Time elapsed: 0.173 s
% 61.16/15.61 % (31660)Instructions burned: 580 (million)
% 61.16/15.61 % (31660)------------------------------
% 61.16/15.61 % (31660)------------------------------
% 61.16/15.62 % (31661)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_31377 for (2921ds/643Mi)
% 61.16/15.64 % (31607)Instruction limit reached!
% 61.16/15.64 % (31607)------------------------------
% 61.16/15.64 % (31607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.16/15.64 % (31607)Termination reason: Unknown
% 61.16/15.64 % (31607)Termination phase: Saturation
% 61.16/15.64
% 61.16/15.64 % (31607)Memory used [KB]: 58208
% 61.16/15.64 % (31607)Time elapsed: 3.989 s
% 61.16/15.64 % (31607)Instructions burned: 8250 (million)
% 61.16/15.64 % (31607)------------------------------
% 61.16/15.64 % (31607)------------------------------
% 61.16/15.65 % (31655)Instruction limit reached!
% 61.16/15.65 % (31655)------------------------------
% 61.16/15.65 % (31655)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.16/15.65 % (31655)Termination reason: Unknown
% 61.16/15.65 % (31655)Termination phase: shuffling
% 61.16/15.65
% 61.16/15.65 % (31655)Memory used [KB]: 26737
% 61.16/15.65 % (31655)Time elapsed: 0.440 s
% 61.16/15.65 % (31655)Instructions burned: 865 (million)
% 61.16/15.65 % (31655)------------------------------
% 61.16/15.65 % (31655)------------------------------
% 61.16/15.65 % (31662)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_31377 for (2920ds/865Mi)
% 61.16/15.68 % (31663)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_31377 for (2920ds/1174Mi)
% 61.84/15.72 % (31659)Instruction limit reached!
% 61.84/15.72 % (31659)------------------------------
% 61.84/15.72 % (31659)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.84/15.72 % (31659)Termination reason: Unknown
% 61.84/15.72 % (31659)Termination phase: Property scanning
% 61.84/15.72
% 61.84/15.72 % (31659)Memory used [KB]: 26481
% 61.84/15.72 % (31659)Time elapsed: 0.313 s
% 61.84/15.72 % (31659)Instructions burned: 954 (million)
% 61.84/15.72 % (31659)------------------------------
% 61.84/15.72 % (31659)------------------------------
% 61.84/15.73 % (31664)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_31377 for (2920ds/644Mi)
% 61.84/15.75 % (31657)Instruction limit reached!
% 61.84/15.75 % (31657)------------------------------
% 61.84/15.75 % (31657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.84/15.75 % (31657)Termination reason: Unknown
% 61.84/15.75 % (31657)Termination phase: Property scanning
% 61.84/15.75
% 61.84/15.75 % (31657)Memory used [KB]: 26481
% 61.84/15.75 % (31657)Time elapsed: 0.519 s
% 61.84/15.75 % (31657)Instructions burned: 1002 (million)
% 61.84/15.75 % (31657)------------------------------
% 61.84/15.75 % (31657)------------------------------
% 61.84/15.77 % (31665)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_31377 for (2919ds/1201Mi)
% 62.70/15.81 % (31661)Instruction limit reached!
% 62.70/15.81 % (31661)------------------------------
% 62.70/15.81 % (31661)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.70/15.81 % (31661)Termination reason: Unknown
% 62.70/15.81 % (31661)Termination phase: shuffling
% 62.70/15.81
% 62.70/15.81 % (31661)Memory used [KB]: 24562
% 62.70/15.81 % (31661)Time elapsed: 0.189 s
% 62.70/15.81 % (31661)Instructions burned: 644 (million)
% 62.70/15.81 % (31661)------------------------------
% 62.70/15.81 % (31661)------------------------------
% 62.70/15.82 % (31666)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_31377 for (2919ds/1046Mi)
% 62.90/15.95 % (31664)Instruction limit reached!
% 62.90/15.95 % (31664)------------------------------
% 62.90/15.95 % (31664)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.90/15.95 % (31664)Termination reason: Unknown
% 62.90/15.95 % (31664)Termination phase: shuffling
% 62.90/15.95
% 62.90/15.95 % (31664)Memory used [KB]: 23922
% 62.90/15.95 % (31664)Time elapsed: 0.218 s
% 62.90/15.95 % (31664)Instructions burned: 644 (million)
% 62.90/15.95 % (31664)------------------------------
% 62.90/15.95 % (31664)------------------------------
% 62.90/15.96 % (31662)Instruction limit reached!
% 62.90/15.96 % (31662)------------------------------
% 62.90/15.96 % (31662)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.90/15.96 % (31662)Termination reason: Unknown
% 62.90/15.96 % (31662)Termination phase: shuffling
% 62.90/15.96
% 62.90/15.96 % (31662)Memory used [KB]: 26865
% 62.90/15.96 % (31662)Time elapsed: 0.295 s
% 62.90/15.96 % (31662)Instructions burned: 867 (million)
% 62.90/15.96 % (31662)------------------------------
% 62.90/15.96 % (31662)------------------------------
% 62.90/15.96 % (31667)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_31377 for (2917ds/222Mi)
% 62.90/15.97 % (31668)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_31377 for (2917ds/68Mi)
% 62.90/16.01 % (31668)Instruction limit reached!
% 62.90/16.01 % (31668)------------------------------
% 62.90/16.01 % (31668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.02/16.02 % (31668)Termination reason: Unknown
% 66.02/16.02 % (31668)Termination phase: shuffling
% 66.02/16.02
% 66.02/16.02 % (31668)Memory used [KB]: 14583
% 66.02/16.02 % (31668)Time elapsed: 0.033 s
% 66.02/16.02 % (31668)Instructions burned: 71 (million)
% 66.02/16.02 % (31668)------------------------------
% 66.02/16.02 % (31668)------------------------------
% 66.02/16.03 % (31669)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_31377 for (2917ds/454Mi)
% 66.02/16.03 % (31667)Instruction limit reached!
% 66.02/16.03 % (31667)------------------------------
% 66.02/16.03 % (31667)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.02/16.03 % (31667)Termination reason: Unknown
% 66.02/16.03 % (31667)Termination phase: shuffling
% 66.02/16.03
% 66.02/16.03 % (31667)Memory used [KB]: 16758
% 66.02/16.03 % (31667)Time elapsed: 0.071 s
% 66.02/16.03 % (31667)Instructions burned: 224 (million)
% 66.02/16.03 % (31667)------------------------------
% 66.02/16.03 % (31667)------------------------------
% 66.02/16.04 % (31670)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_31377 for (2917ds/360Mi)
% 66.57/16.12 % (31666)Instruction limit reached!
% 66.57/16.12 % (31666)------------------------------
% 66.57/16.12 % (31666)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.57/16.12 % (31666)Termination reason: Unknown
% 66.57/16.12 % (31666)Termination phase: shuffling
% 66.57/16.12
% 66.57/16.12 % (31666)Memory used [KB]: 34157
% 66.57/16.12 % (31666)Time elapsed: 0.306 s
% 66.57/16.12 % (31666)Instructions burned: 1048 (million)
% 66.57/16.12 % (31666)------------------------------
% 66.57/16.12 % (31666)------------------------------
% 66.57/16.14 % (31671)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_31377 for (2916ds/157Mi)
% 66.57/16.15 % (31656)First to succeed.
% 66.57/16.15 % (31670)Instruction limit reached!
% 66.57/16.15 % (31670)------------------------------
% 66.57/16.15 % (31670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.57/16.15 % (31670)Termination reason: Unknown
% 66.57/16.15 % (31670)Termination phase: shuffling
% 66.57/16.15
% 66.57/16.15 % (31670)Memory used [KB]: 20724
% 66.57/16.15 % (31670)Time elapsed: 0.110 s
% 66.57/16.15 % (31670)Instructions burned: 360 (million)
% 66.57/16.15 % (31670)------------------------------
% 66.57/16.15 % (31670)------------------------------
% 66.57/16.16 % (31656)Refutation found. Thanks to Tanya!
% 66.57/16.16 % SZS status Theorem for DTF2THF_31377
% 66.57/16.16 % SZS output start Proof for DTF2THF_31377
% 66.57/16.16 thf(type_def_5, type, real: $tType).
% 66.57/16.16 thf(type_def_6, type, int: $tType).
% 66.57/16.16 thf(type_def_8, type, rat: $tType).
% 66.57/16.16 thf(type_def_9, type, nat: $tType).
% 66.57/16.16 thf(type_def_10, type, num: $tType).
% 66.57/16.16 thf(type_def_11, type, product_prod_int_int: $tType).
% 66.57/16.16 thf(type_def_12, type, product_prod_nat_nat: $tType).
% 66.57/16.16 thf(type_def_13, type, option_num: $tType).
% 66.57/16.16 thf(type_def_14, type, product_prod_num_num: $tType).
% 66.57/16.16 thf(type_def_15, type, code_integer: $tType).
% 66.57/16.16 thf(type_def_16, type, produc6271795597528267376eger_o: $tType).
% 66.57/16.16 thf(type_def_17, type, produc8923325533196201883nteger: $tType).
% 66.57/16.16 thf(type_def_18, type, set_fi4554929511873752355omplex: $tType).
% 66.57/16.16 thf(type_def_19, type, filter6041513312241820739omplex: $tType).
% 66.57/16.16 thf(type_def_20, type, set_fi7789364187291644575l_real: $tType).
% 66.57/16.16 thf(type_def_21, type, filter2146258269922977983l_real: $tType).
% 66.57/16.16 thf(type_def_22, type, set_real: $tType).
% 66.57/16.16 thf(type_def_23, type, set_int: $tType).
% 66.57/16.16 thf(type_def_24, type, complex: $tType).
% 66.57/16.16 thf(type_def_25, type, filter_real: $tType).
% 66.57/16.16 thf(type_def_26, type, extended_enat: $tType).
% 66.57/16.16 thf(type_def_27, type, filter_nat: $tType).
% 66.57/16.16 thf(type_def_28, type, set_Pr5085853215250843933omplex: $tType).
% 66.57/16.16 thf(type_def_29, type, set_Pr6218003697084177305l_real: $tType).
% 66.57/16.16 thf(type_def_30, type, set_o: $tType).
% 66.57/16.16 thf(type_def_31, type, set_complex: $tType).
% 66.57/16.16 thf(type_def_32, type, set_list_nat: $tType).
% 66.57/16.16 thf(type_def_33, type, set_nat: $tType).
% 66.57/16.16 thf(type_def_34, type, set_Product_unit: $tType).
% 66.57/16.16 thf(type_def_35, type, set_char: $tType).
% 66.57/16.16 thf(type_def_36, type, set_list_o: $tType).
% 66.57/16.16 thf(type_def_37, type, set_list_complex: $tType).
% 66.57/16.16 thf(type_def_38, type, set_list_int: $tType).
% 66.57/16.16 thf(type_def_39, type, set_list_VEBT_VEBT: $tType).
% 66.57/16.16 thf(type_def_40, type, set_num: $tType).
% 66.57/16.16 thf(type_def_41, type, set_rat: $tType).
% 66.57/16.16 thf(type_def_42, type, set_set_complex: $tType).
% 66.57/16.16 thf(type_def_43, type, set_set_int: $tType).
% 66.57/16.16 thf(type_def_44, type, set_set_nat: $tType).
% 66.57/16.16 thf(type_def_45, type, set_VEBT_VEBT: $tType).
% 66.57/16.16 thf(type_def_46, type, char: $tType).
% 66.57/16.16 thf(type_def_47, type, set_Pr8693737435421807431at_nat: $tType).
% 66.57/16.16 thf(type_def_48, type, list_nat: $tType).
% 66.57/16.16 thf(type_def_49, type, produc859450856879609959at_nat: $tType).
% 66.57/16.16 thf(type_def_50, type, list_o: $tType).
% 66.57/16.16 thf(type_def_51, type, list_int: $tType).
% 66.57/16.16 thf(type_def_52, type, option_nat: $tType).
% 66.57/16.16 thf(type_def_53, type, vEBT_VEBT: $tType).
% 66.57/16.16 thf(type_def_54, type, set_Pr958786334691620121nt_int: $tType).
% 66.57/16.16 thf(type_def_55, type, list_complex: $tType).
% 66.57/16.16 thf(type_def_56, type, list_list_nat: $tType).
% 66.57/16.16 thf(type_def_57, type, list_P8469869581646625389at_nat: $tType).
% 66.57/16.16 thf(type_def_58, type, list_real: $tType).
% 66.57/16.16 thf(type_def_59, type, list_VEBT_VEBT: $tType).
% 66.57/16.16 thf(type_def_60, type, list_num: $tType).
% 66.57/16.16 thf(type_def_61, type, list_P4002435161011370285od_o_o: $tType).
% 66.57/16.16 thf(type_def_62, type, product_prod_o_o: $tType).
% 66.57/16.16 thf(type_def_63, type, list_P3795440434834930179_o_int: $tType).
% 66.57/16.16 thf(type_def_64, type, product_prod_o_int: $tType).
% 66.57/16.16 thf(type_def_65, type, list_P6285523579766656935_o_nat: $tType).
% 66.57/16.16 thf(type_def_66, type, product_prod_o_nat: $tType).
% 66.57/16.16 thf(type_def_67, type, list_P7495141550334521929T_VEBT: $tType).
% 66.57/16.16 thf(type_def_68, type, produc2504756804600209347T_VEBT: $tType).
% 66.57/16.16 thf(type_def_69, type, list_P1726324292696863441at_num: $tType).
% 66.57/16.16 thf(type_def_70, type, product_prod_nat_num: $tType).
% 66.57/16.16 thf(type_def_71, type, list_P3744719386663036955um_num: $tType).
% 66.57/16.16 thf(type_def_72, type, list_P3126845725202233233VEBT_o: $tType).
% 66.57/16.16 thf(type_def_73, type, produc334124729049499915VEBT_o: $tType).
% 66.57/16.16 thf(type_def_74, type, list_P4547456442757143711BT_int: $tType).
% 66.57/16.16 thf(type_def_75, type, produc4894624898956917775BT_int: $tType).
% 66.57/16.16 thf(type_def_76, type, list_P7037539587688870467BT_nat: $tType).
% 66.57/16.16 thf(type_def_77, type, produc9072475918466114483BT_nat: $tType).
% 66.57/16.16 thf(type_def_78, type, list_P7413028617227757229T_VEBT: $tType).
% 66.57/16.16 thf(type_def_79, type, produc8243902056947475879T_VEBT: $tType).
% 66.57/16.16 thf(type_def_80, type, list_P7333126701944960589_nat_o: $tType).
% 66.57/16.16 thf(type_def_81, type, list_P5647936690300460905T_VEBT: $tType).
% 66.57/16.16 thf(type_def_82, type, option4927543243414619207at_nat: $tType).
% 66.57/16.16 thf(type_def_83, type, set_Code_integer: $tType).
% 66.57/16.16 thf(type_def_84, type, produc4953844613479565601on_nat: $tType).
% 66.57/16.16 thf(type_def_85, type, produc2233624965454879586on_nat: $tType).
% 66.57/16.16 thf(type_def_86, type, produc8306885398267862888on_nat: $tType).
% 66.57/16.16 thf(type_def_87, type, produc3447558737645232053on_num: $tType).
% 66.57/16.16 thf(type_def_88, type, produc7036089656553540234on_num: $tType).
% 66.57/16.16 thf(type_def_89, type, produc1193250871479095198on_num: $tType).
% 66.57/16.16 thf(type_def_90, type, produc6121120109295599847at_nat: $tType).
% 66.57/16.16 thf(type_def_91, type, produc5491161045314408544at_nat: $tType).
% 66.57/16.16 thf(type_def_92, type, produc5542196010084753463at_nat: $tType).
% 66.57/16.16 thf(type_def_93, type, produc4411394909380815293omplex: $tType).
% 66.57/16.16 thf(type_def_94, type, produc2422161461964618553l_real: $tType).
% 66.57/16.16 thf(func_def_0, type, produc5542196010084753463at_nat: $tType).
% 66.57/16.16 thf(func_def_1, type, produc5491161045314408544at_nat: $tType).
% 66.57/16.16 thf(func_def_2, type, produc1193250871479095198on_num: $tType).
% 66.57/16.16 thf(func_def_3, type, produc8306885398267862888on_nat: $tType).
% 66.57/16.16 thf(func_def_4, type, produc6121120109295599847at_nat: $tType).
% 66.57/16.16 thf(func_def_5, type, produc7036089656553540234on_num: $tType).
% 66.57/16.16 thf(func_def_6, type, produc2233624965454879586on_nat: $tType).
% 66.57/16.16 thf(func_def_7, type, list_P8469869581646625389at_nat: $tType).
% 66.57/16.16 thf(func_def_8, type, set_Pr8693737435421807431at_nat: $tType).
% 66.57/16.16 thf(func_def_9, type, produc859450856879609959at_nat: $tType).
% 66.57/16.16 thf(func_def_10, type, set_fi4554929511873752355omplex: $tType).
% 66.57/16.16 thf(func_def_11, type, list_P7413028617227757229T_VEBT: $tType).
% 66.57/16.16 thf(func_def_12, type, produc3447558737645232053on_num: $tType).
% 66.57/16.16 thf(func_def_13, type, produc4953844613479565601on_nat: $tType).
% 66.57/16.16 thf(func_def_14, type, set_fi7789364187291644575l_real: $tType).
% 66.57/16.16 thf(func_def_15, type, filter6041513312241820739omplex: $tType).
% 66.57/16.16 thf(func_def_16, type, list_P7037539587688870467BT_nat: $tType).
% 66.57/16.16 thf(func_def_17, type, list_P4547456442757143711BT_int: $tType).
% 66.57/16.16 thf(func_def_18, type, list_P5647936690300460905T_VEBT: $tType).
% 66.57/16.16 thf(func_def_19, type, produc8243902056947475879T_VEBT: $tType).
% 66.57/16.16 thf(func_def_20, type, set_Pr5085853215250843933omplex: $tType).
% 66.57/16.16 thf(func_def_21, type, produc8923325533196201883nteger: $tType).
% 66.57/16.16 thf(func_def_22, type, list_P3126845725202233233VEBT_o: $tType).
% 66.57/16.16 thf(func_def_23, type, list_P7495141550334521929T_VEBT: $tType).
% 66.57/16.16 thf(func_def_24, type, filter2146258269922977983l_real: $tType).
% 66.57/16.16 thf(func_def_25, type, option4927543243414619207at_nat: $tType).
% 66.57/16.16 thf(func_def_26, type, set_Pr6218003697084177305l_real: $tType).
% 66.57/16.16 thf(func_def_27, type, list_P3744719386663036955um_num: $tType).
% 66.57/16.16 thf(func_def_28, type, list_P1726324292696863441at_num: $tType).
% 66.57/16.16 thf(func_def_29, type, produc9072475918466114483BT_nat: $tType).
% 66.57/16.16 thf(func_def_30, type, produc4894624898956917775BT_int: $tType).
% 66.57/16.16 thf(func_def_31, type, set_Pr958786334691620121nt_int: $tType).
% 66.57/16.16 thf(func_def_32, type, produc4411394909380815293omplex: $tType).
% 66.57/16.16 thf(func_def_33, type, list_P7333126701944960589_nat_o: $tType).
% 66.57/16.16 thf(func_def_34, type, list_P6285523579766656935_o_nat: $tType).
% 66.57/16.16 thf(func_def_35, type, list_P3795440434834930179_o_int: $tType).
% 66.57/16.16 thf(func_def_36, type, set_list_VEBT_VEBT: $tType).
% 66.57/16.16 thf(func_def_37, type, produc334124729049499915VEBT_o: $tType).
% 66.57/16.16 thf(func_def_38, type, produc2504756804600209347T_VEBT: $tType).
% 66.57/16.16 thf(func_def_39, type, produc6271795597528267376eger_o: $tType).
% 66.57/16.16 thf(func_def_40, type, produc2422161461964618553l_real: $tType).
% 66.57/16.16 thf(func_def_41, type, product_prod_num_num: $tType).
% 66.57/16.16 thf(func_def_42, type, product_prod_nat_num: $tType).
% 66.57/16.16 thf(func_def_43, type, product_prod_nat_nat: $tType).
% 66.57/16.16 thf(func_def_44, type, product_prod_int_int: $tType).
% 66.57/16.16 thf(func_def_45, type, list_P4002435161011370285od_o_o: $tType).
% 66.57/16.16 thf(func_def_46, type, set_list_complex: $tType).
% 66.57/16.16 thf(func_def_47, type, set_set_complex: $tType).
% 66.57/16.16 thf(func_def_48, type, list_list_nat: $tType).
% 66.57/16.16 thf(func_def_49, type, list_VEBT_VEBT: $tType).
% 66.57/16.16 thf(func_def_50, type, set_list_nat: $tType).
% 66.57/16.16 thf(func_def_51, type, set_list_int: $tType).
% 66.57/16.16 thf(func_def_52, type, product_prod_o_nat: $tType).
% 66.57/16.16 thf(func_def_53, type, product_prod_o_int: $tType).
% 66.57/16.16 thf(func_def_54, type, set_VEBT_VEBT: $tType).
% 66.57/16.16 thf(func_def_55, type, set_set_nat: $tType).
% 66.57/16.16 thf(func_def_56, type, set_set_int: $tType).
% 66.57/16.16 thf(func_def_57, type, set_Code_integer: $tType).
% 66.57/16.16 thf(func_def_58, type, set_Product_unit: $tType).
% 66.57/16.16 thf(func_def_59, type, list_complex: $tType).
% 66.57/16.16 thf(func_def_60, type, set_list_o: $tType).
% 66.57/16.16 thf(func_def_61, type, product_prod_o_o: $tType).
% 66.57/16.16 thf(func_def_62, type, set_complex: $tType).
% 66.57/16.16 thf(func_def_63, type, filter_real: $tType).
% 66.57/16.16 thf(func_def_64, type, option_num: $tType).
% 66.57/16.16 thf(func_def_65, type, option_nat: $tType).
% 66.57/16.16 thf(func_def_66, type, filter_nat: $tType).
% 66.57/16.16 thf(func_def_67, type, set_char: $tType).
% 66.57/16.16 thf(func_def_68, type, list_real: $tType).
% 66.57/16.16 thf(func_def_69, type, set_real: $tType).
% 66.57/16.16 thf(func_def_70, type, list_num: $tType).
% 66.57/16.16 thf(func_def_71, type, list_nat: $tType).
% 66.57/16.16 thf(func_def_72, type, list_int: $tType).
% 66.57/16.16 thf(func_def_73, type, vEBT_VEBT: $tType).
% 66.57/16.16 thf(func_def_74, type, set_rat: $tType).
% 66.57/16.16 thf(func_def_75, type, set_num: $tType).
% 66.57/16.16 thf(func_def_76, type, set_nat: $tType).
% 66.57/16.16 thf(func_def_77, type, set_int: $tType).
% 66.57/16.16 thf(func_def_78, type, code_integer: $tType).
% 66.57/16.16 thf(func_def_79, type, extended_enat: $tType).
% 66.57/16.16 thf(func_def_80, type, list_o: $tType).
% 66.57/16.16 thf(func_def_81, type, complex: $tType).
% 66.57/16.16 thf(func_def_82, type, set_o: $tType).
% 66.57/16.16 thf(func_def_83, type, char: $tType).
% 66.57/16.16 thf(func_def_84, type, real: $tType).
% 66.57/16.16 thf(func_def_85, type, rat: $tType).
% 66.57/16.16 thf(func_def_86, type, num: $tType).
% 66.57/16.16 thf(func_def_87, type, nat: $tType).
% 66.57/16.16 thf(func_def_88, type, int: $tType).
% 66.57/16.16 thf(func_def_89, type, archim7802044766580827645g_real: real > int).
% 66.57/16.16 thf(func_def_90, type, archim3151403230148437115or_rat: rat > int).
% 66.57/16.16 thf(func_def_91, type, archim6058952711729229775r_real: real > int).
% 66.57/16.16 thf(func_def_92, type, archim2898591450579166408c_real: real > real).
% 66.57/16.16 thf(func_def_93, type, archim7778729529865785530nd_rat: rat > int).
% 66.57/16.16 thf(func_def_94, type, archim8280529875227126926d_real: real > int).
% 66.57/16.16 thf(func_def_95, 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).
% 66.57/16.16 thf(func_def_96, type, bNF_re895249473297799549at_rat: ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > (nat > rat) > $o) > ((nat > rat) > nat > rat) > ((nat > rat) > nat > rat) > $o).
% 66.57/16.16 thf(func_def_97, type, bNF_re728719798268516973at_o_o: ((nat > rat) > (nat > rat) > $o) > ($o > $o > $o) > ((nat > rat) > $o) > ((nat > rat) > $o) > $o).
% 66.57/16.16 thf(func_def_98, 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).
% 66.57/16.16 thf(func_def_99, type, bNF_re4521903465945308077real_o: ((nat > rat) > real > $o) > (((nat > rat) > $o) > (real > $o) > $o) > ((nat > rat) > (nat > rat) > $o) > (real > real > $o) > $o).
% 66.57/16.16 thf(func_def_100, type, bNF_re3023117138289059399t_real: ((nat > rat) > real > $o) > ((nat > rat) > real > $o) > ((nat > rat) > nat > rat) > (real > real) > $o).
% 66.57/16.16 thf(func_def_101, type, bNF_re4297313714947099218al_o_o: ((nat > rat) > real > $o) > ($o > $o > $o) > ((nat > rat) > $o) > (real > $o) > $o).
% 66.57/16.16 thf(func_def_102, type, bNF_re3403563459893282935_int_o: (int > int > $o) > ((int > $o) > (int > $o) > $o) > (int > int > $o) > (int > int > $o) > $o).
% 66.57/16.16 thf(func_def_103, type, bNF_re711492959462206631nt_int: (int > int > $o) > ((int > int) > (int > int) > $o) > (int > int > int) > (int > int > int) > $o).
% 66.57/16.16 thf(func_def_104, type, bNF_re5089333283451836215nt_o_o: (int > int > $o) > ($o > $o > $o) > (int > $o) > (int > $o) > $o).
% 66.57/16.16 thf(func_def_105, type, bNF_re4712519889275205905nt_int: (int > int > $o) > (int > int > $o) > (int > int) > (int > int) > $o).
% 66.57/16.16 thf(func_def_106, type, bNF_re578469030762574527_nat_o: (nat > nat > $o) > ((nat > $o) > (nat > $o) > $o) > (nat > nat > $o) > (nat > nat > $o) > $o).
% 66.57/16.16 thf(func_def_107, type, bNF_re1345281282404953727at_nat: (nat > nat > $o) > ((nat > nat) > (nat > nat) > $o) > (nat > nat > nat) > (nat > nat > nat) > $o).
% 66.57/16.16 thf(func_def_108, type, bNF_re4705727531993890431at_o_o: (nat > nat > $o) > ($o > $o > $o) > (nat > $o) > (nat > $o) > $o).
% 66.57/16.16 thf(func_def_109, type, bNF_re5653821019739307937at_nat: (nat > nat > $o) > (nat > nat > $o) > (nat > nat) > (nat > nat) > $o).
% 66.57/16.16 thf(func_def_110, type, bNF_re8402795839162346335um_int: (num > num > $o) > ((num > int) > (num > int) > $o) > (num > num > int) > (num > num > int) > $o).
% 66.57/16.16 thf(func_def_111, type, bNF_re1822329894187522285nt_int: (num > num > $o) > (int > int > $o) > (num > int) > (num > int) > $o).
% 66.57/16.16 thf(func_def_112, 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).
% 66.57/16.16 thf(func_def_113, 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).
% 66.57/16.16 thf(func_def_114, 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).
% 66.57/16.16 thf(func_def_115, 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).
% 66.57/16.16 thf(func_def_116, type, bNF_re1494630372529172596at_o_o: (product_prod_int_int > rat > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (rat > $o) > $o).
% 66.57/16.16 thf(func_def_117, 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).
% 66.57/16.16 thf(func_def_118, 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).
% 66.57/16.16 thf(func_def_119, 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).
% 66.57/16.16 thf(func_def_120, type, bNF_re6644619430987730960nt_o_o: (product_prod_nat_nat > int > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (int > $o) > $o).
% 66.57/16.16 thf(func_def_121, type, bNF_re4555766996558763186at_nat: (product_prod_nat_nat > int > $o) > (nat > nat > $o) > (product_prod_nat_nat > nat) > (int > nat) > $o).
% 66.57/16.16 thf(func_def_122, 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).
% 66.57/16.16 thf(func_def_123, 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).
% 66.57/16.16 thf(func_def_124, 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).
% 66.57/16.16 thf(func_def_125, 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).
% 66.57/16.16 thf(func_def_126, 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).
% 66.57/16.16 thf(func_def_127, 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).
% 66.57/16.16 thf(func_def_128, type, binomial: nat > nat > nat).
% 66.57/16.16 thf(func_def_129, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 66.57/16.16 thf(func_def_130, type, bit_and_not_num: num > num > option_num).
% 66.57/16.16 thf(func_def_131, type, bit_and_not_num_rel: product_prod_num_num > product_prod_num_num > $o).
% 66.57/16.16 thf(func_def_132, type, bit_concat_bit: nat > int > int > int).
% 66.57/16.16 thf(func_def_133, type, bit_or_not_num_neg: num > num > num).
% 66.57/16.16 thf(func_def_134, type, bit_or3848514188828904588eg_rel: product_prod_num_num > product_prod_num_num > $o).
% 66.57/16.16 thf(func_def_135, type, bit_ri7919022796975470100ot_int: int > int).
% 66.57/16.16 thf(func_def_136, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 66.57/16.16 thf(func_def_137, type, bit_ri631733984087533419it_int: nat > int > int).
% 66.57/16.16 thf(func_def_138, type, bit_se725231765392027082nd_int: int > int > int).
% 66.57/16.16 thf(func_def_139, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_140, type, bit_se8568078237143864401it_int: nat > int > int).
% 66.57/16.16 thf(func_def_141, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_142, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 66.57/16.16 thf(func_def_143, type, bit_se2159334234014336723it_int: nat > int > int).
% 66.57/16.16 thf(func_def_144, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_145, type, bit_se2000444600071755411sk_int: nat > int).
% 66.57/16.16 thf(func_def_146, type, bit_se2002935070580805687sk_nat: nat > nat).
% 66.57/16.16 thf(func_def_147, type, bit_se1409905431419307370or_int: int > int > int).
% 66.57/16.16 thf(func_def_148, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_149, type, bit_se545348938243370406it_int: nat > int > int).
% 66.57/16.16 thf(func_def_150, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_151, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 66.57/16.16 thf(func_def_152, type, bit_se7879613467334960850it_int: nat > int > int).
% 66.57/16.16 thf(func_def_153, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_154, type, bit_se1745604003318907178nteger: nat > code_integer > code_integer).
% 66.57/16.16 thf(func_def_155, type, bit_se2923211474154528505it_int: nat > int > int).
% 66.57/16.16 thf(func_def_156, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_157, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 66.57/16.16 thf(func_def_158, type, bit_se4203085406695923979it_int: nat > int > int).
% 66.57/16.16 thf(func_def_159, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_160, type, bit_se6526347334894502574or_int: int > int > int).
% 66.57/16.16 thf(func_def_161, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_162, type, bit_se9216721137139052372nteger: code_integer > nat > $o).
% 66.57/16.16 thf(func_def_163, type, bit_se1146084159140164899it_int: int > nat > $o).
% 66.57/16.16 thf(func_def_164, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 66.57/16.16 thf(func_def_165, type, bit_take_bit_num: nat > num > option_num).
% 66.57/16.16 thf(func_def_166, type, bit_un1837492267222099188nd_num: num > num > option_num).
% 66.57/16.16 thf(func_def_167, type, bit_un5425074673868309765um_rel: product_prod_num_num > product_prod_num_num > $o).
% 66.57/16.16 thf(func_def_168, type, bit_un6178654185764691216or_num: num > num > option_num).
% 66.57/16.16 thf(func_def_169, type, bit_un3595099601533988841um_rel: product_prod_num_num > product_prod_num_num > $o).
% 66.57/16.16 thf(func_def_170, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 66.57/16.16 thf(func_def_171, type, bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o).
% 66.57/16.16 thf(func_def_172, type, bit_un2480387367778600638or_num: num > num > option_num).
% 66.57/16.16 thf(func_def_173, type, bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o).
% 66.57/16.16 thf(func_def_174, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 66.57/16.16 thf(func_def_175, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_176, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_177, type, code_int_of_integer: code_integer > int).
% 66.57/16.16 thf(func_def_178, type, code_integer_of_int: int > code_integer).
% 66.57/16.16 thf(func_def_179, type, code_integer_of_num: num > code_integer).
% 66.57/16.16 thf(func_def_180, type, code_nat_of_integer: code_integer > nat).
% 66.57/16.16 thf(func_def_181, type, code_negative: num > code_integer).
% 66.57/16.16 thf(func_def_182, type, code_num_of_integer: code_integer > num).
% 66.57/16.16 thf(func_def_183, type, code_positive: num > code_integer).
% 66.57/16.16 thf(func_def_184, type, code_Target_negative: num > int).
% 66.57/16.16 thf(func_def_185, type, code_Target_positive: num > int).
% 66.57/16.16 thf(func_def_186, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 66.57/16.16 thf(func_def_187, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 66.57/16.16 thf(func_def_188, type, comple4887499456419720421f_real: set_real > real).
% 66.57/16.16 thf(func_def_189, type, complete_Sup_Sup_int: set_int > int).
% 66.57/16.16 thf(func_def_190, type, comple1385675409528146559p_real: set_real > real).
% 66.57/16.16 thf(func_def_191, type, arg: complex > real).
% 66.57/16.16 thf(func_def_192, type, cis: real > complex).
% 66.57/16.16 thf(func_def_193, type, cnj: complex > complex).
% 66.57/16.16 thf(func_def_194, type, complex2: real > real > complex).
% 66.57/16.16 thf(func_def_195, type, im: complex > real).
% 66.57/16.16 thf(func_def_196, type, re: complex > real).
% 66.57/16.16 thf(func_def_197, type, csqrt: complex > complex).
% 66.57/16.16 thf(func_def_198, type, imaginary_unit: complex).
% 66.57/16.16 thf(func_def_199, type, rcis: real > real > complex).
% 66.57/16.16 thf(func_def_200, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 66.57/16.16 thf(func_def_201, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 66.57/16.16 thf(func_def_202, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 66.57/16.16 thf(func_def_203, type, has_ve631408500373753343e_real: (real > real) > real > filter_real > $o).
% 66.57/16.16 thf(func_def_204, type, adjust_div: product_prod_int_int > int).
% 66.57/16.16 thf(func_def_205, type, adjust_mod: int > int > int).
% 66.57/16.16 thf(func_def_206, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_207, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 66.57/16.16 thf(func_def_208, type, unique5706413561485394159nteger: produc8923325533196201883nteger > $o).
% 66.57/16.16 thf(func_def_209, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 66.57/16.16 thf(func_def_210, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 66.57/16.16 thf(func_def_211, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_212, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 66.57/16.16 thf(func_def_213, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_214, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_215, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 66.57/16.16 thf(func_def_216, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_217, type, euclid4774559944035922753ze_int: int > nat).
% 66.57/16.16 thf(func_def_218, type, euclid3395696857347342551nt_int: int > int).
% 66.57/16.16 thf(func_def_219, type, euclid3398187327856392827nt_nat: nat > nat).
% 66.57/16.16 thf(func_def_220, type, extended_case_enat_o: (nat > $o) > $o > extended_enat > $o).
% 66.57/16.16 thf(func_def_221, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 66.57/16.16 thf(func_def_222, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 66.57/16.16 thf(func_def_223, type, comm_s4660882817536571857er_int: int > nat > int).
% 66.57/16.16 thf(func_def_224, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_225, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 66.57/16.16 thf(func_def_226, type, comm_s7457072308508201937r_real: real > nat > real).
% 66.57/16.16 thf(func_def_227, type, semiri3624122377584611663nteger: nat > code_integer).
% 66.57/16.16 thf(func_def_228, type, semiri5044797733671781792omplex: nat > complex).
% 66.57/16.16 thf(func_def_229, type, semiri1406184849735516958ct_int: nat > int).
% 66.57/16.16 thf(func_def_230, type, semiri1408675320244567234ct_nat: nat > nat).
% 66.57/16.16 thf(func_def_231, type, semiri773545260158071498ct_rat: nat > rat).
% 66.57/16.16 thf(func_def_232, type, semiri2265585572941072030t_real: nat > real).
% 66.57/16.16 thf(func_def_233, type, invers8013647133539491842omplex: complex > complex).
% 66.57/16.16 thf(func_def_234, type, inverse_inverse_rat: rat > rat).
% 66.57/16.16 thf(func_def_235, type, inverse_inverse_real: real > real).
% 66.57/16.16 thf(func_def_236, type, at_bot_real: filter_real).
% 66.57/16.16 thf(func_def_237, type, at_top_nat: filter_nat).
% 66.57/16.16 thf(func_def_238, type, at_top_real: filter_real).
% 66.57/16.16 thf(func_def_239, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 66.57/16.16 thf(func_def_240, type, eventually_real: (real > $o) > filter_real > $o).
% 66.57/16.16 thf(func_def_241, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 66.57/16.16 thf(func_def_242, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 66.57/16.16 thf(func_def_243, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 66.57/16.16 thf(func_def_244, type, filtermap_real_real: (real > real) > filter_real > filter_real).
% 66.57/16.16 thf(func_def_245, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 66.57/16.16 thf(func_def_246, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 66.57/16.16 thf(func_def_247, type, finite_card_o: set_o > nat).
% 66.57/16.16 thf(func_def_248, type, finite_card_complex: set_complex > nat).
% 66.57/16.16 thf(func_def_249, type, finite_card_int: set_int > nat).
% 66.57/16.16 thf(func_def_250, type, finite_card_list_nat: set_list_nat > nat).
% 66.57/16.16 thf(func_def_251, type, finite_card_nat: set_nat > nat).
% 66.57/16.16 thf(func_def_252, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 66.57/16.16 thf(func_def_253, type, finite_card_char: set_char > nat).
% 66.57/16.16 thf(func_def_254, type, finite_finite_o: set_o > $o).
% 66.57/16.16 thf(func_def_255, type, finite3207457112153483333omplex: set_complex > $o).
% 66.57/16.16 thf(func_def_256, type, finite_finite_int: set_int > $o).
% 66.57/16.16 thf(func_def_257, type, finite_finite_list_o: set_list_o > $o).
% 66.57/16.16 thf(func_def_258, type, finite8712137658972009173omplex: set_list_complex > $o).
% 66.57/16.16 thf(func_def_259, type, finite3922522038869484883st_int: set_list_int > $o).
% 66.57/16.16 thf(func_def_260, type, finite8100373058378681591st_nat: set_list_nat > $o).
% 66.57/16.16 thf(func_def_261, type, finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_262, type, finite_finite_nat: set_nat > $o).
% 66.57/16.16 thf(func_def_263, type, finite_finite_num: set_num > $o).
% 66.57/16.16 thf(func_def_264, type, finite_finite_rat: set_rat > $o).
% 66.57/16.16 thf(func_def_265, type, finite_finite_real: set_real > $o).
% 66.57/16.16 thf(func_def_266, type, finite6551019134538273531omplex: set_set_complex > $o).
% 66.57/16.16 thf(func_def_267, type, finite6197958912794628473et_int: set_set_int > $o).
% 66.57/16.16 thf(func_def_268, type, finite1152437895449049373et_nat: set_set_nat > $o).
% 66.57/16.16 thf(func_def_269, type, finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_270, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 66.57/16.16 thf(func_def_271, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 66.57/16.16 thf(func_def_272, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 66.57/16.16 thf(func_def_273, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_274, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_275, type, comp_C3531382070062128313er_num: (code_integer > code_integer) > (num > code_integer) > num > code_integer).
% 66.57/16.16 thf(func_def_276, type, comp_int_int_num: (int > int) > (num > int) > num > int).
% 66.57/16.16 thf(func_def_277, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 66.57/16.16 thf(func_def_278, type, comp_int_real_real: (int > real) > (real > int) > real > real).
% 66.57/16.16 thf(func_def_279, type, comp_nat_real_nat: (nat > real) > (nat > nat) > nat > real).
% 66.57/16.16 thf(func_def_280, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 66.57/16.16 thf(func_def_281, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 66.57/16.16 thf(func_def_282, type, inj_on_real_real: (real > real) > set_real > $o).
% 66.57/16.16 thf(func_def_283, type, map_fu1532550112467129777l_real: (real > nat > rat) > (((nat > rat) > nat > rat) > real > real) > ((nat > rat) > (nat > rat) > nat > rat) > real > real > real).
% 66.57/16.16 thf(func_def_284, type, map_fu7146612038024189824t_real: (real > nat > rat) > ((nat > rat) > real) > ((nat > rat) > nat > rat) > real > real).
% 66.57/16.16 thf(func_def_285, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 66.57/16.16 thf(func_def_286, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 66.57/16.16 thf(func_def_287, type, fun_pair_leq: set_Pr8693737435421807431at_nat).
% 66.57/16.16 thf(func_def_288, type, fun_pair_less: set_Pr8693737435421807431at_nat).
% 66.57/16.16 thf(func_def_289, type, gcd_Gcd_int: set_int > int).
% 66.57/16.16 thf(func_def_290, type, gcd_Gcd_nat: set_nat > nat).
% 66.57/16.16 thf(func_def_291, type, bezw: nat > nat > product_prod_int_int).
% 66.57/16.16 thf(func_def_292, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 66.57/16.16 thf(func_def_293, type, gcd_gcd_int: int > int > int).
% 66.57/16.16 thf(func_def_294, type, gcd_gcd_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_295, type, abs_abs_Code_integer: code_integer > code_integer).
% 66.57/16.16 thf(func_def_296, type, abs_abs_complex: complex > complex).
% 66.57/16.16 thf(func_def_297, type, abs_abs_int: int > int).
% 66.57/16.16 thf(func_def_298, type, abs_abs_rat: rat > rat).
% 66.57/16.16 thf(func_def_299, type, abs_abs_real: real > real).
% 66.57/16.16 thf(func_def_300, type, minus_8727706125548526216plex_o: (complex > $o) > (complex > $o) > complex > $o).
% 66.57/16.16 thf(func_def_301, type, minus_minus_int_o: (int > $o) > (int > $o) > int > $o).
% 66.57/16.16 thf(func_def_302, type, minus_1139252259498527702_nat_o: (list_nat > $o) > (list_nat > $o) > list_nat > $o).
% 66.57/16.16 thf(func_def_303, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 66.57/16.16 thf(func_def_304, type, minus_4666906530369457071_nat_o: (produc859450856879609959at_nat > $o) > (produc859450856879609959at_nat > $o) > produc859450856879609959at_nat > $o).
% 66.57/16.16 thf(func_def_305, type, minus_minus_real_o: (real > $o) > (real > $o) > real > $o).
% 66.57/16.16 thf(func_def_306, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 66.57/16.16 thf(func_def_307, type, minus_minus_complex: complex > complex > complex).
% 66.57/16.16 thf(func_def_308, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 66.57/16.16 thf(func_def_309, type, minus_minus_int: int > int > int).
% 66.57/16.16 thf(func_def_310, type, minus_minus_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_311, type, minus_minus_rat: rat > rat > rat).
% 66.57/16.16 thf(func_def_312, type, minus_minus_real: real > real > real).
% 66.57/16.16 thf(func_def_313, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 66.57/16.16 thf(func_def_314, type, minus_minus_set_int: set_int > set_int > set_int).
% 66.57/16.16 thf(func_def_315, type, minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat).
% 66.57/16.16 thf(func_def_316, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 66.57/16.16 thf(func_def_317, type, minus_8321449233255521966at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat).
% 66.57/16.16 thf(func_def_318, type, minus_minus_set_real: set_real > set_real > set_real).
% 66.57/16.16 thf(func_def_319, type, one_one_Code_integer: code_integer).
% 66.57/16.16 thf(func_def_320, type, one_one_complex: complex).
% 66.57/16.16 thf(func_def_321, type, one_on7984719198319812577d_enat: extended_enat).
% 66.57/16.16 thf(func_def_322, type, one_one_int: int).
% 66.57/16.16 thf(func_def_323, type, one_one_nat: nat).
% 66.57/16.16 thf(func_def_324, type, one_one_rat: rat).
% 66.57/16.16 thf(func_def_325, type, one_one_real: real).
% 66.57/16.16 thf(func_def_326, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 66.57/16.16 thf(func_def_327, type, plus_plus_complex: complex > complex > complex).
% 66.57/16.16 thf(func_def_328, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 66.57/16.16 thf(func_def_329, type, plus_plus_int: int > int > int).
% 66.57/16.16 thf(func_def_330, type, plus_plus_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_331, type, plus_plus_num: num > num > num).
% 66.57/16.16 thf(func_def_332, type, plus_plus_rat: rat > rat > rat).
% 66.57/16.16 thf(func_def_333, type, plus_plus_real: real > real > real).
% 66.57/16.16 thf(func_def_334, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 66.57/16.16 thf(func_def_335, type, sgn_sgn_complex: complex > complex).
% 66.57/16.16 thf(func_def_336, type, sgn_sgn_int: int > int).
% 66.57/16.16 thf(func_def_337, type, sgn_sgn_rat: rat > rat).
% 66.57/16.16 thf(func_def_338, type, sgn_sgn_real: real > real).
% 66.57/16.16 thf(func_def_339, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 66.57/16.16 thf(func_def_340, type, times_times_complex: complex > complex > complex).
% 66.57/16.16 thf(func_def_341, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 66.57/16.16 thf(func_def_342, type, times_times_int: int > int > int).
% 66.57/16.16 thf(func_def_343, type, times_times_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_344, type, times_times_num: num > num > num).
% 66.57/16.16 thf(func_def_345, type, times_times_rat: rat > rat > rat).
% 66.57/16.16 thf(func_def_346, type, times_times_real: real > real > real).
% 66.57/16.16 thf(func_def_347, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 66.57/16.16 thf(func_def_348, type, uminus1482373934393186551omplex: complex > complex).
% 66.57/16.16 thf(func_def_349, type, uminus_uminus_int: int > int).
% 66.57/16.16 thf(func_def_350, type, uminus_uminus_rat: rat > rat).
% 66.57/16.16 thf(func_def_351, type, uminus_uminus_real: real > real).
% 66.57/16.16 thf(func_def_352, type, uminus1532241313380277803et_int: set_int > set_int).
% 66.57/16.16 thf(func_def_353, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 66.57/16.16 thf(func_def_354, type, uminus612125837232591019t_real: set_real > set_real).
% 66.57/16.16 thf(func_def_355, type, zero_z3403309356797280102nteger: code_integer).
% 66.57/16.16 thf(func_def_356, type, zero_zero_complex: complex).
% 66.57/16.16 thf(func_def_357, type, zero_z5237406670263579293d_enat: extended_enat).
% 66.57/16.16 thf(func_def_358, type, zero_zero_int: int).
% 66.57/16.16 thf(func_def_359, type, zero_zero_nat: nat).
% 66.57/16.16 thf(func_def_360, type, zero_zero_rat: rat).
% 66.57/16.16 thf(func_def_361, type, zero_zero_real: real).
% 66.57/16.16 thf(func_def_362, type, groups6621422865394947399nteger: (complex > code_integer) > set_complex > code_integer).
% 66.57/16.16 thf(func_def_363, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 66.57/16.16 thf(func_def_364, type, groups5690904116761175830ex_int: (complex > int) > set_complex > int).
% 66.57/16.16 thf(func_def_365, type, groups5693394587270226106ex_nat: (complex > nat) > set_complex > nat).
% 66.57/16.16 thf(func_def_366, type, groups5058264527183730370ex_rat: (complex > rat) > set_complex > rat).
% 66.57/16.16 thf(func_def_367, type, groups5808333547571424918x_real: (complex > real) > set_complex > real).
% 66.57/16.16 thf(func_def_368, type, groups7873554091576472773nteger: (int > code_integer) > set_int > code_integer).
% 66.57/16.16 thf(func_def_369, type, groups3049146728041665814omplex: (int > complex) > set_int > complex).
% 66.57/16.16 thf(func_def_370, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 66.57/16.16 thf(func_def_371, type, groups4541462559716669496nt_nat: (int > nat) > set_int > nat).
% 66.57/16.16 thf(func_def_372, type, groups3906332499630173760nt_rat: (int > rat) > set_int > rat).
% 66.57/16.16 thf(func_def_373, type, groups8778361861064173332t_real: (int > real) > set_int > real).
% 66.57/16.16 thf(func_def_374, type, groups7501900531339628137nteger: (nat > code_integer) > set_nat > code_integer).
% 66.57/16.16 thf(func_def_375, type, groups2073611262835488442omplex: (nat > complex) > set_nat > complex).
% 66.57/16.16 thf(func_def_376, type, groups3539618377306564664at_int: (nat > int) > set_nat > int).
% 66.57/16.16 thf(func_def_377, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 66.57/16.16 thf(func_def_378, type, groups2906978787729119204at_rat: (nat > rat) > set_nat > rat).
% 66.57/16.16 thf(func_def_379, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 66.57/16.16 thf(func_def_380, type, groups7963564606188603611omplex: (produc859450856879609959at_nat > complex) > set_Pr8693737435421807431at_nat > complex).
% 66.57/16.16 thf(func_def_381, type, groups1900718384385340925at_nat: (produc859450856879609959at_nat > nat) > set_Pr8693737435421807431at_nat > nat).
% 66.57/16.16 thf(func_def_382, type, groups7817252763086508633t_real: (produc859450856879609959at_nat > real) > set_Pr8693737435421807431at_nat > real).
% 66.57/16.16 thf(func_def_383, type, groups7713935264441627589nteger: (real > code_integer) > set_real > code_integer).
% 66.57/16.16 thf(func_def_384, type, groups5754745047067104278omplex: (real > complex) > set_real > complex).
% 66.57/16.16 thf(func_def_385, type, groups1932886352136224148al_int: (real > int) > set_real > int).
% 66.57/16.16 thf(func_def_386, type, groups1935376822645274424al_nat: (real > nat) > set_real > nat).
% 66.57/16.16 thf(func_def_387, type, groups1300246762558778688al_rat: (real > rat) > set_real > rat).
% 66.57/16.16 thf(func_def_388, type, groups8097168146408367636l_real: (real > real) > set_real > real).
% 66.57/16.16 thf(func_def_389, type, groups8682486955453173170nteger: (complex > code_integer) > set_complex > code_integer).
% 66.57/16.16 thf(func_def_390, type, groups3708469109370488835omplex: (complex > complex) > set_complex > complex).
% 66.57/16.16 thf(func_def_391, type, groups858564598930262913ex_int: (complex > int) > set_complex > int).
% 66.57/16.16 thf(func_def_392, type, groups861055069439313189ex_nat: (complex > nat) > set_complex > nat).
% 66.57/16.16 thf(func_def_393, type, groups225925009352817453ex_rat: (complex > rat) > set_complex > rat).
% 66.57/16.16 thf(func_def_394, type, groups766887009212190081x_real: (complex > real) > set_complex > real).
% 66.57/16.16 thf(func_def_395, type, groups3827104343326376752nteger: (int > code_integer) > set_int > code_integer).
% 66.57/16.16 thf(func_def_396, type, groups7440179247065528705omplex: (int > complex) > set_int > complex).
% 66.57/16.16 thf(func_def_397, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 66.57/16.16 thf(func_def_398, type, groups1707563613775114915nt_nat: (int > nat) > set_int > nat).
% 66.57/16.16 thf(func_def_399, type, groups1072433553688619179nt_rat: (int > rat) > set_int > rat).
% 66.57/16.16 thf(func_def_400, type, groups2316167850115554303t_real: (int > real) > set_int > real).
% 66.57/16.16 thf(func_def_401, type, groups3455450783089532116nteger: (nat > code_integer) > set_nat > code_integer).
% 66.57/16.16 thf(func_def_402, type, groups6464643781859351333omplex: (nat > complex) > set_nat > complex).
% 66.57/16.16 thf(func_def_403, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 66.57/16.16 thf(func_def_404, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 66.57/16.16 thf(func_def_405, type, groups73079841787564623at_rat: (nat > rat) > set_nat > rat).
% 66.57/16.16 thf(func_def_406, type, groups129246275422532515t_real: (nat > real) > set_nat > real).
% 66.57/16.16 thf(func_def_407, type, groups6225526099057966256nteger: (real > code_integer) > set_real > code_integer).
% 66.57/16.16 thf(func_def_408, type, groups713298508707869441omplex: (real > complex) > set_real > complex).
% 66.57/16.16 thf(func_def_409, type, groups4694064378042380927al_int: (real > int) > set_real > int).
% 66.57/16.16 thf(func_def_410, type, groups4696554848551431203al_nat: (real > nat) > set_real > nat).
% 66.57/16.16 thf(func_def_411, type, groups4061424788464935467al_rat: (real > rat) > set_real > rat).
% 66.57/16.16 thf(func_def_412, type, groups1681761925125756287l_real: (real > real) > set_real > real).
% 66.57/16.16 thf(func_def_413, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 66.57/16.16 thf(func_def_414, type, groups4561878855575611511st_nat: list_nat > nat).
% 66.57/16.16 thf(func_def_415, type, the_int: (int > $o) > int).
% 66.57/16.16 thf(func_def_416, type, the_Pr4378521158711661632nt_int: (product_prod_int_int > $o) > product_prod_int_int).
% 66.57/16.16 thf(func_def_417, type, the_real: (real > $o) > real).
% 66.57/16.16 thf(func_def_418, type, if_int_int: $o > (int > int) > (int > int) > int > int).
% 66.57/16.16 thf(func_def_419, type, if_nat_int_int: $o > (nat > int > int) > (nat > int > int) > nat > int > int).
% 66.57/16.16 thf(func_def_420, type, if_nat_nat_nat: $o > (nat > nat > nat) > (nat > nat > nat) > nat > nat > nat).
% 66.57/16.16 thf(func_def_421, type, if_nat_rat: $o > (nat > rat) > (nat > rat) > nat > rat).
% 66.57/16.16 thf(func_def_422, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 66.57/16.16 thf(func_def_423, type, if_complex: $o > complex > complex > complex).
% 66.57/16.16 thf(func_def_424, type, if_Extended_enat: $o > extended_enat > extended_enat > extended_enat).
% 66.57/16.16 thf(func_def_425, type, if_int: $o > int > int > int).
% 66.57/16.16 thf(func_def_426, type, if_list_int: $o > list_int > list_int > list_int).
% 66.57/16.16 thf(func_def_427, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 66.57/16.16 thf(func_def_428, type, if_nat: $o > nat > nat > nat).
% 66.57/16.16 thf(func_def_429, type, if_num: $o > num > num > num).
% 66.57/16.16 thf(func_def_430, type, if_option_nat: $o > option_nat > option_nat > option_nat).
% 66.57/16.16 thf(func_def_431, type, if_option_num: $o > option_num > option_num > option_num).
% 66.57/16.16 thf(func_def_432, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 66.57/16.16 thf(func_def_433, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_434, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 66.57/16.16 thf(func_def_435, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_436, type, if_rat: $o > rat > rat > rat).
% 66.57/16.16 thf(func_def_437, type, if_real: $o > real > real > real).
% 66.57/16.16 thf(func_def_438, type, if_set_int: $o > set_int > set_int > set_int).
% 66.57/16.16 thf(func_def_439, type, if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 66.57/16.16 thf(func_def_440, type, infini8530281810654367211te_nat: set_nat > nat > nat).
% 66.57/16.16 thf(func_def_441, type, abs_Integ: product_prod_nat_nat > int).
% 66.57/16.16 thf(func_def_442, type, rep_Integ: int > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_443, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 66.57/16.16 thf(func_def_444, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 66.57/16.16 thf(func_def_445, type, intrel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 66.57/16.16 thf(func_def_446, type, nat2: int > nat).
% 66.57/16.16 thf(func_def_447, type, pcr_int: product_prod_nat_nat > int > $o).
% 66.57/16.16 thf(func_def_448, type, power_int_real: real > int > real).
% 66.57/16.16 thf(func_def_449, type, ring_1_Ints_real: set_real).
% 66.57/16.16 thf(func_def_450, type, ring_18347121197199848620nteger: int > code_integer).
% 66.57/16.16 thf(func_def_451, type, ring_17405671764205052669omplex: int > complex).
% 66.57/16.16 thf(func_def_452, type, ring_1_of_int_int: int > int).
% 66.57/16.16 thf(func_def_453, type, ring_1_of_int_rat: int > rat).
% 66.57/16.16 thf(func_def_454, type, ring_1_of_int_real: int > real).
% 66.57/16.16 thf(func_def_455, type, inf_in1870772243966228564d_enat: extended_enat > extended_enat > extended_enat).
% 66.57/16.16 thf(func_def_456, type, inf_inf_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_457, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 66.57/16.16 thf(func_def_458, type, sup_su3973961784419623482d_enat: extended_enat > extended_enat > extended_enat).
% 66.57/16.16 thf(func_def_459, type, sup_sup_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_460, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 66.57/16.16 thf(func_def_461, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 66.57/16.16 thf(func_def_462, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 66.57/16.16 thf(func_def_463, type, at_infinity_real: filter_real).
% 66.57/16.16 thf(func_def_464, type, append_int: list_int > list_int > list_int).
% 66.57/16.16 thf(func_def_465, type, append_nat: list_nat > list_nat > list_nat).
% 66.57/16.16 thf(func_def_466, type, drop_nat: nat > list_nat > list_nat).
% 66.57/16.16 thf(func_def_467, type, last_nat: list_nat > nat).
% 66.57/16.16 thf(func_def_468, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 66.57/16.16 thf(func_def_469, type, cons_int: int > list_int > list_int).
% 66.57/16.16 thf(func_def_470, type, cons_nat: nat > list_nat > list_nat).
% 66.57/16.16 thf(func_def_471, type, nil_int: list_int).
% 66.57/16.16 thf(func_def_472, type, nil_nat: list_nat).
% 66.57/16.16 thf(func_def_473, type, hd_nat: list_nat > nat).
% 66.57/16.16 thf(func_def_474, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 66.57/16.16 thf(func_def_475, type, set_o2: list_o > set_o).
% 66.57/16.16 thf(func_def_476, type, set_complex2: list_complex > set_complex).
% 66.57/16.16 thf(func_def_477, type, set_int2: list_int > set_int).
% 66.57/16.16 thf(func_def_478, type, set_list_nat2: list_list_nat > set_list_nat).
% 66.57/16.16 thf(func_def_479, type, set_nat2: list_nat > set_nat).
% 66.57/16.16 thf(func_def_480, type, set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat).
% 66.57/16.16 thf(func_def_481, type, set_real2: list_real > set_real).
% 66.57/16.16 thf(func_def_482, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 66.57/16.16 thf(func_def_483, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 66.57/16.16 thf(func_def_484, type, list_update_o: list_o > nat > $o > list_o).
% 66.57/16.16 thf(func_def_485, type, list_update_complex: list_complex > nat > complex > list_complex).
% 66.57/16.16 thf(func_def_486, type, list_update_int: list_int > nat > int > list_int).
% 66.57/16.16 thf(func_def_487, type, list_update_nat: list_nat > nat > nat > list_nat).
% 66.57/16.16 thf(func_def_488, type, list_u5003261594476800725at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat > list_P8469869581646625389at_nat).
% 66.57/16.16 thf(func_def_489, type, list_update_real: list_real > nat > real > list_real).
% 66.57/16.16 thf(func_def_490, type, list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 66.57/16.16 thf(func_def_491, type, nth_o: list_o > nat > $o).
% 66.57/16.16 thf(func_def_492, type, nth_complex: list_complex > nat > complex).
% 66.57/16.16 thf(func_def_493, type, nth_int: list_int > nat > int).
% 66.57/16.16 thf(func_def_494, type, nth_list_nat: list_list_nat > nat > list_nat).
% 66.57/16.16 thf(func_def_495, type, nth_nat: list_nat > nat > nat).
% 66.57/16.16 thf(func_def_496, type, nth_num: list_num > nat > num).
% 66.57/16.16 thf(func_def_497, type, nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o).
% 66.57/16.16 thf(func_def_498, type, nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int).
% 66.57/16.16 thf(func_def_499, type, nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat).
% 66.57/16.16 thf(func_def_500, type, nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT).
% 66.57/16.16 thf(func_def_501, type, nth_Pr8326237132889035090at_num: list_P1726324292696863441at_num > nat > product_prod_nat_num).
% 66.57/16.16 thf(func_def_502, type, nth_Pr6456567536196504476um_num: list_P3744719386663036955um_num > nat > product_prod_num_num).
% 66.57/16.16 thf(func_def_503, type, nth_Pr6744343527793145070at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat).
% 66.57/16.16 thf(func_def_504, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 66.57/16.16 thf(func_def_505, type, nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int).
% 66.57/16.16 thf(func_def_506, type, nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat).
% 66.57/16.16 thf(func_def_507, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 66.57/16.16 thf(func_def_508, type, nth_real: list_real > nat > real).
% 66.57/16.16 thf(func_def_509, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 66.57/16.16 thf(func_def_510, type, product_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 66.57/16.16 thf(func_def_511, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 66.57/16.16 thf(func_def_512, type, product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat).
% 66.57/16.16 thf(func_def_513, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 66.57/16.16 thf(func_def_514, type, product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 66.57/16.16 thf(func_def_515, type, product_nat_num: list_nat > list_num > list_P1726324292696863441at_num).
% 66.57/16.16 thf(func_def_516, type, produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 66.57/16.16 thf(func_def_517, type, product_num_num: list_num > list_num > list_P3744719386663036955um_num).
% 66.57/16.16 thf(func_def_518, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 66.57/16.16 thf(func_def_519, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 66.57/16.16 thf(func_def_520, type, produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 66.57/16.16 thf(func_def_521, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 66.57/16.16 thf(func_def_522, type, replicate_int: nat > int > list_int).
% 66.57/16.16 thf(func_def_523, type, replicate_nat: nat > nat > list_nat).
% 66.57/16.16 thf(func_def_524, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 66.57/16.16 thf(func_def_525, type, sorted_wrt_int: (int > int > $o) > list_int > $o).
% 66.57/16.16 thf(func_def_526, type, sorted_wrt_nat: (nat > nat > $o) > list_nat > $o).
% 66.57/16.16 thf(func_def_527, type, take_nat: nat > list_nat > list_nat).
% 66.57/16.16 thf(func_def_528, type, upt: nat > nat > list_nat).
% 66.57/16.16 thf(func_def_529, type, upto: int > int > list_int).
% 66.57/16.16 thf(func_def_530, type, upto_aux: int > int > list_int > list_int).
% 66.57/16.16 thf(func_def_531, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 66.57/16.16 thf(func_def_532, type, suc: nat > nat).
% 66.57/16.16 thf(func_def_533, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 66.57/16.16 thf(func_def_534, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 66.57/16.16 thf(func_def_535, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 66.57/16.16 thf(func_def_536, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 66.57/16.16 thf(func_def_537, type, pred: nat > nat).
% 66.57/16.16 thf(func_def_538, type, semiri4939895301339042750nteger: nat > code_integer).
% 66.57/16.16 thf(func_def_539, type, semiri8010041392384452111omplex: nat > complex).
% 66.57/16.16 thf(func_def_540, type, semiri4216267220026989637d_enat: nat > extended_enat).
% 66.57/16.16 thf(func_def_541, type, semiri1314217659103216013at_int: nat > int).
% 66.57/16.16 thf(func_def_542, type, semiri1316708129612266289at_nat: nat > nat).
% 66.57/16.16 thf(func_def_543, type, semiri681578069525770553at_rat: nat > rat).
% 66.57/16.16 thf(func_def_544, type, semiri5074537144036343181t_real: nat > real).
% 66.57/16.16 thf(func_def_545, type, semiri2816024913162550771omplex: (complex > complex) > nat > complex > complex).
% 66.57/16.16 thf(func_def_546, type, semiri8420488043553186161ux_int: (int > int) > nat > int > int).
% 66.57/16.16 thf(func_def_547, type, semiri8422978514062236437ux_nat: (nat > nat) > nat > nat > nat).
% 66.57/16.16 thf(func_def_548, type, semiri7787848453975740701ux_rat: (rat > rat) > nat > rat > rat).
% 66.57/16.16 thf(func_def_549, type, semiri7260567687927622513x_real: (real > real) > nat > real > real).
% 66.57/16.16 thf(func_def_550, type, size_size_list_o: list_o > nat).
% 66.57/16.16 thf(func_def_551, type, size_s3451745648224563538omplex: list_complex > nat).
% 66.57/16.16 thf(func_def_552, type, size_size_list_int: list_int > nat).
% 66.57/16.16 thf(func_def_553, type, size_s3023201423986296836st_nat: list_list_nat > nat).
% 66.57/16.16 thf(func_def_554, type, size_size_list_nat: list_nat > nat).
% 66.57/16.16 thf(func_def_555, type, size_size_list_num: list_num > nat).
% 66.57/16.16 thf(func_def_556, type, size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat).
% 66.57/16.16 thf(func_def_557, type, size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat).
% 66.57/16.16 thf(func_def_558, type, size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat).
% 66.57/16.16 thf(func_def_559, type, size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat).
% 66.57/16.16 thf(func_def_560, type, size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat).
% 66.57/16.16 thf(func_def_561, type, size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat).
% 66.57/16.16 thf(func_def_562, type, size_s3679842834875189465at_nat: list_P8469869581646625389at_nat > nat).
% 66.57/16.16 thf(func_def_563, type, size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat).
% 66.57/16.16 thf(func_def_564, type, size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat).
% 66.57/16.16 thf(func_def_565, type, size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat).
% 66.57/16.16 thf(func_def_566, type, size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat).
% 66.57/16.16 thf(func_def_567, type, size_size_list_real: list_real > nat).
% 66.57/16.16 thf(func_def_568, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 66.57/16.16 thf(func_def_569, type, size_size_num: num > nat).
% 66.57/16.16 thf(func_def_570, type, size_size_option_nat: option_nat > nat).
% 66.57/16.16 thf(func_def_571, type, size_size_option_num: option_num > nat).
% 66.57/16.16 thf(func_def_572, type, size_s170228958280169651at_nat: option4927543243414619207at_nat > nat).
% 66.57/16.16 thf(func_def_573, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 66.57/16.16 thf(func_def_574, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_575, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 66.57/16.16 thf(func_def_576, type, nat_prod_encode: product_prod_nat_nat > nat).
% 66.57/16.16 thf(func_def_577, type, nat_set_decode: nat > set_nat).
% 66.57/16.16 thf(func_def_578, type, nat_set_encode: set_nat > nat).
% 66.57/16.16 thf(func_def_579, type, nat_triangle: nat > nat).
% 66.57/16.16 thf(func_def_580, type, root: nat > real > real).
% 66.57/16.16 thf(func_def_581, type, sqrt: real > real).
% 66.57/16.16 thf(func_def_582, type, bitM: num > num).
% 66.57/16.16 thf(func_def_583, type, inc: num > num).
% 66.57/16.16 thf(func_def_584, type, neg_nu8804712462038260780nteger: code_integer > code_integer).
% 66.57/16.16 thf(func_def_585, type, neg_nu7009210354673126013omplex: complex > complex).
% 66.57/16.16 thf(func_def_586, type, neg_numeral_dbl_int: int > int).
% 66.57/16.16 thf(func_def_587, type, neg_numeral_dbl_rat: rat > rat).
% 66.57/16.16 thf(func_def_588, type, neg_numeral_dbl_real: real > real).
% 66.57/16.16 thf(func_def_589, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 66.57/16.16 thf(func_def_590, type, neg_nu6511756317524482435omplex: complex > complex).
% 66.57/16.16 thf(func_def_591, type, neg_nu3811975205180677377ec_int: int > int).
% 66.57/16.16 thf(func_def_592, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 66.57/16.16 thf(func_def_593, type, neg_nu6075765906172075777c_real: real > real).
% 66.57/16.16 thf(func_def_594, type, neg_nu5831290666863070958nteger: code_integer > code_integer).
% 66.57/16.16 thf(func_def_595, type, neg_nu8557863876264182079omplex: complex > complex).
% 66.57/16.16 thf(func_def_596, type, neg_nu5851722552734809277nc_int: int > int).
% 66.57/16.16 thf(func_def_597, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 66.57/16.16 thf(func_def_598, type, neg_nu8295874005876285629c_real: real > real).
% 66.57/16.16 thf(func_def_599, type, neg_numeral_sub_int: num > num > int).
% 66.57/16.16 thf(func_def_600, type, bit0: num > num).
% 66.57/16.16 thf(func_def_601, type, bit1: num > num).
% 66.57/16.16 thf(func_def_602, type, one: num).
% 66.57/16.16 thf(func_def_603, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 66.57/16.16 thf(func_def_604, type, size_num: num > nat).
% 66.57/16.16 thf(func_def_605, type, num_of_nat: nat > num).
% 66.57/16.16 thf(func_def_606, type, numera6620942414471956472nteger: num > code_integer).
% 66.57/16.16 thf(func_def_607, type, numera6690914467698888265omplex: num > complex).
% 66.57/16.16 thf(func_def_608, type, numera1916890842035813515d_enat: num > extended_enat).
% 66.57/16.16 thf(func_def_609, type, numeral_numeral_int: num > int).
% 66.57/16.16 thf(func_def_610, type, numeral_numeral_nat: num > nat).
% 66.57/16.16 thf(func_def_611, type, numeral_numeral_rat: num > rat).
% 66.57/16.16 thf(func_def_612, type, numeral_numeral_real: num > real).
% 66.57/16.16 thf(func_def_613, type, pow: num > num > num).
% 66.57/16.16 thf(func_def_614, type, pred_numeral: num > nat).
% 66.57/16.16 thf(func_def_615, type, sqr: num > num).
% 66.57/16.16 thf(func_def_616, type, none_nat: option_nat).
% 66.57/16.16 thf(func_def_617, type, none_num: option_num).
% 66.57/16.16 thf(func_def_618, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 66.57/16.16 thf(func_def_619, type, some_nat: nat > option_nat).
% 66.57/16.16 thf(func_def_620, type, some_num: num > option_num).
% 66.57/16.16 thf(func_def_621, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 66.57/16.16 thf(func_def_622, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 66.57/16.16 thf(func_def_623, type, case_option_int_num: int > (num > int) > option_num > int).
% 66.57/16.16 thf(func_def_624, type, case_option_num_num: num > (num > num) > option_num > num).
% 66.57/16.16 thf(func_def_625, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 66.57/16.16 thf(func_def_626, type, map_option_num_num: (num > num) > option_num > option_num).
% 66.57/16.16 thf(func_def_627, type, size_option_nat: (nat > nat) > option_nat > nat).
% 66.57/16.16 thf(func_def_628, type, size_option_num: (num > nat) > option_num > nat).
% 66.57/16.16 thf(func_def_629, type, size_o8335143837870341156at_nat: (product_prod_nat_nat > nat) > option4927543243414619207at_nat > nat).
% 66.57/16.16 thf(func_def_630, type, the_nat: option_nat > nat).
% 66.57/16.16 thf(func_def_631, type, the_num: option_num > num).
% 66.57/16.16 thf(func_def_632, type, the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_633, type, bot_bot_nat_o: nat > $o).
% 66.57/16.16 thf(func_def_634, type, bot_bo4199563552545308370d_enat: extended_enat).
% 66.57/16.16 thf(func_def_635, type, bot_bot_nat: nat).
% 66.57/16.16 thf(func_def_636, type, bot_bot_set_complex: set_complex).
% 66.57/16.16 thf(func_def_637, type, bot_bot_set_int: set_int).
% 66.57/16.16 thf(func_def_638, type, bot_bot_set_nat: set_nat).
% 66.57/16.16 thf(func_def_639, type, bot_bot_set_num: set_num).
% 66.57/16.16 thf(func_def_640, type, bot_bot_set_rat: set_rat).
% 66.57/16.16 thf(func_def_641, type, bot_bot_set_real: set_real).
% 66.57/16.16 thf(func_def_642, type, bot_bot_set_set_int: set_set_int).
% 66.57/16.16 thf(func_def_643, type, ord_Least_nat: (nat > $o) > nat).
% 66.57/16.16 thf(func_def_644, type, ord_Least_real: (real > $o) > real).
% 66.57/16.16 thf(func_def_645, type, ord_less_complex_o: (complex > $o) > (complex > $o) > $o).
% 66.57/16.16 thf(func_def_646, type, ord_less_int_o: (int > $o) > (int > $o) > $o).
% 66.57/16.16 thf(func_def_647, type, ord_less_nat_o: (nat > $o) > (nat > $o) > $o).
% 66.57/16.16 thf(func_def_648, type, ord_le7432393201274343594_nat_o: (produc859450856879609959at_nat > $o) > (produc859450856879609959at_nat > $o) > $o).
% 66.57/16.16 thf(func_def_649, type, ord_less_real_o: (real > $o) > (real > $o) > $o).
% 66.57/16.16 thf(func_def_650, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 66.57/16.16 thf(func_def_651, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 66.57/16.16 thf(func_def_652, type, ord_less_int: int > int > $o).
% 66.57/16.16 thf(func_def_653, type, ord_less_nat: nat > nat > $o).
% 66.57/16.16 thf(func_def_654, type, ord_less_num: num > num > $o).
% 66.57/16.16 thf(func_def_655, type, ord_less_rat: rat > rat > $o).
% 66.57/16.16 thf(func_def_656, type, ord_less_real: real > real > $o).
% 66.57/16.16 thf(func_def_657, type, ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o).
% 66.57/16.16 thf(func_def_658, type, ord_less_set_complex: set_complex > set_complex > $o).
% 66.57/16.16 thf(func_def_659, type, ord_less_set_int: set_int > set_int > $o).
% 66.57/16.16 thf(func_def_660, type, ord_less_set_nat: set_nat > set_nat > $o).
% 66.57/16.16 thf(func_def_661, type, ord_less_set_num: set_num > set_num > $o).
% 66.57/16.16 thf(func_def_662, type, ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o).
% 66.57/16.16 thf(func_def_663, type, ord_less_set_rat: set_rat > set_rat > $o).
% 66.57/16.16 thf(func_def_664, type, ord_less_set_real: set_real > set_real > $o).
% 66.57/16.16 thf(func_def_665, type, ord_less_set_set_int: set_set_int > set_set_int > $o).
% 66.57/16.16 thf(func_def_666, type, ord_le4573692005234683329plex_o: (complex > $o) > (complex > $o) > $o).
% 66.57/16.16 thf(func_def_667, type, ord_less_eq_int_o: (int > $o) > (int > $o) > $o).
% 66.57/16.16 thf(func_def_668, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 66.57/16.16 thf(func_def_669, type, ord_le7858099551454983350_nat_o: (produc859450856879609959at_nat > $o) > (produc859450856879609959at_nat > $o) > $o).
% 66.57/16.16 thf(func_def_670, type, ord_less_eq_real_o: (real > $o) > (real > $o) > $o).
% 66.57/16.16 thf(func_def_671, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 66.57/16.16 thf(func_def_672, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 66.57/16.16 thf(func_def_673, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 66.57/16.16 thf(func_def_674, type, ord_le4104064031414453916r_real: filter_real > filter_real > $o).
% 66.57/16.16 thf(func_def_675, type, ord_less_eq_int: int > int > $o).
% 66.57/16.16 thf(func_def_676, type, ord_less_eq_nat: nat > nat > $o).
% 66.57/16.16 thf(func_def_677, type, ord_less_eq_num: num > num > $o).
% 66.57/16.16 thf(func_def_678, type, ord_less_eq_rat: rat > rat > $o).
% 66.57/16.16 thf(func_def_679, type, ord_less_eq_real: real > real > $o).
% 66.57/16.16 thf(func_def_680, type, ord_less_eq_set_o: set_o > set_o > $o).
% 66.57/16.16 thf(func_def_681, type, ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o).
% 66.57/16.16 thf(func_def_682, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 66.57/16.16 thf(func_def_683, type, ord_less_eq_set_int: set_int > set_int > $o).
% 66.57/16.16 thf(func_def_684, type, ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o).
% 66.57/16.16 thf(func_def_685, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 66.57/16.16 thf(func_def_686, type, ord_less_eq_set_num: set_num > set_num > $o).
% 66.57/16.16 thf(func_def_687, type, ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o).
% 66.57/16.16 thf(func_def_688, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 66.57/16.16 thf(func_def_689, type, ord_less_eq_set_real: set_real > set_real > $o).
% 66.57/16.16 thf(func_def_690, type, ord_le4403425263959731960et_int: set_set_int > set_set_int > $o).
% 66.57/16.16 thf(func_def_691, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_692, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 66.57/16.16 thf(func_def_693, type, ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat).
% 66.57/16.16 thf(func_def_694, type, ord_max_int: int > int > int).
% 66.57/16.16 thf(func_def_695, type, ord_max_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_696, type, ord_max_num: num > num > num).
% 66.57/16.16 thf(func_def_697, type, ord_max_rat: rat > rat > rat).
% 66.57/16.16 thf(func_def_698, type, ord_max_real: real > real > real).
% 66.57/16.16 thf(func_def_699, type, ord_max_set_int: set_int > set_int > set_int).
% 66.57/16.16 thf(func_def_700, type, ord_mi8085742599997312461d_enat: extended_enat > extended_enat > extended_enat).
% 66.57/16.16 thf(func_def_701, type, ord_min_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_702, type, order_Greatest_nat: (nat > $o) > nat).
% 66.57/16.16 thf(func_def_703, type, order_9091379641038594480t_real: (nat > real) > $o).
% 66.57/16.16 thf(func_def_704, type, order_mono_nat_nat: (nat > nat) > $o).
% 66.57/16.16 thf(func_def_705, type, order_mono_nat_real: (nat > real) > $o).
% 66.57/16.16 thf(func_def_706, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 66.57/16.16 thf(func_def_707, type, order_7092887310737990675l_real: (real > real) > $o).
% 66.57/16.16 thf(func_def_708, type, ordering_top_nat: (nat > nat > $o) > (nat > nat > $o) > nat > $o).
% 66.57/16.16 thf(func_def_709, type, top_top_set_o: set_o).
% 66.57/16.16 thf(func_def_710, type, top_top_set_nat: set_nat).
% 66.57/16.16 thf(func_def_711, type, top_to1996260823553986621t_unit: set_Product_unit).
% 66.57/16.16 thf(func_def_712, type, top_top_set_real: set_real).
% 66.57/16.16 thf(func_def_713, type, top_top_set_char: set_char).
% 66.57/16.16 thf(func_def_714, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 66.57/16.16 thf(func_def_715, type, power_power_complex: complex > nat > complex).
% 66.57/16.16 thf(func_def_716, type, power_power_int: int > nat > int).
% 66.57/16.16 thf(func_def_717, type, power_power_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_718, type, power_power_rat: rat > nat > rat).
% 66.57/16.16 thf(func_def_719, type, power_power_real: real > nat > real).
% 66.57/16.16 thf(func_def_720, type, produc4035269172776083154on_nat: (nat > nat > $o) > produc4953844613479565601on_nat > produc2233624965454879586on_nat).
% 66.57/16.16 thf(func_def_721, type, produc8929957630744042906on_nat: (nat > nat > nat) > produc4953844613479565601on_nat > produc8306885398267862888on_nat).
% 66.57/16.16 thf(func_def_722, type, produc3576312749637752826on_num: (num > num > $o) > produc3447558737645232053on_num > produc7036089656553540234on_num).
% 66.57/16.16 thf(func_def_723, type, produc5778274026573060048on_num: (num > num > num) > produc3447558737645232053on_num > produc1193250871479095198on_num).
% 66.57/16.16 thf(func_def_724, type, produc3994169339658061776at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > produc6121120109295599847at_nat > produc5491161045314408544at_nat).
% 66.57/16.16 thf(func_def_725, type, produc2899441246263362727at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > produc6121120109295599847at_nat > produc5542196010084753463at_nat).
% 66.57/16.16 thf(func_def_726, type, product_Pair_o_o: $o > $o > product_prod_o_o).
% 66.57/16.16 thf(func_def_727, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 66.57/16.16 thf(func_def_728, type, product_Pair_o_nat: $o > nat > product_prod_o_nat).
% 66.57/16.16 thf(func_def_729, type, produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT).
% 66.57/16.16 thf(func_def_730, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 66.57/16.16 thf(func_def_731, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_732, type, product_Pair_int_int: int > int > product_prod_int_int).
% 66.57/16.16 thf(func_def_733, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_734, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 66.57/16.16 thf(func_def_735, type, product_Pair_num_num: num > num > product_prod_num_num).
% 66.57/16.16 thf(func_def_736, type, produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat).
% 66.57/16.16 thf(func_def_737, type, produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num).
% 66.57/16.16 thf(func_def_738, type, produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat).
% 66.57/16.16 thf(func_def_739, type, produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat).
% 66.57/16.16 thf(func_def_740, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 66.57/16.16 thf(func_def_741, type, produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int).
% 66.57/16.16 thf(func_def_742, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 66.57/16.16 thf(func_def_743, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 66.57/16.16 thf(func_def_744, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_745, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 66.57/16.16 thf(func_def_746, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 66.57/16.16 thf(func_def_747, type, produc7336495610019696514er_num: (code_integer > code_integer > num) > produc8923325533196201883nteger > num).
% 66.57/16.16 thf(func_def_748, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 66.57/16.16 thf(func_def_749, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 66.57/16.16 thf(func_def_750, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 66.57/16.16 thf(func_def_751, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 66.57/16.16 thf(func_def_752, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 66.57/16.16 thf(func_def_753, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 66.57/16.16 thf(func_def_754, 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).
% 66.57/16.16 thf(func_def_755, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 66.57/16.16 thf(func_def_756, type, produc1917071388513777916omplex: (nat > nat > complex) > product_prod_nat_nat > complex).
% 66.57/16.16 thf(func_def_757, type, produc6840382203811409530at_int: (nat > nat > int) > product_prod_nat_nat > int).
% 66.57/16.16 thf(func_def_758, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 66.57/16.16 thf(func_def_759, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 66.57/16.16 thf(func_def_760, type, produc6207742614233964070at_rat: (nat > nat > rat) > product_prod_nat_nat > rat).
% 66.57/16.16 thf(func_def_761, type, produc1703576794950452218t_real: (nat > nat > real) > product_prod_nat_nat > real).
% 66.57/16.16 thf(func_def_762, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 66.57/16.16 thf(func_def_763, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 66.57/16.16 thf(func_def_764, type, produc8508995932063986495nteger: produc8923325533196201883nteger > code_integer).
% 66.57/16.16 thf(func_def_765, type, product_fst_int_int: product_prod_int_int > int).
% 66.57/16.16 thf(func_def_766, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 66.57/16.16 thf(func_def_767, type, product_snd_int_int: product_prod_int_int > int).
% 66.57/16.16 thf(func_def_768, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 66.57/16.16 thf(func_def_769, type, abs_Rat: product_prod_int_int > rat).
% 66.57/16.16 thf(func_def_770, type, fract: int > int > rat).
% 66.57/16.16 thf(func_def_771, type, frct: product_prod_int_int > rat).
% 66.57/16.16 thf(func_def_772, type, rep_Rat: rat > product_prod_int_int).
% 66.57/16.16 thf(func_def_773, type, field_5140801741446780682s_real: set_real).
% 66.57/16.16 thf(func_def_774, type, field_7254667332652039916t_real: rat > real).
% 66.57/16.16 thf(func_def_775, type, normalize: product_prod_int_int > product_prod_int_int).
% 66.57/16.16 thf(func_def_776, type, pcr_rat: product_prod_int_int > rat > $o).
% 66.57/16.16 thf(func_def_777, type, positive: rat > $o).
% 66.57/16.16 thf(func_def_778, type, quotient_of: rat > product_prod_int_int).
% 66.57/16.16 thf(func_def_779, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 66.57/16.16 thf(func_def_780, type, real2: (nat > rat) > real).
% 66.57/16.16 thf(func_def_781, type, cauchy: (nat > rat) > $o).
% 66.57/16.16 thf(func_def_782, type, cr_real: (nat > rat) > real > $o).
% 66.57/16.16 thf(func_def_783, type, pcr_real: (nat > rat) > real > $o).
% 66.57/16.16 thf(func_def_784, type, positive2: real > $o).
% 66.57/16.16 thf(func_def_785, type, realrel: (nat > rat) > (nat > rat) > $o).
% 66.57/16.16 thf(func_def_786, type, rep_real: real > nat > rat).
% 66.57/16.16 thf(func_def_787, type, vanishes: (nat > rat) > $o).
% 66.57/16.16 thf(func_def_788, type, real_V2521375963428798218omplex: set_complex).
% 66.57/16.16 thf(func_def_789, type, real_V5970128139526366754l_real: (real > real) > $o).
% 66.57/16.16 thf(func_def_790, type, real_V3694042436643373181omplex: complex > complex > real).
% 66.57/16.16 thf(func_def_791, type, real_V975177566351809787t_real: real > real > real).
% 66.57/16.16 thf(func_def_792, type, real_V1022390504157884413omplex: complex > real).
% 66.57/16.16 thf(func_def_793, type, real_V7735802525324610683m_real: real > real).
% 66.57/16.16 thf(func_def_794, type, real_V4546457046886955230omplex: real > complex).
% 66.57/16.16 thf(func_def_795, type, real_V2046097035970521341omplex: real > complex > complex).
% 66.57/16.16 thf(func_def_796, type, real_V1485227260804924795R_real: real > real > real).
% 66.57/16.16 thf(func_def_797, type, transp_nat_rat: ((nat > rat) > (nat > rat) > $o) > $o).
% 66.57/16.16 thf(func_def_798, type, algebr932160517623751201me_int: int > int > $o).
% 66.57/16.16 thf(func_def_799, type, algebr934650988132801477me_nat: nat > nat > $o).
% 66.57/16.16 thf(func_def_800, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 66.57/16.16 thf(func_def_801, type, divide1717551699836669952omplex: complex > complex > complex).
% 66.57/16.16 thf(func_def_802, type, divide_divide_int: int > int > int).
% 66.57/16.16 thf(func_def_803, type, divide_divide_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_804, type, divide_divide_rat: rat > rat > rat).
% 66.57/16.16 thf(func_def_805, type, divide_divide_real: real > real > real).
% 66.57/16.16 thf(func_def_806, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 66.57/16.16 thf(func_def_807, type, dvd_dvd_complex: complex > complex > $o).
% 66.57/16.16 thf(func_def_808, type, dvd_dvd_int: int > int > $o).
% 66.57/16.16 thf(func_def_809, type, dvd_dvd_nat: nat > nat > $o).
% 66.57/16.16 thf(func_def_810, type, dvd_dvd_rat: rat > rat > $o).
% 66.57/16.16 thf(func_def_811, type, dvd_dvd_real: real > real > $o).
% 66.57/16.16 thf(func_def_812, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 66.57/16.16 thf(func_def_813, type, modulo_modulo_int: int > int > int).
% 66.57/16.16 thf(func_def_814, type, modulo_modulo_nat: nat > nat > nat).
% 66.57/16.16 thf(func_def_815, type, zero_n356916108424825756nteger: $o > code_integer).
% 66.57/16.16 thf(func_def_816, type, zero_n1201886186963655149omplex: $o > complex).
% 66.57/16.16 thf(func_def_817, type, zero_n2684676970156552555ol_int: $o > int).
% 66.57/16.16 thf(func_def_818, type, zero_n2687167440665602831ol_nat: $o > nat).
% 66.57/16.16 thf(func_def_819, type, zero_n2052037380579107095ol_rat: $o > rat).
% 66.57/16.16 thf(func_def_820, type, zero_n3304061248610475627l_real: $o > real).
% 66.57/16.16 thf(func_def_821, type, suminf_complex: (nat > complex) > complex).
% 66.57/16.16 thf(func_def_822, type, suminf_int: (nat > int) > int).
% 66.57/16.16 thf(func_def_823, type, suminf_nat: (nat > nat) > nat).
% 66.57/16.16 thf(func_def_824, type, suminf_real: (nat > real) > real).
% 66.57/16.16 thf(func_def_825, type, summable_complex: (nat > complex) > $o).
% 66.57/16.16 thf(func_def_826, type, summable_int: (nat > int) > $o).
% 66.57/16.16 thf(func_def_827, type, summable_nat: (nat > nat) > $o).
% 66.57/16.16 thf(func_def_828, type, summable_real: (nat > real) > $o).
% 66.57/16.16 thf(func_def_829, type, sums_complex: (nat > complex) > complex > $o).
% 66.57/16.16 thf(func_def_830, type, sums_int: (nat > int) > int > $o).
% 66.57/16.16 thf(func_def_831, type, sums_nat: (nat > nat) > nat > $o).
% 66.57/16.16 thf(func_def_832, type, sums_real: (nat > real) > real > $o).
% 66.57/16.16 thf(func_def_833, type, collect_o: ($o > $o) > set_o).
% 66.57/16.16 thf(func_def_834, type, collect_Code_integer: (code_integer > $o) > set_Code_integer).
% 66.57/16.16 thf(func_def_835, type, collect_complex: (complex > $o) > set_complex).
% 66.57/16.16 thf(func_def_836, type, collect_int: (int > $o) > set_int).
% 66.57/16.16 thf(func_def_837, type, collect_list_o: (list_o > $o) > set_list_o).
% 66.57/16.16 thf(func_def_838, type, collect_list_complex: (list_complex > $o) > set_list_complex).
% 66.57/16.16 thf(func_def_839, type, collect_list_int: (list_int > $o) > set_list_int).
% 66.57/16.16 thf(func_def_840, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 66.57/16.16 thf(func_def_841, type, collec5608196760682091941T_VEBT: (list_VEBT_VEBT > $o) > set_list_VEBT_VEBT).
% 66.57/16.16 thf(func_def_842, type, collect_nat: (nat > $o) > set_nat).
% 66.57/16.16 thf(func_def_843, type, collect_num: (num > $o) > set_num).
% 66.57/16.16 thf(func_def_844, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 66.57/16.16 thf(func_def_845, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 66.57/16.16 thf(func_def_846, type, collec7088162979684241874at_nat: (produc859450856879609959at_nat > $o) > set_Pr8693737435421807431at_nat).
% 66.57/16.16 thf(func_def_847, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 66.57/16.16 thf(func_def_848, type, collect_rat: (rat > $o) > set_rat).
% 66.57/16.16 thf(func_def_849, type, collect_real: (real > $o) > set_real).
% 66.57/16.16 thf(func_def_850, type, collect_set_complex: (set_complex > $o) > set_set_complex).
% 66.57/16.16 thf(func_def_851, type, collect_set_int: (set_int > $o) > set_set_int).
% 66.57/16.16 thf(func_def_852, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 66.57/16.16 thf(func_def_853, type, collect_VEBT_VEBT: (vEBT_VEBT > $o) > set_VEBT_VEBT).
% 66.57/16.16 thf(func_def_854, type, image_int_int: (int > int) > set_int > set_int).
% 66.57/16.16 thf(func_def_855, type, image_nat_int: (nat > int) > set_nat > set_int).
% 66.57/16.16 thf(func_def_856, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 66.57/16.16 thf(func_def_857, type, image_nat_real: (nat > real) > set_nat > set_real).
% 66.57/16.16 thf(func_def_858, type, image_nat_char: (nat > char) > set_nat > set_char).
% 66.57/16.16 thf(func_def_859, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 66.57/16.16 thf(func_def_860, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 66.57/16.16 thf(func_def_861, type, image_real_real: (real > real) > set_real > set_real).
% 66.57/16.16 thf(func_def_862, type, image_char_nat: (char > nat) > set_char > set_nat).
% 66.57/16.16 thf(func_def_863, type, insert_int: int > set_int > set_int).
% 66.57/16.16 thf(func_def_864, type, insert_nat: nat > set_nat > set_nat).
% 66.57/16.16 thf(func_def_865, type, insert_real: real > set_real > set_real).
% 66.57/16.16 thf(func_def_866, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 66.57/16.16 thf(func_def_867, type, set_fo1517530859248394432omplex: (nat > complex > complex) > nat > nat > complex > complex).
% 66.57/16.16 thf(func_def_868, type, set_fo2581907887559384638at_int: (nat > int > int) > nat > nat > int > int).
% 66.57/16.16 thf(func_def_869, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 66.57/16.16 thf(func_def_870, type, set_fo1949268297981939178at_rat: (nat > rat > rat) > nat > nat > rat > rat).
% 66.57/16.16 thf(func_def_871, type, set_fo3111899725591712190t_real: (nat > real > real) > nat > nat > real > real).
% 66.57/16.16 thf(func_def_872, type, set_or1266510415728281911st_int: int > int > set_int).
% 66.57/16.16 thf(func_def_873, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 66.57/16.16 thf(func_def_874, type, set_or7049704709247886629st_num: num > num > set_num).
% 66.57/16.16 thf(func_def_875, type, set_or633870826150836451st_rat: rat > rat > set_rat).
% 66.57/16.16 thf(func_def_876, type, set_or1222579329274155063t_real: real > real > set_real).
% 66.57/16.16 thf(func_def_877, type, set_or370866239135849197et_int: set_int > set_int > set_set_int).
% 66.57/16.16 thf(func_def_878, type, set_or4662586982721622107an_int: int > int > set_int).
% 66.57/16.16 thf(func_def_879, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 66.57/16.16 thf(func_def_880, type, set_ord_atLeast_nat: nat > set_nat).
% 66.57/16.16 thf(func_def_881, type, set_ord_atLeast_real: real > set_real).
% 66.57/16.16 thf(func_def_882, type, set_ord_atMost_int: int > set_int).
% 66.57/16.16 thf(func_def_883, type, set_ord_atMost_nat: nat > set_nat).
% 66.57/16.16 thf(func_def_884, type, set_or6656581121297822940st_int: int > int > set_int).
% 66.57/16.16 thf(func_def_885, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 66.57/16.16 thf(func_def_886, type, set_or5832277885323065728an_int: int > int > set_int).
% 66.57/16.16 thf(func_def_887, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 66.57/16.16 thf(func_def_888, type, set_or1633881224788618240n_real: real > real > set_real).
% 66.57/16.16 thf(func_def_889, type, set_or1210151606488870762an_nat: nat > set_nat).
% 66.57/16.16 thf(func_def_890, type, set_or5849166863359141190n_real: real > set_real).
% 66.57/16.16 thf(func_def_891, type, set_ord_lessThan_int: int > set_int).
% 66.57/16.16 thf(func_def_892, type, set_ord_lessThan_nat: nat > set_nat).
% 66.57/16.16 thf(func_def_893, type, set_ord_lessThan_num: num > set_num).
% 66.57/16.16 thf(func_def_894, type, set_ord_lessThan_rat: rat > set_rat).
% 66.57/16.16 thf(func_def_895, type, set_or5984915006950818249n_real: real > set_real).
% 66.57/16.16 thf(func_def_896, type, ascii_of: char > char).
% 66.57/16.16 thf(func_def_897, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 66.57/16.16 thf(func_def_898, type, comm_s629917340098488124ar_nat: char > nat).
% 66.57/16.16 thf(func_def_899, type, integer_of_char: char > code_integer).
% 66.57/16.16 thf(func_def_900, type, unique3096191561947761185of_nat: nat > char).
% 66.57/16.16 thf(func_def_901, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 66.57/16.16 thf(func_def_902, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 66.57/16.16 thf(func_def_903, type, topolo4899668324122417113eq_int: (nat > int) > $o).
% 66.57/16.16 thf(func_def_904, type, topolo4902158794631467389eq_nat: (nat > nat) > $o).
% 66.57/16.16 thf(func_def_905, type, topolo1459490580787246023eq_num: (nat > num) > $o).
% 66.57/16.16 thf(func_def_906, type, topolo4267028734544971653eq_rat: (nat > rat) > $o).
% 66.57/16.16 thf(func_def_907, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 66.57/16.16 thf(func_def_908, type, topolo3100542954746470799et_int: (nat > set_int) > $o).
% 66.57/16.16 thf(func_def_909, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 66.57/16.16 thf(func_def_910, type, topolo7531315842566124627t_real: (nat > real) > $o).
% 66.57/16.16 thf(func_def_911, type, topolo2815343760600316023s_real: real > filter_real).
% 66.57/16.16 thf(func_def_912, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 66.57/16.16 thf(func_def_913, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 66.57/16.16 thf(func_def_914, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 66.57/16.16 thf(func_def_915, type, arccos: real > real).
% 66.57/16.16 thf(func_def_916, type, arcosh_real: real > real).
% 66.57/16.16 thf(func_def_917, type, arcsin: real > real).
% 66.57/16.16 thf(func_def_918, type, arctan: real > real).
% 66.57/16.16 thf(func_def_919, type, arsinh_real: real > real).
% 66.57/16.16 thf(func_def_920, type, artanh_real: real > real).
% 66.57/16.16 thf(func_def_921, type, cos_complex: complex > complex).
% 66.57/16.16 thf(func_def_922, type, cos_real: real > real).
% 66.57/16.16 thf(func_def_923, type, cos_coeff: nat > real).
% 66.57/16.16 thf(func_def_924, type, cosh_real: real > real).
% 66.57/16.16 thf(func_def_925, type, cot_real: real > real).
% 66.57/16.16 thf(func_def_926, type, diffs_complex: (nat > complex) > nat > complex).
% 66.57/16.16 thf(func_def_927, type, diffs_int: (nat > int) > nat > int).
% 66.57/16.16 thf(func_def_928, type, diffs_rat: (nat > rat) > nat > rat).
% 66.57/16.16 thf(func_def_929, type, diffs_real: (nat > real) > nat > real).
% 66.57/16.16 thf(func_def_930, type, exp_complex: complex > complex).
% 66.57/16.16 thf(func_def_931, type, exp_real: real > real).
% 66.57/16.16 thf(func_def_932, type, ln_ln_real: real > real).
% 66.57/16.16 thf(func_def_933, type, log: real > real > real).
% 66.57/16.16 thf(func_def_934, type, pi: real).
% 66.57/16.16 thf(func_def_935, type, powr_real: real > real > real).
% 66.57/16.16 thf(func_def_936, type, sin_complex: complex > complex).
% 66.57/16.16 thf(func_def_937, type, sin_real: real > real).
% 66.57/16.16 thf(func_def_938, type, sin_coeff: nat > real).
% 66.57/16.16 thf(func_def_939, type, sinh_real: real > real).
% 66.57/16.16 thf(func_def_940, type, tan_real: real > real).
% 66.57/16.16 thf(func_def_941, type, tanh_real: real > real).
% 66.57/16.16 thf(func_def_942, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 66.57/16.16 thf(func_def_943, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 66.57/16.16 thf(func_def_944, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 66.57/16.16 thf(func_def_945, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 66.57/16.16 thf(func_def_946, type, vEBT_VEBT_high: nat > nat > nat).
% 66.57/16.16 thf(func_def_947, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 66.57/16.16 thf(func_def_948, type, vEBT_VEBT_low: nat > nat > nat).
% 66.57/16.16 thf(func_def_949, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 66.57/16.16 thf(func_def_950, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 66.57/16.16 thf(func_def_951, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 66.57/16.16 thf(func_def_952, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 66.57/16.16 thf(func_def_953, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 66.57/16.16 thf(func_def_954, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 66.57/16.16 thf(func_def_955, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 66.57/16.16 thf(func_def_956, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 66.57/16.16 thf(func_def_957, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 66.57/16.16 thf(func_def_958, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 66.57/16.16 thf(func_def_959, type, vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT).
% 66.57/16.16 thf(func_def_960, type, vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 66.57/16.16 thf(func_def_961, type, vEBT_VEBT_bit_concat: nat > nat > nat > nat).
% 66.57/16.16 thf(func_def_962, type, vEBT_VEBT_minNull: vEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_963, type, vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_964, type, vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat).
% 66.57/16.16 thf(func_def_965, type, vEBT_vebt_member: vEBT_VEBT > nat > $o).
% 66.57/16.16 thf(func_def_966, type, vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 66.57/16.16 thf(func_def_967, type, vEBT_VEBT_add: option_nat > option_nat > option_nat).
% 66.57/16.16 thf(func_def_968, type, vEBT_VEBT_greater: option_nat > option_nat > $o).
% 66.57/16.16 thf(func_def_969, type, vEBT_VEBT_less: option_nat > option_nat > $o).
% 66.57/16.16 thf(func_def_970, type, vEBT_VEBT_lesseq: option_nat > option_nat > $o).
% 66.57/16.16 thf(func_def_971, type, vEBT_VEBT_max_in_set: set_nat > nat > $o).
% 66.57/16.16 thf(func_def_972, type, vEBT_VEBT_min_in_set: set_nat > nat > $o).
% 66.57/16.16 thf(func_def_973, type, vEBT_VEBT_mul: option_nat > option_nat > option_nat).
% 66.57/16.16 thf(func_def_974, type, vEBT_V4262088993061758097ft_nat: (nat > nat > nat) > option_nat > option_nat > option_nat).
% 66.57/16.16 thf(func_def_975, type, vEBT_V819420779217536731ft_num: (num > num > num) > option_num > option_num > option_num).
% 66.57/16.16 thf(func_def_976, type, vEBT_V1502963449132264192at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat).
% 66.57/16.16 thf(func_def_977, type, vEBT_VEBT_power: option_nat > option_nat > option_nat).
% 66.57/16.16 thf(func_def_978, type, vEBT_vebt_maxt: vEBT_VEBT > option_nat).
% 66.57/16.16 thf(func_def_979, type, vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_980, type, vEBT_vebt_mint: vEBT_VEBT > option_nat).
% 66.57/16.16 thf(func_def_981, type, vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_982, type, vEBT_is_succ_in_set: set_nat > nat > nat > $o).
% 66.57/16.16 thf(func_def_983, type, vEBT_vebt_succ: vEBT_VEBT > nat > option_nat).
% 66.57/16.16 thf(func_def_984, type, vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 66.57/16.16 thf(func_def_985, type, accp_nat: (nat > nat > $o) > nat > $o).
% 66.57/16.16 thf(func_def_986, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 66.57/16.16 thf(func_def_987, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 66.57/16.16 thf(func_def_988, type, accp_P3113834385874906142um_num: (product_prod_num_num > product_prod_num_num > $o) > product_prod_num_num > $o).
% 66.57/16.16 thf(func_def_989, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 66.57/16.16 thf(func_def_990, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_991, type, fChoice_real: (real > $o) > real).
% 66.57/16.16 thf(func_def_992, type, member_o: $o > set_o > $o).
% 66.57/16.16 thf(func_def_993, type, member_complex: complex > set_complex > $o).
% 66.57/16.16 thf(func_def_994, type, member_int: int > set_int > $o).
% 66.57/16.16 thf(func_def_995, type, member_list_o: list_o > set_list_o > $o).
% 66.57/16.16 thf(func_def_996, type, member_list_int: list_int > set_list_int > $o).
% 66.57/16.16 thf(func_def_997, type, member_list_nat: list_nat > set_list_nat > $o).
% 66.57/16.16 thf(func_def_998, type, member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_999, type, member_nat: nat > set_nat > $o).
% 66.57/16.16 thf(func_def_1000, type, member_num: num > set_num > $o).
% 66.57/16.16 thf(func_def_1001, type, member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o).
% 66.57/16.16 thf(func_def_1002, type, member_rat: rat > set_rat > $o).
% 66.57/16.16 thf(func_def_1003, type, member_real: real > set_real > $o).
% 66.57/16.16 thf(func_def_1004, type, member_set_int: set_int > set_set_int > $o).
% 66.57/16.16 thf(func_def_1005, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 66.57/16.16 thf(func_def_1006, type, deg: nat).
% 66.57/16.16 thf(func_def_1007, type, m: nat).
% 66.57/16.16 thf(func_def_1008, type, ma: nat).
% 66.57/16.16 thf(func_def_1009, type, mi: nat).
% 66.57/16.16 thf(func_def_1010, type, na: nat).
% 66.57/16.16 thf(func_def_1011, type, sc: nat).
% 66.57/16.16 thf(func_def_1012, type, summary: vEBT_VEBT).
% 66.57/16.16 thf(func_def_1013, type, treeList: list_VEBT_VEBT).
% 66.57/16.16 thf(func_def_1014, type, xa: nat).
% 66.57/16.16 thf(func_def_1033, type, sK0: nat).
% 66.57/16.16 thf(func_def_1034, type, sK1: nat > nat > nat).
% 66.57/16.16 thf(func_def_1035, type, sK2: set_nat > nat > nat).
% 66.57/16.16 thf(func_def_1036, type, sP3: nat > $o).
% 66.57/16.16 thf(func_def_1037, type, sP4: $o > $o).
% 66.57/16.16 thf(func_def_1038, type, sP5: nat > $o).
% 66.57/16.16 thf(func_def_1039, type, sP6: nat > $o).
% 66.57/16.16 thf(func_def_1040, type, sP7: nat > $o).
% 66.57/16.16 thf(func_def_1041, type, sP8: nat > $o).
% 66.57/16.16 thf(func_def_1042, type, sP9: option_nat > $o).
% 66.57/16.16 thf(func_def_1043, type, ph10: !>[X0: $tType]:(X0)).
% 66.57/16.16 thf(f30628,plain,(
% 66.57/16.16 $false),
% 66.57/16.16 inference(trivial_inequality_removal,[status(thm)],[f30627])).
% 66.57/16.16 thf(f30627,plain,(
% 66.57/16.16 ($true = $false)),
% 66.57/16.16 inference(backward_demodulation,[status(thm)],[f30579,f30626])).
% 66.57/16.16 thf(f30626,plain,(
% 66.57/16.16 ((sP4 @ $true) = $false)),
% 66.57/16.16 inference(boolean_simplification,[status(thm)],[f30624])).
% 66.57/16.16 thf(f30624,plain,(
% 66.57/16.16 ($false = (sP4 @ ((some_nat @ sc) = (some_nat @ sc))))),
% 66.57/16.16 inference(superposition,[status(thm)],[f30617,f30599])).
% 66.57/16.16 thf(f30599,plain,(
% 66.57/16.16 ((some_nat @ sc) = (vEBT_vebt_succ @ summary @ (divide_divide_nat @ xa @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))))),
% 66.57/16.16 inference(beta_eta_normalization,[status(thm)],[f30514])).
% 66.57/16.16 thf(f30514,plain,(
% 66.57/16.16 ((some_nat @ sc) = (vEBT_vebt_succ @ summary @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))))),
% 66.57/16.16 inference(definition_unfolding,[status(thm)],[f30387,f30432])).
% 66.57/16.16 thf(f30432,plain,(
% 66.57/16.16 (vEBT_VEBT_high = (^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))))),
% 66.57/16.16 inference(cnf_transformation,[status(thm)],[f10262])).
% 66.57/16.16 thf(f10262,plain,(
% 66.57/16.16 (vEBT_VEBT_high = (^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))))),
% 66.57/16.16 inference(fool_elimination,[status(thm)],[f10261])).
% 66.57/16.16 thf(f10261,plain,(
% 66.57/16.16 ((^[X0 : nat, X1 : nat] : (divide_divide_nat @ X0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X1))) = vEBT_VEBT_high)),
% 66.57/16.16 inference(rectify,[status(thm)],[f11])).
% 66.57/16.16 thf(f11,axiom,(
% 66.57/16.16 ((^[X5 : nat, X6 : nat] : (divide_divide_nat @ X5 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X6))) = vEBT_VEBT_high)),
% 66.57/16.16 file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_10_high__def)).
% 66.57/16.16 thf(f30387,plain,(
% 66.57/16.16 ((vEBT_vebt_succ @ summary @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) = (some_nat @ sc))),
% 66.57/16.16 inference(cnf_transformation,[status(thm)],[f21923])).
% 66.57/16.16 thf(f21923,plain,(
% 66.57/16.16 ((vEBT_vebt_succ @ summary @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) = (some_nat @ sc))),
% 66.57/16.16 inference(fool_elimination,[status(thm)],[f2])).
% 66.57/16.16 thf(f2,axiom,(
% 66.57/16.16 ((vEBT_vebt_succ @ summary @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) = (some_nat @ sc))),
% 66.57/16.16 file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_1__092_060open_062vebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_092_060close_062)).
% 66.57/16.16 thf(f30617,plain,(
% 66.57/16.16 ($false = (sP4 @ ((vEBT_vebt_succ @ summary @ (divide_divide_nat @ xa @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))))) = (some_nat @ sc))))),
% 66.57/16.16 inference(superposition,[status(thm)],[f30607,f30592])).
% 66.57/16.16 thf(f30592,plain,(
% 66.57/16.16 ( ! [X0 : nat,X1 : nat] : (((((member_nat @ X1 @ (collect_nat @ (vEBT_vebt_member @ summary))) & (ord_less_nat @ X0 @ X1)) & (!! @ nat @ (^[Y0 : nat]: ((member_nat @ Y0 @ (collect_nat @ (vEBT_vebt_member @ summary))) => ((ord_less_nat @ X0 @ Y0) => (ord_less_eq_nat @ X1 @ Y0)))))) = ((vEBT_vebt_succ @ summary @ X0) = (some_nat @ X1)))) )),
% 66.57/16.16 inference(beta_eta_normalization,[status(thm)],[f30549])).
% 66.57/16.16 thf(f30549,plain,(
% 66.57/16.16 ( ! [X0 : nat,X1 : nat] : ((((vEBT_vebt_succ @ summary @ X0) = (some_nat @ X1)) = ((^[Y0 : set_nat]: ((^[Y1 : nat]: ((^[Y2 : nat]: (((member_nat @ Y2 @ Y0) & (ord_less_nat @ Y1 @ Y2)) & (!! @ nat @ (^[Y3 : nat]: ((member_nat @ Y3 @ Y0) => ((ord_less_nat @ Y1 @ Y3) => (ord_less_eq_nat @ Y2 @ Y3))))))))))) @ ((^[Y0 : vEBT_VEBT]: (collect_nat @ (vEBT_vebt_member @ Y0))) @ summary) @ X0 @ X1))) )),
% 66.57/16.16 inference(definition_unfolding,[status(thm)],[f30458,f30500,f30501])).
% 66.57/16.16 thf(f30501,plain,(
% 66.57/16.16 (vEBT_VEBT_set_vebt = (^[Y0 : vEBT_VEBT]: (collect_nat @ (vEBT_vebt_member @ Y0))))),
% 66.57/16.16 inference(cnf_transformation,[status(thm)],[f19206])).
% 66.57/16.16 thf(f19206,plain,(
% 66.57/16.16 (vEBT_VEBT_set_vebt = (^[Y0 : vEBT_VEBT]: (collect_nat @ (vEBT_vebt_member @ Y0))))),
% 66.57/16.16 inference(fool_elimination,[status(thm)],[f19205])).
% 66.57/16.16 thf(f19205,plain,(
% 66.57/16.16 ((^[X0 : vEBT_VEBT] : (collect_nat @ (vEBT_vebt_member @ X0))) = vEBT_VEBT_set_vebt)),
% 66.57/16.16 inference(rectify,[status(thm)],[f3177])).
% 66.57/16.16 thf(f3177,axiom,(
% 66.57/16.16 ((^[X170 : vEBT_VEBT] : (collect_nat @ (vEBT_vebt_member @ X170))) = vEBT_VEBT_set_vebt)),
% 66.57/16.16 file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_3176_set__vebt_H__def)).
% 66.57/16.16 thf(f30500,plain,(
% 66.57/16.16 (vEBT_is_succ_in_set = (^[Y0 : set_nat]: ((^[Y1 : nat]: ((^[Y2 : nat]: (((member_nat @ Y2 @ Y0) & (ord_less_nat @ Y1 @ Y2)) & (!! @ nat @ (^[Y3 : nat]: ((member_nat @ Y3 @ Y0) => ((ord_less_nat @ Y1 @ Y3) => (ord_less_eq_nat @ Y2 @ Y3))))))))))))),
% 66.57/16.16 inference(cnf_transformation,[status(thm)],[f21418])).
% 66.57/16.16 thf(f21418,plain,(
% 66.57/16.16 (vEBT_is_succ_in_set = (^[Y0 : set_nat]: ((^[Y1 : nat]: ((^[Y2 : nat]: (((member_nat @ Y2 @ Y0) & (ord_less_nat @ Y1 @ Y2)) & (!! @ nat @ (^[Y3 : nat]: ((member_nat @ Y3 @ Y0) => ((ord_less_nat @ Y1 @ Y3) => (ord_less_eq_nat @ Y2 @ Y3))))))))))))),
% 66.57/16.16 inference(fool_elimination,[status(thm)],[f21417])).
% 66.57/16.16 thf(f21417,plain,(
% 66.57/16.16 ((^[X0 : set_nat, X1 : nat, X2 : nat] : (! [X3 : nat] : ((member_nat @ X3 @ X0) => ((ord_less_nat @ X1 @ X3) => (ord_less_eq_nat @ X2 @ X3))) & (ord_less_nat @ X1 @ X2) & (member_nat @ X2 @ X0))) = vEBT_is_succ_in_set)),
% 66.57/16.16 inference(rectify,[status(thm)],[f230])).
% 66.57/16.16 thf(f230,axiom,(
% 66.57/16.16 ((^[X10 : set_nat, X5 : nat, X11 : nat] : (! [X29 : nat] : ((member_nat @ X29 @ X10) => ((ord_less_nat @ X5 @ X29) => (ord_less_eq_nat @ X11 @ X29))) & (ord_less_nat @ X5 @ X11) & (member_nat @ X11 @ X10))) = vEBT_is_succ_in_set)),
% 66.57/16.16 file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_229_is__succ__in__set__def)).
% 66.57/16.16 thf(f30458,plain,(
% 66.57/16.16 ( ! [X0 : nat,X1 : nat] : (((vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ X0 @ X1) = ((vEBT_vebt_succ @ summary @ X0) = (some_nat @ X1)))) )),
% 66.57/16.16 inference(cnf_transformation,[status(thm)],[f26650])).
% 66.57/16.16 thf(f26650,plain,(
% 66.57/16.16 ! [X0 : nat,X1 : nat] : ((vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ X0 @ X1) = ((vEBT_vebt_succ @ summary @ X0) = (some_nat @ X1)))),
% 66.57/16.16 inference(fool_elimination,[status(thm)],[f1])).
% 66.57/16.16 thf(f1,axiom,(
% 66.57/16.16 ! [X0 : nat,X1 : nat] : (((vEBT_vebt_succ @ summary @ X0) = (some_nat @ X1)) = (vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ X0 @ X1))),
% 66.57/16.16 file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',fact_0__C4_Ohyps_C_I3_J)).
% 66.57/16.16 thf(f30607,plain,(
% 66.57/16.16 ((sP4 @ (((member_nat @ sc @ (collect_nat @ (vEBT_vebt_member @ summary))) & (ord_less_nat @ (divide_divide_nat @ xa @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ sc)) & (!! @ nat @ (^[Y0 : nat]: ((member_nat @ Y0 @ (collect_nat @ (vEBT_vebt_member @ summary))) => ((ord_less_nat @ (divide_divide_nat @ xa @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one))))) @ Y0) => (ord_less_eq_nat @ sc @ Y0))))))) = $false)),
% 66.57/16.16 inference(beta_eta_normalization,[status(thm)],[f30578])).
% 66.57/16.16 thf(f30578,plain,(
% 66.57/16.16 ($false = (sP4 @ ((^[Y0 : set_nat]: ((^[Y1 : nat]: ((^[Y2 : nat]: (((member_nat @ Y2 @ Y0) & (ord_less_nat @ Y1 @ Y2)) & (!! @ nat @ (^[Y3 : nat]: ((member_nat @ Y3 @ Y0) => ((ord_less_nat @ Y1 @ Y3) => (ord_less_eq_nat @ Y2 @ Y3))))))))))) @ ((^[Y0 : vEBT_VEBT]: (collect_nat @ (vEBT_vebt_member @ Y0))) @ summary) @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ sc)))),
% 66.57/16.16 introduced(definition,[new_symbols(naming,[sP4])],[inequality_splitting_name_introduction])).
% 66.57/16.16 thf(f30579,plain,(
% 66.57/16.16 ($true = (sP4 @ $true))),
% 66.57/16.16 inference(inequality_splitting,[status(thm)],[f30509,f30578])).
% 66.57/16.16 thf(f30509,plain,(
% 66.57/16.16 ($true != ((^[Y0 : set_nat]: ((^[Y1 : nat]: ((^[Y2 : nat]: (((member_nat @ Y2 @ Y0) & (ord_less_nat @ Y1 @ Y2)) & (!! @ nat @ (^[Y3 : nat]: ((member_nat @ Y3 @ Y0) => ((ord_less_nat @ Y1 @ Y3) => (ord_less_eq_nat @ Y2 @ Y3))))))))))) @ ((^[Y0 : vEBT_VEBT]: (collect_nat @ (vEBT_vebt_member @ Y0))) @ summary) @ ((^[Y0 : nat]: ((^[Y1 : nat]: (divide_divide_nat @ Y0 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Y1))))) @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ sc))),
% 66.57/16.16 inference(definition_unfolding,[status(thm)],[f30378,f30500,f30501,f30432])).
% 66.57/16.16 thf(f30378,plain,(
% 66.57/16.16 ((vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ sc) != $true)),
% 66.57/16.16 inference(cnf_transformation,[status(thm)],[f30237])).
% 66.57/16.16 thf(f30237,plain,(
% 66.57/16.16 ((vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ sc) != $true)),
% 66.57/16.16 inference(flattening,[status(thm)],[f12065])).
% 66.57/16.16 thf(f12065,plain,(
% 66.57/16.16 ~((vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ sc) = $true)),
% 66.57/16.16 inference(fool_elimination,[status(thm)],[f12064])).
% 66.57/16.16 thf(f12064,plain,(
% 66.57/16.16 ~(vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ sc)),
% 66.57/16.16 inference(rectify,[status(thm)],[f10257])).
% 66.57/16.16 thf(f10257,negated_conjecture,(
% 66.57/16.16 ~(vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ sc)),
% 66.57/16.16 inference(negated_conjecture,[status(cth)],[f10256])).
% 66.57/16.16 thf(f10256,conjecture,(
% 66.57/16.16 (vEBT_is_succ_in_set @ (vEBT_VEBT_set_vebt @ summary) @ (vEBT_VEBT_high @ xa @ (divide_divide_nat @ deg @ (numeral_numeral_nat @ (bit0 @ one)))) @ sc)),
% 66.57/16.16 file('/export/starexec/sandbox/tmp/tmp.9dOYh6TSNa/DTF2THF_31377.p',conj_0)).
% 66.57/16.16 % SZS output end Proof for DTF2THF_31377
% 66.57/16.16 % (31656)------------------------------
% 66.57/16.16 % (31656)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 66.57/16.16 % (31656)Termination reason: Refutation
% 66.57/16.16
% 66.57/16.16 % (31656)Memory used [KB]: 29679
% 66.57/16.16 % (31656)Time elapsed: 0.923 s
% 66.57/16.16 % (31656)Instructions burned: 1893 (million)
% 66.57/16.16 % (31656)------------------------------
% 66.57/16.16 % (31656)------------------------------
% 66.57/16.16 % (31485)Success in time 8.404 s
%------------------------------------------------------------------------------